diff --git "a/llmeval-env/lib/python3.10/site-packages/scipy/signal/_signaltools.py" "b/llmeval-env/lib/python3.10/site-packages/scipy/signal/_signaltools.py" new file mode 100644--- /dev/null +++ "b/llmeval-env/lib/python3.10/site-packages/scipy/signal/_signaltools.py" @@ -0,0 +1,4628 @@ +# Author: Travis Oliphant +# 1999 -- 2002 + +import operator +import math +from math import prod as _prod +import timeit +import warnings + +from scipy.spatial import cKDTree +from . import _sigtools +from ._ltisys import dlti +from ._upfirdn import upfirdn, _output_len, _upfirdn_modes +from scipy import linalg, fft as sp_fft +from scipy import ndimage +from scipy.fft._helper import _init_nd_shape_and_axes +import numpy as np +from scipy.special import lambertw +from .windows import get_window +from ._arraytools import axis_slice, axis_reverse, odd_ext, even_ext, const_ext +from ._filter_design import cheby1, _validate_sos, zpk2sos +from ._fir_filter_design import firwin +from ._sosfilt import _sosfilt + + +__all__ = ['correlate', 'correlation_lags', 'correlate2d', + 'convolve', 'convolve2d', 'fftconvolve', 'oaconvolve', + 'order_filter', 'medfilt', 'medfilt2d', 'wiener', 'lfilter', + 'lfiltic', 'sosfilt', 'deconvolve', 'hilbert', 'hilbert2', + 'cmplx_sort', 'unique_roots', 'invres', 'invresz', 'residue', + 'residuez', 'resample', 'resample_poly', 'detrend', + 'lfilter_zi', 'sosfilt_zi', 'sosfiltfilt', 'choose_conv_method', + 'filtfilt', 'decimate', 'vectorstrength'] + + +_modedict = {'valid': 0, 'same': 1, 'full': 2} + +_boundarydict = {'fill': 0, 'pad': 0, 'wrap': 2, 'circular': 2, 'symm': 1, + 'symmetric': 1, 'reflect': 4} + + +def _valfrommode(mode): + try: + return _modedict[mode] + except KeyError as e: + raise ValueError("Acceptable mode flags are 'valid'," + " 'same', or 'full'.") from e + + +def _bvalfromboundary(boundary): + try: + return _boundarydict[boundary] << 2 + except KeyError as e: + raise ValueError("Acceptable boundary flags are 'fill', 'circular' " + "(or 'wrap'), and 'symmetric' (or 'symm').") from e + + +def _inputs_swap_needed(mode, shape1, shape2, axes=None): + """Determine if inputs arrays need to be swapped in `"valid"` mode. + + If in `"valid"` mode, returns whether or not the input arrays need to be + swapped depending on whether `shape1` is at least as large as `shape2` in + every calculated dimension. + + This is important for some of the correlation and convolution + implementations in this module, where the larger array input needs to come + before the smaller array input when operating in this mode. + + Note that if the mode provided is not 'valid', False is immediately + returned. + + """ + if mode != 'valid': + return False + + if not shape1: + return False + + if axes is None: + axes = range(len(shape1)) + + ok1 = all(shape1[i] >= shape2[i] for i in axes) + ok2 = all(shape2[i] >= shape1[i] for i in axes) + + if not (ok1 or ok2): + raise ValueError("For 'valid' mode, one must be at least " + "as large as the other in every dimension") + + return not ok1 + + +def correlate(in1, in2, mode='full', method='auto'): + r""" + Cross-correlate two N-dimensional arrays. + + Cross-correlate `in1` and `in2`, with the output size determined by the + `mode` argument. + + Parameters + ---------- + in1 : array_like + First input. + in2 : array_like + Second input. Should have the same number of dimensions as `in1`. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output: + + ``full`` + The output is the full discrete linear cross-correlation + of the inputs. (Default) + ``valid`` + The output consists only of those elements that do not + rely on the zero-padding. In 'valid' mode, either `in1` or `in2` + must be at least as large as the other in every dimension. + ``same`` + The output is the same size as `in1`, centered + with respect to the 'full' output. + method : str {'auto', 'direct', 'fft'}, optional + A string indicating which method to use to calculate the correlation. + + ``direct`` + The correlation is determined directly from sums, the definition of + correlation. + ``fft`` + The Fast Fourier Transform is used to perform the correlation more + quickly (only available for numerical arrays.) + ``auto`` + Automatically chooses direct or Fourier method based on an estimate + of which is faster (default). See `convolve` Notes for more detail. + + .. versionadded:: 0.19.0 + + Returns + ------- + correlate : array + An N-dimensional array containing a subset of the discrete linear + cross-correlation of `in1` with `in2`. + + See Also + -------- + choose_conv_method : contains more documentation on `method`. + correlation_lags : calculates the lag / displacement indices array for 1D + cross-correlation. + + Notes + ----- + The correlation z of two d-dimensional arrays x and y is defined as:: + + z[...,k,...] = sum[..., i_l, ...] x[..., i_l,...] * conj(y[..., i_l - k,...]) + + This way, if x and y are 1-D arrays and ``z = correlate(x, y, 'full')`` + then + + .. math:: + + z[k] = (x * y)(k - N + 1) + = \sum_{l=0}^{||x||-1}x_l y_{l-k+N-1}^{*} + + for :math:`k = 0, 1, ..., ||x|| + ||y|| - 2` + + where :math:`||x||` is the length of ``x``, :math:`N = \max(||x||,||y||)`, + and :math:`y_m` is 0 when m is outside the range of y. + + ``method='fft'`` only works for numerical arrays as it relies on + `fftconvolve`. In certain cases (i.e., arrays of objects or when + rounding integers can lose precision), ``method='direct'`` is always used. + + When using "same" mode with even-length inputs, the outputs of `correlate` + and `correlate2d` differ: There is a 1-index offset between them. + + Examples + -------- + Implement a matched filter using cross-correlation, to recover a signal + that has passed through a noisy channel. + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + + >>> sig = np.repeat([0., 1., 1., 0., 1., 0., 0., 1.], 128) + >>> sig_noise = sig + rng.standard_normal(len(sig)) + >>> corr = signal.correlate(sig_noise, np.ones(128), mode='same') / 128 + + >>> clock = np.arange(64, len(sig), 128) + >>> fig, (ax_orig, ax_noise, ax_corr) = plt.subplots(3, 1, sharex=True) + >>> ax_orig.plot(sig) + >>> ax_orig.plot(clock, sig[clock], 'ro') + >>> ax_orig.set_title('Original signal') + >>> ax_noise.plot(sig_noise) + >>> ax_noise.set_title('Signal with noise') + >>> ax_corr.plot(corr) + >>> ax_corr.plot(clock, corr[clock], 'ro') + >>> ax_corr.axhline(0.5, ls=':') + >>> ax_corr.set_title('Cross-correlated with rectangular pulse') + >>> ax_orig.margins(0, 0.1) + >>> fig.tight_layout() + >>> plt.show() + + Compute the cross-correlation of a noisy signal with the original signal. + + >>> x = np.arange(128) / 128 + >>> sig = np.sin(2 * np.pi * x) + >>> sig_noise = sig + rng.standard_normal(len(sig)) + >>> corr = signal.correlate(sig_noise, sig) + >>> lags = signal.correlation_lags(len(sig), len(sig_noise)) + >>> corr /= np.max(corr) + + >>> fig, (ax_orig, ax_noise, ax_corr) = plt.subplots(3, 1, figsize=(4.8, 4.8)) + >>> ax_orig.plot(sig) + >>> ax_orig.set_title('Original signal') + >>> ax_orig.set_xlabel('Sample Number') + >>> ax_noise.plot(sig_noise) + >>> ax_noise.set_title('Signal with noise') + >>> ax_noise.set_xlabel('Sample Number') + >>> ax_corr.plot(lags, corr) + >>> ax_corr.set_title('Cross-correlated signal') + >>> ax_corr.set_xlabel('Lag') + >>> ax_orig.margins(0, 0.1) + >>> ax_noise.margins(0, 0.1) + >>> ax_corr.margins(0, 0.1) + >>> fig.tight_layout() + >>> plt.show() + + """ + in1 = np.asarray(in1) + in2 = np.asarray(in2) + + if in1.ndim == in2.ndim == 0: + return in1 * in2.conj() + elif in1.ndim != in2.ndim: + raise ValueError("in1 and in2 should have the same dimensionality") + + # Don't use _valfrommode, since correlate should not accept numeric modes + try: + val = _modedict[mode] + except KeyError as e: + raise ValueError("Acceptable mode flags are 'valid'," + " 'same', or 'full'.") from e + + # this either calls fftconvolve or this function with method=='direct' + if method in ('fft', 'auto'): + return convolve(in1, _reverse_and_conj(in2), mode, method) + + elif method == 'direct': + # fastpath to faster numpy.correlate for 1d inputs when possible + if _np_conv_ok(in1, in2, mode): + return np.correlate(in1, in2, mode) + + # _correlateND is far slower when in2.size > in1.size, so swap them + # and then undo the effect afterward if mode == 'full'. Also, it fails + # with 'valid' mode if in2 is larger than in1, so swap those, too. + # Don't swap inputs for 'same' mode, since shape of in1 matters. + swapped_inputs = ((mode == 'full') and (in2.size > in1.size) or + _inputs_swap_needed(mode, in1.shape, in2.shape)) + + if swapped_inputs: + in1, in2 = in2, in1 + + if mode == 'valid': + ps = [i - j + 1 for i, j in zip(in1.shape, in2.shape)] + out = np.empty(ps, in1.dtype) + + z = _sigtools._correlateND(in1, in2, out, val) + + else: + ps = [i + j - 1 for i, j in zip(in1.shape, in2.shape)] + + # zero pad input + in1zpadded = np.zeros(ps, in1.dtype) + sc = tuple(slice(0, i) for i in in1.shape) + in1zpadded[sc] = in1.copy() + + if mode == 'full': + out = np.empty(ps, in1.dtype) + elif mode == 'same': + out = np.empty(in1.shape, in1.dtype) + + z = _sigtools._correlateND(in1zpadded, in2, out, val) + + if swapped_inputs: + # Reverse and conjugate to undo the effect of swapping inputs + z = _reverse_and_conj(z) + + return z + + else: + raise ValueError("Acceptable method flags are 'auto'," + " 'direct', or 'fft'.") + + +def correlation_lags(in1_len, in2_len, mode='full'): + r""" + Calculates the lag / displacement indices array for 1D cross-correlation. + + Parameters + ---------- + in1_len : int + First input size. + in2_len : int + Second input size. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output. + See the documentation `correlate` for more information. + + Returns + ------- + lags : array + Returns an array containing cross-correlation lag/displacement indices. + Indices can be indexed with the np.argmax of the correlation to return + the lag/displacement. + + See Also + -------- + correlate : Compute the N-dimensional cross-correlation. + + Notes + ----- + Cross-correlation for continuous functions :math:`f` and :math:`g` is + defined as: + + .. math:: + + \left ( f\star g \right )\left ( \tau \right ) + \triangleq \int_{t_0}^{t_0 +T} + \overline{f\left ( t \right )}g\left ( t+\tau \right )dt + + Where :math:`\tau` is defined as the displacement, also known as the lag. + + Cross correlation for discrete functions :math:`f` and :math:`g` is + defined as: + + .. math:: + \left ( f\star g \right )\left [ n \right ] + \triangleq \sum_{-\infty}^{\infty} + \overline{f\left [ m \right ]}g\left [ m+n \right ] + + Where :math:`n` is the lag. + + Examples + -------- + Cross-correlation of a signal with its time-delayed self. + + >>> import numpy as np + >>> from scipy import signal + >>> rng = np.random.default_rng() + >>> x = rng.standard_normal(1000) + >>> y = np.concatenate([rng.standard_normal(100), x]) + >>> correlation = signal.correlate(x, y, mode="full") + >>> lags = signal.correlation_lags(x.size, y.size, mode="full") + >>> lag = lags[np.argmax(correlation)] + """ + + # calculate lag ranges in different modes of operation + if mode == "full": + # the output is the full discrete linear convolution + # of the inputs. (Default) + lags = np.arange(-in2_len + 1, in1_len) + elif mode == "same": + # the output is the same size as `in1`, centered + # with respect to the 'full' output. + # calculate the full output + lags = np.arange(-in2_len + 1, in1_len) + # determine the midpoint in the full output + mid = lags.size // 2 + # determine lag_bound to be used with respect + # to the midpoint + lag_bound = in1_len // 2 + # calculate lag ranges for even and odd scenarios + if in1_len % 2 == 0: + lags = lags[(mid-lag_bound):(mid+lag_bound)] + else: + lags = lags[(mid-lag_bound):(mid+lag_bound)+1] + elif mode == "valid": + # the output consists only of those elements that do not + # rely on the zero-padding. In 'valid' mode, either `in1` or `in2` + # must be at least as large as the other in every dimension. + + # the lag_bound will be either negative or positive + # this let's us infer how to present the lag range + lag_bound = in1_len - in2_len + if lag_bound >= 0: + lags = np.arange(lag_bound + 1) + else: + lags = np.arange(lag_bound, 1) + return lags + + +def _centered(arr, newshape): + # Return the center newshape portion of the array. + newshape = np.asarray(newshape) + currshape = np.array(arr.shape) + startind = (currshape - newshape) // 2 + endind = startind + newshape + myslice = [slice(startind[k], endind[k]) for k in range(len(endind))] + return arr[tuple(myslice)] + + +def _init_freq_conv_axes(in1, in2, mode, axes, sorted_axes=False): + """Handle the axes argument for frequency-domain convolution. + + Returns the inputs and axes in a standard form, eliminating redundant axes, + swapping the inputs if necessary, and checking for various potential + errors. + + Parameters + ---------- + in1 : array + First input. + in2 : array + Second input. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output. + See the documentation `fftconvolve` for more information. + axes : list of ints + Axes over which to compute the FFTs. + sorted_axes : bool, optional + If `True`, sort the axes. + Default is `False`, do not sort. + + Returns + ------- + in1 : array + The first input, possible swapped with the second input. + in2 : array + The second input, possible swapped with the first input. + axes : list of ints + Axes over which to compute the FFTs. + + """ + s1 = in1.shape + s2 = in2.shape + noaxes = axes is None + + _, axes = _init_nd_shape_and_axes(in1, shape=None, axes=axes) + + if not noaxes and not len(axes): + raise ValueError("when provided, axes cannot be empty") + + # Axes of length 1 can rely on broadcasting rules for multiply, + # no fft needed. + axes = [a for a in axes if s1[a] != 1 and s2[a] != 1] + + if sorted_axes: + axes.sort() + + if not all(s1[a] == s2[a] or s1[a] == 1 or s2[a] == 1 + for a in range(in1.ndim) if a not in axes): + raise ValueError("incompatible shapes for in1 and in2:" + f" {s1} and {s2}") + + # Check that input sizes are compatible with 'valid' mode. + if _inputs_swap_needed(mode, s1, s2, axes=axes): + # Convolution is commutative; order doesn't have any effect on output. + in1, in2 = in2, in1 + + return in1, in2, axes + + +def _freq_domain_conv(in1, in2, axes, shape, calc_fast_len=False): + """Convolve two arrays in the frequency domain. + + This function implements only base the FFT-related operations. + Specifically, it converts the signals to the frequency domain, multiplies + them, then converts them back to the time domain. Calculations of axes, + shapes, convolution mode, etc. are implemented in higher level-functions, + such as `fftconvolve` and `oaconvolve`. Those functions should be used + instead of this one. + + Parameters + ---------- + in1 : array_like + First input. + in2 : array_like + Second input. Should have the same number of dimensions as `in1`. + axes : array_like of ints + Axes over which to compute the FFTs. + shape : array_like of ints + The sizes of the FFTs. + calc_fast_len : bool, optional + If `True`, set each value of `shape` to the next fast FFT length. + Default is `False`, use `axes` as-is. + + Returns + ------- + out : array + An N-dimensional array containing the discrete linear convolution of + `in1` with `in2`. + + """ + if not len(axes): + return in1 * in2 + + complex_result = (in1.dtype.kind == 'c' or in2.dtype.kind == 'c') + + if calc_fast_len: + # Speed up FFT by padding to optimal size. + fshape = [ + sp_fft.next_fast_len(shape[a], not complex_result) for a in axes] + else: + fshape = shape + + if not complex_result: + fft, ifft = sp_fft.rfftn, sp_fft.irfftn + else: + fft, ifft = sp_fft.fftn, sp_fft.ifftn + + sp1 = fft(in1, fshape, axes=axes) + sp2 = fft(in2, fshape, axes=axes) + + ret = ifft(sp1 * sp2, fshape, axes=axes) + + if calc_fast_len: + fslice = tuple([slice(sz) for sz in shape]) + ret = ret[fslice] + + return ret + + +def _apply_conv_mode(ret, s1, s2, mode, axes): + """Calculate the convolution result shape based on the `mode` argument. + + Returns the result sliced to the correct size for the given mode. + + Parameters + ---------- + ret : array + The result array, with the appropriate shape for the 'full' mode. + s1 : list of int + The shape of the first input. + s2 : list of int + The shape of the second input. + mode : str {'full', 'valid', 'same'} + A string indicating the size of the output. + See the documentation `fftconvolve` for more information. + axes : list of ints + Axes over which to compute the convolution. + + Returns + ------- + ret : array + A copy of `res`, sliced to the correct size for the given `mode`. + + """ + if mode == "full": + return ret.copy() + elif mode == "same": + return _centered(ret, s1).copy() + elif mode == "valid": + shape_valid = [ret.shape[a] if a not in axes else s1[a] - s2[a] + 1 + for a in range(ret.ndim)] + return _centered(ret, shape_valid).copy() + else: + raise ValueError("acceptable mode flags are 'valid'," + " 'same', or 'full'") + + +def fftconvolve(in1, in2, mode="full", axes=None): + """Convolve two N-dimensional arrays using FFT. + + Convolve `in1` and `in2` using the fast Fourier transform method, with + the output size determined by the `mode` argument. + + This is generally much faster than `convolve` for large arrays (n > ~500), + but can be slower when only a few output values are needed, and can only + output float arrays (int or object array inputs will be cast to float). + + As of v0.19, `convolve` automatically chooses this method or the direct + method based on an estimation of which is faster. + + Parameters + ---------- + in1 : array_like + First input. + in2 : array_like + Second input. Should have the same number of dimensions as `in1`. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output: + + ``full`` + The output is the full discrete linear convolution + of the inputs. (Default) + ``valid`` + The output consists only of those elements that do not + rely on the zero-padding. In 'valid' mode, either `in1` or `in2` + must be at least as large as the other in every dimension. + ``same`` + The output is the same size as `in1`, centered + with respect to the 'full' output. + axes : int or array_like of ints or None, optional + Axes over which to compute the convolution. + The default is over all axes. + + Returns + ------- + out : array + An N-dimensional array containing a subset of the discrete linear + convolution of `in1` with `in2`. + + See Also + -------- + convolve : Uses the direct convolution or FFT convolution algorithm + depending on which is faster. + oaconvolve : Uses the overlap-add method to do convolution, which is + generally faster when the input arrays are large and + significantly different in size. + + Examples + -------- + Autocorrelation of white noise is an impulse. + + >>> import numpy as np + >>> from scipy import signal + >>> rng = np.random.default_rng() + >>> sig = rng.standard_normal(1000) + >>> autocorr = signal.fftconvolve(sig, sig[::-1], mode='full') + + >>> import matplotlib.pyplot as plt + >>> fig, (ax_orig, ax_mag) = plt.subplots(2, 1) + >>> ax_orig.plot(sig) + >>> ax_orig.set_title('White noise') + >>> ax_mag.plot(np.arange(-len(sig)+1,len(sig)), autocorr) + >>> ax_mag.set_title('Autocorrelation') + >>> fig.tight_layout() + >>> fig.show() + + Gaussian blur implemented using FFT convolution. Notice the dark borders + around the image, due to the zero-padding beyond its boundaries. + The `convolve2d` function allows for other types of image boundaries, + but is far slower. + + >>> from scipy import datasets + >>> face = datasets.face(gray=True) + >>> kernel = np.outer(signal.windows.gaussian(70, 8), + ... signal.windows.gaussian(70, 8)) + >>> blurred = signal.fftconvolve(face, kernel, mode='same') + + >>> fig, (ax_orig, ax_kernel, ax_blurred) = plt.subplots(3, 1, + ... figsize=(6, 15)) + >>> ax_orig.imshow(face, cmap='gray') + >>> ax_orig.set_title('Original') + >>> ax_orig.set_axis_off() + >>> ax_kernel.imshow(kernel, cmap='gray') + >>> ax_kernel.set_title('Gaussian kernel') + >>> ax_kernel.set_axis_off() + >>> ax_blurred.imshow(blurred, cmap='gray') + >>> ax_blurred.set_title('Blurred') + >>> ax_blurred.set_axis_off() + >>> fig.show() + + """ + in1 = np.asarray(in1) + in2 = np.asarray(in2) + + if in1.ndim == in2.ndim == 0: # scalar inputs + return in1 * in2 + elif in1.ndim != in2.ndim: + raise ValueError("in1 and in2 should have the same dimensionality") + elif in1.size == 0 or in2.size == 0: # empty arrays + return np.array([]) + + in1, in2, axes = _init_freq_conv_axes(in1, in2, mode, axes, + sorted_axes=False) + + s1 = in1.shape + s2 = in2.shape + + shape = [max((s1[i], s2[i])) if i not in axes else s1[i] + s2[i] - 1 + for i in range(in1.ndim)] + + ret = _freq_domain_conv(in1, in2, axes, shape, calc_fast_len=True) + + return _apply_conv_mode(ret, s1, s2, mode, axes) + + +def _calc_oa_lens(s1, s2): + """Calculate the optimal FFT lengths for overlapp-add convolution. + + The calculation is done for a single dimension. + + Parameters + ---------- + s1 : int + Size of the dimension for the first array. + s2 : int + Size of the dimension for the second array. + + Returns + ------- + block_size : int + The size of the FFT blocks. + overlap : int + The amount of overlap between two blocks. + in1_step : int + The size of each step for the first array. + in2_step : int + The size of each step for the first array. + + """ + # Set up the arguments for the conventional FFT approach. + fallback = (s1+s2-1, None, s1, s2) + + # Use conventional FFT convolve if sizes are same. + if s1 == s2 or s1 == 1 or s2 == 1: + return fallback + + if s2 > s1: + s1, s2 = s2, s1 + swapped = True + else: + swapped = False + + # There cannot be a useful block size if s2 is more than half of s1. + if s2 >= s1/2: + return fallback + + # Derivation of optimal block length + # For original formula see: + # https://en.wikipedia.org/wiki/Overlap-add_method + # + # Formula: + # K = overlap = s2-1 + # N = block_size + # C = complexity + # e = exponential, exp(1) + # + # C = (N*(log2(N)+1))/(N-K) + # C = (N*log2(2N))/(N-K) + # C = N/(N-K) * log2(2N) + # C1 = N/(N-K) + # C2 = log2(2N) = ln(2N)/ln(2) + # + # dC1/dN = (1*(N-K)-N)/(N-K)^2 = -K/(N-K)^2 + # dC2/dN = 2/(2*N*ln(2)) = 1/(N*ln(2)) + # + # dC/dN = dC1/dN*C2 + dC2/dN*C1 + # dC/dN = -K*ln(2N)/(ln(2)*(N-K)^2) + N/(N*ln(2)*(N-K)) + # dC/dN = -K*ln(2N)/(ln(2)*(N-K)^2) + 1/(ln(2)*(N-K)) + # dC/dN = -K*ln(2N)/(ln(2)*(N-K)^2) + (N-K)/(ln(2)*(N-K)^2) + # dC/dN = (-K*ln(2N) + (N-K)/(ln(2)*(N-K)^2) + # dC/dN = (N - K*ln(2N) - K)/(ln(2)*(N-K)^2) + # + # Solve for minimum, where dC/dN = 0 + # 0 = (N - K*ln(2N) - K)/(ln(2)*(N-K)^2) + # 0 * ln(2)*(N-K)^2 = N - K*ln(2N) - K + # 0 = N - K*ln(2N) - K + # 0 = N - K*(ln(2N) + 1) + # 0 = N - K*ln(2Ne) + # N = K*ln(2Ne) + # N/K = ln(2Ne) + # + # e^(N/K) = e^ln(2Ne) + # e^(N/K) = 2Ne + # 1/e^(N/K) = 1/(2*N*e) + # e^(N/-K) = 1/(2*N*e) + # e^(N/-K) = K/N*1/(2*K*e) + # N/K*e^(N/-K) = 1/(2*e*K) + # N/-K*e^(N/-K) = -1/(2*e*K) + # + # Using Lambert W function + # https://en.wikipedia.org/wiki/Lambert_W_function + # x = W(y) It is the solution to y = x*e^x + # x = N/-K + # y = -1/(2*e*K) + # + # N/-K = W(-1/(2*e*K)) + # + # N = -K*W(-1/(2*e*K)) + overlap = s2-1 + opt_size = -overlap*lambertw(-1/(2*math.e*overlap), k=-1).real + block_size = sp_fft.next_fast_len(math.ceil(opt_size)) + + # Use conventional FFT convolve if there is only going to be one block. + if block_size >= s1: + return fallback + + if not swapped: + in1_step = block_size-s2+1 + in2_step = s2 + else: + in1_step = s2 + in2_step = block_size-s2+1 + + return block_size, overlap, in1_step, in2_step + + +def oaconvolve(in1, in2, mode="full", axes=None): + """Convolve two N-dimensional arrays using the overlap-add method. + + Convolve `in1` and `in2` using the overlap-add method, with + the output size determined by the `mode` argument. + + This is generally much faster than `convolve` for large arrays (n > ~500), + and generally much faster than `fftconvolve` when one array is much + larger than the other, but can be slower when only a few output values are + needed or when the arrays are very similar in shape, and can only + output float arrays (int or object array inputs will be cast to float). + + Parameters + ---------- + in1 : array_like + First input. + in2 : array_like + Second input. Should have the same number of dimensions as `in1`. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output: + + ``full`` + The output is the full discrete linear convolution + of the inputs. (Default) + ``valid`` + The output consists only of those elements that do not + rely on the zero-padding. In 'valid' mode, either `in1` or `in2` + must be at least as large as the other in every dimension. + ``same`` + The output is the same size as `in1`, centered + with respect to the 'full' output. + axes : int or array_like of ints or None, optional + Axes over which to compute the convolution. + The default is over all axes. + + Returns + ------- + out : array + An N-dimensional array containing a subset of the discrete linear + convolution of `in1` with `in2`. + + See Also + -------- + convolve : Uses the direct convolution or FFT convolution algorithm + depending on which is faster. + fftconvolve : An implementation of convolution using FFT. + + Notes + ----- + .. versionadded:: 1.4.0 + + References + ---------- + .. [1] Wikipedia, "Overlap-add_method". + https://en.wikipedia.org/wiki/Overlap-add_method + .. [2] Richard G. Lyons. Understanding Digital Signal Processing, + Third Edition, 2011. Chapter 13.10. + ISBN 13: 978-0137-02741-5 + + Examples + -------- + Convolve a 100,000 sample signal with a 512-sample filter. + + >>> import numpy as np + >>> from scipy import signal + >>> rng = np.random.default_rng() + >>> sig = rng.standard_normal(100000) + >>> filt = signal.firwin(512, 0.01) + >>> fsig = signal.oaconvolve(sig, filt) + + >>> import matplotlib.pyplot as plt + >>> fig, (ax_orig, ax_mag) = plt.subplots(2, 1) + >>> ax_orig.plot(sig) + >>> ax_orig.set_title('White noise') + >>> ax_mag.plot(fsig) + >>> ax_mag.set_title('Filtered noise') + >>> fig.tight_layout() + >>> fig.show() + + """ + in1 = np.asarray(in1) + in2 = np.asarray(in2) + + if in1.ndim == in2.ndim == 0: # scalar inputs + return in1 * in2 + elif in1.ndim != in2.ndim: + raise ValueError("in1 and in2 should have the same dimensionality") + elif in1.size == 0 or in2.size == 0: # empty arrays + return np.array([]) + elif in1.shape == in2.shape: # Equivalent to fftconvolve + return fftconvolve(in1, in2, mode=mode, axes=axes) + + in1, in2, axes = _init_freq_conv_axes(in1, in2, mode, axes, + sorted_axes=True) + + s1 = in1.shape + s2 = in2.shape + + if not axes: + ret = in1 * in2 + return _apply_conv_mode(ret, s1, s2, mode, axes) + + # Calculate this now since in1 is changed later + shape_final = [None if i not in axes else + s1[i] + s2[i] - 1 for i in range(in1.ndim)] + + # Calculate the block sizes for the output, steps, first and second inputs. + # It is simpler to calculate them all together than doing them in separate + # loops due to all the special cases that need to be handled. + optimal_sizes = ((-1, -1, s1[i], s2[i]) if i not in axes else + _calc_oa_lens(s1[i], s2[i]) for i in range(in1.ndim)) + block_size, overlaps, \ + in1_step, in2_step = zip(*optimal_sizes) + + # Fall back to fftconvolve if there is only one block in every dimension. + if in1_step == s1 and in2_step == s2: + return fftconvolve(in1, in2, mode=mode, axes=axes) + + # Figure out the number of steps and padding. + # This would get too complicated in a list comprehension. + nsteps1 = [] + nsteps2 = [] + pad_size1 = [] + pad_size2 = [] + for i in range(in1.ndim): + if i not in axes: + pad_size1 += [(0, 0)] + pad_size2 += [(0, 0)] + continue + + if s1[i] > in1_step[i]: + curnstep1 = math.ceil((s1[i]+1)/in1_step[i]) + if (block_size[i] - overlaps[i])*curnstep1 < shape_final[i]: + curnstep1 += 1 + + curpad1 = curnstep1*in1_step[i] - s1[i] + else: + curnstep1 = 1 + curpad1 = 0 + + if s2[i] > in2_step[i]: + curnstep2 = math.ceil((s2[i]+1)/in2_step[i]) + if (block_size[i] - overlaps[i])*curnstep2 < shape_final[i]: + curnstep2 += 1 + + curpad2 = curnstep2*in2_step[i] - s2[i] + else: + curnstep2 = 1 + curpad2 = 0 + + nsteps1 += [curnstep1] + nsteps2 += [curnstep2] + pad_size1 += [(0, curpad1)] + pad_size2 += [(0, curpad2)] + + # Pad the array to a size that can be reshaped to the desired shape + # if necessary. + if not all(curpad == (0, 0) for curpad in pad_size1): + in1 = np.pad(in1, pad_size1, mode='constant', constant_values=0) + + if not all(curpad == (0, 0) for curpad in pad_size2): + in2 = np.pad(in2, pad_size2, mode='constant', constant_values=0) + + # Reshape the overlap-add parts to input block sizes. + split_axes = [iax+i for i, iax in enumerate(axes)] + fft_axes = [iax+1 for iax in split_axes] + + # We need to put each new dimension before the corresponding dimension + # being reshaped in order to get the data in the right layout at the end. + reshape_size1 = list(in1_step) + reshape_size2 = list(in2_step) + for i, iax in enumerate(split_axes): + reshape_size1.insert(iax, nsteps1[i]) + reshape_size2.insert(iax, nsteps2[i]) + + in1 = in1.reshape(*reshape_size1) + in2 = in2.reshape(*reshape_size2) + + # Do the convolution. + fft_shape = [block_size[i] for i in axes] + ret = _freq_domain_conv(in1, in2, fft_axes, fft_shape, calc_fast_len=False) + + # Do the overlap-add. + for ax, ax_fft, ax_split in zip(axes, fft_axes, split_axes): + overlap = overlaps[ax] + if overlap is None: + continue + + ret, overpart = np.split(ret, [-overlap], ax_fft) + overpart = np.split(overpart, [-1], ax_split)[0] + + ret_overpart = np.split(ret, [overlap], ax_fft)[0] + ret_overpart = np.split(ret_overpart, [1], ax_split)[1] + ret_overpart += overpart + + # Reshape back to the correct dimensionality. + shape_ret = [ret.shape[i] if i not in fft_axes else + ret.shape[i]*ret.shape[i-1] + for i in range(ret.ndim) if i not in split_axes] + ret = ret.reshape(*shape_ret) + + # Slice to the correct size. + slice_final = tuple([slice(islice) for islice in shape_final]) + ret = ret[slice_final] + + return _apply_conv_mode(ret, s1, s2, mode, axes) + + +def _numeric_arrays(arrays, kinds='buifc'): + """ + See if a list of arrays are all numeric. + + Parameters + ---------- + arrays : array or list of arrays + arrays to check if numeric. + kinds : string-like + The dtypes of the arrays to be checked. If the dtype.kind of + the ndarrays are not in this string the function returns False and + otherwise returns True. + """ + if type(arrays) == np.ndarray: + return arrays.dtype.kind in kinds + for array_ in arrays: + if array_.dtype.kind not in kinds: + return False + return True + + +def _conv_ops(x_shape, h_shape, mode): + """ + Find the number of operations required for direct/fft methods of + convolution. The direct operations were recorded by making a dummy class to + record the number of operations by overriding ``__mul__`` and ``__add__``. + The FFT operations rely on the (well-known) computational complexity of the + FFT (and the implementation of ``_freq_domain_conv``). + + """ + if mode == "full": + out_shape = [n + k - 1 for n, k in zip(x_shape, h_shape)] + elif mode == "valid": + out_shape = [abs(n - k) + 1 for n, k in zip(x_shape, h_shape)] + elif mode == "same": + out_shape = x_shape + else: + raise ValueError("Acceptable mode flags are 'valid'," + f" 'same', or 'full', not mode={mode}") + + s1, s2 = x_shape, h_shape + if len(x_shape) == 1: + s1, s2 = s1[0], s2[0] + if mode == "full": + direct_ops = s1 * s2 + elif mode == "valid": + direct_ops = (s2 - s1 + 1) * s1 if s2 >= s1 else (s1 - s2 + 1) * s2 + elif mode == "same": + direct_ops = (s1 * s2 if s1 < s2 else + s1 * s2 - (s2 // 2) * ((s2 + 1) // 2)) + else: + if mode == "full": + direct_ops = min(_prod(s1), _prod(s2)) * _prod(out_shape) + elif mode == "valid": + direct_ops = min(_prod(s1), _prod(s2)) * _prod(out_shape) + elif mode == "same": + direct_ops = _prod(s1) * _prod(s2) + + full_out_shape = [n + k - 1 for n, k in zip(x_shape, h_shape)] + N = _prod(full_out_shape) + fft_ops = 3 * N * np.log(N) # 3 separate FFTs of size full_out_shape + return fft_ops, direct_ops + + +def _fftconv_faster(x, h, mode): + """ + See if using fftconvolve or convolve is faster. + + Parameters + ---------- + x : np.ndarray + Signal + h : np.ndarray + Kernel + mode : str + Mode passed to convolve + + Returns + ------- + fft_faster : bool + + Notes + ----- + See docstring of `choose_conv_method` for details on tuning hardware. + + See pull request 11031 for more detail: + https://github.com/scipy/scipy/pull/11031. + + """ + fft_ops, direct_ops = _conv_ops(x.shape, h.shape, mode) + offset = -1e-3 if x.ndim == 1 else -1e-4 + constants = { + "valid": (1.89095737e-9, 2.1364985e-10, offset), + "full": (1.7649070e-9, 2.1414831e-10, offset), + "same": (3.2646654e-9, 2.8478277e-10, offset) + if h.size <= x.size + else (3.21635404e-9, 1.1773253e-8, -1e-5), + } if x.ndim == 1 else { + "valid": (1.85927e-9, 2.11242e-8, offset), + "full": (1.99817e-9, 1.66174e-8, offset), + "same": (2.04735e-9, 1.55367e-8, offset), + } + O_fft, O_direct, O_offset = constants[mode] + return O_fft * fft_ops < O_direct * direct_ops + O_offset + + +def _reverse_and_conj(x): + """ + Reverse array `x` in all dimensions and perform the complex conjugate + """ + reverse = (slice(None, None, -1),) * x.ndim + return x[reverse].conj() + + +def _np_conv_ok(volume, kernel, mode): + """ + See if numpy supports convolution of `volume` and `kernel` (i.e. both are + 1D ndarrays and of the appropriate shape). NumPy's 'same' mode uses the + size of the larger input, while SciPy's uses the size of the first input. + + Invalid mode strings will return False and be caught by the calling func. + """ + if volume.ndim == kernel.ndim == 1: + if mode in ('full', 'valid'): + return True + elif mode == 'same': + return volume.size >= kernel.size + else: + return False + + +def _timeit_fast(stmt="pass", setup="pass", repeat=3): + """ + Returns the time the statement/function took, in seconds. + + Faster, less precise version of IPython's timeit. `stmt` can be a statement + written as a string or a callable. + + Will do only 1 loop (like IPython's timeit) with no repetitions + (unlike IPython) for very slow functions. For fast functions, only does + enough loops to take 5 ms, which seems to produce similar results (on + Windows at least), and avoids doing an extraneous cycle that isn't + measured. + + """ + timer = timeit.Timer(stmt, setup) + + # determine number of calls per rep so total time for 1 rep >= 5 ms + x = 0 + for p in range(0, 10): + number = 10**p + x = timer.timeit(number) # seconds + if x >= 5e-3 / 10: # 5 ms for final test, 1/10th that for this one + break + if x > 1: # second + # If it's macroscopic, don't bother with repetitions + best = x + else: + number *= 10 + r = timer.repeat(repeat, number) + best = min(r) + + sec = best / number + return sec + + +def choose_conv_method(in1, in2, mode='full', measure=False): + """ + Find the fastest convolution/correlation method. + + This primarily exists to be called during the ``method='auto'`` option in + `convolve` and `correlate`. It can also be used to determine the value of + ``method`` for many different convolutions of the same dtype/shape. + In addition, it supports timing the convolution to adapt the value of + ``method`` to a particular set of inputs and/or hardware. + + Parameters + ---------- + in1 : array_like + The first argument passed into the convolution function. + in2 : array_like + The second argument passed into the convolution function. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output: + + ``full`` + The output is the full discrete linear convolution + of the inputs. (Default) + ``valid`` + The output consists only of those elements that do not + rely on the zero-padding. + ``same`` + The output is the same size as `in1`, centered + with respect to the 'full' output. + measure : bool, optional + If True, run and time the convolution of `in1` and `in2` with both + methods and return the fastest. If False (default), predict the fastest + method using precomputed values. + + Returns + ------- + method : str + A string indicating which convolution method is fastest, either + 'direct' or 'fft' + times : dict, optional + A dictionary containing the times (in seconds) needed for each method. + This value is only returned if ``measure=True``. + + See Also + -------- + convolve + correlate + + Notes + ----- + Generally, this method is 99% accurate for 2D signals and 85% accurate + for 1D signals for randomly chosen input sizes. For precision, use + ``measure=True`` to find the fastest method by timing the convolution. + This can be used to avoid the minimal overhead of finding the fastest + ``method`` later, or to adapt the value of ``method`` to a particular set + of inputs. + + Experiments were run on an Amazon EC2 r5a.2xlarge machine to test this + function. These experiments measured the ratio between the time required + when using ``method='auto'`` and the time required for the fastest method + (i.e., ``ratio = time_auto / min(time_fft, time_direct)``). In these + experiments, we found: + + * There is a 95% chance of this ratio being less than 1.5 for 1D signals + and a 99% chance of being less than 2.5 for 2D signals. + * The ratio was always less than 2.5/5 for 1D/2D signals respectively. + * This function is most inaccurate for 1D convolutions that take between 1 + and 10 milliseconds with ``method='direct'``. A good proxy for this + (at least in our experiments) is ``1e6 <= in1.size * in2.size <= 1e7``. + + The 2D results almost certainly generalize to 3D/4D/etc because the + implementation is the same (the 1D implementation is different). + + All the numbers above are specific to the EC2 machine. However, we did find + that this function generalizes fairly decently across hardware. The speed + tests were of similar quality (and even slightly better) than the same + tests performed on the machine to tune this function's numbers (a mid-2014 + 15-inch MacBook Pro with 16GB RAM and a 2.5GHz Intel i7 processor). + + There are cases when `fftconvolve` supports the inputs but this function + returns `direct` (e.g., to protect against floating point integer + precision). + + .. versionadded:: 0.19 + + Examples + -------- + Estimate the fastest method for a given input: + + >>> import numpy as np + >>> from scipy import signal + >>> rng = np.random.default_rng() + >>> img = rng.random((32, 32)) + >>> filter = rng.random((8, 8)) + >>> method = signal.choose_conv_method(img, filter, mode='same') + >>> method + 'fft' + + This can then be applied to other arrays of the same dtype and shape: + + >>> img2 = rng.random((32, 32)) + >>> filter2 = rng.random((8, 8)) + >>> corr2 = signal.correlate(img2, filter2, mode='same', method=method) + >>> conv2 = signal.convolve(img2, filter2, mode='same', method=method) + + The output of this function (``method``) works with `correlate` and + `convolve`. + + """ + volume = np.asarray(in1) + kernel = np.asarray(in2) + + if measure: + times = {} + for method in ['fft', 'direct']: + times[method] = _timeit_fast(lambda: convolve(volume, kernel, + mode=mode, method=method)) + + chosen_method = 'fft' if times['fft'] < times['direct'] else 'direct' + return chosen_method, times + + # for integer input, + # catch when more precision required than float provides (representing an + # integer as float can lose precision in fftconvolve if larger than 2**52) + if any([_numeric_arrays([x], kinds='ui') for x in [volume, kernel]]): + max_value = int(np.abs(volume).max()) * int(np.abs(kernel).max()) + max_value *= int(min(volume.size, kernel.size)) + if max_value > 2**np.finfo('float').nmant - 1: + return 'direct' + + if _numeric_arrays([volume, kernel], kinds='b'): + return 'direct' + + if _numeric_arrays([volume, kernel]): + if _fftconv_faster(volume, kernel, mode): + return 'fft' + + return 'direct' + + +def convolve(in1, in2, mode='full', method='auto'): + """ + Convolve two N-dimensional arrays. + + Convolve `in1` and `in2`, with the output size determined by the + `mode` argument. + + Parameters + ---------- + in1 : array_like + First input. + in2 : array_like + Second input. Should have the same number of dimensions as `in1`. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output: + + ``full`` + The output is the full discrete linear convolution + of the inputs. (Default) + ``valid`` + The output consists only of those elements that do not + rely on the zero-padding. In 'valid' mode, either `in1` or `in2` + must be at least as large as the other in every dimension. + ``same`` + The output is the same size as `in1`, centered + with respect to the 'full' output. + method : str {'auto', 'direct', 'fft'}, optional + A string indicating which method to use to calculate the convolution. + + ``direct`` + The convolution is determined directly from sums, the definition of + convolution. + ``fft`` + The Fourier Transform is used to perform the convolution by calling + `fftconvolve`. + ``auto`` + Automatically chooses direct or Fourier method based on an estimate + of which is faster (default). See Notes for more detail. + + .. versionadded:: 0.19.0 + + Returns + ------- + convolve : array + An N-dimensional array containing a subset of the discrete linear + convolution of `in1` with `in2`. + + Warns + ----- + RuntimeWarning + Use of the FFT convolution on input containing NAN or INF will lead + to the entire output being NAN or INF. Use method='direct' when your + input contains NAN or INF values. + + See Also + -------- + numpy.polymul : performs polynomial multiplication (same operation, but + also accepts poly1d objects) + choose_conv_method : chooses the fastest appropriate convolution method + fftconvolve : Always uses the FFT method. + oaconvolve : Uses the overlap-add method to do convolution, which is + generally faster when the input arrays are large and + significantly different in size. + + Notes + ----- + By default, `convolve` and `correlate` use ``method='auto'``, which calls + `choose_conv_method` to choose the fastest method using pre-computed + values (`choose_conv_method` can also measure real-world timing with a + keyword argument). Because `fftconvolve` relies on floating point numbers, + there are certain constraints that may force `method=direct` (more detail + in `choose_conv_method` docstring). + + Examples + -------- + Smooth a square pulse using a Hann window: + + >>> import numpy as np + >>> from scipy import signal + >>> sig = np.repeat([0., 1., 0.], 100) + >>> win = signal.windows.hann(50) + >>> filtered = signal.convolve(sig, win, mode='same') / sum(win) + + >>> import matplotlib.pyplot as plt + >>> fig, (ax_orig, ax_win, ax_filt) = plt.subplots(3, 1, sharex=True) + >>> ax_orig.plot(sig) + >>> ax_orig.set_title('Original pulse') + >>> ax_orig.margins(0, 0.1) + >>> ax_win.plot(win) + >>> ax_win.set_title('Filter impulse response') + >>> ax_win.margins(0, 0.1) + >>> ax_filt.plot(filtered) + >>> ax_filt.set_title('Filtered signal') + >>> ax_filt.margins(0, 0.1) + >>> fig.tight_layout() + >>> fig.show() + + """ + volume = np.asarray(in1) + kernel = np.asarray(in2) + + if volume.ndim == kernel.ndim == 0: + return volume * kernel + elif volume.ndim != kernel.ndim: + raise ValueError("volume and kernel should have the same " + "dimensionality") + + if _inputs_swap_needed(mode, volume.shape, kernel.shape): + # Convolution is commutative; order doesn't have any effect on output + volume, kernel = kernel, volume + + if method == 'auto': + method = choose_conv_method(volume, kernel, mode=mode) + + if method == 'fft': + out = fftconvolve(volume, kernel, mode=mode) + result_type = np.result_type(volume, kernel) + if result_type.kind in {'u', 'i'}: + out = np.around(out) + + if np.isnan(out.flat[0]) or np.isinf(out.flat[0]): + warnings.warn("Use of fft convolution on input with NAN or inf" + " results in NAN or inf output. Consider using" + " method='direct' instead.", + category=RuntimeWarning, stacklevel=2) + + return out.astype(result_type) + elif method == 'direct': + # fastpath to faster numpy.convolve for 1d inputs when possible + if _np_conv_ok(volume, kernel, mode): + return np.convolve(volume, kernel, mode) + + return correlate(volume, _reverse_and_conj(kernel), mode, 'direct') + else: + raise ValueError("Acceptable method flags are 'auto'," + " 'direct', or 'fft'.") + + +def order_filter(a, domain, rank): + """ + Perform an order filter on an N-D array. + + Perform an order filter on the array in. The domain argument acts as a + mask centered over each pixel. The non-zero elements of domain are + used to select elements surrounding each input pixel which are placed + in a list. The list is sorted, and the output for that pixel is the + element corresponding to rank in the sorted list. + + Parameters + ---------- + a : ndarray + The N-dimensional input array. + domain : array_like + A mask array with the same number of dimensions as `a`. + Each dimension should have an odd number of elements. + rank : int + A non-negative integer which selects the element from the + sorted list (0 corresponds to the smallest element, 1 is the + next smallest element, etc.). + + Returns + ------- + out : ndarray + The results of the order filter in an array with the same + shape as `a`. + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> x = np.arange(25).reshape(5, 5) + >>> domain = np.identity(3) + >>> x + array([[ 0, 1, 2, 3, 4], + [ 5, 6, 7, 8, 9], + [10, 11, 12, 13, 14], + [15, 16, 17, 18, 19], + [20, 21, 22, 23, 24]]) + >>> signal.order_filter(x, domain, 0) + array([[ 0., 0., 0., 0., 0.], + [ 0., 0., 1., 2., 0.], + [ 0., 5., 6., 7., 0.], + [ 0., 10., 11., 12., 0.], + [ 0., 0., 0., 0., 0.]]) + >>> signal.order_filter(x, domain, 2) + array([[ 6., 7., 8., 9., 4.], + [ 11., 12., 13., 14., 9.], + [ 16., 17., 18., 19., 14.], + [ 21., 22., 23., 24., 19.], + [ 20., 21., 22., 23., 24.]]) + + """ + domain = np.asarray(domain) + for dimsize in domain.shape: + if (dimsize % 2) != 1: + raise ValueError("Each dimension of domain argument " + "should have an odd number of elements.") + + a = np.asarray(a) + if a.dtype in [object, 'float128']: + mesg = (f"Using order_filter with arrays of dtype {a.dtype} is " + f"deprecated in SciPy 1.11 and will be removed in SciPy 1.14") + warnings.warn(mesg, DeprecationWarning, stacklevel=2) + + result = _sigtools._order_filterND(a, domain, rank) + else: + result = ndimage.rank_filter(a, rank, footprint=domain, mode='constant') + + return result + + +def medfilt(volume, kernel_size=None): + """ + Perform a median filter on an N-dimensional array. + + Apply a median filter to the input array using a local window-size + given by `kernel_size`. The array will automatically be zero-padded. + + Parameters + ---------- + volume : array_like + An N-dimensional input array. + kernel_size : array_like, optional + A scalar or an N-length list giving the size of the median filter + window in each dimension. Elements of `kernel_size` should be odd. + If `kernel_size` is a scalar, then this scalar is used as the size in + each dimension. Default size is 3 for each dimension. + + Returns + ------- + out : ndarray + An array the same size as input containing the median filtered + result. + + Warns + ----- + UserWarning + If array size is smaller than kernel size along any dimension + + See Also + -------- + scipy.ndimage.median_filter + scipy.signal.medfilt2d + + Notes + ----- + The more general function `scipy.ndimage.median_filter` has a more + efficient implementation of a median filter and therefore runs much faster. + + For 2-dimensional images with ``uint8``, ``float32`` or ``float64`` dtypes, + the specialised function `scipy.signal.medfilt2d` may be faster. + + """ + volume = np.atleast_1d(volume) + if kernel_size is None: + kernel_size = [3] * volume.ndim + kernel_size = np.asarray(kernel_size) + if kernel_size.shape == (): + kernel_size = np.repeat(kernel_size.item(), volume.ndim) + + for k in range(volume.ndim): + if (kernel_size[k] % 2) != 1: + raise ValueError("Each element of kernel_size should be odd.") + if any(k > s for k, s in zip(kernel_size, volume.shape)): + warnings.warn('kernel_size exceeds volume extent: the volume will be ' + 'zero-padded.', + stacklevel=2) + + domain = np.ones(kernel_size, dtype=volume.dtype) + + numels = np.prod(kernel_size, axis=0) + order = numels // 2 + + if volume.dtype in [np.bool_, np.complex64, np.complex128, np.clongdouble, + np.float16]: + raise ValueError(f"dtype={volume.dtype} is not supported by medfilt") + + if volume.dtype.char in ['O', 'g']: + mesg = (f"Using medfilt with arrays of dtype {volume.dtype} is " + f"deprecated in SciPy 1.11 and will be removed in SciPy 1.14") + warnings.warn(mesg, DeprecationWarning, stacklevel=2) + + result = _sigtools._order_filterND(volume, domain, order) + else: + size = math.prod(kernel_size) + result = ndimage.rank_filter(volume, size // 2, size=kernel_size, + mode='constant') + + return result + + +def wiener(im, mysize=None, noise=None): + """ + Perform a Wiener filter on an N-dimensional array. + + Apply a Wiener filter to the N-dimensional array `im`. + + Parameters + ---------- + im : ndarray + An N-dimensional array. + mysize : int or array_like, optional + A scalar or an N-length list giving the size of the Wiener filter + window in each dimension. Elements of mysize should be odd. + If mysize is a scalar, then this scalar is used as the size + in each dimension. + noise : float, optional + The noise-power to use. If None, then noise is estimated as the + average of the local variance of the input. + + Returns + ------- + out : ndarray + Wiener filtered result with the same shape as `im`. + + Notes + ----- + This implementation is similar to wiener2 in Matlab/Octave. + For more details see [1]_ + + References + ---------- + .. [1] Lim, Jae S., Two-Dimensional Signal and Image Processing, + Englewood Cliffs, NJ, Prentice Hall, 1990, p. 548. + + Examples + -------- + >>> from scipy.datasets import face + >>> from scipy.signal import wiener + >>> import matplotlib.pyplot as plt + >>> import numpy as np + >>> rng = np.random.default_rng() + >>> img = rng.random((40, 40)) #Create a random image + >>> filtered_img = wiener(img, (5, 5)) #Filter the image + >>> f, (plot1, plot2) = plt.subplots(1, 2) + >>> plot1.imshow(img) + >>> plot2.imshow(filtered_img) + >>> plt.show() + + """ + im = np.asarray(im) + if mysize is None: + mysize = [3] * im.ndim + mysize = np.asarray(mysize) + if mysize.shape == (): + mysize = np.repeat(mysize.item(), im.ndim) + + # Estimate the local mean + lMean = correlate(im, np.ones(mysize), 'same') / np.prod(mysize, axis=0) + + # Estimate the local variance + lVar = (correlate(im ** 2, np.ones(mysize), 'same') / + np.prod(mysize, axis=0) - lMean ** 2) + + # Estimate the noise power if needed. + if noise is None: + noise = np.mean(np.ravel(lVar), axis=0) + + res = (im - lMean) + res *= (1 - noise / lVar) + res += lMean + out = np.where(lVar < noise, lMean, res) + + return out + + +def convolve2d(in1, in2, mode='full', boundary='fill', fillvalue=0): + """ + Convolve two 2-dimensional arrays. + + Convolve `in1` and `in2` with output size determined by `mode`, and + boundary conditions determined by `boundary` and `fillvalue`. + + Parameters + ---------- + in1 : array_like + First input. + in2 : array_like + Second input. Should have the same number of dimensions as `in1`. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output: + + ``full`` + The output is the full discrete linear convolution + of the inputs. (Default) + ``valid`` + The output consists only of those elements that do not + rely on the zero-padding. In 'valid' mode, either `in1` or `in2` + must be at least as large as the other in every dimension. + ``same`` + The output is the same size as `in1`, centered + with respect to the 'full' output. + boundary : str {'fill', 'wrap', 'symm'}, optional + A flag indicating how to handle boundaries: + + ``fill`` + pad input arrays with fillvalue. (default) + ``wrap`` + circular boundary conditions. + ``symm`` + symmetrical boundary conditions. + + fillvalue : scalar, optional + Value to fill pad input arrays with. Default is 0. + + Returns + ------- + out : ndarray + A 2-dimensional array containing a subset of the discrete linear + convolution of `in1` with `in2`. + + Examples + -------- + Compute the gradient of an image by 2D convolution with a complex Scharr + operator. (Horizontal operator is real, vertical is imaginary.) Use + symmetric boundary condition to avoid creating edges at the image + boundaries. + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy import datasets + >>> ascent = datasets.ascent() + >>> scharr = np.array([[ -3-3j, 0-10j, +3 -3j], + ... [-10+0j, 0+ 0j, +10 +0j], + ... [ -3+3j, 0+10j, +3 +3j]]) # Gx + j*Gy + >>> grad = signal.convolve2d(ascent, scharr, boundary='symm', mode='same') + + >>> import matplotlib.pyplot as plt + >>> fig, (ax_orig, ax_mag, ax_ang) = plt.subplots(3, 1, figsize=(6, 15)) + >>> ax_orig.imshow(ascent, cmap='gray') + >>> ax_orig.set_title('Original') + >>> ax_orig.set_axis_off() + >>> ax_mag.imshow(np.absolute(grad), cmap='gray') + >>> ax_mag.set_title('Gradient magnitude') + >>> ax_mag.set_axis_off() + >>> ax_ang.imshow(np.angle(grad), cmap='hsv') # hsv is cyclic, like angles + >>> ax_ang.set_title('Gradient orientation') + >>> ax_ang.set_axis_off() + >>> fig.show() + + """ + in1 = np.asarray(in1) + in2 = np.asarray(in2) + + if not in1.ndim == in2.ndim == 2: + raise ValueError('convolve2d inputs must both be 2-D arrays') + + if _inputs_swap_needed(mode, in1.shape, in2.shape): + in1, in2 = in2, in1 + + val = _valfrommode(mode) + bval = _bvalfromboundary(boundary) + out = _sigtools._convolve2d(in1, in2, 1, val, bval, fillvalue) + return out + + +def correlate2d(in1, in2, mode='full', boundary='fill', fillvalue=0): + """ + Cross-correlate two 2-dimensional arrays. + + Cross correlate `in1` and `in2` with output size determined by `mode`, and + boundary conditions determined by `boundary` and `fillvalue`. + + Parameters + ---------- + in1 : array_like + First input. + in2 : array_like + Second input. Should have the same number of dimensions as `in1`. + mode : str {'full', 'valid', 'same'}, optional + A string indicating the size of the output: + + ``full`` + The output is the full discrete linear cross-correlation + of the inputs. (Default) + ``valid`` + The output consists only of those elements that do not + rely on the zero-padding. In 'valid' mode, either `in1` or `in2` + must be at least as large as the other in every dimension. + ``same`` + The output is the same size as `in1`, centered + with respect to the 'full' output. + boundary : str {'fill', 'wrap', 'symm'}, optional + A flag indicating how to handle boundaries: + + ``fill`` + pad input arrays with fillvalue. (default) + ``wrap`` + circular boundary conditions. + ``symm`` + symmetrical boundary conditions. + + fillvalue : scalar, optional + Value to fill pad input arrays with. Default is 0. + + Returns + ------- + correlate2d : ndarray + A 2-dimensional array containing a subset of the discrete linear + cross-correlation of `in1` with `in2`. + + Notes + ----- + When using "same" mode with even-length inputs, the outputs of `correlate` + and `correlate2d` differ: There is a 1-index offset between them. + + Examples + -------- + Use 2D cross-correlation to find the location of a template in a noisy + image: + + >>> import numpy as np + >>> from scipy import signal + >>> from scipy import datasets + >>> rng = np.random.default_rng() + >>> face = datasets.face(gray=True) - datasets.face(gray=True).mean() + >>> template = np.copy(face[300:365, 670:750]) # right eye + >>> template -= template.mean() + >>> face = face + rng.standard_normal(face.shape) * 50 # add noise + >>> corr = signal.correlate2d(face, template, boundary='symm', mode='same') + >>> y, x = np.unravel_index(np.argmax(corr), corr.shape) # find the match + + >>> import matplotlib.pyplot as plt + >>> fig, (ax_orig, ax_template, ax_corr) = plt.subplots(3, 1, + ... figsize=(6, 15)) + >>> ax_orig.imshow(face, cmap='gray') + >>> ax_orig.set_title('Original') + >>> ax_orig.set_axis_off() + >>> ax_template.imshow(template, cmap='gray') + >>> ax_template.set_title('Template') + >>> ax_template.set_axis_off() + >>> ax_corr.imshow(corr, cmap='gray') + >>> ax_corr.set_title('Cross-correlation') + >>> ax_corr.set_axis_off() + >>> ax_orig.plot(x, y, 'ro') + >>> fig.show() + + """ + in1 = np.asarray(in1) + in2 = np.asarray(in2) + + if not in1.ndim == in2.ndim == 2: + raise ValueError('correlate2d inputs must both be 2-D arrays') + + swapped_inputs = _inputs_swap_needed(mode, in1.shape, in2.shape) + if swapped_inputs: + in1, in2 = in2, in1 + + val = _valfrommode(mode) + bval = _bvalfromboundary(boundary) + out = _sigtools._convolve2d(in1, in2.conj(), 0, val, bval, fillvalue) + + if swapped_inputs: + out = out[::-1, ::-1] + + return out + + +def medfilt2d(input, kernel_size=3): + """ + Median filter a 2-dimensional array. + + Apply a median filter to the `input` array using a local window-size + given by `kernel_size` (must be odd). The array is zero-padded + automatically. + + Parameters + ---------- + input : array_like + A 2-dimensional input array. + kernel_size : array_like, optional + A scalar or a list of length 2, giving the size of the + median filter window in each dimension. Elements of + `kernel_size` should be odd. If `kernel_size` is a scalar, + then this scalar is used as the size in each dimension. + Default is a kernel of size (3, 3). + + Returns + ------- + out : ndarray + An array the same size as input containing the median filtered + result. + + See Also + -------- + scipy.ndimage.median_filter + + Notes + ----- + This is faster than `medfilt` when the input dtype is ``uint8``, + ``float32``, or ``float64``; for other types, this falls back to + `medfilt`. In some situations, `scipy.ndimage.median_filter` may be + faster than this function. + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> x = np.arange(25).reshape(5, 5) + >>> x + array([[ 0, 1, 2, 3, 4], + [ 5, 6, 7, 8, 9], + [10, 11, 12, 13, 14], + [15, 16, 17, 18, 19], + [20, 21, 22, 23, 24]]) + + # Replaces i,j with the median out of 5*5 window + + >>> signal.medfilt2d(x, kernel_size=5) + array([[ 0, 0, 2, 0, 0], + [ 0, 3, 7, 4, 0], + [ 2, 8, 12, 9, 4], + [ 0, 8, 12, 9, 0], + [ 0, 0, 12, 0, 0]]) + + # Replaces i,j with the median out of default 3*3 window + + >>> signal.medfilt2d(x) + array([[ 0, 1, 2, 3, 0], + [ 1, 6, 7, 8, 4], + [ 6, 11, 12, 13, 9], + [11, 16, 17, 18, 14], + [ 0, 16, 17, 18, 0]]) + + # Replaces i,j with the median out of default 5*3 window + + >>> signal.medfilt2d(x, kernel_size=[5,3]) + array([[ 0, 1, 2, 3, 0], + [ 0, 6, 7, 8, 3], + [ 5, 11, 12, 13, 8], + [ 5, 11, 12, 13, 8], + [ 0, 11, 12, 13, 0]]) + + # Replaces i,j with the median out of default 3*5 window + + >>> signal.medfilt2d(x, kernel_size=[3,5]) + array([[ 0, 0, 2, 1, 0], + [ 1, 5, 7, 6, 3], + [ 6, 10, 12, 11, 8], + [11, 15, 17, 16, 13], + [ 0, 15, 17, 16, 0]]) + + # As seen in the examples, + # kernel numbers must be odd and not exceed original array dim + + """ + image = np.asarray(input) + + # checking dtype.type, rather than just dtype, is necessary for + # excluding np.longdouble with MS Visual C. + if image.dtype.type not in (np.ubyte, np.float32, np.float64): + return medfilt(image, kernel_size) + + if kernel_size is None: + kernel_size = [3] * 2 + kernel_size = np.asarray(kernel_size) + if kernel_size.shape == (): + kernel_size = np.repeat(kernel_size.item(), 2) + + for size in kernel_size: + if (size % 2) != 1: + raise ValueError("Each element of kernel_size should be odd.") + + return _sigtools._medfilt2d(image, kernel_size) + + +def lfilter(b, a, x, axis=-1, zi=None): + """ + Filter data along one-dimension with an IIR or FIR filter. + + Filter a data sequence, `x`, using a digital filter. This works for many + fundamental data types (including Object type). The filter is a direct + form II transposed implementation of the standard difference equation + (see Notes). + + The function `sosfilt` (and filter design using ``output='sos'``) should be + preferred over `lfilter` for most filtering tasks, as second-order sections + have fewer numerical problems. + + Parameters + ---------- + b : array_like + The numerator coefficient vector in a 1-D sequence. + a : array_like + The denominator coefficient vector in a 1-D sequence. If ``a[0]`` + is not 1, then both `a` and `b` are normalized by ``a[0]``. + x : array_like + An N-dimensional input array. + axis : int, optional + The axis of the input data array along which to apply the + linear filter. The filter is applied to each subarray along + this axis. Default is -1. + zi : array_like, optional + Initial conditions for the filter delays. It is a vector + (or array of vectors for an N-dimensional input) of length + ``max(len(a), len(b)) - 1``. If `zi` is None or is not given then + initial rest is assumed. See `lfiltic` for more information. + + Returns + ------- + y : array + The output of the digital filter. + zf : array, optional + If `zi` is None, this is not returned, otherwise, `zf` holds the + final filter delay values. + + See Also + -------- + lfiltic : Construct initial conditions for `lfilter`. + lfilter_zi : Compute initial state (steady state of step response) for + `lfilter`. + filtfilt : A forward-backward filter, to obtain a filter with zero phase. + savgol_filter : A Savitzky-Golay filter. + sosfilt: Filter data using cascaded second-order sections. + sosfiltfilt: A forward-backward filter using second-order sections. + + Notes + ----- + The filter function is implemented as a direct II transposed structure. + This means that the filter implements:: + + a[0]*y[n] = b[0]*x[n] + b[1]*x[n-1] + ... + b[M]*x[n-M] + - a[1]*y[n-1] - ... - a[N]*y[n-N] + + where `M` is the degree of the numerator, `N` is the degree of the + denominator, and `n` is the sample number. It is implemented using + the following difference equations (assuming M = N):: + + a[0]*y[n] = b[0] * x[n] + d[0][n-1] + d[0][n] = b[1] * x[n] - a[1] * y[n] + d[1][n-1] + d[1][n] = b[2] * x[n] - a[2] * y[n] + d[2][n-1] + ... + d[N-2][n] = b[N-1]*x[n] - a[N-1]*y[n] + d[N-1][n-1] + d[N-1][n] = b[N] * x[n] - a[N] * y[n] + + where `d` are the state variables. + + The rational transfer function describing this filter in the + z-transform domain is:: + + -1 -M + b[0] + b[1]z + ... + b[M] z + Y(z) = -------------------------------- X(z) + -1 -N + a[0] + a[1]z + ... + a[N] z + + Examples + -------- + Generate a noisy signal to be filtered: + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + >>> t = np.linspace(-1, 1, 201) + >>> x = (np.sin(2*np.pi*0.75*t*(1-t) + 2.1) + + ... 0.1*np.sin(2*np.pi*1.25*t + 1) + + ... 0.18*np.cos(2*np.pi*3.85*t)) + >>> xn = x + rng.standard_normal(len(t)) * 0.08 + + Create an order 3 lowpass butterworth filter: + + >>> b, a = signal.butter(3, 0.05) + + Apply the filter to xn. Use lfilter_zi to choose the initial condition of + the filter: + + >>> zi = signal.lfilter_zi(b, a) + >>> z, _ = signal.lfilter(b, a, xn, zi=zi*xn[0]) + + Apply the filter again, to have a result filtered at an order the same as + filtfilt: + + >>> z2, _ = signal.lfilter(b, a, z, zi=zi*z[0]) + + Use filtfilt to apply the filter: + + >>> y = signal.filtfilt(b, a, xn) + + Plot the original signal and the various filtered versions: + + >>> plt.figure + >>> plt.plot(t, xn, 'b', alpha=0.75) + >>> plt.plot(t, z, 'r--', t, z2, 'r', t, y, 'k') + >>> plt.legend(('noisy signal', 'lfilter, once', 'lfilter, twice', + ... 'filtfilt'), loc='best') + >>> plt.grid(True) + >>> plt.show() + + """ + b = np.atleast_1d(b) + a = np.atleast_1d(a) + if len(a) == 1: + # This path only supports types fdgFDGO to mirror _linear_filter below. + # Any of b, a, x, or zi can set the dtype, but there is no default + # casting of other types; instead a NotImplementedError is raised. + b = np.asarray(b) + a = np.asarray(a) + if b.ndim != 1 and a.ndim != 1: + raise ValueError('object of too small depth for desired array') + x = _validate_x(x) + inputs = [b, a, x] + if zi is not None: + # _linear_filter does not broadcast zi, but does do expansion of + # singleton dims. + zi = np.asarray(zi) + if zi.ndim != x.ndim: + raise ValueError('object of too small depth for desired array') + expected_shape = list(x.shape) + expected_shape[axis] = b.shape[0] - 1 + expected_shape = tuple(expected_shape) + # check the trivial case where zi is the right shape first + if zi.shape != expected_shape: + strides = zi.ndim * [None] + if axis < 0: + axis += zi.ndim + for k in range(zi.ndim): + if k == axis and zi.shape[k] == expected_shape[k]: + strides[k] = zi.strides[k] + elif k != axis and zi.shape[k] == expected_shape[k]: + strides[k] = zi.strides[k] + elif k != axis and zi.shape[k] == 1: + strides[k] = 0 + else: + raise ValueError('Unexpected shape for zi: expected ' + f'{expected_shape}, found {zi.shape}.') + zi = np.lib.stride_tricks.as_strided(zi, expected_shape, + strides) + inputs.append(zi) + dtype = np.result_type(*inputs) + + if dtype.char not in 'fdgFDGO': + raise NotImplementedError("input type '%s' not supported" % dtype) + + b = np.array(b, dtype=dtype) + a = np.asarray(a, dtype=dtype) + b /= a[0] + x = np.asarray(x, dtype=dtype) + + out_full = np.apply_along_axis(lambda y: np.convolve(b, y), axis, x) + ind = out_full.ndim * [slice(None)] + if zi is not None: + ind[axis] = slice(zi.shape[axis]) + out_full[tuple(ind)] += zi + + ind[axis] = slice(out_full.shape[axis] - len(b) + 1) + out = out_full[tuple(ind)] + + if zi is None: + return out + else: + ind[axis] = slice(out_full.shape[axis] - len(b) + 1, None) + zf = out_full[tuple(ind)] + return out, zf + else: + if zi is None: + return _sigtools._linear_filter(b, a, x, axis) + else: + return _sigtools._linear_filter(b, a, x, axis, zi) + + +def lfiltic(b, a, y, x=None): + """ + Construct initial conditions for lfilter given input and output vectors. + + Given a linear filter (b, a) and initial conditions on the output `y` + and the input `x`, return the initial conditions on the state vector zi + which is used by `lfilter` to generate the output given the input. + + Parameters + ---------- + b : array_like + Linear filter term. + a : array_like + Linear filter term. + y : array_like + Initial conditions. + + If ``N = len(a) - 1``, then ``y = {y[-1], y[-2], ..., y[-N]}``. + + If `y` is too short, it is padded with zeros. + x : array_like, optional + Initial conditions. + + If ``M = len(b) - 1``, then ``x = {x[-1], x[-2], ..., x[-M]}``. + + If `x` is not given, its initial conditions are assumed zero. + + If `x` is too short, it is padded with zeros. + + Returns + ------- + zi : ndarray + The state vector ``zi = {z_0[-1], z_1[-1], ..., z_K-1[-1]}``, + where ``K = max(M, N)``. + + See Also + -------- + lfilter, lfilter_zi + + """ + N = np.size(a) - 1 + M = np.size(b) - 1 + K = max(M, N) + y = np.asarray(y) + + if x is None: + result_type = np.result_type(np.asarray(b), np.asarray(a), y) + if result_type.kind in 'bui': + result_type = np.float64 + x = np.zeros(M, dtype=result_type) + else: + x = np.asarray(x) + + result_type = np.result_type(np.asarray(b), np.asarray(a), y, x) + if result_type.kind in 'bui': + result_type = np.float64 + x = x.astype(result_type) + + L = np.size(x) + if L < M: + x = np.r_[x, np.zeros(M - L)] + + y = y.astype(result_type) + zi = np.zeros(K, result_type) + + L = np.size(y) + if L < N: + y = np.r_[y, np.zeros(N - L)] + + for m in range(M): + zi[m] = np.sum(b[m + 1:] * x[:M - m], axis=0) + + for m in range(N): + zi[m] -= np.sum(a[m + 1:] * y[:N - m], axis=0) + + return zi + + +def deconvolve(signal, divisor): + """Deconvolves ``divisor`` out of ``signal`` using inverse filtering. + + Returns the quotient and remainder such that + ``signal = convolve(divisor, quotient) + remainder`` + + Parameters + ---------- + signal : (N,) array_like + Signal data, typically a recorded signal + divisor : (N,) array_like + Divisor data, typically an impulse response or filter that was + applied to the original signal + + Returns + ------- + quotient : ndarray + Quotient, typically the recovered original signal + remainder : ndarray + Remainder + + See Also + -------- + numpy.polydiv : performs polynomial division (same operation, but + also accepts poly1d objects) + + Examples + -------- + Deconvolve a signal that's been filtered: + + >>> from scipy import signal + >>> original = [0, 1, 0, 0, 1, 1, 0, 0] + >>> impulse_response = [2, 1] + >>> recorded = signal.convolve(impulse_response, original) + >>> recorded + array([0, 2, 1, 0, 2, 3, 1, 0, 0]) + >>> recovered, remainder = signal.deconvolve(recorded, impulse_response) + >>> recovered + array([ 0., 1., 0., 0., 1., 1., 0., 0.]) + + """ + num = np.atleast_1d(signal) + den = np.atleast_1d(divisor) + if num.ndim > 1: + raise ValueError("signal must be 1-D.") + if den.ndim > 1: + raise ValueError("divisor must be 1-D.") + N = len(num) + D = len(den) + if D > N: + quot = [] + rem = num + else: + input = np.zeros(N - D + 1, float) + input[0] = 1 + quot = lfilter(num, den, input) + rem = num - convolve(den, quot, mode='full') + return quot, rem + + +def hilbert(x, N=None, axis=-1): + """ + Compute the analytic signal, using the Hilbert transform. + + The transformation is done along the last axis by default. + + Parameters + ---------- + x : array_like + Signal data. Must be real. + N : int, optional + Number of Fourier components. Default: ``x.shape[axis]`` + axis : int, optional + Axis along which to do the transformation. Default: -1. + + Returns + ------- + xa : ndarray + Analytic signal of `x`, of each 1-D array along `axis` + + Notes + ----- + The analytic signal ``x_a(t)`` of signal ``x(t)`` is: + + .. math:: x_a = F^{-1}(F(x) 2U) = x + i y + + where `F` is the Fourier transform, `U` the unit step function, + and `y` the Hilbert transform of `x`. [1]_ + + In other words, the negative half of the frequency spectrum is zeroed + out, turning the real-valued signal into a complex signal. The Hilbert + transformed signal can be obtained from ``np.imag(hilbert(x))``, and the + original signal from ``np.real(hilbert(x))``. + + References + ---------- + .. [1] Wikipedia, "Analytic signal". + https://en.wikipedia.org/wiki/Analytic_signal + .. [2] Leon Cohen, "Time-Frequency Analysis", 1995. Chapter 2. + .. [3] Alan V. Oppenheim, Ronald W. Schafer. Discrete-Time Signal + Processing, Third Edition, 2009. Chapter 12. + ISBN 13: 978-1292-02572-8 + + Examples + -------- + In this example we use the Hilbert transform to determine the amplitude + envelope and instantaneous frequency of an amplitude-modulated signal. + + >>> import numpy as np + >>> import matplotlib.pyplot as plt + >>> from scipy.signal import hilbert, chirp + + >>> duration = 1.0 + >>> fs = 400.0 + >>> samples = int(fs*duration) + >>> t = np.arange(samples) / fs + + We create a chirp of which the frequency increases from 20 Hz to 100 Hz and + apply an amplitude modulation. + + >>> signal = chirp(t, 20.0, t[-1], 100.0) + >>> signal *= (1.0 + 0.5 * np.sin(2.0*np.pi*3.0*t) ) + + The amplitude envelope is given by magnitude of the analytic signal. The + instantaneous frequency can be obtained by differentiating the + instantaneous phase in respect to time. The instantaneous phase corresponds + to the phase angle of the analytic signal. + + >>> analytic_signal = hilbert(signal) + >>> amplitude_envelope = np.abs(analytic_signal) + >>> instantaneous_phase = np.unwrap(np.angle(analytic_signal)) + >>> instantaneous_frequency = (np.diff(instantaneous_phase) / + ... (2.0*np.pi) * fs) + + >>> fig, (ax0, ax1) = plt.subplots(nrows=2) + >>> ax0.plot(t, signal, label='signal') + >>> ax0.plot(t, amplitude_envelope, label='envelope') + >>> ax0.set_xlabel("time in seconds") + >>> ax0.legend() + >>> ax1.plot(t[1:], instantaneous_frequency) + >>> ax1.set_xlabel("time in seconds") + >>> ax1.set_ylim(0.0, 120.0) + >>> fig.tight_layout() + + """ + x = np.asarray(x) + if np.iscomplexobj(x): + raise ValueError("x must be real.") + if N is None: + N = x.shape[axis] + if N <= 0: + raise ValueError("N must be positive.") + + Xf = sp_fft.fft(x, N, axis=axis) + h = np.zeros(N, dtype=Xf.dtype) + if N % 2 == 0: + h[0] = h[N // 2] = 1 + h[1:N // 2] = 2 + else: + h[0] = 1 + h[1:(N + 1) // 2] = 2 + + if x.ndim > 1: + ind = [np.newaxis] * x.ndim + ind[axis] = slice(None) + h = h[tuple(ind)] + x = sp_fft.ifft(Xf * h, axis=axis) + return x + + +def hilbert2(x, N=None): + """ + Compute the '2-D' analytic signal of `x` + + Parameters + ---------- + x : array_like + 2-D signal data. + N : int or tuple of two ints, optional + Number of Fourier components. Default is ``x.shape`` + + Returns + ------- + xa : ndarray + Analytic signal of `x` taken along axes (0,1). + + References + ---------- + .. [1] Wikipedia, "Analytic signal", + https://en.wikipedia.org/wiki/Analytic_signal + + """ + x = np.atleast_2d(x) + if x.ndim > 2: + raise ValueError("x must be 2-D.") + if np.iscomplexobj(x): + raise ValueError("x must be real.") + if N is None: + N = x.shape + elif isinstance(N, int): + if N <= 0: + raise ValueError("N must be positive.") + N = (N, N) + elif len(N) != 2 or np.any(np.asarray(N) <= 0): + raise ValueError("When given as a tuple, N must hold exactly " + "two positive integers") + + Xf = sp_fft.fft2(x, N, axes=(0, 1)) + h1 = np.zeros(N[0], dtype=Xf.dtype) + h2 = np.zeros(N[1], dtype=Xf.dtype) + for h in (h1, h2): + N1 = h.shape[0] + if N1 % 2 == 0: + h[0] = h[N1 // 2] = 1 + h[1:N1 // 2] = 2 + else: + h[0] = 1 + h[1:(N1 + 1) // 2] = 2 + + h = h1[:, np.newaxis] * h2[np.newaxis, :] + k = x.ndim + while k > 2: + h = h[:, np.newaxis] + k -= 1 + x = sp_fft.ifft2(Xf * h, axes=(0, 1)) + return x + + +_msg_cplx_sort="""cmplx_sort was deprecated in SciPy 1.12 and will be removed +in SciPy 1.15. The exact equivalent for a numpy array argument is +>>> def cmplx_sort(p): +... idx = np.argsort(abs(p)) +... return np.take(p, idx, 0), idx +""" + +def cmplx_sort(p): + warnings.warn(_msg_cplx_sort, DeprecationWarning, stacklevel=2) + return _cmplx_sort(p) + + +def _cmplx_sort(p): + """Sort roots based on magnitude. + + Parameters + ---------- + p : array_like + The roots to sort, as a 1-D array. + + Returns + ------- + p_sorted : ndarray + Sorted roots. + indx : ndarray + Array of indices needed to sort the input `p`. + + Examples + -------- + >>> from scipy import signal + >>> vals = [1, 4, 1+1.j, 3] + >>> p_sorted, indx = signal.cmplx_sort(vals) + >>> p_sorted + array([1.+0.j, 1.+1.j, 3.+0.j, 4.+0.j]) + >>> indx + array([0, 2, 3, 1]) + """ + p = np.asarray(p) + indx = np.argsort(abs(p)) + return np.take(p, indx, 0), indx + + +def unique_roots(p, tol=1e-3, rtype='min'): + """Determine unique roots and their multiplicities from a list of roots. + + Parameters + ---------- + p : array_like + The list of roots. + tol : float, optional + The tolerance for two roots to be considered equal in terms of + the distance between them. Default is 1e-3. Refer to Notes about + the details on roots grouping. + rtype : {'max', 'maximum', 'min', 'minimum', 'avg', 'mean'}, optional + How to determine the returned root if multiple roots are within + `tol` of each other. + + - 'max', 'maximum': pick the maximum of those roots + - 'min', 'minimum': pick the minimum of those roots + - 'avg', 'mean': take the average of those roots + + When finding minimum or maximum among complex roots they are compared + first by the real part and then by the imaginary part. + + Returns + ------- + unique : ndarray + The list of unique roots. + multiplicity : ndarray + The multiplicity of each root. + + Notes + ----- + If we have 3 roots ``a``, ``b`` and ``c``, such that ``a`` is close to + ``b`` and ``b`` is close to ``c`` (distance is less than `tol`), then it + doesn't necessarily mean that ``a`` is close to ``c``. It means that roots + grouping is not unique. In this function we use "greedy" grouping going + through the roots in the order they are given in the input `p`. + + This utility function is not specific to roots but can be used for any + sequence of values for which uniqueness and multiplicity has to be + determined. For a more general routine, see `numpy.unique`. + + Examples + -------- + >>> from scipy import signal + >>> vals = [0, 1.3, 1.31, 2.8, 1.25, 2.2, 10.3] + >>> uniq, mult = signal.unique_roots(vals, tol=2e-2, rtype='avg') + + Check which roots have multiplicity larger than 1: + + >>> uniq[mult > 1] + array([ 1.305]) + """ + if rtype in ['max', 'maximum']: + reduce = np.max + elif rtype in ['min', 'minimum']: + reduce = np.min + elif rtype in ['avg', 'mean']: + reduce = np.mean + else: + raise ValueError("`rtype` must be one of " + "{'max', 'maximum', 'min', 'minimum', 'avg', 'mean'}") + + p = np.asarray(p) + + points = np.empty((len(p), 2)) + points[:, 0] = np.real(p) + points[:, 1] = np.imag(p) + tree = cKDTree(points) + + p_unique = [] + p_multiplicity = [] + used = np.zeros(len(p), dtype=bool) + for i in range(len(p)): + if used[i]: + continue + + group = tree.query_ball_point(points[i], tol) + group = [x for x in group if not used[x]] + + p_unique.append(reduce(p[group])) + p_multiplicity.append(len(group)) + + used[group] = True + + return np.asarray(p_unique), np.asarray(p_multiplicity) + + +def invres(r, p, k, tol=1e-3, rtype='avg'): + """Compute b(s) and a(s) from partial fraction expansion. + + If `M` is the degree of numerator `b` and `N` the degree of denominator + `a`:: + + b(s) b[0] s**(M) + b[1] s**(M-1) + ... + b[M] + H(s) = ------ = ------------------------------------------ + a(s) a[0] s**(N) + a[1] s**(N-1) + ... + a[N] + + then the partial-fraction expansion H(s) is defined as:: + + r[0] r[1] r[-1] + = -------- + -------- + ... + --------- + k(s) + (s-p[0]) (s-p[1]) (s-p[-1]) + + If there are any repeated roots (closer together than `tol`), then H(s) + has terms like:: + + r[i] r[i+1] r[i+n-1] + -------- + ----------- + ... + ----------- + (s-p[i]) (s-p[i])**2 (s-p[i])**n + + This function is used for polynomials in positive powers of s or z, + such as analog filters or digital filters in controls engineering. For + negative powers of z (typical for digital filters in DSP), use `invresz`. + + Parameters + ---------- + r : array_like + Residues corresponding to the poles. For repeated poles, the residues + must be ordered to correspond to ascending by power fractions. + p : array_like + Poles. Equal poles must be adjacent. + k : array_like + Coefficients of the direct polynomial term. + tol : float, optional + The tolerance for two roots to be considered equal in terms of + the distance between them. Default is 1e-3. See `unique_roots` + for further details. + rtype : {'avg', 'min', 'max'}, optional + Method for computing a root to represent a group of identical roots. + Default is 'avg'. See `unique_roots` for further details. + + Returns + ------- + b : ndarray + Numerator polynomial coefficients. + a : ndarray + Denominator polynomial coefficients. + + See Also + -------- + residue, invresz, unique_roots + + """ + r = np.atleast_1d(r) + p = np.atleast_1d(p) + k = np.trim_zeros(np.atleast_1d(k), 'f') + + unique_poles, multiplicity = _group_poles(p, tol, rtype) + factors, denominator = _compute_factors(unique_poles, multiplicity, + include_powers=True) + + if len(k) == 0: + numerator = 0 + else: + numerator = np.polymul(k, denominator) + + for residue, factor in zip(r, factors): + numerator = np.polyadd(numerator, residue * factor) + + return numerator, denominator + + +def _compute_factors(roots, multiplicity, include_powers=False): + """Compute the total polynomial divided by factors for each root.""" + current = np.array([1]) + suffixes = [current] + for pole, mult in zip(roots[-1:0:-1], multiplicity[-1:0:-1]): + monomial = np.array([1, -pole]) + for _ in range(mult): + current = np.polymul(current, monomial) + suffixes.append(current) + suffixes = suffixes[::-1] + + factors = [] + current = np.array([1]) + for pole, mult, suffix in zip(roots, multiplicity, suffixes): + monomial = np.array([1, -pole]) + block = [] + for i in range(mult): + if i == 0 or include_powers: + block.append(np.polymul(current, suffix)) + current = np.polymul(current, monomial) + factors.extend(reversed(block)) + + return factors, current + + +def _compute_residues(poles, multiplicity, numerator): + denominator_factors, _ = _compute_factors(poles, multiplicity) + numerator = numerator.astype(poles.dtype) + + residues = [] + for pole, mult, factor in zip(poles, multiplicity, + denominator_factors): + if mult == 1: + residues.append(np.polyval(numerator, pole) / + np.polyval(factor, pole)) + else: + numer = numerator.copy() + monomial = np.array([1, -pole]) + factor, d = np.polydiv(factor, monomial) + + block = [] + for _ in range(mult): + numer, n = np.polydiv(numer, monomial) + r = n[0] / d[0] + numer = np.polysub(numer, r * factor) + block.append(r) + + residues.extend(reversed(block)) + + return np.asarray(residues) + + +def residue(b, a, tol=1e-3, rtype='avg'): + """Compute partial-fraction expansion of b(s) / a(s). + + If `M` is the degree of numerator `b` and `N` the degree of denominator + `a`:: + + b(s) b[0] s**(M) + b[1] s**(M-1) + ... + b[M] + H(s) = ------ = ------------------------------------------ + a(s) a[0] s**(N) + a[1] s**(N-1) + ... + a[N] + + then the partial-fraction expansion H(s) is defined as:: + + r[0] r[1] r[-1] + = -------- + -------- + ... + --------- + k(s) + (s-p[0]) (s-p[1]) (s-p[-1]) + + If there are any repeated roots (closer together than `tol`), then H(s) + has terms like:: + + r[i] r[i+1] r[i+n-1] + -------- + ----------- + ... + ----------- + (s-p[i]) (s-p[i])**2 (s-p[i])**n + + This function is used for polynomials in positive powers of s or z, + such as analog filters or digital filters in controls engineering. For + negative powers of z (typical for digital filters in DSP), use `residuez`. + + See Notes for details about the algorithm. + + Parameters + ---------- + b : array_like + Numerator polynomial coefficients. + a : array_like + Denominator polynomial coefficients. + tol : float, optional + The tolerance for two roots to be considered equal in terms of + the distance between them. Default is 1e-3. See `unique_roots` + for further details. + rtype : {'avg', 'min', 'max'}, optional + Method for computing a root to represent a group of identical roots. + Default is 'avg'. See `unique_roots` for further details. + + Returns + ------- + r : ndarray + Residues corresponding to the poles. For repeated poles, the residues + are ordered to correspond to ascending by power fractions. + p : ndarray + Poles ordered by magnitude in ascending order. + k : ndarray + Coefficients of the direct polynomial term. + + See Also + -------- + invres, residuez, numpy.poly, unique_roots + + Notes + ----- + The "deflation through subtraction" algorithm is used for + computations --- method 6 in [1]_. + + The form of partial fraction expansion depends on poles multiplicity in + the exact mathematical sense. However there is no way to exactly + determine multiplicity of roots of a polynomial in numerical computing. + Thus you should think of the result of `residue` with given `tol` as + partial fraction expansion computed for the denominator composed of the + computed poles with empirically determined multiplicity. The choice of + `tol` can drastically change the result if there are close poles. + + References + ---------- + .. [1] J. F. Mahoney, B. D. Sivazlian, "Partial fractions expansion: a + review of computational methodology and efficiency", Journal of + Computational and Applied Mathematics, Vol. 9, 1983. + """ + b = np.asarray(b) + a = np.asarray(a) + if (np.issubdtype(b.dtype, np.complexfloating) + or np.issubdtype(a.dtype, np.complexfloating)): + b = b.astype(complex) + a = a.astype(complex) + else: + b = b.astype(float) + a = a.astype(float) + + b = np.trim_zeros(np.atleast_1d(b), 'f') + a = np.trim_zeros(np.atleast_1d(a), 'f') + + if a.size == 0: + raise ValueError("Denominator `a` is zero.") + + poles = np.roots(a) + if b.size == 0: + return np.zeros(poles.shape), _cmplx_sort(poles)[0], np.array([]) + + if len(b) < len(a): + k = np.empty(0) + else: + k, b = np.polydiv(b, a) + + unique_poles, multiplicity = unique_roots(poles, tol=tol, rtype=rtype) + unique_poles, order = _cmplx_sort(unique_poles) + multiplicity = multiplicity[order] + + residues = _compute_residues(unique_poles, multiplicity, b) + + index = 0 + for pole, mult in zip(unique_poles, multiplicity): + poles[index:index + mult] = pole + index += mult + + return residues / a[0], poles, k + + +def residuez(b, a, tol=1e-3, rtype='avg'): + """Compute partial-fraction expansion of b(z) / a(z). + + If `M` is the degree of numerator `b` and `N` the degree of denominator + `a`:: + + b(z) b[0] + b[1] z**(-1) + ... + b[M] z**(-M) + H(z) = ------ = ------------------------------------------ + a(z) a[0] + a[1] z**(-1) + ... + a[N] z**(-N) + + then the partial-fraction expansion H(z) is defined as:: + + r[0] r[-1] + = --------------- + ... + ---------------- + k[0] + k[1]z**(-1) ... + (1-p[0]z**(-1)) (1-p[-1]z**(-1)) + + If there are any repeated roots (closer than `tol`), then the partial + fraction expansion has terms like:: + + r[i] r[i+1] r[i+n-1] + -------------- + ------------------ + ... + ------------------ + (1-p[i]z**(-1)) (1-p[i]z**(-1))**2 (1-p[i]z**(-1))**n + + This function is used for polynomials in negative powers of z, + such as digital filters in DSP. For positive powers, use `residue`. + + See Notes of `residue` for details about the algorithm. + + Parameters + ---------- + b : array_like + Numerator polynomial coefficients. + a : array_like + Denominator polynomial coefficients. + tol : float, optional + The tolerance for two roots to be considered equal in terms of + the distance between them. Default is 1e-3. See `unique_roots` + for further details. + rtype : {'avg', 'min', 'max'}, optional + Method for computing a root to represent a group of identical roots. + Default is 'avg'. See `unique_roots` for further details. + + Returns + ------- + r : ndarray + Residues corresponding to the poles. For repeated poles, the residues + are ordered to correspond to ascending by power fractions. + p : ndarray + Poles ordered by magnitude in ascending order. + k : ndarray + Coefficients of the direct polynomial term. + + See Also + -------- + invresz, residue, unique_roots + """ + b = np.asarray(b) + a = np.asarray(a) + if (np.issubdtype(b.dtype, np.complexfloating) + or np.issubdtype(a.dtype, np.complexfloating)): + b = b.astype(complex) + a = a.astype(complex) + else: + b = b.astype(float) + a = a.astype(float) + + b = np.trim_zeros(np.atleast_1d(b), 'b') + a = np.trim_zeros(np.atleast_1d(a), 'b') + + if a.size == 0: + raise ValueError("Denominator `a` is zero.") + elif a[0] == 0: + raise ValueError("First coefficient of determinant `a` must be " + "non-zero.") + + poles = np.roots(a) + if b.size == 0: + return np.zeros(poles.shape), _cmplx_sort(poles)[0], np.array([]) + + b_rev = b[::-1] + a_rev = a[::-1] + + if len(b_rev) < len(a_rev): + k_rev = np.empty(0) + else: + k_rev, b_rev = np.polydiv(b_rev, a_rev) + + unique_poles, multiplicity = unique_roots(poles, tol=tol, rtype=rtype) + unique_poles, order = _cmplx_sort(unique_poles) + multiplicity = multiplicity[order] + + residues = _compute_residues(1 / unique_poles, multiplicity, b_rev) + + index = 0 + powers = np.empty(len(residues), dtype=int) + for pole, mult in zip(unique_poles, multiplicity): + poles[index:index + mult] = pole + powers[index:index + mult] = 1 + np.arange(mult) + index += mult + + residues *= (-poles) ** powers / a_rev[0] + + return residues, poles, k_rev[::-1] + + +def _group_poles(poles, tol, rtype): + if rtype in ['max', 'maximum']: + reduce = np.max + elif rtype in ['min', 'minimum']: + reduce = np.min + elif rtype in ['avg', 'mean']: + reduce = np.mean + else: + raise ValueError("`rtype` must be one of " + "{'max', 'maximum', 'min', 'minimum', 'avg', 'mean'}") + + unique = [] + multiplicity = [] + + pole = poles[0] + block = [pole] + for i in range(1, len(poles)): + if abs(poles[i] - pole) <= tol: + block.append(pole) + else: + unique.append(reduce(block)) + multiplicity.append(len(block)) + pole = poles[i] + block = [pole] + + unique.append(reduce(block)) + multiplicity.append(len(block)) + + return np.asarray(unique), np.asarray(multiplicity) + + +def invresz(r, p, k, tol=1e-3, rtype='avg'): + """Compute b(z) and a(z) from partial fraction expansion. + + If `M` is the degree of numerator `b` and `N` the degree of denominator + `a`:: + + b(z) b[0] + b[1] z**(-1) + ... + b[M] z**(-M) + H(z) = ------ = ------------------------------------------ + a(z) a[0] + a[1] z**(-1) + ... + a[N] z**(-N) + + then the partial-fraction expansion H(z) is defined as:: + + r[0] r[-1] + = --------------- + ... + ---------------- + k[0] + k[1]z**(-1) ... + (1-p[0]z**(-1)) (1-p[-1]z**(-1)) + + If there are any repeated roots (closer than `tol`), then the partial + fraction expansion has terms like:: + + r[i] r[i+1] r[i+n-1] + -------------- + ------------------ + ... + ------------------ + (1-p[i]z**(-1)) (1-p[i]z**(-1))**2 (1-p[i]z**(-1))**n + + This function is used for polynomials in negative powers of z, + such as digital filters in DSP. For positive powers, use `invres`. + + Parameters + ---------- + r : array_like + Residues corresponding to the poles. For repeated poles, the residues + must be ordered to correspond to ascending by power fractions. + p : array_like + Poles. Equal poles must be adjacent. + k : array_like + Coefficients of the direct polynomial term. + tol : float, optional + The tolerance for two roots to be considered equal in terms of + the distance between them. Default is 1e-3. See `unique_roots` + for further details. + rtype : {'avg', 'min', 'max'}, optional + Method for computing a root to represent a group of identical roots. + Default is 'avg'. See `unique_roots` for further details. + + Returns + ------- + b : ndarray + Numerator polynomial coefficients. + a : ndarray + Denominator polynomial coefficients. + + See Also + -------- + residuez, unique_roots, invres + + """ + r = np.atleast_1d(r) + p = np.atleast_1d(p) + k = np.trim_zeros(np.atleast_1d(k), 'b') + + unique_poles, multiplicity = _group_poles(p, tol, rtype) + factors, denominator = _compute_factors(unique_poles, multiplicity, + include_powers=True) + + if len(k) == 0: + numerator = 0 + else: + numerator = np.polymul(k[::-1], denominator[::-1]) + + for residue, factor in zip(r, factors): + numerator = np.polyadd(numerator, residue * factor[::-1]) + + return numerator[::-1], denominator + + +def resample(x, num, t=None, axis=0, window=None, domain='time'): + """ + Resample `x` to `num` samples using Fourier method along the given axis. + + The resampled signal starts at the same value as `x` but is sampled + with a spacing of ``len(x) / num * (spacing of x)``. Because a + Fourier method is used, the signal is assumed to be periodic. + + Parameters + ---------- + x : array_like + The data to be resampled. + num : int + The number of samples in the resampled signal. + t : array_like, optional + If `t` is given, it is assumed to be the equally spaced sample + positions associated with the signal data in `x`. + axis : int, optional + The axis of `x` that is resampled. Default is 0. + window : array_like, callable, string, float, or tuple, optional + Specifies the window applied to the signal in the Fourier + domain. See below for details. + domain : string, optional + A string indicating the domain of the input `x`: + ``time`` Consider the input `x` as time-domain (Default), + ``freq`` Consider the input `x` as frequency-domain. + + Returns + ------- + resampled_x or (resampled_x, resampled_t) + Either the resampled array, or, if `t` was given, a tuple + containing the resampled array and the corresponding resampled + positions. + + See Also + -------- + decimate : Downsample the signal after applying an FIR or IIR filter. + resample_poly : Resample using polyphase filtering and an FIR filter. + + Notes + ----- + The argument `window` controls a Fourier-domain window that tapers + the Fourier spectrum before zero-padding to alleviate ringing in + the resampled values for sampled signals you didn't intend to be + interpreted as band-limited. + + If `window` is a function, then it is called with a vector of inputs + indicating the frequency bins (i.e. fftfreq(x.shape[axis]) ). + + If `window` is an array of the same length as `x.shape[axis]` it is + assumed to be the window to be applied directly in the Fourier + domain (with dc and low-frequency first). + + For any other type of `window`, the function `scipy.signal.get_window` + is called to generate the window. + + The first sample of the returned vector is the same as the first + sample of the input vector. The spacing between samples is changed + from ``dx`` to ``dx * len(x) / num``. + + If `t` is not None, then it is used solely to calculate the resampled + positions `resampled_t` + + As noted, `resample` uses FFT transformations, which can be very + slow if the number of input or output samples is large and prime; + see `scipy.fft.fft`. + + Examples + -------- + Note that the end of the resampled data rises to meet the first + sample of the next cycle: + + >>> import numpy as np + >>> from scipy import signal + + >>> x = np.linspace(0, 10, 20, endpoint=False) + >>> y = np.cos(-x**2/6.0) + >>> f = signal.resample(y, 100) + >>> xnew = np.linspace(0, 10, 100, endpoint=False) + + >>> import matplotlib.pyplot as plt + >>> plt.plot(x, y, 'go-', xnew, f, '.-', 10, y[0], 'ro') + >>> plt.legend(['data', 'resampled'], loc='best') + >>> plt.show() + """ + + if domain not in ('time', 'freq'): + raise ValueError("Acceptable domain flags are 'time' or" + f" 'freq', not domain={domain}") + + x = np.asarray(x) + Nx = x.shape[axis] + + # Check if we can use faster real FFT + real_input = np.isrealobj(x) + + if domain == 'time': + # Forward transform + if real_input: + X = sp_fft.rfft(x, axis=axis) + else: # Full complex FFT + X = sp_fft.fft(x, axis=axis) + else: # domain == 'freq' + X = x + + # Apply window to spectrum + if window is not None: + if callable(window): + W = window(sp_fft.fftfreq(Nx)) + elif isinstance(window, np.ndarray): + if window.shape != (Nx,): + raise ValueError('window must have the same length as data') + W = window + else: + W = sp_fft.ifftshift(get_window(window, Nx)) + + newshape_W = [1] * x.ndim + newshape_W[axis] = X.shape[axis] + if real_input: + # Fold the window back on itself to mimic complex behavior + W_real = W.copy() + W_real[1:] += W_real[-1:0:-1] + W_real[1:] *= 0.5 + X *= W_real[:newshape_W[axis]].reshape(newshape_W) + else: + X *= W.reshape(newshape_W) + + # Copy each half of the original spectrum to the output spectrum, either + # truncating high frequencies (downsampling) or zero-padding them + # (upsampling) + + # Placeholder array for output spectrum + newshape = list(x.shape) + if real_input: + newshape[axis] = num // 2 + 1 + else: + newshape[axis] = num + Y = np.zeros(newshape, X.dtype) + + # Copy positive frequency components (and Nyquist, if present) + N = min(num, Nx) + nyq = N // 2 + 1 # Slice index that includes Nyquist if present + sl = [slice(None)] * x.ndim + sl[axis] = slice(0, nyq) + Y[tuple(sl)] = X[tuple(sl)] + if not real_input: + # Copy negative frequency components + if N > 2: # (slice expression doesn't collapse to empty array) + sl[axis] = slice(nyq - N, None) + Y[tuple(sl)] = X[tuple(sl)] + + # Split/join Nyquist component(s) if present + # So far we have set Y[+N/2]=X[+N/2] + if N % 2 == 0: + if num < Nx: # downsampling + if real_input: + sl[axis] = slice(N//2, N//2 + 1) + Y[tuple(sl)] *= 2. + else: + # select the component of Y at frequency +N/2, + # add the component of X at -N/2 + sl[axis] = slice(-N//2, -N//2 + 1) + Y[tuple(sl)] += X[tuple(sl)] + elif Nx < num: # upsampling + # select the component at frequency +N/2 and halve it + sl[axis] = slice(N//2, N//2 + 1) + Y[tuple(sl)] *= 0.5 + if not real_input: + temp = Y[tuple(sl)] + # set the component at -N/2 equal to the component at +N/2 + sl[axis] = slice(num-N//2, num-N//2 + 1) + Y[tuple(sl)] = temp + + # Inverse transform + if real_input: + y = sp_fft.irfft(Y, num, axis=axis) + else: + y = sp_fft.ifft(Y, axis=axis, overwrite_x=True) + + y *= (float(num) / float(Nx)) + + if t is None: + return y + else: + new_t = np.arange(0, num) * (t[1] - t[0]) * Nx / float(num) + t[0] + return y, new_t + + +def resample_poly(x, up, down, axis=0, window=('kaiser', 5.0), + padtype='constant', cval=None): + """ + Resample `x` along the given axis using polyphase filtering. + + The signal `x` is upsampled by the factor `up`, a zero-phase low-pass + FIR filter is applied, and then it is downsampled by the factor `down`. + The resulting sample rate is ``up / down`` times the original sample + rate. By default, values beyond the boundary of the signal are assumed + to be zero during the filtering step. + + Parameters + ---------- + x : array_like + The data to be resampled. + up : int + The upsampling factor. + down : int + The downsampling factor. + axis : int, optional + The axis of `x` that is resampled. Default is 0. + window : string, tuple, or array_like, optional + Desired window to use to design the low-pass filter, or the FIR filter + coefficients to employ. See below for details. + padtype : string, optional + `constant`, `line`, `mean`, `median`, `maximum`, `minimum` or any of + the other signal extension modes supported by `scipy.signal.upfirdn`. + Changes assumptions on values beyond the boundary. If `constant`, + assumed to be `cval` (default zero). If `line` assumed to continue a + linear trend defined by the first and last points. `mean`, `median`, + `maximum` and `minimum` work as in `np.pad` and assume that the values + beyond the boundary are the mean, median, maximum or minimum + respectively of the array along the axis. + + .. versionadded:: 1.4.0 + cval : float, optional + Value to use if `padtype='constant'`. Default is zero. + + .. versionadded:: 1.4.0 + + Returns + ------- + resampled_x : array + The resampled array. + + See Also + -------- + decimate : Downsample the signal after applying an FIR or IIR filter. + resample : Resample up or down using the FFT method. + + Notes + ----- + This polyphase method will likely be faster than the Fourier method + in `scipy.signal.resample` when the number of samples is large and + prime, or when the number of samples is large and `up` and `down` + share a large greatest common denominator. The length of the FIR + filter used will depend on ``max(up, down) // gcd(up, down)``, and + the number of operations during polyphase filtering will depend on + the filter length and `down` (see `scipy.signal.upfirdn` for details). + + The argument `window` specifies the FIR low-pass filter design. + + If `window` is an array_like it is assumed to be the FIR filter + coefficients. Note that the FIR filter is applied after the upsampling + step, so it should be designed to operate on a signal at a sampling + frequency higher than the original by a factor of `up//gcd(up, down)`. + This function's output will be centered with respect to this array, so it + is best to pass a symmetric filter with an odd number of samples if, as + is usually the case, a zero-phase filter is desired. + + For any other type of `window`, the functions `scipy.signal.get_window` + and `scipy.signal.firwin` are called to generate the appropriate filter + coefficients. + + The first sample of the returned vector is the same as the first + sample of the input vector. The spacing between samples is changed + from ``dx`` to ``dx * down / float(up)``. + + Examples + -------- + By default, the end of the resampled data rises to meet the first + sample of the next cycle for the FFT method, and gets closer to zero + for the polyphase method: + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> x = np.linspace(0, 10, 20, endpoint=False) + >>> y = np.cos(-x**2/6.0) + >>> f_fft = signal.resample(y, 100) + >>> f_poly = signal.resample_poly(y, 100, 20) + >>> xnew = np.linspace(0, 10, 100, endpoint=False) + + >>> plt.plot(xnew, f_fft, 'b.-', xnew, f_poly, 'r.-') + >>> plt.plot(x, y, 'ko-') + >>> plt.plot(10, y[0], 'bo', 10, 0., 'ro') # boundaries + >>> plt.legend(['resample', 'resamp_poly', 'data'], loc='best') + >>> plt.show() + + This default behaviour can be changed by using the padtype option: + + >>> N = 5 + >>> x = np.linspace(0, 1, N, endpoint=False) + >>> y = 2 + x**2 - 1.7*np.sin(x) + .2*np.cos(11*x) + >>> y2 = 1 + x**3 + 0.1*np.sin(x) + .1*np.cos(11*x) + >>> Y = np.stack([y, y2], axis=-1) + >>> up = 4 + >>> xr = np.linspace(0, 1, N*up, endpoint=False) + + >>> y2 = signal.resample_poly(Y, up, 1, padtype='constant') + >>> y3 = signal.resample_poly(Y, up, 1, padtype='mean') + >>> y4 = signal.resample_poly(Y, up, 1, padtype='line') + + >>> for i in [0,1]: + ... plt.figure() + ... plt.plot(xr, y4[:,i], 'g.', label='line') + ... plt.plot(xr, y3[:,i], 'y.', label='mean') + ... plt.plot(xr, y2[:,i], 'r.', label='constant') + ... plt.plot(x, Y[:,i], 'k-') + ... plt.legend() + >>> plt.show() + + """ + x = np.asarray(x) + if up != int(up): + raise ValueError("up must be an integer") + if down != int(down): + raise ValueError("down must be an integer") + up = int(up) + down = int(down) + if up < 1 or down < 1: + raise ValueError('up and down must be >= 1') + if cval is not None and padtype != 'constant': + raise ValueError('cval has no effect when padtype is ', padtype) + + # Determine our up and down factors + # Use a rational approximation to save computation time on really long + # signals + g_ = math.gcd(up, down) + up //= g_ + down //= g_ + if up == down == 1: + return x.copy() + n_in = x.shape[axis] + n_out = n_in * up + n_out = n_out // down + bool(n_out % down) + + if isinstance(window, (list, np.ndarray)): + window = np.array(window) # use array to force a copy (we modify it) + if window.ndim > 1: + raise ValueError('window must be 1-D') + half_len = (window.size - 1) // 2 + h = window + else: + # Design a linear-phase low-pass FIR filter + max_rate = max(up, down) + f_c = 1. / max_rate # cutoff of FIR filter (rel. to Nyquist) + half_len = 10 * max_rate # reasonable cutoff for sinc-like function + h = firwin(2 * half_len + 1, f_c, + window=window).astype(x.dtype) # match dtype of x + h *= up + + # Zero-pad our filter to put the output samples at the center + n_pre_pad = (down - half_len % down) + n_post_pad = 0 + n_pre_remove = (half_len + n_pre_pad) // down + # We should rarely need to do this given our filter lengths... + while _output_len(len(h) + n_pre_pad + n_post_pad, n_in, + up, down) < n_out + n_pre_remove: + n_post_pad += 1 + h = np.concatenate((np.zeros(n_pre_pad, dtype=h.dtype), h, + np.zeros(n_post_pad, dtype=h.dtype))) + n_pre_remove_end = n_pre_remove + n_out + + # Remove background depending on the padtype option + funcs = {'mean': np.mean, 'median': np.median, + 'minimum': np.amin, 'maximum': np.amax} + upfirdn_kwargs = {'mode': 'constant', 'cval': 0} + if padtype in funcs: + background_values = funcs[padtype](x, axis=axis, keepdims=True) + elif padtype in _upfirdn_modes: + upfirdn_kwargs = {'mode': padtype} + if padtype == 'constant': + if cval is None: + cval = 0 + upfirdn_kwargs['cval'] = cval + else: + raise ValueError( + 'padtype must be one of: maximum, mean, median, minimum, ' + + ', '.join(_upfirdn_modes)) + + if padtype in funcs: + x = x - background_values + + # filter then remove excess + y = upfirdn(h, x, up, down, axis=axis, **upfirdn_kwargs) + keep = [slice(None), ]*x.ndim + keep[axis] = slice(n_pre_remove, n_pre_remove_end) + y_keep = y[tuple(keep)] + + # Add background back + if padtype in funcs: + y_keep += background_values + + return y_keep + + +def vectorstrength(events, period): + ''' + Determine the vector strength of the events corresponding to the given + period. + + The vector strength is a measure of phase synchrony, how well the + timing of the events is synchronized to a single period of a periodic + signal. + + If multiple periods are used, calculate the vector strength of each. + This is called the "resonating vector strength". + + Parameters + ---------- + events : 1D array_like + An array of time points containing the timing of the events. + period : float or array_like + The period of the signal that the events should synchronize to. + The period is in the same units as `events`. It can also be an array + of periods, in which case the outputs are arrays of the same length. + + Returns + ------- + strength : float or 1D array + The strength of the synchronization. 1.0 is perfect synchronization + and 0.0 is no synchronization. If `period` is an array, this is also + an array with each element containing the vector strength at the + corresponding period. + phase : float or array + The phase that the events are most strongly synchronized to in radians. + If `period` is an array, this is also an array with each element + containing the phase for the corresponding period. + + References + ---------- + van Hemmen, JL, Longtin, A, and Vollmayr, AN. Testing resonating vector + strength: Auditory system, electric fish, and noise. + Chaos 21, 047508 (2011); + :doi:`10.1063/1.3670512`. + van Hemmen, JL. Vector strength after Goldberg, Brown, and von Mises: + biological and mathematical perspectives. Biol Cybern. + 2013 Aug;107(4):385-96. :doi:`10.1007/s00422-013-0561-7`. + van Hemmen, JL and Vollmayr, AN. Resonating vector strength: what happens + when we vary the "probing" frequency while keeping the spike times + fixed. Biol Cybern. 2013 Aug;107(4):491-94. + :doi:`10.1007/s00422-013-0560-8`. + ''' + events = np.asarray(events) + period = np.asarray(period) + if events.ndim > 1: + raise ValueError('events cannot have dimensions more than 1') + if period.ndim > 1: + raise ValueError('period cannot have dimensions more than 1') + + # we need to know later if period was originally a scalar + scalarperiod = not period.ndim + + events = np.atleast_2d(events) + period = np.atleast_2d(period) + if (period <= 0).any(): + raise ValueError('periods must be positive') + + # this converts the times to vectors + vectors = np.exp(np.dot(2j*np.pi/period.T, events)) + + # the vector strength is just the magnitude of the mean of the vectors + # the vector phase is the angle of the mean of the vectors + vectormean = np.mean(vectors, axis=1) + strength = abs(vectormean) + phase = np.angle(vectormean) + + # if the original period was a scalar, return scalars + if scalarperiod: + strength = strength[0] + phase = phase[0] + return strength, phase + + +def detrend(data, axis=-1, type='linear', bp=0, overwrite_data=False): + """ + Remove linear trend along axis from data. + + Parameters + ---------- + data : array_like + The input data. + axis : int, optional + The axis along which to detrend the data. By default this is the + last axis (-1). + type : {'linear', 'constant'}, optional + The type of detrending. If ``type == 'linear'`` (default), + the result of a linear least-squares fit to `data` is subtracted + from `data`. + If ``type == 'constant'``, only the mean of `data` is subtracted. + bp : array_like of ints, optional + A sequence of break points. If given, an individual linear fit is + performed for each part of `data` between two break points. + Break points are specified as indices into `data`. This parameter + only has an effect when ``type == 'linear'``. + overwrite_data : bool, optional + If True, perform in place detrending and avoid a copy. Default is False + + Returns + ------- + ret : ndarray + The detrended input data. + + Examples + -------- + >>> import numpy as np + >>> from scipy import signal + >>> rng = np.random.default_rng() + >>> npoints = 1000 + >>> noise = rng.standard_normal(npoints) + >>> x = 3 + 2*np.linspace(0, 1, npoints) + noise + >>> (signal.detrend(x) - noise).max() + 0.06 # random + + """ + if type not in ['linear', 'l', 'constant', 'c']: + raise ValueError("Trend type must be 'linear' or 'constant'.") + data = np.asarray(data) + dtype = data.dtype.char + if dtype not in 'dfDF': + dtype = 'd' + if type in ['constant', 'c']: + ret = data - np.mean(data, axis, keepdims=True) + return ret + else: + dshape = data.shape + N = dshape[axis] + bp = np.sort(np.unique(np.concatenate(np.atleast_1d(0, bp, N)))) + if np.any(bp > N): + raise ValueError("Breakpoints must be less than length " + "of data along given axis.") + + # Restructure data so that axis is along first dimension and + # all other dimensions are collapsed into second dimension + rnk = len(dshape) + if axis < 0: + axis = axis + rnk + newdata = np.moveaxis(data, axis, 0) + newdata_shape = newdata.shape + newdata = newdata.reshape(N, -1) + + if not overwrite_data: + newdata = newdata.copy() # make sure we have a copy + if newdata.dtype.char not in 'dfDF': + newdata = newdata.astype(dtype) + +# Nreg = len(bp) - 1 + # Find leastsq fit and remove it for each piece + for m in range(len(bp) - 1): + Npts = bp[m + 1] - bp[m] + A = np.ones((Npts, 2), dtype) + A[:, 0] = np.arange(1, Npts + 1, dtype=dtype) / Npts + sl = slice(bp[m], bp[m + 1]) + coef, resids, rank, s = linalg.lstsq(A, newdata[sl]) + newdata[sl] = newdata[sl] - A @ coef + + # Put data back in original shape. + newdata = newdata.reshape(newdata_shape) + ret = np.moveaxis(newdata, 0, axis) + return ret + + +def lfilter_zi(b, a): + """ + Construct initial conditions for lfilter for step response steady-state. + + Compute an initial state `zi` for the `lfilter` function that corresponds + to the steady state of the step response. + + A typical use of this function is to set the initial state so that the + output of the filter starts at the same value as the first element of + the signal to be filtered. + + Parameters + ---------- + b, a : array_like (1-D) + The IIR filter coefficients. See `lfilter` for more + information. + + Returns + ------- + zi : 1-D ndarray + The initial state for the filter. + + See Also + -------- + lfilter, lfiltic, filtfilt + + Notes + ----- + A linear filter with order m has a state space representation (A, B, C, D), + for which the output y of the filter can be expressed as:: + + z(n+1) = A*z(n) + B*x(n) + y(n) = C*z(n) + D*x(n) + + where z(n) is a vector of length m, A has shape (m, m), B has shape + (m, 1), C has shape (1, m) and D has shape (1, 1) (assuming x(n) is + a scalar). lfilter_zi solves:: + + zi = A*zi + B + + In other words, it finds the initial condition for which the response + to an input of all ones is a constant. + + Given the filter coefficients `a` and `b`, the state space matrices + for the transposed direct form II implementation of the linear filter, + which is the implementation used by scipy.signal.lfilter, are:: + + A = scipy.linalg.companion(a).T + B = b[1:] - a[1:]*b[0] + + assuming `a[0]` is 1.0; if `a[0]` is not 1, `a` and `b` are first + divided by a[0]. + + Examples + -------- + The following code creates a lowpass Butterworth filter. Then it + applies that filter to an array whose values are all 1.0; the + output is also all 1.0, as expected for a lowpass filter. If the + `zi` argument of `lfilter` had not been given, the output would have + shown the transient signal. + + >>> from numpy import array, ones + >>> from scipy.signal import lfilter, lfilter_zi, butter + >>> b, a = butter(5, 0.25) + >>> zi = lfilter_zi(b, a) + >>> y, zo = lfilter(b, a, ones(10), zi=zi) + >>> y + array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1.]) + + Another example: + + >>> x = array([0.5, 0.5, 0.5, 0.0, 0.0, 0.0, 0.0]) + >>> y, zf = lfilter(b, a, x, zi=zi*x[0]) + >>> y + array([ 0.5 , 0.5 , 0.5 , 0.49836039, 0.48610528, + 0.44399389, 0.35505241]) + + Note that the `zi` argument to `lfilter` was computed using + `lfilter_zi` and scaled by `x[0]`. Then the output `y` has no + transient until the input drops from 0.5 to 0.0. + + """ + + # FIXME: Can this function be replaced with an appropriate + # use of lfiltic? For example, when b,a = butter(N,Wn), + # lfiltic(b, a, y=numpy.ones_like(a), x=numpy.ones_like(b)). + # + + # We could use scipy.signal.normalize, but it uses warnings in + # cases where a ValueError is more appropriate, and it allows + # b to be 2D. + b = np.atleast_1d(b) + if b.ndim != 1: + raise ValueError("Numerator b must be 1-D.") + a = np.atleast_1d(a) + if a.ndim != 1: + raise ValueError("Denominator a must be 1-D.") + + while len(a) > 1 and a[0] == 0.0: + a = a[1:] + if a.size < 1: + raise ValueError("There must be at least one nonzero `a` coefficient.") + + if a[0] != 1.0: + # Normalize the coefficients so a[0] == 1. + b = b / a[0] + a = a / a[0] + + n = max(len(a), len(b)) + + # Pad a or b with zeros so they are the same length. + if len(a) < n: + a = np.r_[a, np.zeros(n - len(a), dtype=a.dtype)] + elif len(b) < n: + b = np.r_[b, np.zeros(n - len(b), dtype=b.dtype)] + + IminusA = np.eye(n - 1, dtype=np.result_type(a, b)) - linalg.companion(a).T + B = b[1:] - a[1:] * b[0] + # Solve zi = A*zi + B + zi = np.linalg.solve(IminusA, B) + + # For future reference: we could also use the following + # explicit formulas to solve the linear system: + # + # zi = np.zeros(n - 1) + # zi[0] = B.sum() / IminusA[:,0].sum() + # asum = 1.0 + # csum = 0.0 + # for k in range(1,n-1): + # asum += a[k] + # csum += b[k] - a[k]*b[0] + # zi[k] = asum*zi[0] - csum + + return zi + + +def sosfilt_zi(sos): + """ + Construct initial conditions for sosfilt for step response steady-state. + + Compute an initial state `zi` for the `sosfilt` function that corresponds + to the steady state of the step response. + + A typical use of this function is to set the initial state so that the + output of the filter starts at the same value as the first element of + the signal to be filtered. + + Parameters + ---------- + sos : array_like + Array of second-order filter coefficients, must have shape + ``(n_sections, 6)``. See `sosfilt` for the SOS filter format + specification. + + Returns + ------- + zi : ndarray + Initial conditions suitable for use with ``sosfilt``, shape + ``(n_sections, 2)``. + + See Also + -------- + sosfilt, zpk2sos + + Notes + ----- + .. versionadded:: 0.16.0 + + Examples + -------- + Filter a rectangular pulse that begins at time 0, with and without + the use of the `zi` argument of `scipy.signal.sosfilt`. + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + >>> sos = signal.butter(9, 0.125, output='sos') + >>> zi = signal.sosfilt_zi(sos) + >>> x = (np.arange(250) < 100).astype(int) + >>> f1 = signal.sosfilt(sos, x) + >>> f2, zo = signal.sosfilt(sos, x, zi=zi) + + >>> plt.plot(x, 'k--', label='x') + >>> plt.plot(f1, 'b', alpha=0.5, linewidth=2, label='filtered') + >>> plt.plot(f2, 'g', alpha=0.25, linewidth=4, label='filtered with zi') + >>> plt.legend(loc='best') + >>> plt.show() + + """ + sos = np.asarray(sos) + if sos.ndim != 2 or sos.shape[1] != 6: + raise ValueError('sos must be shape (n_sections, 6)') + + if sos.dtype.kind in 'bui': + sos = sos.astype(np.float64) + + n_sections = sos.shape[0] + zi = np.empty((n_sections, 2), dtype=sos.dtype) + scale = 1.0 + for section in range(n_sections): + b = sos[section, :3] + a = sos[section, 3:] + zi[section] = scale * lfilter_zi(b, a) + # If H(z) = B(z)/A(z) is this section's transfer function, then + # b.sum()/a.sum() is H(1), the gain at omega=0. That's the steady + # state value of this section's step response. + scale *= b.sum() / a.sum() + + return zi + + +def _filtfilt_gust(b, a, x, axis=-1, irlen=None): + """Forward-backward IIR filter that uses Gustafsson's method. + + Apply the IIR filter defined by `(b,a)` to `x` twice, first forward + then backward, using Gustafsson's initial conditions [1]_. + + Let ``y_fb`` be the result of filtering first forward and then backward, + and let ``y_bf`` be the result of filtering first backward then forward. + Gustafsson's method is to compute initial conditions for the forward + pass and the backward pass such that ``y_fb == y_bf``. + + Parameters + ---------- + b : scalar or 1-D ndarray + Numerator coefficients of the filter. + a : scalar or 1-D ndarray + Denominator coefficients of the filter. + x : ndarray + Data to be filtered. + axis : int, optional + Axis of `x` to be filtered. Default is -1. + irlen : int or None, optional + The length of the nonnegligible part of the impulse response. + If `irlen` is None, or if the length of the signal is less than + ``2 * irlen``, then no part of the impulse response is ignored. + + Returns + ------- + y : ndarray + The filtered data. + x0 : ndarray + Initial condition for the forward filter. + x1 : ndarray + Initial condition for the backward filter. + + Notes + ----- + Typically the return values `x0` and `x1` are not needed by the + caller. The intended use of these return values is in unit tests. + + References + ---------- + .. [1] F. Gustaffson. Determining the initial states in forward-backward + filtering. Transactions on Signal Processing, 46(4):988-992, 1996. + + """ + # In the comments, "Gustafsson's paper" and [1] refer to the + # paper referenced in the docstring. + + b = np.atleast_1d(b) + a = np.atleast_1d(a) + + order = max(len(b), len(a)) - 1 + if order == 0: + # The filter is just scalar multiplication, with no state. + scale = (b[0] / a[0])**2 + y = scale * x + return y, np.array([]), np.array([]) + + if axis != -1 or axis != x.ndim - 1: + # Move the axis containing the data to the end. + x = np.swapaxes(x, axis, x.ndim - 1) + + # n is the number of samples in the data to be filtered. + n = x.shape[-1] + + if irlen is None or n <= 2*irlen: + m = n + else: + m = irlen + + # Create Obs, the observability matrix (called O in the paper). + # This matrix can be interpreted as the operator that propagates + # an arbitrary initial state to the output, assuming the input is + # zero. + # In Gustafsson's paper, the forward and backward filters are not + # necessarily the same, so he has both O_f and O_b. We use the same + # filter in both directions, so we only need O. The same comment + # applies to S below. + Obs = np.zeros((m, order)) + zi = np.zeros(order) + zi[0] = 1 + Obs[:, 0] = lfilter(b, a, np.zeros(m), zi=zi)[0] + for k in range(1, order): + Obs[k:, k] = Obs[:-k, 0] + + # Obsr is O^R (Gustafsson's notation for row-reversed O) + Obsr = Obs[::-1] + + # Create S. S is the matrix that applies the filter to the reversed + # propagated initial conditions. That is, + # out = S.dot(zi) + # is the same as + # tmp, _ = lfilter(b, a, zeros(), zi=zi) # Propagate ICs. + # out = lfilter(b, a, tmp[::-1]) # Reverse and filter. + + # Equations (5) & (6) of [1] + S = lfilter(b, a, Obs[::-1], axis=0) + + # Sr is S^R (row-reversed S) + Sr = S[::-1] + + # M is [(S^R - O), (O^R - S)] + if m == n: + M = np.hstack((Sr - Obs, Obsr - S)) + else: + # Matrix described in section IV of [1]. + M = np.zeros((2*m, 2*order)) + M[:m, :order] = Sr - Obs + M[m:, order:] = Obsr - S + + # Naive forward-backward and backward-forward filters. + # These have large transients because the filters use zero initial + # conditions. + y_f = lfilter(b, a, x) + y_fb = lfilter(b, a, y_f[..., ::-1])[..., ::-1] + + y_b = lfilter(b, a, x[..., ::-1])[..., ::-1] + y_bf = lfilter(b, a, y_b) + + delta_y_bf_fb = y_bf - y_fb + if m == n: + delta = delta_y_bf_fb + else: + start_m = delta_y_bf_fb[..., :m] + end_m = delta_y_bf_fb[..., -m:] + delta = np.concatenate((start_m, end_m), axis=-1) + + # ic_opt holds the "optimal" initial conditions. + # The following code computes the result shown in the formula + # of the paper between equations (6) and (7). + if delta.ndim == 1: + ic_opt = linalg.lstsq(M, delta)[0] + else: + # Reshape delta so it can be used as an array of multiple + # right-hand-sides in linalg.lstsq. + delta2d = delta.reshape(-1, delta.shape[-1]).T + ic_opt0 = linalg.lstsq(M, delta2d)[0].T + ic_opt = ic_opt0.reshape(delta.shape[:-1] + (M.shape[-1],)) + + # Now compute the filtered signal using equation (7) of [1]. + # First, form [S^R, O^R] and call it W. + if m == n: + W = np.hstack((Sr, Obsr)) + else: + W = np.zeros((2*m, 2*order)) + W[:m, :order] = Sr + W[m:, order:] = Obsr + + # Equation (7) of [1] says + # Y_fb^opt = Y_fb^0 + W * [x_0^opt; x_{N-1}^opt] + # `wic` is (almost) the product on the right. + # W has shape (m, 2*order), and ic_opt has shape (..., 2*order), + # so we can't use W.dot(ic_opt). Instead, we dot ic_opt with W.T, + # so wic has shape (..., m). + wic = ic_opt.dot(W.T) + + # `wic` is "almost" the product of W and the optimal ICs in equation + # (7)--if we're using a truncated impulse response (m < n), `wic` + # contains only the adjustments required for the ends of the signal. + # Here we form y_opt, taking this into account if necessary. + y_opt = y_fb + if m == n: + y_opt += wic + else: + y_opt[..., :m] += wic[..., :m] + y_opt[..., -m:] += wic[..., -m:] + + x0 = ic_opt[..., :order] + x1 = ic_opt[..., -order:] + if axis != -1 or axis != x.ndim - 1: + # Restore the data axis to its original position. + x0 = np.swapaxes(x0, axis, x.ndim - 1) + x1 = np.swapaxes(x1, axis, x.ndim - 1) + y_opt = np.swapaxes(y_opt, axis, x.ndim - 1) + + return y_opt, x0, x1 + + +def filtfilt(b, a, x, axis=-1, padtype='odd', padlen=None, method='pad', + irlen=None): + """ + Apply a digital filter forward and backward to a signal. + + This function applies a linear digital filter twice, once forward and + once backwards. The combined filter has zero phase and a filter order + twice that of the original. + + The function provides options for handling the edges of the signal. + + The function `sosfiltfilt` (and filter design using ``output='sos'``) + should be preferred over `filtfilt` for most filtering tasks, as + second-order sections have fewer numerical problems. + + Parameters + ---------- + b : (N,) array_like + The numerator coefficient vector of the filter. + a : (N,) array_like + The denominator coefficient vector of the filter. If ``a[0]`` + is not 1, then both `a` and `b` are normalized by ``a[0]``. + x : array_like + The array of data to be filtered. + axis : int, optional + The axis of `x` to which the filter is applied. + Default is -1. + padtype : str or None, optional + Must be 'odd', 'even', 'constant', or None. This determines the + type of extension to use for the padded signal to which the filter + is applied. If `padtype` is None, no padding is used. The default + is 'odd'. + padlen : int or None, optional + The number of elements by which to extend `x` at both ends of + `axis` before applying the filter. This value must be less than + ``x.shape[axis] - 1``. ``padlen=0`` implies no padding. + The default value is ``3 * max(len(a), len(b))``. + method : str, optional + Determines the method for handling the edges of the signal, either + "pad" or "gust". When `method` is "pad", the signal is padded; the + type of padding is determined by `padtype` and `padlen`, and `irlen` + is ignored. When `method` is "gust", Gustafsson's method is used, + and `padtype` and `padlen` are ignored. + irlen : int or None, optional + When `method` is "gust", `irlen` specifies the length of the + impulse response of the filter. If `irlen` is None, no part + of the impulse response is ignored. For a long signal, specifying + `irlen` can significantly improve the performance of the filter. + + Returns + ------- + y : ndarray + The filtered output with the same shape as `x`. + + See Also + -------- + sosfiltfilt, lfilter_zi, lfilter, lfiltic, savgol_filter, sosfilt + + Notes + ----- + When `method` is "pad", the function pads the data along the given axis + in one of three ways: odd, even or constant. The odd and even extensions + have the corresponding symmetry about the end point of the data. The + constant extension extends the data with the values at the end points. On + both the forward and backward passes, the initial condition of the + filter is found by using `lfilter_zi` and scaling it by the end point of + the extended data. + + When `method` is "gust", Gustafsson's method [1]_ is used. Initial + conditions are chosen for the forward and backward passes so that the + forward-backward filter gives the same result as the backward-forward + filter. + + The option to use Gustaffson's method was added in scipy version 0.16.0. + + References + ---------- + .. [1] F. Gustaffson, "Determining the initial states in forward-backward + filtering", Transactions on Signal Processing, Vol. 46, pp. 988-992, + 1996. + + Examples + -------- + The examples will use several functions from `scipy.signal`. + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + First we create a one second signal that is the sum of two pure sine + waves, with frequencies 5 Hz and 250 Hz, sampled at 2000 Hz. + + >>> t = np.linspace(0, 1.0, 2001) + >>> xlow = np.sin(2 * np.pi * 5 * t) + >>> xhigh = np.sin(2 * np.pi * 250 * t) + >>> x = xlow + xhigh + + Now create a lowpass Butterworth filter with a cutoff of 0.125 times + the Nyquist frequency, or 125 Hz, and apply it to ``x`` with `filtfilt`. + The result should be approximately ``xlow``, with no phase shift. + + >>> b, a = signal.butter(8, 0.125) + >>> y = signal.filtfilt(b, a, x, padlen=150) + >>> np.abs(y - xlow).max() + 9.1086182074789912e-06 + + We get a fairly clean result for this artificial example because + the odd extension is exact, and with the moderately long padding, + the filter's transients have dissipated by the time the actual data + is reached. In general, transient effects at the edges are + unavoidable. + + The following example demonstrates the option ``method="gust"``. + + First, create a filter. + + >>> b, a = signal.ellip(4, 0.01, 120, 0.125) # Filter to be applied. + + `sig` is a random input signal to be filtered. + + >>> rng = np.random.default_rng() + >>> n = 60 + >>> sig = rng.standard_normal(n)**3 + 3*rng.standard_normal(n).cumsum() + + Apply `filtfilt` to `sig`, once using the Gustafsson method, and + once using padding, and plot the results for comparison. + + >>> fgust = signal.filtfilt(b, a, sig, method="gust") + >>> fpad = signal.filtfilt(b, a, sig, padlen=50) + >>> plt.plot(sig, 'k-', label='input') + >>> plt.plot(fgust, 'b-', linewidth=4, label='gust') + >>> plt.plot(fpad, 'c-', linewidth=1.5, label='pad') + >>> plt.legend(loc='best') + >>> plt.show() + + The `irlen` argument can be used to improve the performance + of Gustafsson's method. + + Estimate the impulse response length of the filter. + + >>> z, p, k = signal.tf2zpk(b, a) + >>> eps = 1e-9 + >>> r = np.max(np.abs(p)) + >>> approx_impulse_len = int(np.ceil(np.log(eps) / np.log(r))) + >>> approx_impulse_len + 137 + + Apply the filter to a longer signal, with and without the `irlen` + argument. The difference between `y1` and `y2` is small. For long + signals, using `irlen` gives a significant performance improvement. + + >>> x = rng.standard_normal(4000) + >>> y1 = signal.filtfilt(b, a, x, method='gust') + >>> y2 = signal.filtfilt(b, a, x, method='gust', irlen=approx_impulse_len) + >>> print(np.max(np.abs(y1 - y2))) + 2.875334415008979e-10 + + """ + b = np.atleast_1d(b) + a = np.atleast_1d(a) + x = np.asarray(x) + + if method not in ["pad", "gust"]: + raise ValueError("method must be 'pad' or 'gust'.") + + if method == "gust": + y, z1, z2 = _filtfilt_gust(b, a, x, axis=axis, irlen=irlen) + return y + + # method == "pad" + edge, ext = _validate_pad(padtype, padlen, x, axis, + ntaps=max(len(a), len(b))) + + # Get the steady state of the filter's step response. + zi = lfilter_zi(b, a) + + # Reshape zi and create x0 so that zi*x0 broadcasts + # to the correct value for the 'zi' keyword argument + # to lfilter. + zi_shape = [1] * x.ndim + zi_shape[axis] = zi.size + zi = np.reshape(zi, zi_shape) + x0 = axis_slice(ext, stop=1, axis=axis) + + # Forward filter. + (y, zf) = lfilter(b, a, ext, axis=axis, zi=zi * x0) + + # Backward filter. + # Create y0 so zi*y0 broadcasts appropriately. + y0 = axis_slice(y, start=-1, axis=axis) + (y, zf) = lfilter(b, a, axis_reverse(y, axis=axis), axis=axis, zi=zi * y0) + + # Reverse y. + y = axis_reverse(y, axis=axis) + + if edge > 0: + # Slice the actual signal from the extended signal. + y = axis_slice(y, start=edge, stop=-edge, axis=axis) + + return y + + +def _validate_pad(padtype, padlen, x, axis, ntaps): + """Helper to validate padding for filtfilt""" + if padtype not in ['even', 'odd', 'constant', None]: + raise ValueError(("Unknown value '%s' given to padtype. padtype " + "must be 'even', 'odd', 'constant', or None.") % + padtype) + + if padtype is None: + padlen = 0 + + if padlen is None: + # Original padding; preserved for backwards compatibility. + edge = ntaps * 3 + else: + edge = padlen + + # x's 'axis' dimension must be bigger than edge. + if x.shape[axis] <= edge: + raise ValueError("The length of the input vector x must be greater " + "than padlen, which is %d." % edge) + + if padtype is not None and edge > 0: + # Make an extension of length `edge` at each + # end of the input array. + if padtype == 'even': + ext = even_ext(x, edge, axis=axis) + elif padtype == 'odd': + ext = odd_ext(x, edge, axis=axis) + else: + ext = const_ext(x, edge, axis=axis) + else: + ext = x + return edge, ext + + +def _validate_x(x): + x = np.asarray(x) + if x.ndim == 0: + raise ValueError('x must be at least 1-D') + return x + + +def sosfilt(sos, x, axis=-1, zi=None): + """ + Filter data along one dimension using cascaded second-order sections. + + Filter a data sequence, `x`, using a digital IIR filter defined by + `sos`. + + Parameters + ---------- + sos : array_like + Array of second-order filter coefficients, must have shape + ``(n_sections, 6)``. Each row corresponds to a second-order + section, with the first three columns providing the numerator + coefficients and the last three providing the denominator + coefficients. + x : array_like + An N-dimensional input array. + axis : int, optional + The axis of the input data array along which to apply the + linear filter. The filter is applied to each subarray along + this axis. Default is -1. + zi : array_like, optional + Initial conditions for the cascaded filter delays. It is a (at + least 2D) vector of shape ``(n_sections, ..., 2, ...)``, where + ``..., 2, ...`` denotes the shape of `x`, but with ``x.shape[axis]`` + replaced by 2. If `zi` is None or is not given then initial rest + (i.e. all zeros) is assumed. + Note that these initial conditions are *not* the same as the initial + conditions given by `lfiltic` or `lfilter_zi`. + + Returns + ------- + y : ndarray + The output of the digital filter. + zf : ndarray, optional + If `zi` is None, this is not returned, otherwise, `zf` holds the + final filter delay values. + + See Also + -------- + zpk2sos, sos2zpk, sosfilt_zi, sosfiltfilt, sosfreqz + + Notes + ----- + The filter function is implemented as a series of second-order filters + with direct-form II transposed structure. It is designed to minimize + numerical precision errors for high-order filters. + + .. versionadded:: 0.16.0 + + Examples + -------- + Plot a 13th-order filter's impulse response using both `lfilter` and + `sosfilt`, showing the instability that results from trying to do a + 13th-order filter in a single stage (the numerical error pushes some poles + outside of the unit circle): + + >>> import matplotlib.pyplot as plt + >>> from scipy import signal + >>> b, a = signal.ellip(13, 0.009, 80, 0.05, output='ba') + >>> sos = signal.ellip(13, 0.009, 80, 0.05, output='sos') + >>> x = signal.unit_impulse(700) + >>> y_tf = signal.lfilter(b, a, x) + >>> y_sos = signal.sosfilt(sos, x) + >>> plt.plot(y_tf, 'r', label='TF') + >>> plt.plot(y_sos, 'k', label='SOS') + >>> plt.legend(loc='best') + >>> plt.show() + + """ + x = _validate_x(x) + sos, n_sections = _validate_sos(sos) + x_zi_shape = list(x.shape) + x_zi_shape[axis] = 2 + x_zi_shape = tuple([n_sections] + x_zi_shape) + inputs = [sos, x] + if zi is not None: + inputs.append(np.asarray(zi)) + dtype = np.result_type(*inputs) + if dtype.char not in 'fdgFDGO': + raise NotImplementedError("input type '%s' not supported" % dtype) + if zi is not None: + zi = np.array(zi, dtype) # make a copy so that we can operate in place + if zi.shape != x_zi_shape: + raise ValueError('Invalid zi shape. With axis=%r, an input with ' + 'shape %r, and an sos array with %d sections, zi ' + 'must have shape %r, got %r.' % + (axis, x.shape, n_sections, x_zi_shape, zi.shape)) + return_zi = True + else: + zi = np.zeros(x_zi_shape, dtype=dtype) + return_zi = False + axis = axis % x.ndim # make positive + x = np.moveaxis(x, axis, -1) + zi = np.moveaxis(zi, [0, axis + 1], [-2, -1]) + x_shape, zi_shape = x.shape, zi.shape + x = np.reshape(x, (-1, x.shape[-1])) + x = np.array(x, dtype, order='C') # make a copy, can modify in place + zi = np.ascontiguousarray(np.reshape(zi, (-1, n_sections, 2))) + sos = sos.astype(dtype, copy=False) + _sosfilt(sos, x, zi) + x.shape = x_shape + x = np.moveaxis(x, -1, axis) + if return_zi: + zi.shape = zi_shape + zi = np.moveaxis(zi, [-2, -1], [0, axis + 1]) + out = (x, zi) + else: + out = x + return out + + +def sosfiltfilt(sos, x, axis=-1, padtype='odd', padlen=None): + """ + A forward-backward digital filter using cascaded second-order sections. + + See `filtfilt` for more complete information about this method. + + Parameters + ---------- + sos : array_like + Array of second-order filter coefficients, must have shape + ``(n_sections, 6)``. Each row corresponds to a second-order + section, with the first three columns providing the numerator + coefficients and the last three providing the denominator + coefficients. + x : array_like + The array of data to be filtered. + axis : int, optional + The axis of `x` to which the filter is applied. + Default is -1. + padtype : str or None, optional + Must be 'odd', 'even', 'constant', or None. This determines the + type of extension to use for the padded signal to which the filter + is applied. If `padtype` is None, no padding is used. The default + is 'odd'. + padlen : int or None, optional + The number of elements by which to extend `x` at both ends of + `axis` before applying the filter. This value must be less than + ``x.shape[axis] - 1``. ``padlen=0`` implies no padding. + The default value is:: + + 3 * (2 * len(sos) + 1 - min((sos[:, 2] == 0).sum(), + (sos[:, 5] == 0).sum())) + + The extra subtraction at the end attempts to compensate for poles + and zeros at the origin (e.g. for odd-order filters) to yield + equivalent estimates of `padlen` to those of `filtfilt` for + second-order section filters built with `scipy.signal` functions. + + Returns + ------- + y : ndarray + The filtered output with the same shape as `x`. + + See Also + -------- + filtfilt, sosfilt, sosfilt_zi, sosfreqz + + Notes + ----- + .. versionadded:: 0.18.0 + + Examples + -------- + >>> import numpy as np + >>> from scipy.signal import sosfiltfilt, butter + >>> import matplotlib.pyplot as plt + >>> rng = np.random.default_rng() + + Create an interesting signal to filter. + + >>> n = 201 + >>> t = np.linspace(0, 1, n) + >>> x = 1 + (t < 0.5) - 0.25*t**2 + 0.05*rng.standard_normal(n) + + Create a lowpass Butterworth filter, and use it to filter `x`. + + >>> sos = butter(4, 0.125, output='sos') + >>> y = sosfiltfilt(sos, x) + + For comparison, apply an 8th order filter using `sosfilt`. The filter + is initialized using the mean of the first four values of `x`. + + >>> from scipy.signal import sosfilt, sosfilt_zi + >>> sos8 = butter(8, 0.125, output='sos') + >>> zi = x[:4].mean() * sosfilt_zi(sos8) + >>> y2, zo = sosfilt(sos8, x, zi=zi) + + Plot the results. Note that the phase of `y` matches the input, while + `y2` has a significant phase delay. + + >>> plt.plot(t, x, alpha=0.5, label='x(t)') + >>> plt.plot(t, y, label='y(t)') + >>> plt.plot(t, y2, label='y2(t)') + >>> plt.legend(framealpha=1, shadow=True) + >>> plt.grid(alpha=0.25) + >>> plt.xlabel('t') + >>> plt.show() + + """ + sos, n_sections = _validate_sos(sos) + x = _validate_x(x) + + # `method` is "pad"... + ntaps = 2 * n_sections + 1 + ntaps -= min((sos[:, 2] == 0).sum(), (sos[:, 5] == 0).sum()) + edge, ext = _validate_pad(padtype, padlen, x, axis, + ntaps=ntaps) + + # These steps follow the same form as filtfilt with modifications + zi = sosfilt_zi(sos) # shape (n_sections, 2) --> (n_sections, ..., 2, ...) + zi_shape = [1] * x.ndim + zi_shape[axis] = 2 + zi.shape = [n_sections] + zi_shape + x_0 = axis_slice(ext, stop=1, axis=axis) + (y, zf) = sosfilt(sos, ext, axis=axis, zi=zi * x_0) + y_0 = axis_slice(y, start=-1, axis=axis) + (y, zf) = sosfilt(sos, axis_reverse(y, axis=axis), axis=axis, zi=zi * y_0) + y = axis_reverse(y, axis=axis) + if edge > 0: + y = axis_slice(y, start=edge, stop=-edge, axis=axis) + return y + + +def decimate(x, q, n=None, ftype='iir', axis=-1, zero_phase=True): + """ + Downsample the signal after applying an anti-aliasing filter. + + By default, an order 8 Chebyshev type I filter is used. A 30 point FIR + filter with Hamming window is used if `ftype` is 'fir'. + + Parameters + ---------- + x : array_like + The signal to be downsampled, as an N-dimensional array. + q : int + The downsampling factor. When using IIR downsampling, it is recommended + to call `decimate` multiple times for downsampling factors higher than + 13. + n : int, optional + The order of the filter (1 less than the length for 'fir'). Defaults to + 8 for 'iir' and 20 times the downsampling factor for 'fir'. + ftype : str {'iir', 'fir'} or ``dlti`` instance, optional + If 'iir' or 'fir', specifies the type of lowpass filter. If an instance + of an `dlti` object, uses that object to filter before downsampling. + axis : int, optional + The axis along which to decimate. + zero_phase : bool, optional + Prevent phase shift by filtering with `filtfilt` instead of `lfilter` + when using an IIR filter, and shifting the outputs back by the filter's + group delay when using an FIR filter. The default value of ``True`` is + recommended, since a phase shift is generally not desired. + + .. versionadded:: 0.18.0 + + Returns + ------- + y : ndarray + The down-sampled signal. + + See Also + -------- + resample : Resample up or down using the FFT method. + resample_poly : Resample using polyphase filtering and an FIR filter. + + Notes + ----- + The ``zero_phase`` keyword was added in 0.18.0. + The possibility to use instances of ``dlti`` as ``ftype`` was added in + 0.18.0. + + Examples + -------- + + >>> import numpy as np + >>> from scipy import signal + >>> import matplotlib.pyplot as plt + + Define wave parameters. + + >>> wave_duration = 3 + >>> sample_rate = 100 + >>> freq = 2 + >>> q = 5 + + Calculate number of samples. + + >>> samples = wave_duration*sample_rate + >>> samples_decimated = int(samples/q) + + Create cosine wave. + + >>> x = np.linspace(0, wave_duration, samples, endpoint=False) + >>> y = np.cos(x*np.pi*freq*2) + + Decimate cosine wave. + + >>> ydem = signal.decimate(y, q) + >>> xnew = np.linspace(0, wave_duration, samples_decimated, endpoint=False) + + Plot original and decimated waves. + + >>> plt.plot(x, y, '.-', xnew, ydem, 'o-') + >>> plt.xlabel('Time, Seconds') + >>> plt.legend(['data', 'decimated'], loc='best') + >>> plt.show() + + """ + + x = np.asarray(x) + q = operator.index(q) + + if n is not None: + n = operator.index(n) + + result_type = x.dtype + if not np.issubdtype(result_type, np.inexact) \ + or result_type.type == np.float16: + # upcast integers and float16 to float64 + result_type = np.float64 + + if ftype == 'fir': + if n is None: + half_len = 10 * q # reasonable cutoff for our sinc-like function + n = 2 * half_len + b, a = firwin(n+1, 1. / q, window='hamming'), 1. + b = np.asarray(b, dtype=result_type) + a = np.asarray(a, dtype=result_type) + elif ftype == 'iir': + iir_use_sos = True + if n is None: + n = 8 + sos = cheby1(n, 0.05, 0.8 / q, output='sos') + sos = np.asarray(sos, dtype=result_type) + elif isinstance(ftype, dlti): + system = ftype._as_zpk() + if system.poles.shape[0] == 0: + # FIR + system = ftype._as_tf() + b, a = system.num, system.den + ftype = 'fir' + elif (any(np.iscomplex(system.poles)) + or any(np.iscomplex(system.poles)) + or np.iscomplex(system.gain)): + # sosfilt & sosfiltfilt don't handle complex coeffs + iir_use_sos = False + system = ftype._as_tf() + b, a = system.num, system.den + else: + iir_use_sos = True + sos = zpk2sos(system.zeros, system.poles, system.gain) + sos = np.asarray(sos, dtype=result_type) + else: + raise ValueError('invalid ftype') + + sl = [slice(None)] * x.ndim + + if ftype == 'fir': + b = b / a + if zero_phase: + y = resample_poly(x, 1, q, axis=axis, window=b) + else: + # upfirdn is generally faster than lfilter by a factor equal to the + # downsampling factor, since it only calculates the needed outputs + n_out = x.shape[axis] // q + bool(x.shape[axis] % q) + y = upfirdn(b, x, up=1, down=q, axis=axis) + sl[axis] = slice(None, n_out, None) + + else: # IIR case + if zero_phase: + if iir_use_sos: + y = sosfiltfilt(sos, x, axis=axis) + else: + y = filtfilt(b, a, x, axis=axis) + else: + if iir_use_sos: + y = sosfilt(sos, x, axis=axis) + else: + y = lfilter(b, a, x, axis=axis) + + sl[axis] = slice(None, None, q) + + return y[tuple(sl)]