a BCCfg@sddlZddlZddlmZddlZddlZddlmZddlm Z ddl m Z ddl m Z mZmZddlmZmZdd lmZdd lmZddlZdd lmZdd lmZdd lmZmZm Z m!Z!m"Z"ddl#m$Z$m%Z%m&Z&ddl'm(Z(ddl)m)Z)gdZ*ddddZ+ddddddddZ,ddZ-ddZ.dddZ/dddZ0dd d!Z1d"d#Z2dd%d&Z3dd'd(Z4d)d*Z5dd+d,Z6d-d.Z7dd/d0Z8dd2d3Z9d4d5Z:d6d7Z;d8d9Zd?Z>dd@dAZ?ddBdCZ@dDdEZAddFdGZBddHdIZCddKdLZDddMdNZEddOdPZFddRdSZGddTdUZHdVdWZIddXdYZJddZd[ZKd\ZLd]d^ZMd_d`ZNddcddZOddfdgZPddhdiZQdjdkZRddldmZSddndoZTdpdqZUddrdsZVddudvZWddydzZXd{d|ZYdd~dZZddZ[ddZ\dddZ]dddZ^ddZ_ddZ`dddZadddZbdddZcdS)N)prod)cKDTree) _sigtools)dlti)upfirdn _output_len_upfirdn_modes)linalgfft)ndimage)_init_nd_shape_and_axes)lambertw) get_window) axis_slice axis_reverseodd_exteven_ext const_ext)cheby1 _validate_soszpk2sos)firwin)_sosfilt)! correlatecorrelation_lags correlate2dconvolve convolve2d fftconvolve oaconvolve order_filtermedfilt medfilt2dwienerlfilterlfilticsosfilt deconvolvehilberthilbert2 cmplx_sort unique_rootsinvresinvreszresidueresiduezresample resample_polydetrend lfilter_zi sosfilt_zi sosfiltfiltchoose_conv_methodfiltfiltdecimatevectorstrength)validsamefull)fillpadwrapZcircularZsymmZ symmetricZreflectc Cs>z t|WSty8}ztd|WYd}~n d}~00dS)N5Acceptable mode flags are 'valid', 'same', or 'full'.) _modedictKeyError ValueError)modeerIU/var/www/html/django/DPS/env/lib/python3.9/site-packages/scipy/signal/_signaltools.py _valfrommode*s  rKc CsBzt|d>WSty<}ztd|WYd}~n d}~00dS)Nr;zZAcceptable boundary flags are 'fill', 'circular' (or 'wrap'), and 'symmetric' (or 'symm').) _boundarydictrErF)boundaryrHrIrIrJ_bvalfromboundary2s rNcsn|dkr dSsdS|dur(tt}tfdd|D}tfdd|D}|sh|shtd| S)a+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. r<FNc3s|]}||kVqdSNrI.0ishape1shape2rIrJ Rz&_inputs_swap_needed..c3s|]}||kVqdSrOrIrPrSrIrJrVSrWzOFor 'valid' mode, one must be at least as large as the other in every dimension)rangelenallrF)rGrTrUaxesZok1Zok2rIrSrJ_inputs_swap_needed:s r\r>autoc Cst|}t|}|j|jkr,dkr>> 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() r/in1 and in2 should have the same dimensionalityrCN)r r]directr>r<cSsg|]\}}||dqSrrIrQrRjrIrIrJ rWzcorrelate..cSsg|]\}}||dqSr`rIrarIrIrJrc rWcss|]}td|VqdS)rNslicerPrIrIrJrVrWzcorrelate..r=7Acceptable method flags are 'auto', 'direct', or 'fft'.)npasarrayndimconjrFrDrEr_reverse_and_conj _np_conv_okrsizer\shapezipemptydtyperZ _correlateNDzerostuplecopy) in1in2rGmethodvalrHswapped_inputsZpsoutzZ in1zpaddedscrIrIrJr\sP            rcCs|dkrt| d|}n|dkrt| d|}|jd}|d}|ddkrj|||||}q|||||d}n4|dkr||}|dkrt|d}n t|d}|S)a 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)] r>rr=r;rr<)rgarangerm)Zin1_lenZin2_lenrGZlagsmidZ lag_boundrIrIrJr$s?   rcsRt|}t|j}||d|fddttD}|t|S)Nr;csg|]}t||qSrIrd)rQkZendindZstartindrIrJrcrWz_centered..)rgrharrayrnrXrYrs)ZarrnewshapeZ currshapeZmyslicerIrrJ _centereds    rFcs|j|jdu}t|dd\}|s:ts:tdfddD|rZtfddt|jDstdd t|d r||}}||fS) aHandle 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. N)rnr[z#when provided, axes cannot be emptycs(g|] }|dkr|dkr|qSr`rIrQa)s1s2rIrJrcrWz(_init_freq_conv_axes..c3s>|]6}|vr||kp4|dkp4|dkVqdS)rNrIrr[rrrIrJrVs z'_init_freq_conv_axes..z%incompatible shapes for in1 and in2: z and r[) rnr rYrFsortrZrXrir\)rurvrGr[ sorted_axesZnoaxes_rIrrJ_init_freq_conv_axess(   rc st|s||S|jjdkp&|jjdk|rBfdd|D}n}sZtjtj}}ntjtj}}||||d}||||d} ||| ||d} |rtddD} | | } | S)a 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`. ccsg|]}t| qSrI)sp_fft next_fast_lenrZcomplex_resultrnrIrJrcsz%_freq_domain_conv..rcSsg|] }t|qSrIrd)rQszrIrIrJrcrW) rYrqkindrZrfftnZirfftnZfftnZifftnrs) rurvr[rn calc_fast_lenZfshaper ifftZsp1Zsp2retZfslicerIrrJ_freq_domain_convs$ rcsf|dkrS|dkr&tS|dkrZfddtjD}t|StddS)aCalculate 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`. r>r=r<cs2g|]*}|vrj|n||dqSr`rnrr[rrrrIrJrc&sz$_apply_conv_mode..z4acceptable mode flags are 'valid', 'same', or 'full'N)rtrrXrirF)rrrrGr[Z shape_validrIrrJ_apply_conv_modesrcst|}t|}|j|jkr,dkr8nn||S|j|jkrNtdn|jdksb|jdkrltgSt|||dd\}}|j|jfddt|jD}t |||dd}t ||S) a 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() rr^Frcs<g|]4}|vr$t||fn||dqSr`)maxrPrrIrJrcszfftconvolve..Tr) rgrhrirFrmrrrnrXrr)rurvrGr[rnrrIrrJr.s$^       rc Cs||dd||f}||ks,|dks,|dkr0|S||krH||}}d}nd}||dkr\|S|d}| tddtj|ddj}tt|}||kr|S|s||d}|}n|}||d}||||fS)a<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. rNTFr;)r)rmathrHrealrrceil) rrfallbackZswappedoverlapZopt_size block_sizein1_stepin2_steprIrIrJ _calc_oa_lenss(  6"  rcsLt|}t|}|j|jkr,dkr8nn||S|j|jkrNtdn:|jdksb|jdkrltgS|j|jkrt|||dSt|||dd\}}|j|js||t |Sfddt |jD}fdd t |jD}t |\}}}|kr4|kr4t|||dSg} g} g} g} t |jD]8} | vrv| d g7} | d g7} qN| || krt | d || }| || ||| kr|d 7}||| | }nd }d}| || krNt | d || }| || ||| kr8|d 7}||| | }nd }d}| |g7} | |g7} | d|fg7} | d|fg7} qNtd d | Dstj|| d dd}tdd | Dstj|| d dd}ddtDddDt|}t|}tD]*\} }||| | ||| | q|j|}|j|}fddD}t|||ddt D]|\}}}||}|durq~t| g|\}t|dg|d}t|g|d}t|d g|d }||7}q~fddt jD}j|tdd|D}|t |S)a 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() rr^)rGr[Trcs,g|]$}|vrdn||dqS)NrrIrPrrIrJrczs zoaconvolve..c3s<|]4}|vr"dd||fnt||VqdS)rN)rrPrrIrJrVszoaconvolve..rrrcss|]}|dkVqdSrNrIrQZcurpadrIrIrJrVrWconstant)rGZconstant_valuescss|]}|dkVqdSrrIrrIrIrJrVrWcSsg|]\}}||qSrIrI)rQrRiaxrIrIrJrcrWcSsg|] }|dqSr`rI)rQrrIrIrJrcrWcsg|] }|qSrIrIrP)rrIrJrcrWFrNrcs>g|]6}|vr|vr"j|nj|j|dqSr`rrP)fft_axesr split_axesrIrJrcs cSsg|] }t|qSrIrd)rQislicerIrIrJrcrW)rgrhrirFrmrrnrrrrXrorrrZrA enumeratelistinsertreshapersplitrs)rurvrGr[Z shape_finalZ optimal_sizesoverlapsrrZnsteps1Znsteps2Z pad_size1Z pad_size2rRZ curnstep1Zcurpad1Z curnstep2Zcurpad2Z reshape_size1Z reshape_size2rZ fft_shapeaxZax_fftZax_splitrZoverpartZ ret_overpartZ shape_retZ slice_finalrI)r[rrrrrrrJr sP                  r buifccCs:t|tjkr|jj|vS|D]}|jj|vrdSqdS)ab 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. FT)typergndarrayrqr)ZarrayskindsZarray_rIrIrJ_numeric_arrayss   rc Cs|dkrddt||D}n:|dkrcSsg|]\}}||dqSr`rIrQnrrIrIrJrcrWz_conv_ops..r<cSs g|]\}}t||dqSr`)absrrIrIrJrcrWr=z?Acceptable mode flags are 'valid', 'same', or 'full', not mode=rrr;cSsg|]\}}||dqSr`rIrrIrIrJrcrW)rorFrYmin_prodrglog) x_shapeZh_shaperGZ out_shaperr direct_opsZfull_out_shapeNfft_opsrIrIrJ _conv_opss<    *   rc Cst|j|j|\}}|jdkr"dnd}|jdkr^dd|fdd|f|j|jkrVdd |fnd d nd d |fdd|fdd|fd }||\}}} ||||| kS)a 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. rgMbPg-C6g^]B> >g "]=g{$R>gen=gr0 < ,>gg=)gd[֠+>g@jHI>gh㈵)r<r>r=g.w'>gvlV>gG[*!>gʑQ>gVN+!>g4RP>)rrnrirm) xhrGrroffset constantsZO_fftZO_directZO_offsetrIrIrJ_fftconv_faster#s"  rcCs tdddf|j}||S)zO Reverse array `x` in all dimensions and perform the complex conjugate Nr)rerirj)rreverserIrIrJrkMsrkcCsF|j|jkrdkr>nn"|dvr(dS|dkrB|j|jkSndSdS)a= 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. r)r>r<Tr=FN)rirm)volumekernelrGrIrIrJrlUs rlpassrc Cstt||}d}tddD]"}d|}||}|dkrq>q|dkrL|}n|d9}|||}t|}||} | S)a 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. r gMb@?r)timeitTimerrXrepeatr) stmtsetuprtimerrpnumberbestrsecrIrIrJ _timeit_fastfs   rcst|t||r`i}dD]tfdd|<q |d|dkrTdnd}||fStddfDrtttt}|ttjj9}|dt d j d krdSt gd d rdSt grt rdSdS) a 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`. )r r_cstdS)N)rGrw)rrIrrwrGrrIrJsz$choose_conv_method..r r_cSsg|]}t|gddqS)Zuir)rrQrrIrIrJrcrWz&choose_conv_method..r;floatrbr) rgrhranyintrrrrmZfinfoZnmantrr)rurvrGmeasuretimesZ chosen_methodZ max_valuerIrrJr7s&l  $  r7cCs0t|}t|}|j|jkr,dkr8nn||S|j|jkrLtdt||j|jrf||}}|dkr|t|||d}|dkrt|||d}t||}|j dvrt |}t |j dst |j drtjdtdd ||S|d kr$t|||rt|||St|t||d Std d S) a 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() rz5volume and kernel should have the same dimensionalityr]rGr >urRzuUse of fft convolution on input with NAN or inf results in NAN or inf output. Consider using method='direct' instead.r;)category stacklevelr_rfN)rgrhrirFr\rnr7r result_typerZaroundisnanZflatisinfwarningswarnRuntimeWarningastyperlrrrk)rurvrGrwrrrzrrIrIrJrs2b          rcCst|}|jD]}|ddkrtdqt|}|jtdfvrpd|jd}tj|tddt |||}nt j |||dd }|S) a  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.]]) r;rzHEach dimension of domain argument should have an odd number of elements.Zfloat128z(Using order_filter with arrays of dtype > is deprecated in SciPy 1.11 and will be removed in SciPy 1.14rr)Z footprintrG) rgrhrnrFrqobjectrrDeprecationWarningr_order_filterNDr rank_filter)rdomainrankZdimsizemesgresultrIrIrJr!s6     r!c CsJt|}|durdg|j}t|}|jdkrDt||j}t|jD]}||ddkrNtdqNt ddt ||jDrt j d dd tj ||jd }tj|d d }|d}|jtjtjtjtjtjfvrtd|jd|jjdvr&d|jd}t j |tdd t|||}n t|}tj||d|dd}|S)a 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. NrrIr;r*Each element of kernel_size should be odd.css|]\}}||kVqdSrOrI)rQrsrIrIrJrVrWzmedfilt..zBkernel_size exceeds volume extent: the volume will be zero-padded.rrqraxiszdtype=z is not supported by medfilt)Ogz#Using medfilt with arrays of dtype rr)rmrG)rg atleast_1drirhrnritemrXrFrrorronesrqrZbool_Z complex64Z complex128Z clongdoublefloat16charrrrrr r) r kernel_sizerrZnumelsorderrrrmrIrIrJr"s:*      r"cCst|}|durdg|j}t|}|jdkrDt||j}t|t|dtj|dd}t|dt|dtj|dd|d}|durtj t |dd}||}|d||9}||7}t ||k||}|S) a 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() NrrIr=rrr;r) rgrhrirnrrrrrmeanZravelwhere)ZimZmysizenoiseZlMeanZlVarresrzrIrIrJr$7s&1      r$r@cCsxt|}t|}|j|jkr,dks6ntdt||j|jrP||}}t|}t|}t ||d|||}|S)a^ 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() r;z)convolve2d inputs must both be 2-D arraysr) rgrhrirFr\rnrKrNr _convolve2d)rurvrGrM fillvaluerxbvalrzrIrIrJrsK   rc Cst|}t|}|j|jkr,dks6ntdt||j|j}|rT||}}t|}t|}t || d|||}|r|ddddddf}|S)ag 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() r;z*correlate2d inputs must both be 2-D arraysrNr) rgrhrirFr\rnrKrNrrrj) rurvrGrMrryrxrrzrIrIrJrsR   rcCst|}|jjtjtjtjfvr,t||S|dur>dgd}t|}|jdkrbt | d}|D]}|ddkrft dqft ||S)a 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 Nrr;rIrr)rgrhrqrZubyteZfloat32float64r"rnrrrFrZ _medfilt2d)inputr imagermrIrIrJr#BsX       r#rcstt|}t|dkrtt|}jdkrR|jdkrRtdt|}||g}|durt|}|j|jkrtdt|j}jdd||<t |}|j|kr|jdg}|dkr||j7}t |jD]}||kr|j|||kr|j |||<q||krD|j|||krD|j |||<q||krh|j|dkrhd||<qtd|d|jdqtj j |||}||tj|} | jdvrtd | tj| d tj|| d }|dtj|| d }tfd d ||} | jtdg} |durLt|j|| |<| t | |7<t| j|td| |<| t | } |dur| St| j|tdd| |<| t | } | | fSn,|durt|||St||||SdS) aF 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() rz+object of too small depth for desired arrayNrz"Unexpected shape for zi: expected z, found .fdgFDGOinput type '%s' not supportedrcs t|SrO)rgr)yrrIrJr[rWzlfilter..)rgrrYrhrirF _validate_xrrnrsrXstrideslibZ stride_tricksZ as_stridedappendrrNotImplementedErrorrZapply_along_axisrerZ_linear_filter)rrrrziinputsZexpected_shaperrrqZout_fullindrzzfrIrrJr%st|                           r%c Cst|d}t|d}t||}t|}|durrtt|t||}|jdvrbtj}tj||d}njt|}tt|t|||}|jdvrtj}||}t|}||krtj |t||f}||}t||} t|}||krtj |t||f}t |D]4} tj || dd|d|| dd| | <q&t |D]<} | | tj || dd|d|| dd8<qd| S)a! 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 rNbuirrr) rgrmrrhrrrrrrr_rXsum) rrrrrMKrLrmrIrIrJr&qs6(            2 :r&c Cst|}t|}|jdkr&td|jdkr8tdt|}t|}||krZg}|}n:t||dt}d|d<t|||}|t||dd}||fS)aDeconvolves ``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.]) rzsignal must be 1-D.zdivisor must be 1-D.rr>r) rgrrirFrYrrrr%r) signalZdivisornumdenrDquotremrrIrIrJr(s )     r(cCst|}t|rtd|dur.|j|}|dkr>tdtj|||d}tj||jd}|ddkrd|d<||d<d|d|d<nd|d<d|d|dd<|j dkrtj g|j }t d||<|t |}tj |||d}|S) ao 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 must be real.NrN must be positive.rrr;r)rgrh iscomplexobjrFrnrr rrrqrinewaxisrersr)rrrXfrr!rIrIrJr)s(U       r)cCst|}|jdkrtdt|r.td|dur>|j}nLt|trb|dkrXtd||f}n(t|dkst t |dkrtdt j ||dd }tj |d|jd }tj |d |jd }||fD]^}|jd}|ddkr d |d<||d<d|d |d<qd |d<d|d |d d<q|ddtjf|tjddf}|j}|dkr||ddtjf}|d 8}qTt j||dd }|S) a 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 r;zx must be 2-D.r0Nrr1z@When given as a tuple, N must hold exactly two positive integers)rrrrr)rg atleast_2drirFr2rn isinstancerrYrrhrZfft2rrrqr3Zifft2)rrr4Zh1Zh2rZN1rrIrIrJr*i s<        $  r*zcmplx_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 cCstjttddt|S)Nr;r)rr_msg_cplx_sortr _cmplx_sort)rrIrIrJr+ sr+cCs*t|}tt|}t||d|fS)aSort 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]) r)rgrhZargsortrZtake)rZindxrIrIrJr8 s r8MbP?rc s|dvrtj}n(|dvr tj}n|dvr0tj}ntdt|}tt|df}t||dddf<t ||dddf<t |}g}g}tj t|t d t t|D]X}|rq||||} fd d | D} |||| |t| d | <qt|t|fS) ahDetermine 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]) rmaximumrminimumavgr J`rtype` must be one of {'max', 'maximum', 'min', 'minimum', 'avg', 'mean'}r;Nrrrcsg|]}|s|qSrIrIrusedrIrJrc rWz unique_roots..T)rgrrr rFrhrprYrimagrrrboolrXZquery_ball_pointr) rtolrtypereduceZpointstreeZp_uniqueZp_multiplicityrRgrouprIrArJr, s04  r,r?c Cst|}t|}tt|d}t|||\}}t||dd\}}t|dkrZd} n t||} t||D]\} } t| | | } qp| |fS)aCompute 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 fTinclude_powersr rgr trim_zeros _group_poles_compute_factorsrYpolymulroZpolyadd rrrrErF unique_poles multiplicityfactors denominator numeratorr/factorrIrIrJr-$ s8     r-c Cstdg}|g}t|ddd|dddD]<\}}td| g}t|D]}t||}qP||q0|ddd}g} tdg}t|||D]d\}}} td| g}g} t|D].} | dks|r| t|| t||}q| t| q| |fS)z>Compute the total polynomial divided by factors for each root.rrrN)rgrrorXrQrextendreversed) rootsrTrLcurrentsuffixespolemultmonomialrrUsuffixblockrRrIrIrJrPo s& &     rPcCst||\}}||j}g}t|||D]\}}}|dkr\|t||t||q*|} td| g} t || \}} g} t |D]>}t | | \} } | d| d}t | ||} | |q| t | q*t|S)Nrr)rPrrqrorrgZpolyvalrtrpolydivrXZpolysubrYrZrh)polesrTrWZdenominator_factorsrresiduesr^r_rXZnumerr`drbrrrIrIrJ_compute_residues s*    rgc Csjt|}t|}t|jtjs4t|jtjrJ|t}|t}n|t}|t}tt |d}tt |d}|j dkrt dt |}|j dkrt |jt|dtgfSt|t|krtd}nt||\}}t|||d\}}t|\}}||}t|||} d} t||D]"\} } | || | | <| | 7} q4| |d||fS)aZ 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. rJrDenominator `a` is zero.rErF)rgrh issubdtyperqcomplexfloatingrcomplexrrNrrmrFr[rrrnr8rrYrprcr,rgro) rrrErFrdrrSrTr reindexr^r_rIrIrJr/ s8L              r/cCst|}t|}t|jtjs4t|jtjrJ|t}|t}n|t}|t}tt |d}tt |d}|j dkrt dn|ddkrt dt |}|j dkrt |jt|dtgfS|ddd}|ddd}t|t|krtd}nt||\}}t|||d\}} t|\}} | | } td|| |} d} tjt| td } t|| D]<\}}||| | |<dt|| | | |<| |7} q~| | | |d9} | ||dddfS) aCompute 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 rrrhz6First coefficient of determinant `a` must be non-zero.Nrrirr)rgrhrjrqrkrrlrrNrrmrFr[rrrnr8rrYrprcr,rgrror})rrrErFrdZb_revZa_revZk_revrSrTr rermZpowersr^r_rIrIrJr0 sF8               r0c Cs|dvrtj}n(|dvr tj}n|dvr0tj}ntdg}g}|d}|g}tdt|D]N}t||||kr||q\||||t|||}|g}q\||||t|t |t |fS)Nr:r<r>r@rr) rgrrr rFrXrYrrrh) rdrErFrGuniquerTr^rbrRrIrIrJrO s* rOc Cst|}t|}tt|d}t|||\}}t||dd\}}t|dkrZd} n t|ddd|ddd} t||D]"\} } t| | | ddd} q| ddd|fS)aCompute 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 rTrKrNrrMrRrIrIrJr. s7     r.timecCs~|dvrtd|t|}|j|}t|}|dkr`|rPtj||d}qdtj||d}n|}|dur@t|r|t |} n6t |tj r|j|fkrtd|} nt t ||} dg|j} |j|| |<|r2| } | dd| dd d7<| ddd 9<|| d| || 9}n|| | 9}t|j} |rb|d d| |<n|| |<t| |j} t||}|d d}tdg|j}td |||<|t|| t|<|s|d krt||d||<|t|| t|<|d d kr||kr|rDt|d |d d||<| t|d 9<n:t| d | d d||<| t||t|7<nx||krt|d |d d||<| t|d 9<|s| t|}t||d ||d d||<|| t|<|rtj| ||d}ntj| |d d}|t|t|9}|durB|Std ||d|d |t||d }||fSdS)a[ 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() )rofreqz9Acceptable domain flags are 'time' or 'freq', not domain=rorNz(window must have the same length as datarrrg?r;g@T)rZ overwrite_x)rFrgrhrnZ isrealobjrZrfftr callableZfftfreqr6rZ ifftshiftrrirtrrrrrqrrersZirfftrrr})rr+trwindowrZNxZ real_inputXWZ newshape_WZW_realrYrZnyqsltemprZnew_trIrIrJr1 sV              "  0r1Zkaiserg@rcCst|}|t|krtd|t|kr2tdt|}t|}|dksR|dkrZtd|durt|dkrttd|t||}||}||}||krdkrnn|S|j|}||} | |t| |} t |t tj frt |}|j dkrtd|jdd } |} n8t||} d | } d | } td | d| |d |j} | |9} || |}d }| ||}tt| |||||| |kr|d7}qvttj|| jd| tj|| jdf} || }tjtjtjtjd}dd d}||vr||||dd}nF|tvrHd|i}|dkrZ|dur>d }||d<ntddt||vrl||}t| |||fd|i|}tdg|j }t||||<|t|}||vr||7}|S)a/ 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() zup must be an integerzdown must be an integerrzup and down must be >= 1Nrz#cval has no effect when padtype is zwindow must be 1-Dr;?rrsrr)r medianr=r;)rGcvalT)rkeepdimsrGr}z8padtype must be one of: maximum, mean, median, minimum, z, r) rgrhrrFrgcdrtrnrDr6rrrrirmrrrrqrrY concatenaterrr r|ZaminZamaxr joinrrers)rupdownrrspadtyper}Zg_Zn_inn_outhalf_lenrZmax_rateZf_cZ n_pre_padZ n_post_padZ n_pre_removeZn_pre_remove_endfuncsZupfirdn_kwargsZbackground_valuesrZkeepZy_keeprIrIrJr2 s|                     r2cCst|}t|}|jdkr&td|jdkr8td|j }t|}t|}|dkrhtdttdtj|j |}tj |dd}t |}t |}|r|d}|d}||fS)a 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`. rz)events cannot have dimensions more than 1z)period cannot have dimensions more than 1rzperiods must be positivey@r) rgrhrirFr5rexpdotpiTr rZangle)eventsZperiodZ scalarperiodZvectorsZ vectormeanZstrengthZphaserIrIrJr:t s&/        r:linearc Cs|dvrtdt|}|jj}|dvr.d}|dvrN|tj||dd}|S|j}||}ttt t d||}t ||krtd t |} |dkr|| }t ||d} | j} | |d } |s| } | jjdvr| |} tt |d D]} || d || } t| d f|}tjd | d |d | |dddf<t|| || d }t|| |\}}}}| |||| |<q| | } t | d|}|SdS)a: 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 )rlrrz*Trend type must be 'linear' or 'constant'.ZdfDFrf)rrT)r~rz>Breakpoints must be less than length of data along given axis.rrr;rN)rFrgrhrqrr rnrrnrrrrYmoveaxisrrtrrXrr}rer lstsq)datarrbpZoverwrite_datarqrZdshaperZrnknewdataZ newdata_shaper)ZNptsArwZcoefZresidsrrrIrIrJr3 sD)     $ r3cCs^t|}|jdkrtdt|}|jdkr8tdt|dkr^|ddkr^|dd}q8|jdkrptd|ddkr||d}||d}tt|t|}t||krtj|tj|t||j d f}n0t||krtj|tj|t||j d f}tj |dt ||d t |j}|dd|dd|d}tj ||}|S) 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. rzNumerator b must be 1-D.zDenominator a must be 1-D.rgNz3There must be at least one nonzero `a` coefficient.rzr)rgrrirFrYrmrr$rrrqeyerr Z companionrZsolve)rrrZIminusABrrIrIrJr4s,[         $"& r4cCst|}|jdks"|jddkr*td|jjdvrB|tj}|jd}tj |df|jd}d}t |D]J}||d d f}||d d f}|t ||||<|| | 9}ql|S) a  Construct initial conditions for sosfilt for step response steady-state. Compute an initial state `zi` for the `sosfilt` function that corresponds to the steady state of the step response. A typical use of this function is to set the initial state so that the output of the filter starts at the same value as the first element of the signal to be filtered. Parameters ---------- sos : array_like Array of second-order filter coefficients, must have shape ``(n_sections, 6)``. See `sosfilt` for the SOS filter format specification. Returns ------- zi : ndarray Initial conditions suitable for use with ``sosfilt``, shape ``(n_sections, 2)``. See Also -------- sosfilt, zpk2sos Notes ----- .. versionadded:: 0.16.0 Examples -------- Filter a rectangular pulse that begins at time 0, with and without the use of the `zi` argument of `scipy.signal.sosfilt`. >>> import numpy as np >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> sos = signal.butter(9, 0.125, output='sos') >>> zi = signal.sosfilt_zi(sos) >>> x = (np.arange(250) < 100).astype(int) >>> f1 = signal.sosfilt(sos, x) >>> f2, zo = signal.sosfilt(sos, x, zi=zi) >>> plt.plot(x, 'k--', label='x') >>> plt.plot(f1, 'b', alpha=0.5, linewidth=2, label='filtered') >>> plt.plot(f2, 'g', alpha=0.25, linewidth=4, label='filtered with zi') >>> plt.legend(loc='best') >>> plt.show() r;rz!sos must be shape (n_sections, 6)r#rrrzNr) rgrhrirnrFrqrrrrprXr4r%)sos n_sectionsrscalesectionrrrIrIrJr5s6     r5c!Cs,t|}t|}tt|t|d}|dkrd|d|dd}||}|tgtgfS|dksz||jdkrt|||jd}|jd}|dus|d|kr|} n|} t| |f} t|} d| d<t ||t| | dd| dddf<t d|D]$} | d| df| | d| f<q| ddd} t ||| ddddd}|ddd}| |krt || | |f}nFtd| d|f}|| |d| d|f<| ||| d|df<t |||}t |||ddddfddddf}t |||ddddfddddf}t |||}||}| |krJ|}n4|dd| f}|d| df}tj ||fdd}|jdkrt ||d}nF|d|jdj}t ||dj}||jdd|jdf}| |krt || f}n>td| d|f}||d| d|f<| || d|df<||j}|}| |kr^||7}nL|dd| f|dd| f7<|d| df|d| df7<|dd|f}|d| df} |dks||jdkr"t|||jd}t| ||jd} t|||jd}||| fS) a7Forward-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. rrr;rN)rr.)rgrrrYrriZswapaxesrnrrr%rXZhstackrr rrrr)!rrrrirlenr rrrr)ZObsrrZObsrSZSrr&Zy_fZy_fbZy_bZy_bfZ delta_y_bf_fbdeltaZstart_mZend_mZic_optZdelta2dZic_opt0ruZwicZy_optx0x1rIrIrJ_filtfilt_gustsx1     &"   ((        $(roddrAc Cs&t|}t|}t|}|dvr.td|dkrRt|||||d\}} } |St||||tt|t|d\} } t||} dg|j }| j ||<t | |} t | d|d}t ||| || |d\}}t |d |d }t ||t||d || |d\}}t||d }| d kr"t || | |d }|S)a+ 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 )rAgustzmethod must be 'pad' or 'gust'.r)rrntapsrstoprrrrstartrrrrrr)rgrrhrFr _validate_padrrYr4rirmrrr%r)rrrrrpadlenrwrrZz1Zz2edgeextrzi_shaperr"Zy0rIrIrJr8s0         "  r8cCs|dvrtd||dur d}|dur2|d}n|}|j||krPtd||dur|dkr|dkrxt|||d}q|d krt|||d}qt|||d}n|}||fS) z'Helper to validate padding for filtfilt)evenrrNzYUnknown value '%s' given to padtype. padtype must be 'even', 'odd', 'constant', or None.NrrzJThe length of the input vector x must be greater than padlen, which is %d.rrr)rFrnrrr)rrrrrrrrIrIrJrks* rcCs t|}|jdkrtd|S)Nrzx must be at least 1-D)rgrhrirF)rrIrIrJrs  rc Cst|}t|\}}t|j}d||<t|g|}||g}|durT|t|tj|}|j dvrtt d||durt ||}|j|krt d||j|||jfd}ntj ||d}d}||j}t||d }t|d |d gd d g}|j|j} } t|d |jd f}tj ||d d}tt|d |df}|j|dd}t|||| |_t|d |}|r| |_t|d d gd |d g}||f} n|} | S)a 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() r;NrrzyInvalid zi shape. With axis=%r, an input with shape %r, and an sos array with %d sections, zi must have shape %r, got %r.TrFrrrC)r )rt)rrrrnrsrrgrhrrrrrFrrrirrZascontiguousarrayrr) rrrrrZ x_zi_shaper rqZ return_zirrrzrIrIrJr'sJF          r'cCst|\}}t|}d|d}|t|dddfdk|dddfdk8}t|||||d\}}t|} dg|j} d| |<|g| | _t|d|d} t |||| | d\} } t| d |d }t |t | |d || |d\} } t | |d } |dkrt| || |d } | S) a 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. 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