### Table Of Contents

- Signal processing (`scipy.signal`)
  - Convolution
  - B-splines
  - Filtering
  - Filter design
  - Matlab-style IIR filter design
  - Continuous-Time Linear Systems
  - Discrete-Time Linear Systems
  - LTI Representations
  - Waveforms
  - Window functions
  - Wavelets
  - Peak finding
  - Spectral Analysis

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# Signal processing (`scipy.signal`)

## Convolution

`convolve`(in1, in2[, mode]) | Convolve two N-dimensional arrays.
`correlate`(in1, in2[, mode]) | Cross-correlate two N-dimensional arrays.
`fftconvolve`(in1, in2[, mode]) | Convolve two N-dimensional arrays using FFT.
`convolve2d`(in1, in2[, mode, boundary, fillvalue]) | Convolve two 2-dimensional arrays.
`correlate2d`(in1, in2[, mode, boundary, ...]) | Cross-correlate two 2-dimensional arrays.
`sepfir2d`((input, hrow, hcol) -> output) | Description: Convolve the rank-2 input array with the separable filter defined by the rank-1 arrays hrow, and hcol.

## B-splines

`bspline`(x, n) | B-spline basis function of order n.
`cubic`(x) | A cubic B-spline.
`quadratic`(x) | A quadratic B-spline.
`gauss_spline`(x, n) | Gaussian approximation to B-spline basis function of order n.
`cspline1d`(signal[, lamb]) | Compute cubic spline coefficients for rank-1 array.
`qspline1d`(signal[, lamb]) | Compute quadratic spline coefficients for rank-1 array.
`cspline2d`((input {, lambda, precision}) -> ck) | Description: Return the third-order B-spline coefficients over a regularly spacedi input grid for the two-dimensional input image.
`qspline2d`((input {, lambda, precision}) -> qk) | Description: Return the second-order B-spline coefficients over a regularly spaced input grid for the two-dimensional input image.
`cspline1d_eval`(cj, newx[, dx, x0]) | Evaluate a spline at the new set of points.
`qspline1d_eval`(cj, newx[, dx, x0]) | Evaluate a quadratic spline at the new set of points.
`spline_filter`(Iin[, lmbda]) | Smoothing spline (cubic) filtering of a rank-2 array.

## Filtering

`order_filter`(a, domain, rank) | Perform an order filter on an N-dimensional array.
`medfilt`(volume[, kernel_size]) | Perform a median filter on an N-dimensional array.
`medfilt2d`(input[, kernel_size]) | Median filter a 2-dimensional array.
`wiener`(im[, mysize, noise]) | Perform a Wiener filter on an N-dimensional array.
`symiirorder1`((input, c0, z1 {, ...) | Implement a smoothing IIR filter with mirror-symmetric boundary conditions using a cascade of first-order sections.
`symiirorder2`((input, r, omega {, ...) | Implement a smoothing IIR filter with mirror-symmetric boundary conditions using a cascade of second-order sections.
`lfilter`(b, a, x[, axis, zi]) | Filter data along one-dimension with an IIR or FIR filter.
`lfiltic`(b, a, y[, x]) | Construct initial conditions for lfilter.
`lfilter_zi`(b, a) | Compute an initial state zi for the lfilter function that corresponds to the steady state of the step response.
`filtfilt`(b, a, x[, axis, padtype, padlen, ...]) | A forward-backward filter.
`savgol_filter`(x, window_length, polyorder[, ...]) | Apply a Savitzky-Golay filter to an array.
`deconvolve`(signal, divisor) | Deconvolves `divisor` out of `signal`.
`sosfilt`(sos, x[, axis, zi]) | Filter data along one dimension using cascaded second-order sections Filter a data sequence, x, using a digital IIR filter defined by sos.
`sosfilt_zi`(sos) | Compute an initial state zi for the sosfilt function that corresponds to the steady state of the step response.
`hilbert`(x[, N, axis]) | Compute the analytic signal, using the Hilbert transform.
`hilbert2`(x[, N]) | Compute the ‘2-D’ analytic signal of x :Parameters: x : array_like 2-D signal data.
`decimate`(x, q[, n, ftype, axis]) | Downsample the signal by using a filter.
`detrend`(data[, axis, type, bp]) | Remove linear trend along axis from data.
`resample`(x, num[, t, axis, window]) | Resample x to num samples using Fourier method along the given axis.

## Filter design

`bilinear`(b, a[, fs]) | Return a digital filter from an analog one using a bilinear transform.
`findfreqs`(num, den, N) | Find an array of frequencies for computing the response of a filter.
`firwin`(numtaps, cutoff[, width, window, ...]) | FIR filter design using the window method.
`firwin2`(numtaps, freq, gain[, nfreqs, ...]) | FIR filter design using the window method.
`freqs`(b, a[, worN, plot]) | Compute frequency response of analog filter.
`freqz`(b[, a, worN, whole, plot]) | Compute the frequency response of a digital filter.
`group_delay`(system[, w, whole]) | Compute the group delay of a digital filter.
`iirdesign`(wp, ws, gpass, gstop[, analog, ...]) | Complete IIR digital and analog filter design.
`iirfilter`(N, Wn[, rp, rs, btype, analog, ...]) | IIR digital and analog filter design given order and critical points.
`kaiser_atten`(numtaps, width) | Compute the attenuation of a Kaiser FIR filter.
`kaiser_beta`(a) | Compute the Kaiser parameter beta, given the attenuation a.
`kaiserord`(ripple, width) | Design a Kaiser window to limit ripple and width of transition region.
`savgol_coeffs`(window_length, polyorder[, ...]) | Compute the coefficients for a 1-d Savitzky-Golay FIR filter.
`remez`(numtaps, bands, desired[, weight, Hz, ...]) | Calculate the minimax optimal filter using the Remez exchange algorithm.
`unique_roots`(p[, tol, rtype]) | Determine unique roots and their multiplicities from a list of roots.
`residue`(b, a[, tol, rtype]) | Compute partial-fraction expansion of b(s) / a(s).
`residuez`(b, a[, tol, rtype]) | Compute partial-fraction expansion of b(z) / a(z).
`invres`(r, p, k[, tol, rtype]) | Compute b(s) and a(s) from partial fraction expansion.
`invresz`(r, p, k[, tol, rtype]) | Compute b(z) and a(z) from partial fraction expansion.

Lower-level filter design functions:

`abcd_normalize`([A, B, C, D]) | Check state-space matrices and ensure they are two-dimensional.
`band_stop_obj`(wp, ind, passb, stopb, gpass, ...) | Band Stop Objective Function for order minimization.
`besselap`(N) | Return (z,p,k) for analog prototype of an Nth order Bessel filter.
`buttap`(N) | Return (z,p,k) for analog prototype of Nth order Butterworth filter.
`cheb1ap`(N, rp) | Return (z,p,k) for Nth order Chebyshev type I analog lowpass filter.
`cheb2ap`(N, rs) | Return (z,p,k) for Nth order Chebyshev type I analog lowpass filter.
`cmplx_sort`(p) | Sort roots based on magnitude.
`ellipap`(N, rp, rs) | Return (z,p,k) of Nth order elliptic analog lowpass filter.
`lp2bp`(b, a[, wo, bw]) | Transform a lowpass filter prototype to a bandpass filter.
`lp2bs`(b, a[, wo, bw]) | Transform a lowpass filter prototype to a bandstop filter.
`lp2hp`(b, a[, wo]) | Transform a lowpass filter prototype to a highpass filter.
`lp2lp`(b, a[, wo]) | Transform a lowpass filter prototype to a different frequency.
`normalize`(b, a) | Normalize polynomial representation of a transfer function.

## Matlab-style IIR filter design

`butter`(N, Wn[, btype, analog, output]) | Butterworth digital and analog filter design.
`buttord`(wp, ws, gpass, gstop[, analog]) | Butterworth filter order selection.
`cheby1`(N, rp, Wn[, btype, analog, output]) | Chebyshev type I digital and analog filter design.
`cheb1ord`(wp, ws, gpass, gstop[, analog]) | Chebyshev type I filter order selection.
`cheby2`(N, rs, Wn[, btype, analog, output]) | Chebyshev type II digital and analog filter design.
`cheb2ord`(wp, ws, gpass, gstop[, analog]) | Chebyshev type II filter order selection.
`ellip`(N, rp, rs, Wn[, btype, analog, output]) | Elliptic (Cauer) digital and analog filter design.
`ellipord`(wp, ws, gpass, gstop[, analog]) | Elliptic (Cauer) filter order selection.
`bessel`(N, Wn[, btype, analog, output]) | Bessel/Thomson digital and analog filter design.

## Continuous-Time Linear Systems

`freqresp`(system[, w, n]) | Calculate the frequency response of a continuous-time system.
`lti`(*system) | Linear Time Invariant system base class.
`StateSpace`(*system) | Linear Time Invariant system class in state-space form.
`TransferFunction`(*system) | Linear Time Invariant system class in transfer function form.
`ZerosPolesGain`(*system) | Linear Time Invariant system class in zeros, poles, gain form.
`lsim`(system, U, T[, X0, interp]) | Simulate output of a continuous-time linear system.
`lsim2`(system[, U, T, X0]) | Simulate output of a continuous-time linear system, by using the ODE solver `scipy.integrate.odeint`.
`impulse`(system[, X0, T, N]) | Impulse response of continuous-time system.
`impulse2`(system[, X0, T, N]) | Impulse response of a single-input, continuous-time linear system.
`step`(system[, X0, T, N]) | Step response of continuous-time system.
`step2`(system[, X0, T, N]) | Step response of continuous-time system.
`bode`(system[, w, n]) | Calculate Bode magnitude and phase data of a continuous-time system.

## Discrete-Time Linear Systems

`dlsim`(system, u[, t, x0]) | Simulate output of a discrete-time linear system.
`dimpulse`(system[, x0, t, n]) | Impulse response of discrete-time system.
`dstep`(system[, x0, t, n]) | Step response of discrete-time system.

## LTI Representations

`tf2zpk`(b, a) | Return zero, pole, gain (z, p, k) representation from a numerator, denominator representation of a linear filter.
`tf2sos`(b, a[, pairing]) | Return second-order sections from transfer function representation :Parameters: b : array_like Numerator polynomial coefficients.
`tf2ss`(num, den) | Transfer function to state-space representation.
`zpk2tf`(z, p, k) | Return polynomial transfer function representation from zeros and poles :Parameters: z : array_like Zeros of the transfer function.
`zpk2sos`(z, p, k[, pairing]) | Return second-order sections from zeros, poles, and gain of a system :Parameters: z : array_like Zeros of the transfer function.
`zpk2ss`(z, p, k) | Zero-pole-gain representation to state-space representation :Parameters: z, p : sequence Zeros and poles.
`ss2tf`(A, B, C, D[, input]) | State-space to transfer function.
`ss2zpk`(A, B, C, D[, input]) | State-space representation to zero-pole-gain representation.
`sos2zpk`(sos) | Return zeros, poles, and gain of a series of second-order sections :Parameters: sos : array_like Array of second-order filter coefficients, must have shape `(n_sections, 6)`.
`sos2tf`(sos) | Return a single transfer function from a series of second-order sections :Parameters: sos : array_like Array of second-order filter coefficients, must have shape `(n_sections, 6)`.
`cont2discrete`(sys, dt[, method, alpha]) | Transform a continuous to a discrete state-space system.
`place_poles`(A, B, poles[, method, rtol, maxiter]) | Compute K such that eigenvalues (A - dot(B, K))=poles.

## Waveforms

`chirp`(t, f0, t1, f1[, method, phi, vertex_zero]) | Frequency-swept cosine generator.
`gausspulse`(t[, fc, bw, bwr, tpr, retquad, ...]) | Return a Gaussian modulated sinusoid: `exp(-a t^2) exp(1j*2*pi*fc*t).` If retquad is True, then return the real and imaginary parts (in-phase and quadrature).
`max_len_seq`(nbits[, state, length, taps]) | Maximum Length Sequence (MLS) generator :Parameters: nbits : int Number of bits to use.
`sawtooth`(t[, width]) | Return a periodic sawtooth or triangle waveform.
`square`(t[, duty]) | Return a periodic square-wave waveform.
`sweep_poly`(t, poly[, phi]) | Frequency-swept cosine generator, with a time-dependent frequency.

## Window functions

`get_window`(window, Nx[, fftbins]) | Return a window.
`barthann`(M[, sym]) | Return a modified Bartlett-Hann window.
`bartlett`(M[, sym]) | Return a Bartlett window.
`blackman`(M[, sym]) | Return a Blackman window.
`blackmanharris`(M[, sym]) | Return a minimum 4-term Blackman-Harris window.
`bohman`(M[, sym]) | Return a Bohman window.
`boxcar`(M[, sym]) | Return a boxcar or rectangular window.
`chebwin`(M, at[, sym]) | Return a Dolph-Chebyshev window.
`cosine`(M[, sym]) | Return a window with a simple cosine shape.
`exponential`(M[, center, tau, sym]) | Return an exponential (or Poisson) window.
`flattop`(M[, sym]) | Return a flat top window.
`gaussian`(M, std[, sym]) | Return a Gaussian window.
`general_gaussian`(M, p, sig[, sym]) | Return a window with a generalized Gaussian shape.
`hamming`(M[, sym]) | Return a Hamming window.
`hann`(M[, sym]) | Return a Hann window.
`kaiser`(M, beta[, sym]) | Return a Kaiser window.
`nuttall`(M[, sym]) | Return a minimum 4-term Blackman-Harris window according to Nuttall.
`parzen`(M[, sym]) | Return a Parzen window.
`slepian`(M, width[, sym]) | Return a digital Slepian (DPSS) window.
`triang`(M[, sym]) | Return a triangular window.
`tukey`(M[, alpha, sym]) | Return a Tukey window, also known as a tapered cosine window.

## Wavelets

`cascade`(hk[, J]) | Return (x, phi, psi) at dyadic points `K/2**J` from filter coefficients.
`daub`(p) | The coefficients for the FIR low-pass filter producing Daubechies wavelets.
`morlet`(M[, w, s, complete]) | Complex Morlet wavelet.
`qmf`(hk) | Return high-pass qmf filter from low-pass :Parameters: hk : array_like Coefficients of high-pass filter.
`ricker`(points, a) | Return a Ricker wavelet, also known as the “Mexican hat wavelet”.
`cwt`(data, wavelet, widths) | Continuous wavelet transform.

## Peak finding

`find_peaks_cwt`(vector, widths[, wavelet, ...]) | Attempt to find the peaks in a 1-D array.
`argrelmin`(data[, axis, order, mode]) | Calculate the relative minima of data.
`argrelmax`(data[, axis, order, mode]) | Calculate the relative maxima of data.
`argrelextrema`(data, comparator[, axis, ...]) | Calculate the relative extrema of data.

## Spectral Analysis

`periodogram`(x[, fs, window, nfft, detrend, ...]) | Estimate power spectral density using a periodogram.
`welch`(x[, fs, window, nperseg, noverlap, ...]) | Estimate power spectral density using Welch’s method.
`csd`(x, y[, fs, window, nperseg, noverlap, ...]) | Estimate the cross power spectral density, Pxy, using Welch’s method.
`coherence`(x, y[, fs, window, nperseg, ...]) | Estimate the magnitude squared coherence estimate, Cxy, of discrete-time signals X and Y using Welch’s method.
`spectrogram`(x[, fs, window, nperseg, ...]) | Compute a spectrogram with consecutive Fourier transforms.
`lombscargle`(x, y, freqs) | Computes the Lomb-Scargle periodogram.
`vectorstrength`(events, period) | Determine the vector strength of the events corresponding to the given period.