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# -*- coding: utf-8 -*-
'firwin', 'firwin2', 'remez', 'firls', 'minimum_phase']
""" Utility for replacing the argument 'nyq' (with default 1) with 'fs'. """ elif nyq is not None: if fs is not None: raise ValueError("Values cannot be given for both 'nyq' and 'fs'.") fs = 2*nyq
# Some notes on function parameters: # # `cutoff` and `width` are given as numbers between 0 and 1. These are # relative frequencies, expressed as a fraction of the Nyquist frequency. # For example, if the Nyquist frequency is 2 KHz, then width=0.15 is a width # of 300 Hz. # # The `order` of a FIR filter is one less than the number of taps. # This is a potential source of confusion, so in the following code, # we will always use the number of taps as the parameterization of # the 'size' of the filter. The "number of taps" means the number # of coefficients, which is the same as the length of the impulse # response of the filter.
"""Compute the Kaiser parameter `beta`, given the attenuation `a`.
Parameters ---------- a : float The desired attenuation in the stopband and maximum ripple in the passband, in dB. This should be a *positive* number.
Returns ------- beta : float The `beta` parameter to be used in the formula for a Kaiser window.
References ---------- Oppenheim, Schafer, "Discrete-Time Signal Processing", p.475-476.
Examples -------- Suppose we want to design a lowpass filter, with 65 dB attenuation in the stop band. The Kaiser window parameter to be used in the window method is computed by `kaiser_beta(65)`:
>>> from scipy.signal import kaiser_beta >>> kaiser_beta(65) 6.20426
""" if a > 50: beta = 0.1102 * (a - 8.7) elif a > 21: beta = 0.5842 * (a - 21) ** 0.4 + 0.07886 * (a - 21) else: beta = 0.0 return beta
"""Compute the attenuation of a Kaiser FIR filter.
Given the number of taps `N` and the transition width `width`, compute the attenuation `a` in dB, given by Kaiser's formula:
a = 2.285 * (N - 1) * pi * width + 7.95
Parameters ---------- numtaps : int The number of taps in the FIR filter. width : float The desired width of the transition region between passband and stopband (or, in general, at any discontinuity) for the filter, expressed as a fraction of the Nyquist frequency.
Returns ------- a : float The attenuation of the ripple, in dB.
See Also -------- kaiserord, kaiser_beta
Examples -------- Suppose we want to design a FIR filter using the Kaiser window method that will have 211 taps and a transition width of 9 Hz for a signal that is sampled at 480 Hz. Expressed as a fraction of the Nyquist frequency, the width is 9/(0.5*480) = 0.0375. The approximate attenuation (in dB) is computed as follows:
>>> from scipy.signal import kaiser_atten >>> kaiser_atten(211, 0.0375) 64.48099630593983
""" a = 2.285 * (numtaps - 1) * np.pi * width + 7.95 return a
""" Determine the filter window parameters for the Kaiser window method.
The parameters returned by this function are generally used to create a finite impulse response filter using the window method, with either `firwin` or `firwin2`.
Parameters ---------- ripple : float Upper bound for the deviation (in dB) of the magnitude of the filter's frequency response from that of the desired filter (not including frequencies in any transition intervals). That is, if w is the frequency expressed as a fraction of the Nyquist frequency, A(w) is the actual frequency response of the filter and D(w) is the desired frequency response, the design requirement is that::
abs(A(w) - D(w))) < 10**(-ripple/20)
for 0 <= w <= 1 and w not in a transition interval. width : float Width of transition region, normalized so that 1 corresponds to pi radians / sample. That is, the frequency is expressed as a fraction of the Nyquist frequency.
Returns ------- numtaps : int The length of the Kaiser window. beta : float The beta parameter for the Kaiser window.
See Also -------- kaiser_beta, kaiser_atten
Notes ----- There are several ways to obtain the Kaiser window:
- ``signal.kaiser(numtaps, beta, sym=True)`` - ``signal.get_window(beta, numtaps)`` - ``signal.get_window(('kaiser', beta), numtaps)``
The empirical equations discovered by Kaiser are used.
References ---------- Oppenheim, Schafer, "Discrete-Time Signal Processing", p.475-476.
Examples -------- We will use the Kaiser window method to design a lowpass FIR filter for a signal that is sampled at 1000 Hz.
We want at least 65 dB rejection in the stop band, and in the pass band the gain should vary no more than 0.5%.
We want a cutoff frequency of 175 Hz, with a transition between the pass band and the stop band of 24 Hz. That is, in the band [0, 163], the gain varies no more than 0.5%, and in the band [187, 500], the signal is attenuated by at least 65 dB.
>>> from scipy.signal import kaiserord, firwin, freqz >>> import matplotlib.pyplot as plt >>> fs = 1000.0 >>> cutoff = 175 >>> width = 24
The Kaiser method accepts just a single parameter to control the pass band ripple and the stop band rejection, so we use the more restrictive of the two. In this case, the pass band ripple is 0.005, or 46.02 dB, so we will use 65 dB as the design parameter.
Use `kaiserord` to determine the length of the filter and the parameter for the Kaiser window.
>>> numtaps, beta = kaiserord(65, width/(0.5*fs)) >>> numtaps 167 >>> beta 6.20426
Use `firwin` to create the FIR filter.
>>> taps = firwin(numtaps, cutoff, window=('kaiser', beta), ... scale=False, nyq=0.5*fs)
Compute the frequency response of the filter. ``w`` is the array of frequencies, and ``h`` is the corresponding complex array of frequency responses.
>>> w, h = freqz(taps, worN=8000) >>> w *= 0.5*fs/np.pi # Convert w to Hz.
Compute the deviation of the magnitude of the filter's response from that of the ideal lowpass filter. Values in the transition region are set to ``nan``, so they won't appear in the plot.
>>> ideal = w < cutoff # The "ideal" frequency response. >>> deviation = np.abs(np.abs(h) - ideal) >>> deviation[(w > cutoff - 0.5*width) & (w < cutoff + 0.5*width)] = np.nan
Plot the deviation. A close look at the left end of the stop band shows that the requirement for 65 dB attenuation is violated in the first lobe by about 0.125 dB. This is not unusual for the Kaiser window method.
>>> plt.plot(w, 20*np.log10(np.abs(deviation))) >>> plt.xlim(0, 0.5*fs) >>> plt.ylim(-90, -60) >>> plt.grid(alpha=0.25) >>> plt.axhline(-65, color='r', ls='--', alpha=0.3) >>> plt.xlabel('Frequency (Hz)') >>> plt.ylabel('Deviation from ideal (dB)') >>> plt.title('Lowpass Filter Frequency Response') >>> plt.show()
""" A = abs(ripple) # in case somebody is confused as to what's meant if A < 8: # Formula for N is not valid in this range. raise ValueError("Requested maximum ripple attentuation %f is too " "small for the Kaiser formula." % A) beta = kaiser_beta(A)
# Kaiser's formula (as given in Oppenheim and Schafer) is for the filter # order, so we have to add 1 to get the number of taps. numtaps = (A - 7.95) / 2.285 / (np.pi * width) + 1
return int(ceil(numtaps)), beta
scale=True, nyq=None, fs=None): """ FIR filter design using the window method.
This function computes the coefficients of a finite impulse response filter. The filter will have linear phase; it will be Type I if `numtaps` is odd and Type II if `numtaps` is even.
Type II filters always have zero response at the Nyquist frequency, so a ValueError exception is raised if firwin is called with `numtaps` even and having a passband whose right end is at the Nyquist frequency.
Parameters ---------- numtaps : int Length of the filter (number of coefficients, i.e. the filter order + 1). `numtaps` must be even if a passband includes the Nyquist frequency. cutoff : float or 1D array_like Cutoff frequency of filter (expressed in the same units as `nyq`) OR an array of cutoff frequencies (that is, band edges). In the latter case, the frequencies in `cutoff` should be positive and monotonically increasing between 0 and `nyq`. The values 0 and `nyq` must not be included in `cutoff`. width : float or None, optional If `width` is not None, then assume it is the approximate width of the transition region (expressed in the same units as `nyq`) for use in Kaiser FIR filter design. In this case, the `window` argument is ignored. window : string or tuple of string and parameter values, optional Desired window to use. See `scipy.signal.get_window` for a list of windows and required parameters. pass_zero : bool, optional If True, the gain at the frequency 0 (i.e. the "DC gain") is 1. Otherwise the DC gain is 0. scale : bool, optional Set to True to scale the coefficients so that the frequency response is exactly unity at a certain frequency. That frequency is either:
- 0 (DC) if the first passband starts at 0 (i.e. pass_zero is True) - `nyq` (the Nyquist frequency) if the first passband ends at `nyq` (i.e the filter is a single band highpass filter); center of first passband otherwise
nyq : float, optional *Deprecated. Use `fs` instead.* This is the Nyquist frequency. Each frequency in `cutoff` must be between 0 and `nyq`. Default is 1. fs : float, optional The sampling frequency of the signal. Each frequency in `cutoff` must be between 0 and ``fs/2``. Default is 2.
Returns ------- h : (numtaps,) ndarray Coefficients of length `numtaps` FIR filter.
Raises ------ ValueError If any value in `cutoff` is less than or equal to 0 or greater than or equal to ``fs/2``, if the values in `cutoff` are not strictly monotonically increasing, or if `numtaps` is even but a passband includes the Nyquist frequency.
See Also -------- firwin2 firls minimum_phase remez
Examples -------- Low-pass from 0 to f:
>>> from scipy import signal >>> numtaps = 3 >>> f = 0.1 >>> signal.firwin(numtaps, f) array([ 0.06799017, 0.86401967, 0.06799017])
Use a specific window function:
>>> signal.firwin(numtaps, f, window='nuttall') array([ 3.56607041e-04, 9.99286786e-01, 3.56607041e-04])
High-pass ('stop' from 0 to f):
>>> signal.firwin(numtaps, f, pass_zero=False) array([-0.00859313, 0.98281375, -0.00859313])
Band-pass:
>>> f1, f2 = 0.1, 0.2 >>> signal.firwin(numtaps, [f1, f2], pass_zero=False) array([ 0.06301614, 0.88770441, 0.06301614])
Band-stop:
>>> signal.firwin(numtaps, [f1, f2]) array([-0.00801395, 1.0160279 , -0.00801395])
Multi-band (passbands are [0, f1], [f2, f3] and [f4, 1]):
>>> f3, f4 = 0.3, 0.4 >>> signal.firwin(numtaps, [f1, f2, f3, f4]) array([-0.01376344, 1.02752689, -0.01376344])
Multi-band (passbands are [f1, f2] and [f3,f4]):
>>> signal.firwin(numtaps, [f1, f2, f3, f4], pass_zero=False) array([ 0.04890915, 0.91284326, 0.04890915])
""" # The major enhancements to this function added in November 2010 were # developed by Tom Krauss (see ticket #902).
# Check for invalid input. raise ValueError("The cutoff argument must be at most " "one-dimensional.") raise ValueError("At least one cutoff frequency must be given.") raise ValueError("Invalid cutoff frequency: frequencies must be " "greater than 0 and less than fs/2.") raise ValueError("Invalid cutoff frequencies: the frequencies " "must be strictly increasing.")
# A width was given. Find the beta parameter of the Kaiser window # and set `window`. This overrides the value of `window` passed in. atten = kaiser_atten(numtaps, float(width) / nyq) beta = kaiser_beta(atten) window = ('kaiser', beta)
raise ValueError("A filter with an even number of coefficients must " "have zero response at the Nyquist frequency.")
# Insert 0 and/or 1 at the ends of cutoff so that the length of cutoff # is even, and each pair in cutoff corresponds to passband.
# `bands` is a 2D array; each row gives the left and right edges of # a passband.
# Build up the coefficients.
# Get and apply the window function.
# Now handle scaling if desired. # Get the first passband. elif right == 1: scale_frequency = 1.0 else: scale_frequency = 0.5 * (left + right)
# Original version of firwin2 from scipy ticket #457, submitted by "tash". # # Rewritten by Warren Weckesser, 2010.
antisymmetric=False, fs=None): """ FIR filter design using the window method.
From the given frequencies `freq` and corresponding gains `gain`, this function constructs an FIR filter with linear phase and (approximately) the given frequency response.
Parameters ---------- numtaps : int The number of taps in the FIR filter. `numtaps` must be less than `nfreqs`. freq : array_like, 1D The frequency sampling points. Typically 0.0 to 1.0 with 1.0 being Nyquist. The Nyquist frequency is half `fs`. The values in `freq` must be nondecreasing. A value can be repeated once to implement a discontinuity. The first value in `freq` must be 0, and the last value must be ``fs/2``. gain : array_like The filter gains at the frequency sampling points. Certain constraints to gain values, depending on the filter type, are applied, see Notes for details. nfreqs : int, optional The size of the interpolation mesh used to construct the filter. For most efficient behavior, this should be a power of 2 plus 1 (e.g, 129, 257, etc). The default is one more than the smallest power of 2 that is not less than `numtaps`. `nfreqs` must be greater than `numtaps`. window : string or (string, float) or float, or None, optional Window function to use. Default is "hamming". See `scipy.signal.get_window` for the complete list of possible values. If None, no window function is applied. nyq : float, optional *Deprecated. Use `fs` instead.* This is the Nyquist frequency. Each frequency in `freq` must be between 0 and `nyq`. Default is 1. antisymmetric : bool, optional Whether resulting impulse response is symmetric/antisymmetric. See Notes for more details. fs : float, optional The sampling frequency of the signal. Each frequency in `cutoff` must be between 0 and ``fs/2``. Default is 2.
Returns ------- taps : ndarray The filter coefficients of the FIR filter, as a 1-D array of length `numtaps`.
See also -------- firls firwin minimum_phase remez
Notes ----- From the given set of frequencies and gains, the desired response is constructed in the frequency domain. The inverse FFT is applied to the desired response to create the associated convolution kernel, and the first `numtaps` coefficients of this kernel, scaled by `window`, are returned.
The FIR filter will have linear phase. The type of filter is determined by the value of 'numtaps` and `antisymmetric` flag. There are four possible combinations:
- odd `numtaps`, `antisymmetric` is False, type I filter is produced - even `numtaps`, `antisymmetric` is False, type II filter is produced - odd `numtaps`, `antisymmetric` is True, type III filter is produced - even `numtaps`, `antisymmetric` is True, type IV filter is produced
Magnitude response of all but type I filters are subjects to following constraints:
- type II -- zero at the Nyquist frequency - type III -- zero at zero and Nyquist frequencies - type IV -- zero at zero frequency
.. versionadded:: 0.9.0
References ---------- .. [1] Oppenheim, A. V. and Schafer, R. W., "Discrete-Time Signal Processing", Prentice-Hall, Englewood Cliffs, New Jersey (1989). (See, for example, Section 7.4.)
.. [2] Smith, Steven W., "The Scientist and Engineer's Guide to Digital Signal Processing", Ch. 17. http://www.dspguide.com/ch17/1.htm
Examples -------- A lowpass FIR filter with a response that is 1 on [0.0, 0.5], and that decreases linearly on [0.5, 1.0] from 1 to 0:
>>> from scipy import signal >>> taps = signal.firwin2(150, [0.0, 0.5, 1.0], [1.0, 1.0, 0.0]) >>> print(taps[72:78]) [-0.02286961 -0.06362756 0.57310236 0.57310236 -0.06362756 -0.02286961]
""" nyq = 0.5 * _get_fs(fs, nyq)
if len(freq) != len(gain): raise ValueError('freq and gain must be of same length.')
if nfreqs is not None and numtaps >= nfreqs: raise ValueError(('ntaps must be less than nfreqs, but firwin2 was ' 'called with ntaps=%d and nfreqs=%s') % (numtaps, nfreqs))
if freq[0] != 0 or freq[-1] != nyq: raise ValueError('freq must start with 0 and end with fs/2.') d = np.diff(freq) if (d < 0).any(): raise ValueError('The values in freq must be nondecreasing.') d2 = d[:-1] + d[1:] if (d2 == 0).any(): raise ValueError('A value in freq must not occur more than twice.')
if antisymmetric: if numtaps % 2 == 0: ftype = 4 else: ftype = 3 else: if numtaps % 2 == 0: ftype = 2 else: ftype = 1
if ftype == 2 and gain[-1] != 0.0: raise ValueError("A Type II filter must have zero gain at the " "Nyquist frequency.") elif ftype == 3 and (gain[0] != 0.0 or gain[-1] != 0.0): raise ValueError("A Type III filter must have zero gain at zero " "and Nyquist frequencies.") elif ftype == 4 and gain[0] != 0.0: raise ValueError("A Type IV filter must have zero gain at zero " "frequency.")
if nfreqs is None: nfreqs = 1 + 2 ** int(ceil(log(numtaps, 2)))
# Tweak any repeated values in freq so that interp works. eps = np.finfo(float).eps for k in range(len(freq)): if k < len(freq) - 1 and freq[k] == freq[k + 1]: freq[k] = freq[k] - eps freq[k + 1] = freq[k + 1] + eps
# Linearly interpolate the desired response on a uniform mesh `x`. x = np.linspace(0.0, nyq, nfreqs) fx = np.interp(x, freq, gain)
# Adjust the phases of the coefficients so that the first `ntaps` of the # inverse FFT are the desired filter coefficients. shift = np.exp(-(numtaps - 1) / 2. * 1.j * np.pi * x / nyq) if ftype > 2: shift *= 1j
fx2 = fx * shift
# Use irfft to compute the inverse FFT. out_full = irfft(fx2)
if window is not None: # Create the window to apply to the filter coefficients. from .signaltools import get_window wind = get_window(window, numtaps, fftbins=False) else: wind = 1
# Keep only the first `numtaps` coefficients in `out`, and multiply by # the window. out = out_full[:numtaps] * wind
if ftype == 3: out[out.size // 2] = 0.0
return out
maxiter=25, grid_density=16, fs=None): """ Calculate the minimax optimal filter using the Remez exchange algorithm.
Calculate the filter-coefficients for the finite impulse response (FIR) filter whose transfer function minimizes the maximum error between the desired gain and the realized gain in the specified frequency bands using the Remez exchange algorithm.
Parameters ---------- numtaps : int The desired number of taps in the filter. The number of taps is the number of terms in the filter, or the filter order plus one. bands : array_like A monotonic sequence containing the band edges. All elements must be non-negative and less than half the sampling frequency as given by `fs`. desired : array_like A sequence half the size of bands containing the desired gain in each of the specified bands. weight : array_like, optional A relative weighting to give to each band region. The length of `weight` has to be half the length of `bands`. Hz : scalar, optional *Deprecated. Use `fs` instead.* The sampling frequency in Hz. Default is 1. type : {'bandpass', 'differentiator', 'hilbert'}, optional The type of filter:
* 'bandpass' : flat response in bands. This is the default.
* 'differentiator' : frequency proportional response in bands.
* 'hilbert' : filter with odd symmetry, that is, type III (for even order) or type IV (for odd order) linear phase filters.
maxiter : int, optional Maximum number of iterations of the algorithm. Default is 25. grid_density : int, optional Grid density. The dense grid used in `remez` is of size ``(numtaps + 1) * grid_density``. Default is 16. fs : float, optional The sampling frequency of the signal. Default is 1.
Returns ------- out : ndarray A rank-1 array containing the coefficients of the optimal (in a minimax sense) filter.
See Also -------- firls firwin firwin2 minimum_phase
References ---------- .. [1] J. H. McClellan and T. W. Parks, "A unified approach to the design of optimum FIR linear phase digital filters", IEEE Trans. Circuit Theory, vol. CT-20, pp. 697-701, 1973. .. [2] J. H. McClellan, T. W. Parks and L. R. Rabiner, "A Computer Program for Designing Optimum FIR Linear Phase Digital Filters", IEEE Trans. Audio Electroacoust., vol. AU-21, pp. 506-525, 1973.
Examples -------- For a signal sampled at 100 Hz, we want to construct a filter with a passband at 20-40 Hz, and stop bands at 0-10 Hz and 45-50 Hz. Note that this means that the behavior in the frequency ranges between those bands is unspecified and may overshoot.
>>> from scipy import signal >>> fs = 100 >>> bpass = signal.remez(72, [0, 10, 20, 40, 45, 50], [0, 1, 0], fs=fs) >>> freq, response = signal.freqz(bpass)
>>> import matplotlib.pyplot as plt >>> plt.semilogy(0.5*fs*freq/np.pi, np.abs(response), 'b-') >>> plt.grid(alpha=0.25) >>> plt.xlabel('Frequency (Hz)') >>> plt.ylabel('Gain') >>> plt.show()
""" if Hz is None and fs is None: fs = 1.0 elif Hz is not None: if fs is not None: raise ValueError("Values cannot be given for both 'Hz' and 'fs'.") fs = Hz
# Convert type try: tnum = {'bandpass': 1, 'differentiator': 2, 'hilbert': 3}[type] except KeyError: raise ValueError("Type must be 'bandpass', 'differentiator', " "or 'hilbert'")
# Convert weight if weight is None: weight = [1] * len(desired)
bands = np.asarray(bands).copy() return sigtools._remez(numtaps, bands, desired, weight, tnum, fs, maxiter, grid_density)
""" FIR filter design using least-squares error minimization.
Calculate the filter coefficients for the linear-phase finite impulse response (FIR) filter which has the best approximation to the desired frequency response described by `bands` and `desired` in the least squares sense (i.e., the integral of the weighted mean-squared error within the specified bands is minimized).
Parameters ---------- numtaps : int The number of taps in the FIR filter. `numtaps` must be odd. bands : array_like A monotonic nondecreasing sequence containing the band edges in Hz. All elements must be non-negative and less than or equal to the Nyquist frequency given by `nyq`. desired : array_like A sequence the same size as `bands` containing the desired gain at the start and end point of each band. weight : array_like, optional A relative weighting to give to each band region when solving the least squares problem. `weight` has to be half the size of `bands`. nyq : float, optional *Deprecated. Use `fs` instead.* Nyquist frequency. Each frequency in `bands` must be between 0 and `nyq` (inclusive). Default is 1. fs : float, optional The sampling frequency of the signal. Each frequency in `bands` must be between 0 and ``fs/2`` (inclusive). Default is 2.
Returns ------- coeffs : ndarray Coefficients of the optimal (in a least squares sense) FIR filter.
See also -------- firwin firwin2 minimum_phase remez
Notes ----- This implementation follows the algorithm given in [1]_. As noted there, least squares design has multiple advantages:
1. Optimal in a least-squares sense. 2. Simple, non-iterative method. 3. The general solution can obtained by solving a linear system of equations. 4. Allows the use of a frequency dependent weighting function.
This function constructs a Type I linear phase FIR filter, which contains an odd number of `coeffs` satisfying for :math:`n < numtaps`:
.. math:: coeffs(n) = coeffs(numtaps - 1 - n)
The odd number of coefficients and filter symmetry avoid boundary conditions that could otherwise occur at the Nyquist and 0 frequencies (e.g., for Type II, III, or IV variants).
.. versionadded:: 0.18
References ---------- .. [1] Ivan Selesnick, Linear-Phase Fir Filter Design By Least Squares. OpenStax CNX. Aug 9, 2005. http://cnx.org/contents/eb1ecb35-03a9-4610-ba87-41cd771c95f2@7
Examples -------- We want to construct a band-pass filter. Note that the behavior in the frequency ranges between our stop bands and pass bands is unspecified, and thus may overshoot depending on the parameters of our filter:
>>> from scipy import signal >>> import matplotlib.pyplot as plt >>> fig, axs = plt.subplots(2) >>> fs = 10.0 # Hz >>> desired = (0, 0, 1, 1, 0, 0) >>> for bi, bands in enumerate(((0, 1, 2, 3, 4, 5), (0, 1, 2, 4, 4.5, 5))): ... fir_firls = signal.firls(73, bands, desired, fs=fs) ... fir_remez = signal.remez(73, bands, desired[::2], fs=fs) ... fir_firwin2 = signal.firwin2(73, bands, desired, fs=fs) ... hs = list() ... ax = axs[bi] ... for fir in (fir_firls, fir_remez, fir_firwin2): ... freq, response = signal.freqz(fir) ... hs.append(ax.semilogy(0.5*fs*freq/np.pi, np.abs(response))[0]) ... for band, gains in zip(zip(bands[::2], bands[1::2]), ... zip(desired[::2], desired[1::2])): ... ax.semilogy(band, np.maximum(gains, 1e-7), 'k--', linewidth=2) ... if bi == 0: ... ax.legend(hs, ('firls', 'remez', 'firwin2'), ... loc='lower center', frameon=False) ... else: ... ax.set_xlabel('Frequency (Hz)') ... ax.grid(True) ... ax.set(title='Band-pass %d-%d Hz' % bands[2:4], ylabel='Magnitude') ... >>> fig.tight_layout() >>> plt.show()
""" # noqa nyq = 0.5 * _get_fs(fs, nyq)
numtaps = int(numtaps) if numtaps % 2 == 0 or numtaps < 1: raise ValueError("numtaps must be odd and >= 1") M = (numtaps-1) // 2
# normalize bands 0->1 and make it 2 columns nyq = float(nyq) if nyq <= 0: raise ValueError('nyq must be positive, got %s <= 0.' % nyq) bands = np.asarray(bands).flatten() / nyq if len(bands) % 2 != 0: raise ValueError("bands must contain frequency pairs.") bands.shape = (-1, 2)
# check remaining params desired = np.asarray(desired).flatten() if bands.size != desired.size: raise ValueError("desired must have one entry per frequency, got %s " "gains for %s frequencies." % (desired.size, bands.size)) desired.shape = (-1, 2) if (np.diff(bands) <= 0).any() or (np.diff(bands[:, 0]) < 0).any(): raise ValueError("bands must be monotonically nondecreasing and have " "width > 0.") if (bands[:-1, 1] > bands[1:, 0]).any(): raise ValueError("bands must not overlap.") if (desired < 0).any(): raise ValueError("desired must be non-negative.") if weight is None: weight = np.ones(len(desired)) weight = np.asarray(weight).flatten() if len(weight) != len(desired): raise ValueError("weight must be the same size as the number of " "band pairs (%s)." % (len(bands),)) if (weight < 0).any(): raise ValueError("weight must be non-negative.")
# Set up the linear matrix equation to be solved, Qa = b
# We can express Q(k,n) = 0.5 Q1(k,n) + 0.5 Q2(k,n) # where Q1(k,n)=q(k−n) and Q2(k,n)=q(k+n), i.e. a Toeplitz plus Hankel.
# We omit the factor of 0.5 above, instead adding it during coefficient # calculation.
# We also omit the 1/π from both Q and b equations, as they cancel # during solving.
# We have that: # q(n) = 1/π ∫W(ω)cos(nω)dω (over 0->π) # Using our nomalization ω=πf and with a constant weight W over each # interval f1->f2 we get: # q(n) = W∫cos(πnf)df (0->1) = Wf sin(πnf)/πnf # integrated over each f1->f2 pair (i.e., value at f2 - value at f1). n = np.arange(numtaps)[:, np.newaxis, np.newaxis] q = np.dot(np.diff(np.sinc(bands * n) * bands, axis=2)[:, :, 0], weight)
# Now we assemble our sum of Toeplitz and Hankel Q1 = toeplitz(q[:M+1]) Q2 = hankel(q[:M+1], q[M:]) Q = Q1 + Q2
# Now for b(n) we have that: # b(n) = 1/π ∫ W(ω)D(ω)cos(nω)dω (over 0->π) # Using our normalization ω=πf and with a constant weight W over each # interval and a linear term for D(ω) we get (over each f1->f2 interval): # b(n) = W ∫ (mf+c)cos(πnf)df # = f(mf+c)sin(πnf)/πnf + mf**2 cos(nπf)/(πnf)**2 # integrated over each f1->f2 pair (i.e., value at f2 - value at f1). n = n[:M + 1] # only need this many coefficients here # Choose m and c such that we are at the start and end weights m = (np.diff(desired, axis=1) / np.diff(bands, axis=1)) c = desired[:, [0]] - bands[:, [0]] * m b = bands * (m*bands + c) * np.sinc(bands * n) # Use L'Hospital's rule here for cos(nπf)/(πnf)**2 @ n=0 b[0] -= m * bands * bands / 2. b[1:] += m * np.cos(n[1:] * np.pi * bands) / (np.pi * n[1:]) ** 2 b = np.dot(np.diff(b, axis=2)[:, :, 0], weight)
# Now we can solve the equation (use pinv because Q can be rank deficient) a = np.dot(pinv(Q), b)
# make coefficients symmetric (linear phase) coeffs = np.hstack((a[:0:-1], 2 * a[0], a[1:])) return coeffs
"""Compute the modified 1D discrete Hilbert transform
Parameters ---------- mag : ndarray The magnitude spectrum. Should be 1D with an even length, and preferably a fast length for FFT/IFFT. """ # Adapted based on code by Niranjan Damera-Venkata, # Brian L. Evans and Shawn R. McCaslin (see refs for `minimum_phase`) sig = np.zeros(len(mag)) # Leave Nyquist and DC at 0, knowing np.abs(fftfreq(N)[midpt]) == 0.5 midpt = len(mag) // 2 sig[1:midpt] = 1 sig[midpt+1:] = -1 # eventually if we want to support complex filters, we will need a # np.abs() on the mag inside the log, and should remove the .real recon = ifft(mag * np.exp(fft(sig * ifft(np.log(mag))))).real return recon
"""Convert a linear-phase FIR filter to minimum phase
Parameters ---------- h : array Linear-phase FIR filter coefficients. method : {'hilbert', 'homomorphic'} The method to use:
'homomorphic' (default) This method [4]_ [5]_ works best with filters with an odd number of taps, and the resulting minimum phase filter will have a magnitude response that approximates the square root of the the original filter's magnitude response.
'hilbert' This method [1]_ is designed to be used with equiripple filters (e.g., from `remez`) with unity or zero gain regions.
n_fft : int The number of points to use for the FFT. Should be at least a few times larger than the signal length (see Notes).
Returns ------- h_minimum : array The minimum-phase version of the filter, with length ``(length(h) + 1) // 2``.
See Also -------- firwin firwin2 remez
Notes ----- Both the Hilbert [1]_ or homomorphic [4]_ [5]_ methods require selection of an FFT length to estimate the complex cepstrum of the filter.
In the case of the Hilbert method, the deviation from the ideal spectrum ``epsilon`` is related to the number of stopband zeros ``n_stop`` and FFT length ``n_fft`` as::
epsilon = 2. * n_stop / n_fft
For example, with 100 stopband zeros and a FFT length of 2048, ``epsilon = 0.0976``. If we conservatively assume that the number of stopband zeros is one less than the filter length, we can take the FFT length to be the next power of 2 that satisfies ``epsilon=0.01`` as::
n_fft = 2 ** int(np.ceil(np.log2(2 * (len(h) - 1) / 0.01)))
This gives reasonable results for both the Hilbert and homomorphic methods, and gives the value used when ``n_fft=None``.
Alternative implementations exist for creating minimum-phase filters, including zero inversion [2]_ and spectral factorization [3]_ [4]_. For more information, see:
http://dspguru.com/dsp/howtos/how-to-design-minimum-phase-fir-filters
Examples -------- Create an optimal linear-phase filter, then convert it to minimum phase:
>>> from scipy.signal import remez, minimum_phase, freqz, group_delay >>> import matplotlib.pyplot as plt >>> freq = [0, 0.2, 0.3, 1.0] >>> desired = [1, 0] >>> h_linear = remez(151, freq, desired, Hz=2.)
Convert it to minimum phase:
>>> h_min_hom = minimum_phase(h_linear, method='homomorphic') >>> h_min_hil = minimum_phase(h_linear, method='hilbert')
Compare the three filters:
>>> fig, axs = plt.subplots(4, figsize=(4, 8)) >>> for h, style, color in zip((h_linear, h_min_hom, h_min_hil), ... ('-', '-', '--'), ('k', 'r', 'c')): ... w, H = freqz(h) ... w, gd = group_delay((h, 1)) ... w /= np.pi ... axs[0].plot(h, color=color, linestyle=style) ... axs[1].plot(w, np.abs(H), color=color, linestyle=style) ... axs[2].plot(w, 20 * np.log10(np.abs(H)), color=color, linestyle=style) ... axs[3].plot(w, gd, color=color, linestyle=style) >>> for ax in axs: ... ax.grid(True, color='0.5') ... ax.fill_between(freq[1:3], *ax.get_ylim(), color='#ffeeaa', zorder=1) >>> axs[0].set(xlim=[0, len(h_linear) - 1], ylabel='Amplitude', xlabel='Samples') >>> axs[1].legend(['Linear', 'Min-Hom', 'Min-Hil'], title='Phase') >>> for ax, ylim in zip(axs[1:], ([0, 1.1], [-150, 10], [-60, 60])): ... ax.set(xlim=[0, 1], ylim=ylim, xlabel='Frequency') >>> axs[1].set(ylabel='Magnitude') >>> axs[2].set(ylabel='Magnitude (dB)') >>> axs[3].set(ylabel='Group delay') >>> plt.tight_layout()
References ---------- .. [1] N. Damera-Venkata and B. L. Evans, "Optimal design of real and complex minimum phase digital FIR filters," Acoustics, Speech, and Signal Processing, 1999. Proceedings., 1999 IEEE International Conference on, Phoenix, AZ, 1999, pp. 1145-1148 vol.3. doi: 10.1109/ICASSP.1999.756179 .. [2] X. Chen and T. W. Parks, "Design of optimal minimum phase FIR filters by direct factorization," Signal Processing, vol. 10, no. 4, pp. 369-383, Jun. 1986. .. [3] T. Saramaki, "Finite Impulse Response Filter Design," in Handbook for Digital Signal Processing, chapter 4, New York: Wiley-Interscience, 1993. .. [4] J. S. Lim, Advanced Topics in Signal Processing. Englewood Cliffs, N.J.: Prentice Hall, 1988. .. [5] A. V. Oppenheim, R. W. Schafer, and J. R. Buck, "Discrete-Time Signal Processing," 2nd edition. Upper Saddle River, N.J.: Prentice Hall, 1999. """ # noqa h = np.asarray(h) if np.iscomplexobj(h): raise ValueError('Complex filters not supported') if h.ndim != 1 or h.size <= 2: raise ValueError('h must be 1D and at least 2 samples long') n_half = len(h) // 2 if not np.allclose(h[-n_half:][::-1], h[:n_half]): warnings.warn('h does not appear to by symmetric, conversion may ' 'fail', RuntimeWarning) if not isinstance(method, string_types) or method not in \ ('homomorphic', 'hilbert',): raise ValueError('method must be "homomorphic" or "hilbert", got %r' % (method,)) if n_fft is None: n_fft = 2 ** int(np.ceil(np.log2(2 * (len(h) - 1) / 0.01))) n_fft = int(n_fft) if n_fft < len(h): raise ValueError('n_fft must be at least len(h)==%s' % len(h)) if method == 'hilbert': w = np.arange(n_fft) * (2 * np.pi / n_fft * n_half) H = np.real(fft(h, n_fft) * np.exp(1j * w)) dp = max(H) - 1 ds = 0 - min(H) S = 4. / (np.sqrt(1+dp+ds) + np.sqrt(1-dp+ds)) ** 2 H += ds H *= S H = np.sqrt(H, out=H) H += 1e-10 # ensure that the log does not explode h_minimum = _dhtm(H) else: # method == 'homomorphic' # zero-pad; calculate the DFT h_temp = np.abs(fft(h, n_fft)) # take 0.25*log(|H|**2) = 0.5*log(|H|) h_temp += 1e-7 * h_temp[h_temp > 0].min() # don't let log blow up np.log(h_temp, out=h_temp) h_temp *= 0.5 # IDFT h_temp = ifft(h_temp).real # multiply pointwise by the homomorphic filter # lmin[n] = 2u[n] - d[n] win = np.zeros(n_fft) win[0] = 1 stop = (len(h) + 1) // 2 win[1:stop] = 2 if len(h) % 2: win[stop] = 1 h_temp *= win h_temp = ifft(np.exp(fft(h_temp))) h_minimum = h_temp.real n_out = n_half + len(h) % 2 return h_minimum[:n_out] |