123 lines
3.2 KiB
Python
123 lines
3.2 KiB
Python
"""
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Some signal functions implemented using mpmath.
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"""
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try:
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import mpmath
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except ImportError:
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mpmath = None
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def _prod(seq):
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"""Returns the product of the elements in the sequence `seq`."""
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p = 1
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for elem in seq:
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p *= elem
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return p
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def _relative_degree(z, p):
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"""
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Return relative degree of transfer function from zeros and poles.
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This is simply len(p) - len(z), which must be nonnegative.
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A ValueError is raised if len(p) < len(z).
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"""
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degree = len(p) - len(z)
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if degree < 0:
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raise ValueError("Improper transfer function. "
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"Must have at least as many poles as zeros.")
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return degree
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def _zpkbilinear(z, p, k, fs):
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"""Bilinear transformation to convert a filter from analog to digital."""
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degree = _relative_degree(z, p)
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fs2 = 2*fs
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# Bilinear transform the poles and zeros
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z_z = [(fs2 + z1) / (fs2 - z1) for z1 in z]
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p_z = [(fs2 + p1) / (fs2 - p1) for p1 in p]
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# Any zeros that were at infinity get moved to the Nyquist frequency
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z_z.extend([-1] * degree)
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# Compensate for gain change
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numer = _prod(fs2 - z1 for z1 in z)
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denom = _prod(fs2 - p1 for p1 in p)
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k_z = k * numer / denom
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return z_z, p_z, k_z.real
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def _zpklp2lp(z, p, k, wo=1):
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"""Transform a lowpass filter to a different cutoff frequency."""
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degree = _relative_degree(z, p)
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# Scale all points radially from origin to shift cutoff frequency
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z_lp = [wo * z1 for z1 in z]
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p_lp = [wo * p1 for p1 in p]
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# Each shifted pole decreases gain by wo, each shifted zero increases it.
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# Cancel out the net change to keep overall gain the same
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k_lp = k * wo**degree
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return z_lp, p_lp, k_lp
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def _butter_analog_poles(n):
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"""
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Poles of an analog Butterworth lowpass filter.
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This is the same calculation as scipy.signal.buttap(n) or
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scipy.signal.butter(n, 1, analog=True, output='zpk'), but mpmath is used,
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and only the poles are returned.
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"""
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poles = [-mpmath.exp(1j*mpmath.pi*k/(2*n)) for k in range(-n+1, n, 2)]
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return poles
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def butter_lp(n, Wn):
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"""
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Lowpass Butterworth digital filter design.
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This computes the same result as scipy.signal.butter(n, Wn, output='zpk'),
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but it uses mpmath, and the results are returned in lists instead of NumPy
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arrays.
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"""
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zeros = []
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poles = _butter_analog_poles(n)
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k = 1
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fs = 2
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warped = 2 * fs * mpmath.tan(mpmath.pi * Wn / fs)
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z, p, k = _zpklp2lp(zeros, poles, k, wo=warped)
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z, p, k = _zpkbilinear(z, p, k, fs=fs)
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return z, p, k
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def zpkfreqz(z, p, k, worN=None):
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"""
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Frequency response of a filter in zpk format, using mpmath.
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This is the same calculation as scipy.signal.freqz, but the input is in
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zpk format, the calculation is performed using mpath, and the results are
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returned in lists instead of NumPy arrays.
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"""
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if worN is None or isinstance(worN, int):
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N = worN or 512
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ws = [mpmath.pi * mpmath.mpf(j) / N for j in range(N)]
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else:
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ws = worN
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h = []
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for wk in ws:
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zm1 = mpmath.exp(1j * wk)
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numer = _prod([zm1 - t for t in z])
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denom = _prod([zm1 - t for t in p])
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hk = k * numer / denom
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h.append(hk)
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return ws, h
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