You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
127 lines
3.3 KiB
Python
127 lines
3.3 KiB
Python
"""
|
|
Some signal functions implemented using mpmath.
|
|
"""
|
|
|
|
from __future__ import division
|
|
|
|
try:
|
|
import mpmath
|
|
except ImportError:
|
|
mpmath = None
|
|
|
|
|
|
def _prod(seq):
|
|
"""Returns the product of the elements in the sequence `seq`."""
|
|
p = 1
|
|
for elem in seq:
|
|
p *= elem
|
|
return p
|
|
|
|
|
|
def _relative_degree(z, p):
|
|
"""
|
|
Return relative degree of transfer function from zeros and poles.
|
|
|
|
This is simply len(p) - len(z), which must be nonnegative.
|
|
A ValueError is raised if len(p) < len(z).
|
|
"""
|
|
degree = len(p) - len(z)
|
|
if degree < 0:
|
|
raise ValueError("Improper transfer function. "
|
|
"Must have at least as many poles as zeros.")
|
|
return degree
|
|
|
|
|
|
def _zpkbilinear(z, p, k, fs):
|
|
"""Bilinear transformation to convert a filter from analog to digital."""
|
|
|
|
degree = _relative_degree(z, p)
|
|
|
|
fs2 = 2*fs
|
|
|
|
# Bilinear transform the poles and zeros
|
|
z_z = [(fs2 + z1) / (fs2 - z1) for z1 in z]
|
|
p_z = [(fs2 + p1) / (fs2 - p1) for p1 in p]
|
|
|
|
# Any zeros that were at infinity get moved to the Nyquist frequency
|
|
z_z.extend([-1] * degree)
|
|
|
|
# Compensate for gain change
|
|
numer = _prod(fs2 - z1 for z1 in z)
|
|
denom = _prod(fs2 - p1 for p1 in p)
|
|
k_z = k * numer / denom
|
|
|
|
return z_z, p_z, k_z.real
|
|
|
|
|
|
def _zpklp2lp(z, p, k, wo=1):
|
|
"""Transform a lowpass filter to a different cutoff frequency."""
|
|
|
|
degree = _relative_degree(z, p)
|
|
|
|
# Scale all points radially from origin to shift cutoff frequency
|
|
z_lp = [wo * z1 for z1 in z]
|
|
p_lp = [wo * p1 for p1 in p]
|
|
|
|
# Each shifted pole decreases gain by wo, each shifted zero increases it.
|
|
# Cancel out the net change to keep overall gain the same
|
|
k_lp = k * wo**degree
|
|
|
|
return z_lp, p_lp, k_lp
|
|
|
|
|
|
def _butter_analog_poles(n):
|
|
"""
|
|
Poles of an analog Butterworth lowpass filter.
|
|
|
|
This is the same calculation as scipy.signal.buttap(n) or
|
|
scipy.signal.butter(n, 1, analog=True, output='zpk'), but mpmath is used,
|
|
and only the poles are returned.
|
|
"""
|
|
poles = []
|
|
for k in range(-n+1, n, 2):
|
|
poles.append(-mpmath.exp(1j*mpmath.pi*k/(2*n)))
|
|
return poles
|
|
|
|
|
|
def butter_lp(n, Wn):
|
|
"""
|
|
Lowpass Butterworth digital filter design.
|
|
|
|
This computes the same result as scipy.signal.butter(n, Wn, output='zpk'),
|
|
but it uses mpmath, and the results are returned in lists instead of numpy
|
|
arrays.
|
|
"""
|
|
zeros = []
|
|
poles = _butter_analog_poles(n)
|
|
k = 1
|
|
fs = 2
|
|
warped = 2 * fs * mpmath.tan(mpmath.pi * Wn / fs)
|
|
z, p, k = _zpklp2lp(zeros, poles, k, wo=warped)
|
|
z, p, k = _zpkbilinear(z, p, k, fs=fs)
|
|
return z, p, k
|
|
|
|
|
|
def zpkfreqz(z, p, k, worN=None):
|
|
"""
|
|
Frequency response of a filter in zpk format, using mpmath.
|
|
|
|
This is the same calculation as scipy.signal.freqz, but the input is in
|
|
zpk format, the calculation is performed using mpath, and the results are
|
|
returned in lists instead of numpy arrays.
|
|
"""
|
|
if worN is None or isinstance(worN, int):
|
|
N = worN or 512
|
|
ws = [mpmath.pi * mpmath.mpf(j) / N for j in range(N)]
|
|
else:
|
|
ws = worN
|
|
|
|
h = []
|
|
for wk in ws:
|
|
zm1 = mpmath.exp(1j * wk)
|
|
numer = _prod([zm1 - t for t in z])
|
|
denom = _prod([zm1 - t for t in p])
|
|
hk = k * numer / denom
|
|
h.append(hk)
|
|
return ws, h
|