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czt.py
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czt.py
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"""Calculate the Chirp Z-transform (CZT).
CZT reference:
Lawrence R. Rabiner, Ronald W. Schafer, and Charles M. Rader, "The chirp
z-transform algorithm and its application," Bell Syst. Tech. J. 48,
1249-1292 (1969).
CZT computation reference:
Sukhoy, V., Stoytchev, A. "Generalizing the inverse FFT off the unit
circle," Sci Rep 9, 14443 (2019).
"""
import numpy as np
from scipy.linalg import matmul_toeplitz, toeplitz
# CZT ------------------------------------------------------------------------
def czt(x, M=None, W=None, A=1.0, simple=False, t_method="ce", f_method="numpy"):
"""Calculate the Chirp Z-transform (CZT).
Solves in O(n log n) time.
See algorithm 1 in Sukhoy & Stoytchev 2019.
Args:
x (np.ndarray): input array
M (int): length of output array
W (complex): complex ratio between points
A (complex): complex starting point
simple (bool): use simple algorithm? (very slow)
t_method (str): Toeplitz matrix multiplication method. 'ce' for
circulant embedding, 'pd' for Pustylnikov's decomposition, 'mm'
for simple matrix multiplication, 'scipy' for matmul_toeplitz
from scipy.linalg.
f_method (str): FFT method. 'numpy' for FFT from NumPy,
'recursive' for recursive method.
Returns:
np.ndarray: Chirp Z-transform
"""
# Unpack arguments
N = len(x)
if M is None:
M = N
if W is None:
W = np.exp(-2j * np.pi / M)
A = np.complex128(A)
W = np.complex128(W)
# Simple algorithm (very slow)
if simple:
k = np.arange(M)
X = np.zeros(M, dtype=complex)
z = A * W ** -k
for n in range(N):
X += x[n] * z ** -n
return X
# Algorithm 1 from Sukhoy & Stoytchev 2019
k = np.arange(max(M, N))
Wk22 = W ** (-(k ** 2) / 2)
r = Wk22[:N]
c = Wk22[:M]
X = A ** -k[:N] * x / r
try:
toeplitz_mult = _available_t_methods[t_method]
except KeyError:
raise ValueError(f"t_method {t_method} not recognized. Must be one of {list(_available_t_methods.keys())}")
X = toeplitz_mult(r, c, X, f_method)
return X / c
def iczt(X, N=None, W=None, A=1.0, simple=True, t_method="scipy", f_method="numpy"):
"""Calculate inverse Chirp Z-transform (ICZT).
Solves in O(n log n) time.
See algorithm 2 in Sukhoy & Stoytchev 2019.
Args:
X (np.ndarray): input array
N (int): length of output array
W (complex): complex ratio between points
A (complex): complex starting point
simple (bool): calculate ICZT using simple method (using CZT and
conjugate)
t_method (str): Toeplitz matrix multiplication method. 'ce' for
circulant embedding, 'pd' for Pustylnikov's decomposition, 'mm'
for simple matrix multiplication, 'scipy' for matmul_toeplitz
from scipy.linalg. Ignored if you are not using the simple ICZT
method.
f_method (str): FFT method. 'numpy' for FFT from NumPy,
'recursive' for recursive method.
Returns:
np.ndarray: Inverse Chirp Z-transform
"""
# Unpack arguments
M = len(X)
if N is None:
N = M
if W is None:
W = np.exp(-2j * np.pi / M)
A = np.complex128(A)
W = np.complex128(W)
# Simple algorithm
if simple:
return np.conj(czt(np.conj(X), M=N, W=W, A=A, t_method=t_method, f_method=f_method)) / M
# Algorithm 2 from Sukhoy & Stoytchev 2019
if M != N:
raise ValueError("M must be equal to N")
n = N
k = np.arange(n)
Wk22 = W ** (-(k ** 2) / 2)
x = Wk22 * X
p = np.r_[1, (W ** k[1:] - 1).cumprod()]
u = (-1) ** k * W ** (k * (k - n + 0.5) + (n / 2 - 0.5) * n) / p
# equivalent to:
# u = (-1) ** k * W ** ((2 * k ** 2 - (2 * n - 1) * k + n * (n - 1)) / 2) / p
u /= p[::-1]
z = np.zeros(n, dtype=complex)
uhat = np.r_[0, u[-1:0:-1]]
util = np.r_[u[0], np.zeros(n - 1)]
try:
toeplitz_mult = _available_t_methods[t_method]
except KeyError:
raise ValueError(f"t_method {t_method} not recognized. Must be one of {list(_available_t_methods.keys())}")
# Note: there is difference in accuracy here depending on the method. Have to check.
x1 = toeplitz_mult(uhat, z, x, f_method)
x1 = toeplitz_mult(z, uhat, x1, f_method)
x2 = toeplitz_mult(u, util, x, f_method)
x2 = toeplitz_mult(util, u, x2, f_method)
x = (x2 - x1) / u[0]
x = A ** k * Wk22 * x
return x
# DFT ------------------------------------------------------------------------
def dft(t, x, f=None):
"""Calculate the Discrete Fourier Transform (DFT).
Simple implementation. Used for testing CZT algorithm.
Args:
t (np.ndarray): time
x (np.ndarray): time-domain signal
f (np.ndarray): frequency for output signal
Returns:
np.ndarray: frequency-domain signal
"""
if f is None:
# Default to FFT frequency sweep
nt = len(t)
tspan = t[-1] - t[0]
dt = tspan / (nt - 1) # more accurate than t[1] - t[0]
f = np.fft.fftshift(np.fft.fftfreq(nt, dt))
X = np.empty(len(f), dtype=complex)
for k in range(len(f)):
X[k] = np.sum(x * np.exp(-2j * np.pi * f[k] * t))
return f, X
def idft(f, X, t=None):
"""Calculate the Inverse Discrete Fourier Transform (IDFT).
Simple implementation. Used for testing the ICZT algorithm.
Args:
f (np.ndarray): frequency
X (np.ndarray): frequency-domain signal
t (np.ndarray): time for output signal
Returns:
np.ndarray: time-domain signal
"""
if t is None:
# Default to FFT time sweep
nf = len(f)
fspan = f[-1] - f[0]
df = fspan / (nf - 1) # more accurate than f[1] - f[0]
t = np.fft.fftshift(np.fft.fftfreq(nf, df))
x = np.empty(len(t), dtype=complex)
for n in range(len(t)):
x[n] = np.sum(X * np.exp(2j * np.pi * f * t[n]))
x /= len(t)
return t, x
# TIME-DOMAIN <--> FREQUENCY-DOMAIN ------------------------------------------
def time2freq(t, x, f=None):
"""Transform a time-domain signal to the frequency-domain.
Args:
t (np.ndarray): time
x (np.ndarray): time-domain signal
f (np.ndarray): frequency for output signal, optional, defaults to
standard FFT frequency sweep
Returns:
frequency-domain signal
"""
# Input time array
nt = len(t)
tspan = t[-1] - t[0]
dt = tspan / (nt - 1) # more accurate than t[1] - t[0]
# Output frequency array
if f is None:
# Default to FFT frequency sweep
f = np.fft.fftshift(np.fft.fftfreq(nt, dt))
else:
f = np.copy(f)
nf = len(f)
fspan = f[-1] - f[0]
df = fspan / (nf - 1) # more accurate than f[1] - f[0]
# Starting point
A = np.exp(2j * np.pi * f[0] * dt)
# Step
W = np.exp(-2j * np.pi * df * dt)
# Phase correction
phase = np.exp(-2j * np.pi * t[0] * f)
# Frequency-domain transform
freq_data = czt(x, nf, W, A) * phase
return f, freq_data
def freq2time(f, X, t=None):
"""Transform a frequency-domain signal to the time-domain.
Args:
f (np.ndarray): frequency
X (np.ndarray): frequency-domain signal
t (np.ndarray): time for output signal, optional, defaults to standard
FFT time sweep
Returns:
time-domain signal
"""
# Input frequency array
nf = len(f)
fspan = f[-1] - f[0]
df = fspan / (nf - 1) # more accurate than f[1] - f[0]
# Output time array
if t is None:
# Default to FFT time sweep
t = np.fft.fftshift(np.fft.fftfreq(nf, df))
else:
t = np.copy(t)
nt = len(t)
tspan = t[-1] - t[0]
dt = tspan / (nt - 1) # more accurate than t[1] - t[0]
# Starting point
A = np.exp(-2j * np.pi * t[0] * df)
# Step
W = np.exp(2j * np.pi * df * dt)
# Phase correction
phase = np.exp(2j * np.pi * f[0] * t)
# Time-domain transform
time_data = czt(X, nt, W=W, A=A) * phase / nf
return t, time_data
# HELPER FUNCTIONS -----------------------------------------------------------
def _toeplitz_mult_ce(r, c, x, f_method="numpy"):
"""Multiply Toeplitz matrix by vector using circulant embedding.
See algorithm S1 in Sukhoy & Stoytchev 2019:
Compute the product y = Tx of a Toeplitz matrix T and a vector x, where
T is specified by its first row r = (r[0], r[1], r[2],...,r[N-1]) and
its first column c = (c[0], c[1], c[2],...,c[M-1]), where r[0] = c[0].
Args:
r (np.ndarray): first row of Toeplitz matrix
c (np.ndarray): first column of Toeplitz matrix
x (np.ndarray): vector to multiply the Toeplitz matrix
f_method (str): FFT method. 'numpy' for FFT from NumPy, 'recursive'
for recursive method.
Returns:
np.ndarray: product of Toeplitz matrix and vector x
"""
N = len(r)
M = len(c)
assert r[0] == c[0]
assert len(x) == N
n = int(2 ** np.ceil(np.log2(M + N - 1)))
assert n >= M
assert n >= N
chat = np.r_[c, np.zeros(n - (M + N - 1)), r[-(N - 1):][::-1]]
xhat = _zero_pad(x, n)
yhat = _circulant_multiply(chat, xhat, f_method)
y = yhat[:M]
return y
def _toeplitz_mult_pd(r, c, x, f_method="numpy"):
"""Multiply Toeplitz matrix by vector using Pustylnikov's decomposition.
See algorithm S3 in Sukhoy & Stoytchev 2019:
Compute the product y = Tx of a Toeplitz matrix T and a vector x, where
T is specified by its first row r = (r[0], r[1], r[2],...,r[N-1]) and
its first column c = (c[0], c[1], c[2],...,c[M-1]), where r[0] = c[0].
Args:
r (np.ndarray): first row of Toeplitz matrix
c (np.ndarray): first column of Toeplitz matrix
x (np.ndarray): vector to multiply the Toeplitz matrix
f_method (str): FFT method. 'numpy' for FFT from NumPy, 'recursive'
for recursive method.
Returns:
np.ndarray: product of Toeplitz matrix and vector x
"""
N = len(r)
M = len(c)
assert r[0] == c[0]
assert len(x) == N
n = int(2 ** np.ceil(np.log2(M + N - 1)))
if N != n:
r = _zero_pad(r, n)
x = _zero_pad(x, n)
if M != n:
c = _zero_pad(c, n)
c1 = np.r_[c[0], c[1:]+r[-1:0:-1]]
c2 = np.r_[c[0], c[1:]-r[-1:0:-1]]
y1 = _circulant_multiply(c1 / 2, x, f_method)
y2 = _skew_circulant_multiply(c2 / 2, x, f_method)
y = y1[:M] + y2[:M]
return y
def _zero_pad(x, n):
"""Zero pad an array x to length n by appending zeros.
Args:
x (np.ndarray): array x
n (int): length of output array
Returns:
np.ndarray: array x with padding
"""
return np.pad(x, (0, n - len(x)))
def _circulant_multiply(c, x, f_method="numpy"):
"""Multiply a circulant matrix by a vector.
Runs in O(n log n) time.
See algorithm S4 in Sukhoy & Stoytchev 2019:
Compute the product y = Gx of a circulant matrix G and a vector x,
where G is generated by its first column c=(c[0], c[1],...,c[n-1]).
Args:
c (np.ndarray): first column of circulant matrix G
x (np.ndarray): vector x
f_method (str): FFT method. 'numpy' for FFT from NumPy,
'recursive' for recursive method.
Returns:
np.ndarray: product Gx
"""
n = len(c)
assert len(x) == n
if f_method == "numpy":
C = np.fft.fft(c)
X = np.fft.fft(x)
Y = C * X
return np.fft.ifft(Y)
elif f_method.lower() == "recursive":
C = _fft(c)
X = _fft(x)
Y = C * X
return _ifft(Y)
else:
raise ValueError("f_method not recognized.")
def _skew_circulant_multiply(c, x, f_method="numpy"):
"""Multiply a skew-circulant matrix by a vector.
Runs in O(n log n) time.
See algorithm S7 in Sukhoy & Stoytchev 2019.
Args:
c (np.ndarray): first column of skew-circulant matrix G
x (np.ndarray): vector x
f_method (str): FFT method. 'numpy' for FFT from NumPy, 'recursive'
for recursive method.
Returns:
np.ndarray: product Gx
"""
n = len(c)
assert len(x) == n
k = np.arange(n, dtype=float)
prefac = np.exp(-1j * k * np.pi / n)
chat = c * prefac
xhat = x * prefac
y = _circulant_multiply(chat, xhat, f_method)
y *= prefac.conjugate()
return y
def _fft(x):
"""Recursive FFT algorithm. Runs in O(n log n) time.
Args:
x (np.ndarray): input
Returns:
np.ndarray: FFT of x
"""
n = len(x)
if n == 1:
return x
xe = x[0::2]
xo = x[1::2]
y1 = _fft(xe)
y2 = _fft(xo)
k = np.arange(n // 2)
w = np.exp(-2j * np.pi * k / n)
y = np.empty(n, dtype=complex)
y[: n // 2] = y1 + w * y2
y[n // 2:] = y1 - w * y2
return y
def _ifft(y):
"""Recursive IFFT algorithm. Runs in O(n log n) time.
Args:
y (np.ndarray): input
Returns:
np.ndarray: IFFT of y
"""
n = len(y)
if n == 1:
return y
ye = y[0::2]
yo = y[1::2]
x1 = _ifft(ye)
x2 = _ifft(yo)
k = np.arange(n // 2)
w = np.exp(2j * np.pi * k / n)
x = np.empty(n, dtype=complex)
x[: n // 2] = (x1 + w * x2) / 2
x[n // 2:] = (x1 - w * x2) / 2
return x
_available_t_methods = {
"ce": _toeplitz_mult_ce,
"pd": _toeplitz_mult_pd,
"mm": lambda r, c, x, _: np.matmul(toeplitz(c, r), x),
"scipy": lambda r, c, x, _: matmul_toeplitz((c, r), x),
}