Zoom FFT functionality. Includes implementation of chirpz transform.

zfft.py

""" Zoom FFT function"""

import numpy as np
from time import time
from scipy.fftpack import fft, ifft
from numpy import swapaxes

def chirpz(x, A=None, W=None, M=None):
    """chirpz(x, A, W, M) - Chirp z-transform of variable x

    Keyword Arguments:
    x -- array to evaluate chirp-z transform (along last dimension of array)
    A -- starting point of chirp-z contour
    W -- controls frequency sample spacing and shape of the contour
    M -- number of frequency sample points

    Return values:
    g -- chirp-z tranform coefficients

    From http://www.mail-archive.com/[email protected]/msg01812.html
    Last accessed December-06-2012

    Written by Stefan van der Walt

    Modified by Adam Luchies, 09/27/12
    Added support for 2- and 3- deminsional arrays. For 2-dimensional array,
    returns chirpz along axis = 1. For 3-dimensional array, returns chirpz
    along axis = 2.

    Modified by Adam Luchies 12/06/12
    Added support to allow M > N - compute chirp-z transform containing
    more points than the original sequence.

    Reference:
    Rabiner, L.R., R.W. Schafer and C.M. Rader. The Chirp z-Transform
    Algorithm. IEEE Transactions on Audio and Electroacoustics,
    AU-17(2):86--92, 1969

    The discrete z-transform,
    X(z) = \sum_{n=0}^{N-1} x_n z^{-n}
    is calculated at M points,
    z_k = AW^-k, k = 0,1,...,M-1
    for A and W complex, which gives
    X(z_k) = \sum_{n=0}^{N-1} x_n z_k^{-n}

    """


    # Handle default arguments
    if (A == None) & (W == None) & (M == None):
        M = x.shape[-1]
        A = 1.
        W = np.exp(-2. * np.pi * 1j / M)
    elif (A == None) & (W == None):
        A = 1.
        W = np.exp(-2. * np.pi * 1j / M)

    A = np.complex(A)
    W = np.complex(W)
    if np.issubdtype(np.complex, x.dtype) or np.issubdtype(np.float, x.dtype):
        dtype = x.dtype
    else:
        dtype = float

    x = np.asarray(x, dtype=np.complex)
    P = x.shape

    if len(P) == 1:
        N = P[-1]
        L = int(2 ** np.ceil(np.log2(M + N - 1)))

        n = np.arange(N, dtype=float)
        y = np.power(A, -n) * np.power(W, n ** 2 / 2.) *  x
        Y = fft(y, L)

        n = np.arange(L, dtype=float)
        v = np.zeros(L, dtype=np.complex)
        v[:M] = np.power(W, -n[:M] ** 2 / 2.)
        v[L-N+1:] = np.power(W, -(L - n[L-N+1:]) ** 2 / 2.)
        V = fft(v, L)

        g = ifft(V * Y)[:M]
        k = np.arange(M)
        g = g * np.power(W, k ** 2 / 2.)

    elif len(P) == 2:
        N = P[-1]
        L = int(2 ** np.ceil(np.log2(M + N - 1)))

        n = np.arange(N, dtype=float)
        y = np.power(A, -n) * np.power(W, n ** 2 / 2.)
        y = np.tile(y, (P[0], 1)) * x
        Y = fft(y, L)

        n = np.arange(L, dtype=float)
        v = np.zeros(L, dtype=np.complex)
        v[:M] = np.power(W, -n[:M] ** 2 / 2.)
        v[L-N+1:] = np.power(W, -(L - n[L-N+1:]) ** 2 / 2.)
        V = fft(v)

        g = ifft(np.tile(V, (P[0], 1)) * Y)[:,:M]
        k = np.arange(M)
        g = g * np.tile(np.power(W, k ** 2 / 2.), (P[0],1))

    elif len(P) == 3:
        N = P[-1]
        L = int(2 ** np.ceil(np.log2(M + N - 1)))

        n = np.arange(N,dtype=float)
        y = np.power(A,-n) * np.power(W,n ** 2 / 2.)
        y = np.tile(y, (P[0],P[1],1)) * x
        Y = fft(y, L)

        n = np.arange(L, dtype=float)
        v = np.zeros(L, dtype=np.complex)
        v[:M] = np.power(W, -n[:M] ** 2 / 2.)
        v[L-N+1:] = np.power(W, -(L - n[L-N+1:]) ** 2 / 2.)
        V = fft(v)

        g = ifft(np.tile(V, (P[0], P[1],1)) * Y)[:,:,:M]
        k = np.arange(M)
        g = g * np.tile(np.power(W, k ** 2 / 2.), (P[0],P[1],1))
    # Return result
    return g


def zfft(x, f0=0., f1=1., fs=1., M=None, axis=-1):
    """zfft(x, f0, f1, fs, M) - Zoom FFT function to evaluate the 1DFT
    coefficients for the rows of an array in the frequency range [f0, f1]
    using N points.

    Keyword arguments:
    x -- array to evaluate DFT (along last dimension of array)
    f0 -- lower bound of frequency bandwidth
    f1 -- upper bound of frequency bandwidth
    fs -- sampling frequency
    M -- number of points used when evaluating the 1DFT (N <= signal length)
    axis -- axis along which the fft's are computed (defaults to last axis)

    Return values:
    y -- DFT coefficients

    """

    # Handle default arguments
    if M == None:
        M = x.shape[-1]

    # Swap axes
    x = swapaxes(a=x, axis1=axis, axis2=-1)

    # Normalize frequency range
    f0_norm = f0 / (fs / 2.)
    f1_norm = f1 / (fs / 2.)

    # Determine shape of signal
    A = np.exp(1j * np.pi * f0_norm)
    W = np.exp(-1j * np.pi * (f1_norm - f0_norm) / M)
    y = chirpz(x=x, A=A, W=W, M=M)
    # Return result
    return swapaxes(a=y, axis1=axis, axis2=-1)


def zfftfreq(f0, f1, M):
    """zfftfreq(f0, f1, M) - Return frequency values of the zoom FFT
    coefficients returned by zfft().

    Keyword arguments:
    f0 - lower bound of frequency bandwidth
    f1 - upper bound of frequency bandwidth
    fs = sampling rate

    Return values:
    freq - vector of frequency values

    """

    df = (f1 - f0) / M
    return np.arange(M) * df + f0


def chirpz_original(x,A,W,M):
    """Unmodified Chirp z-Transform from web address listed below.

    From http://www.mail-archive.com/[email protected]/msg01812.html
    Last accessed December-06-2012

    As described in
    Rabiner, L.R., R.W. Schafer and C.M. Rader.
    The Chirp z-Transform Algorithm.
    IEEE Transactions on Audio and Electroacoustics, AU-17(2):86--92, 1969

    Compute the chirp z-transform.
    The discrete z-transform,
    X(z) = \sum_{n=0}^{N-1} x_n z^{-n}
    is calculated at M points,
    z_k = AW^-k, k = 0,1,...,M-1
    for A and W complex, which gives
    X(z_k) = \sum_{n=0}^{N-1} x_n z_k^{-n}

    """
    A = np.complex(A)
    W = np.complex(W)
    if np.issubdtype(np.complex,x.dtype) or np.issubdtype(np.float,x.dtype):
        dtype = x.dtype
    else:
        dtype = float

    x = np.asarray(x,dtype=np.complex)

    N = x.size
    L = int(2**np.ceil(np.log2(M+N-1)))

    n = np.arange(N,dtype=float)
    y = np.power(A,-n) * np.power(W,n**2 / 2.) * x 
    Y = np.fft.fft(y,L)

    v = np.zeros(L,dtype=np.complex)
    v[:M] = np.power(W,-n[:M]**2/2.)
    v[L-N+1:] = np.power(W,-n[N-1:0:-1]**2/2.)
    V = np.fft.fft(v)

    g = np.fft.ifft(V*Y)[:M]
    k = np.arange(M)
    g *= np.power(W,k**2 / 2.)

    return g

Similar functions will be added in scipy 1.8.0: https://scipy.github.io/devdocs/reference/signal.html#chirp-z-transform-and-zoom-fft Anything we missed?