A long time ago I translated a free code of chirp z transform (zoom fft) into python. Attached here the source and two translations (probably one of the is right)
Nadav. On Wed, 2007-03-14 at 14:02 -0800, Ray S wrote: > We'd like to do what most call a "zoom FFT"; we only are interested > in the frequencies of say, 6kHZ to 9kHz with a given N, and so the > computations from DC to 6kHz are wasted CPU time. > Can this be done without additional numpy pre-filtering computations? > > If explicit filtering is needed to "baseband" the data, is it worth > it? It sounds like we need to generate cosine data once for each band > and N, then multiple with the data for each FFT. > > Has anyone coded this up before? I couldn't find any references other > than http://www.dsprelated.com/showmessage/38917/1.php > and http://www.numerix-dsp.com/siglib/examples/test_zft.c (which uses > Numerix). > > Ray > > _______________________________________________ > Numpy-discussion mailing list > Numpy-discussion@scipy.org > http://projects.scipy.org/mailman/listinfo/numpy-discussion
import numarray as N import numarray.fft as F def czt(x, m=None, w=None, a=1.0): """ Copyright (C) 2000 Paul Kienzle This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 US usage y=czt(x, m, w, a) Chirp z-transform. Compute the frequency response starting at a and stepping by w for m steps. a is a point in the complex plane, and w is the ratio between points in each step (i.e., radius increases exponentially, and angle increases linearly). To evaluate the frequency response for the range f1 to f2 in a signal with sampling frequency Fs, use the following: m = 32; ## number of points desired w = exp(-2i*pi*(f2-f1)/(m*Fs)); ## freq. step of f2-f1/m a = exp(2i*pi*f1/Fs); ## starting at frequency f1 y = czt(x, m, w, a); If you don't specify them, then the parameters default to a fourier transform: m=length(x), w=exp(2i*pi/m), a=1 Because it is computed with three FFTs, this will be faster than computing the fourier transform directly for large m (which is otherwise the best you can do with fft(x,n) for n prime). TODO: More testing---particularly when m+N-1 approaches a power of 2 TODO: Consider treating w,a as f1,f2 expressed in radians if w is real """ # Convinience declations ifft = F.inverse_fft fft = F.fft if m is None: m = len(x) if w is None: w = N.exp(2j*N.pi/m) n = len(x) k = N.arange(m, type=N.Float64) Nk = N.arange(-(n-1), m-1, type=N.Float64) nfft = next2pow(min(m,n) + len(Nk) -1) Wk2 = w**(-(Nk**2)/2) AWk2 = a**(-k) * w**((k**2)/2) y = ifft(fft(Wk2,nfft) * fft(x*AWk2, nfft)); y = w**((k**2)/2) * y[n:m+n] return y def next2pow(x): return 2**int(N.ceil(N.log(float(x))/N.log(2.0)))
import numarray as N import numarray.fft as F def czt(x, m=None, w=None, a=1.0, axis = -1): """ Copyright (C) 2000 Paul Kienzle This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 US usage y=czt(x, m, w, a) Chirp z-transform. Compute the frequency response starting at a and stepping by w for m steps. a is a point in the complex plane, and w is the ratio between points in each step (i.e., radius increases exponentially, and angle increases linearly). To evaluate the frequency response for the range f1 to f2 in a signal with sampling frequency Fs, use the following: m = 32; ## number of points desired w = exp(-2i*pi*(f2-f1)/(m*Fs)); ## freq. step of f2-f1/m a = exp(2i*pi*f1/Fs); ## starting at frequency f1 y = czt(x, m, w, a); If you don't specify them, then the parameters default to a fourier transform: m=length(x), w=exp(2i*pi/m), a=1 Because it is computed with three FFTs, this will be faster than computing the fourier transform directly for large m (which is otherwise the best you can do with fft(x,n) for n prime). TODO: More testing---particularly when m+N-1 approaches a power of 2 TODO: Consider treating w,a as f1,f2 expressed in radians if w is real """ # Convinience declations ifft = F.inverse_fft fft = F.fft do_transpose = (axis != -1) and (x.rank > 1) # transpose data to make it eqivalent to axis=-1 if axis < 0: axis += x.rank if do_transpose: axes = N.arange(x.rank) axes[[axis, x.rank-1]] = axes[[x.rank-1, axis]] x = N.transpose(x, axes) if m is None: m = x.shape[-1] if w is None: w = N.exp(2j*N.pi/m) n = x.shape[-1] k = N.arange(m, type=N.Float64) Nk = N.arange(-(n-1), m-1, type=N.Float64) nfft = next2pow(min(m,n) + len(Nk) -1) Wk2 = w**(-(Nk**2)/2) # length = m + len(x) AWk2 = a**(-k) * w**((k**2)/2) # length = m y = ifft(fft(Wk2,nfft) * fft(x * N.resize(AWk2, x.shape), nfft)); y = N.take(y, range(n,m+n), axis=-1) # [n:m+n] y = N.resize(w**((k**2)/2), y.shape) * y if do_transpose: y.transpose(axes) return y def next2pow(x): return 2**int(N.ceil(N.log(float(x))/N.log(2.0)))
## Copyright (C) 2000 Paul Kienzle ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 2 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; if not, write to the Free Software ## Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA ## usage y=czt(x, m, w, a) ## ## Chirp z-transform. Compute the frequency response starting at a and ## stepping by w for m steps. a is a point in the complex plane, and ## w is the ratio between points in each step (i.e., radius increases ## exponentially, and angle increases linearly). ## ## To evaluate the frequency response for the range f1 to f2 in a signal ## with sampling frequency Fs, use the following: ## m = 32; ## number of points desired ## w = exp(-2i*pi*(f2-f1)/(m*Fs)); ## freq. step of f2-f1/m ## a = exp(2i*pi*f1/Fs); ## starting at frequency f1 ## y = czt(x, m, w, a); ## ## If you don't specify them, then the parameters default to a fourier ## transform: ## m=length(x), w=exp(2i*pi/m), a=1 ## Because it is computed with three FFTs, this will be faster than ## computing the fourier transform directly for large m (which is ## otherwise the best you can do with fft(x,n) for n prime). ## TODO: More testing---particularly when m+N-1 approaches a power of 2 ## TODO: Consider treating w,a as f1,f2 expressed in radians if w is real function y = czt(x, m, w, a) if nargin < 1 || nargin > 4, usage("y=czt(x, m, w, a)"); endif if nargin < 2 || isempty(m), m = length(x); endif if nargin < 3 || isempty(w), w = exp(2i*pi/m); endif if nargin < 4 || isempty(a), a = 1; endif N = length(x); if (columns(x) == 1) k = [0:m-1]'; Nk = [-(N-1):m-2]'; else k = [0:m-1]; Nk = [-(N-1):m-2]; endif nfft = 2^nextpow2(min(m,N)+length(Nk)-1); Wk2 = w.^(-(Nk.^2)/2); AWk2 = (a.^-k) .* (w.^((k.^2)/2)); y = ifft(fft(postpad(Wk2,nfft)).*fft(postpad(x,nfft).*postpad(AWk2,nfft))); y = w.^((k.^2)/2).*y(1+N:m+N); endfunction
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