CHIRPZT - Chirped Z-transform

Usage

c = chirpzt(f,K,fdiff)
c = chirpzt(f,K,fdiff,foff)
c = chirpzt(f,K,fdiff,foff,fs)

Input parameters

Output parameters

Description

c = chirpzt(f,K,fdiff,foff) computes K samples of the discrete-time fourier transform DTFT c of f at values \(c(k+1)=F(2\pi(f_{off}+kf_{diff}))\) for \(k=0,\dots,K-1\) where \(F=DTFT(f)\). Values foff and fdiff should be in range of \(0-1\). If foff is ommited or empty, it is considered to be 0. If fdiff is ommited or empty, K equidistant values \(c(k+1)=F(2\pi k/K)\) are computed. If even K is ommited or empty, input length is used instead resulting in the same values as fft does.

c = chirpzt(f,K,fdiff,foff,fs) computes coefficients using frequency values relative to fs \(c(k+1)=F(2\pi(f_{off}+kf_{diff})/fs)\) for \(k=0,\dots,K-1\).

The input f is processed along the first non-singleton dimension or along dimension dim if specified.

Examples:

Calculating DTFT samples of interest (aka zoom FFT):

% Generate input signal
fs = 8000;
L = 2^10;
k = (0:L-1).';
f1 = 400;
f2 = 825;
f = 5*sin(2*pi*k*f1/fs + pi/4) + 2*sin(2*pi*k*f2/fs - pi/3);

% This is equal to fft(f)
ck = chirpzt(f,L);

%chirpzt to FFT error:
norm(ck-fft(f))

% Frequency "resolution" in Hz
fdiff = 0.4;
% Frequency offset in Hz
foff = 803.9;
% Number of frequency values
K = 125;
% DTFT samples. The frequency range of interest is 803.9-853.5 Hz
ckchzt = chirpzt(f,K,fdiff,foff,fs);

% Plot modulus of coefficients
figure(1);
fax=foff+fdiff.*(0:K-1);
hold on;
stem(k/L*fs,abs(ck),'k');
stem(fax,abs(ckchzt),'r:');
set(gca,'XLim',[foff,foff+K*fdiff]);
set(gca,'YLim',[0 1065]);
xlabel('f[Hz]');
ylabel('|ck|');

% Plot phase of coefficients
figure(2);
hold on;
stem(k/L*fs,angle(ck),'k');
stem(fax,angle(ckchzt),'r:');
set(gca,'XLim',[foff,foff+K*fdiff]);
set(gca,'YLim',[-pi pi]);
xlabel('f[Hz]');
ylabel('angle(ck)');