# Instananeous frequency of a chirp signal is halved?

by Luongo
Tags: chirp, frequency, halved, instananeous, signal
 P: 120 I'm doing some research with MATLAB, where i have a simple chirp. y = sin(x.^2) i take the FFT of the chirp, set the upper half of the FT of the signal to 0, and take the IFFT of this to recover the original chirp y with real and imaginary parts. then i use the formula for calculation of the instantaneous frequency: (s1.*ds2 - s2.*ds1)./(s1.^2 +s2.^2) where s1 is the real part of y, s2 is imag of y and ds1 and ds2 are the derivatives of s1 and s2 respectively. It turns out i get my inst. freq to be y = x. but i expected y = 2x. Is there any reason why my inst. frequency is missing the factor of 2? I have no idea what I'm doing wrong that results in the missing factor. thanks! any help is greatly appreciated. here is my MATLAB code: function []=chirp() x = 0:0.01:5; y = sin(x.^2); y = sin(x.^2)-mean(y); %eliminate DC offset subtract mean plot(x,y,'k') hold on y = fft(y); y = 2*y; %double the amplitude y(250:end) = 0; %cut off upper half the signal y = ifft(y); %double amplitude of the chirp signal s1 = real(y); plot(x,s1, 'p') s2 = imag(y); %plot(x,s2, 'r') ds1 = diff(s1)./0.01; ds1(501) = 0; ds2 = diff(s2)./0.01; ds2(501) = 0; %derivative plots: %plot(x,ds1,'y'); %plot(x,ds2,'k'); %obtain instantaneous frequency n = (s1.*ds2 - s2.*ds1); d = (s1.^2 +s2.^2+ (max(s1).^2)); q = n./d; plot(x,q, 'g')
 P: 10 How did you arrive at your formula for the instantaneous frequency? I suspect that it might be off. If I calculate the instantaneous frequency using a different approach, with the analytical signal 'y' using your code: dy = diff(y)/0.01; dy(501) = 0; instfrq = imag(dy./y); I get something that tracks 2*x instead. Here I used the imaginary part of the log-derivative of the analytic signal: $\hat{y} = A e^{i\varphi(t)}$ $\frac{d\hat{y}}{dt}=i \frac{d\phi(t)}{dt} A e^{i\varphi(t)}$ $\rightarrow \frac{d\phi(t)}{dt} = \frac{\frac{d\hat{y}}{dt}}{i \hat{y}}$
 P: 120 it turns out i made a mistake with my epsilion correction factor, it works now though. thanks

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