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12.2. Solved exercises

EXERCISE 12.1.

Write a MATLAB function to calculate the FRFT of a signal for a given order.

```function FRFT = FracFR(x0,a);
% Inputs: x0 - Original signal```
```% a - FRFT order
% Output: FRFT - Transformed signal
deltax = sqrt(length(x0)/2);
phi= a*pi/2; N = fix (length (x0)/2);
deltaxl = 2*deltax;
alpha = 1/tan(phi);
beta = 1/sin(phi);
% Kernel implementation
T = [-N:N-1]/deltax1;
T = T(:);x0=x0(1:2*N);
fl = exp(-i*pi*tan(phi/2)*T.*T);
fl = f1(:);x = x0.*f1;clear T;
t = [-2*N+1:2*N-1]/deltax1;
hlptc = exp(i*pi*beta*t.*t);
clear t; hlptc=hlptc(:);
N2 = length(hlptc);
N3 = 2* (ceil(log(N2+2*N-1) /log(2)));
hlptcz= [hlptc;zeros(N3*N2,1)];
xz = [x;zeros(N3-2*N,1)];
Hcfft = ifft(fft(xz).*fft(hlptcz));
clear hlptcz; clear xz;
Hc = Hcfft(2*N:4*N-1);
clear Hcfft;clear hlptc;
Aphi = exp(-i*(pi*sign(sin(phi))/4-phi/2))/sqrt(abs(sin(phi)));
xx = [-N:N-1]/deltax1;
fl = f1(:); FRFT = (Aphi*f1.*Hc)/deltax1;```

EXERCISE 12.2.

This exercise aims to estimate the linear modulation rate of a chirp signal using the FRFT.

The MATLAB code below generates the analyzed signal.

```t=0:255;
x0=[ zeros (1,128),exp(j*2*pi*(.1*t+.0007*t.*2)),zeros(1,128)].';```

The instantaneous frequency law and the modulation rate corresponding to this signal are calculated as follows:

```LFT=.1+.0014*t;
c=4*(LFT(256)-LFT(1)) ; % the modulation rate
plot(t,LFI);
xlabel('Temps');
ylabel('Normalized frequency');
title('Instantaneous frequency law');```

Figure 12.3. Instantaneous frequency law for a chirp signal

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