Digital Signal Processing Using Matlab

11.2. Solved exercises

EXERCISE 11.1.

A chirp signal, whose instantaneous frequency linearly sweeps the band between f₁ and f₂, is expressed as follows:

s(t) = sin[(2πf₁ + 2πβt)t+Φ₀]

with β = (f₂ – f₁)/(2PulseLength), Pulselength being the signal duration.

1. Generate this signal using a MATLAB code for Φ₀ = 0.
2. Plot the generated signal and its power spectral density.
3. It is difficult to obtain a complete image about the signal structure from these partial representations. In fact, they are not able to clearly indicate the modulation parameters or the time evolution of the signal spectral content. This information can be easily retrieved in the time-frequency plane. Illustrate this capability of the time-frequency distributions using the spectrogram for example.

a.

% Generation of a linear frequency modulated signal
f1=2000; f2=8000;
pulselength=0.025;
Fs=20000; % Sampling frequency
% Warning: Fs should verify the Nyquist constraint: Fs>2*max(f1,f2)
t=(0:1/Fs:pulselength);
beta=(f2-f1)/ (2*pulselength);
chirp1=sin(2*pi*(f1+beta*t).*t);
% Another way to generate the chirp signal
chirp2 = vco(sawtooth((2*pi/pulselength)*t,1), [f1/Fs,f2/Fs]*Fs,Fs);
% chirpl and chirp2 are similar up to a phase term

b.

figure; clf; subplot (211) plot(t,chirp1); xlabel( 'Time [s]'); ylabel('Amplitude'); title(' Time variation of a chirp signal') C=fftshift (abs(fft(chirp1) ).^2); lc=length(chirpl); mc=lc/2; freq=(-mc:1:mc-1)*Fs/lc; subplot (212) plot (freq,C); xlabel('Frequency [Hz]'); ylabel('Power ...