Example 1

Let us consider first a signal with constant amplitude, and with a linear frequency modulation varying from 0 to 0.5 in normalized frequency (ratio of the frequency in Hertz to the sampling frequency, with respect to the Shannon sampling theorem). This signal is called a chirp, and as its frequency content is varying with time, it is a non-stationary signal. To obtain such a signal, we can use the M-file fmlin.m, which generates a linear frequency modulation (see fig. 1.1):

     >> sig1=fmlin(128,0,0.5);
     >> plot(real(sig1));

**Figure 1.1:** Linear frequency modulation (chirp)
$\begin{figure} \epsfxsize =10cm\epsfysize =6cm \centerline{\epsfbox{figure/in2fig1.eps}}\end{figure}$

From this time-domain representation, it is difficult (except for experienced specialists) to say what kind of modulation is contained in this signal : what are the initial and final frequencies, is it a linear, parabolic, hyperbolic...frequency modulation ?

If we now consider the energy spectrum of this signal sig1 by squaring the modulus of its Fourier transform (using the fft function) (see fig. 1.2),

     >> dsp1=fftshift(abs(fft(sig1)).^2); 
     >> plot((-64:63)/128,dsp1);

**Figure 1.2:** Energy spectrum of the chirp
$\begin{figure} \epsfxsize =10cm\epsfysize =6cm \centerline{\epsfbox{figure/in2fig2.eps}}\end{figure}$

we still can not say, from this plot, anything about the evolution in time of the frequency content. This is due to the fact that the Fourier transform is a decomposition on complex exponentials, which are of infinite duration and completely unlocalized in time. Time information is in fact encoded in the phase of the Fourier transform (which is simply ignored by the energy spectrum), but their interpretation is not straightforward and their direct extraction is faced with a number of difficulties such as phase unwrapping. In order to have a more informative description of such signals, it would be better to directly represent their frequency content while still keeping the time description parameter : this is precisely the aim of time-frequency analysis. To illustrate this, let us try the Wigner-Ville distribution on this signal (see fig. 1.3):

     >> tfrwv(sig1);

**Figure 1.3:** Wigner-Ville distribution of the chirp
$\begin{figure} \epsfxsize =10cm\epsfysize =8cm \centerline{\epsfbox{figure/in2fig3.eps}}\end{figure}$

Without going into details about this representation (it will be developed in the following), we can see that the linear progression of the frequency with time, from 0 to 0.5, is clearly shown.

If we now add some complex white gaussian noise on this signal, using the M-files noisecg.m and sigmerge.m, with a 0dB signal to noise ratio (see fig. 1.4),

     >> sig2=sigmerge(sig1,noisecg(128),0);
     >> plot(real(sig2));

**Figure 1.4:** Chirp embedded in a 0 dB white gaussian noise
$\begin{figure} \epsfxsize =10cm\epsfysize =6cm \centerline{\epsfbox{figure/in2fig4.eps}}\end{figure}$

and consider the spectrum of it (see fig. 1.5):

     >> dsp2=fftshift(abs(fft(sig2)).^2); 
     >> plot((-64:63)/128,dsp2);

**Figure 1.5:** Energy spectrum of the noisy chirp
$\begin{figure} \epsfxsize =10cm\epsfysize =6cm \centerline{\epsfbox{figure/in2fig5.eps}}\end{figure}$

it is worse than before to interpret these plots. On the other hand, the Wigner-Ville distribution still show quite clearly the linear progression of the frequency with time (see fig. 1.6):

     >> tfrwv(sig2);

**Figure 1.6:** Wigner-Ville distribution of the noisy chirp
$\begin{figure} \epsfxsize =10cm\epsfysize =8cm \centerline{\epsfbox{figure/in2fig6.eps}}\end{figure}$

Eric Chassande-Mottin 2005-10-26