Example 2

The second example we consider is a bat sonar signal, recorded with a sampling frequency of 230.4kHz and an effective bandwidth of [8kHz, 80kHz] (this recording was part of the research program RCP 445 supported by CNRS (Centre National de la Recherche Scientifique, France) [Fla86]).

First, load the signal from the MAT-file bat.mat (see fig. 1.7):

     >> load bat
     >> t0=linspace(0,2500/2304,2500);   
     >> plot(t0,bat); xlabel('Time [ms]');

**Figure 1.7:** Sonar signal from a bat
$\begin{figure} \epsfxsize =10cm\epsfysize =6cm \centerline{\epsfbox{figure/in2fig7.eps}}\end{figure}$

From this plot, we can not say precisely what is the frequency content at each time instant

; similarly, if we look at its spectrum (see fig. 1.8),

     >> dsp=fftshift(abs(fft(bat)).^2);
     >> f0=(-1250:1249)*230.4/2500;
     >> plot(f0,dsp); xlabel('Frequency [kHz]');

**Figure 1.8:** Energy spectrum of the bat sonar signal
$\begin{figure} \epsfxsize =10cm\epsfysize =6cm \centerline{\epsfbox{figure/in2fig8.eps}}\end{figure}$

we can not say at what time the signal is located around 38kHz, and at what time around 40kHz (you can use the zoom function to see more precisely what is happening around these frequencies ; see the Matlab Reference Guide). Let us now consider a representation called the pseudo Wigner-Ville distribution, applied on the most interesting part of this signal (this distribution was obtained with the M-file tfrpwv.m, stored in the matrix tfr and saved with the signal in the MAT-file bat.mat ; the corresponding time- and frequency- samples t and f where also saved on bat.mat) (see fig. 1.9):

     >> contour(t,f,tfr,5); axis('xy'); 
     >> xlabel('Time [ms]'); ylabel('Frequency [kHz]'); 
     >> title('TFRPWV of a bat signal');

**Figure 1.9:** Pseudo-WVD of the bat sonar signal
$\begin{figure} \epsfxsize =10cm\epsfysize =8cm \centerline{\epsfbox{figure/in2fig9.eps}}\end{figure}$

We then have a nice description of its spectral content varying with time : it is a narrow-band signal, whose frequency content is decreasing from around 55kHz to 38kHz, with a non-linear frequency modulation (approximately of hyperbolic shape).

Eric Chassande-Mottin 2005-10-26