What about multi-component non-stationary signals ?

The notion of instantaneous frequency implicitly assumes that, at each time instant, there exists only a single frequency component. A dual restriction applies to the group delay : the implicit assumption is that a given frequency is concentrated around a single time instant. Thus, if these assumptions are no longer valid, which is the case for most of the multi-component signals, the result obtained using the instantaneous frequency or the group delay is meaningless.

Example

For example, let us consider the superposition of two linear frequency modulations :

     >> N=128; x1=fmlin(N,0,0.2); x2=fmlin(N,0.3,0.5);
     >> x=x1+x2;

At each time instant

, an ideal time-frequency representation should represent two different frequencies with the same amplitude. The results obtained using the instantaneous frequency and the group delay are of course completely different, and therefore irrelevant (see fig. 2.10):

     >> ifr=instfreq(x); subplot(211); plot(ifr);
     >> fn=0:0.01:0.5; gd=sgrpdlay(x,fn); 
     >> subplot(212); plot(gd,fn);

**Figure 2.10:** Estimation of the instantaneous frequency (first plot) and group-delay (second plot) of a multi-component signal
$\begin{figure} \epsfxsize =10cm\epsfysize =6cm \centerline{\epsfbox{figure/ns7fig1.eps}}\end{figure}$

So these one-dimensional representations, instantaneous frequency and group delay, are not sufficient to represent all the non-stationary signals. A further step has to be made towards two-dimensional mixed representations, jointly in time and in frequency. Even if no gain of information can be expected since it is all contained in the time or in the frequency representation, we can obtain a better structuring of this information, and an improvement in the intelligibility of the representation.

To have an idea of what can be made with a time-frequency decomposition, let us anticipate the following and have a look at the result obtained on this signal with the Short Time Fourier Transform (see fig. 2.11):

     >> tfrstft(x);

**Figure 2.11:** Squared modulus of the short-time Fourier transform of the previous multi-component non-stationary signal
$\begin{figure} \epsfxsize =10cm\epsfysize =8cm \centerline{\epsfbox{figure/ns7fig2.eps}}\end{figure}$

Here two ``time-frequency components'' can be clearly seen, located around the locus of the two frequency modulations.

Eric Chassande-Mottin 2005-10-26