Group delay

The instantaneous frequency characterizes a local frequency behavior as a function of time. In a dual way, the local time behavior as a function of frequency is described by the group delay :

$\begin{displaymath}t_x(\nu) = -\frac{1}{2\pi} \frac{d\arg{X_a(\nu)}}{d \nu}.\end{displaymath}$

This quantity measures the average time arrival of the frequency $\nu$ . The M-file sgrpdlay.m of the Time-Frequency Toolbox gives an estimation of the group delay of a signal (do not mistake it for the file grpdelay.m of the signal processing toolbox which gives the group delay of a digital filter). For example, with signal sig of the previous example, we obtain (see fig. 2.4):

     >> sig=fmlin(256); fnorm=0:.05:.5;
     >> gd=sgrpdlay(sig,fnorm); plot(gd,fnorm);

**Figure 2.4:** Estimation of the group delay of the previous chirp
$\begin{figure} \epsfxsize =10cm\epsfysize =6cm \centerline{\epsfbox{figure/ns4fig1.eps}}\end{figure}$

Be careful of the fact that in general, instantaneous frequency and group delay define two different curves in the time-frequency plane. They are approximatively identical only when the time-bandwidth product $T\times B$ is large. To illustrate this point, let us consider a simple example. We calculate the instantaneous frequency and group delay of two signals, the first one having a large $T\times B$ product, and the second one a small $T\times B$ product (see fig. 2.5):

     >> t=2:255; 
     >> sig1=amgauss(256,128,90).*fmlin(256,0,0.5);
     >> [tm,T1]=loctime(sig1); [fm,B1]=locfreq(sig1); 
     >> T1*B1              --->  T1*B1=15.9138
     >> ifr1=instfreq(sig1,t); f1=linspace(0,0.5-1/256,256);
     >> gd1=sgrpdlay(sig1,f1); plot(t,ifr1,'*',gd1,f1,'-')
     >> sig2=amgauss(256,128,30).*fmlin(256,0.2,0.4);
     >> [tm,T2]=loctime(sig2); [fm,B2]=locfreq(sig2); 
     >> T2*B2              --->  T2*B2=1.224
     >> ifr2=instfreq(sig2,t); f2=linspace(0.2,0.4,256);
     >> gd2=sgrpdlay(sig2,f2); plot(t,ifr2,'*',gd2,f2,'-')

**Figure 2.5:** Estimation of the instantaneous frequency (stars) and group delay (line) of two different chirps with different amplitude modulations. The first plot corresponds to a large $T\times B$ product while the second corresponds to a small one
$\begin{figure} \epsfxsize =10cm\epsfysize =8cm \centerline{\epsfbox{figure/ns4fig2.eps}}\end{figure}$

On the first plot, the two curves are almost superimposed (i.e. the instantaneous frequency is the inverse transform of the group delay), whereas on the second plot, the two curves are clearly different.

Eric Chassande-Mottin 2005-10-26