ADDITIONAL NOTES FOR SEPTEMBER 13 LECTURE

In Figure 1, we see the signal x(t) and its Fourier transform X(f). Clearly, x(t) is narrowband with center frequency f_c of about 0.8KHz (or 1.6 \pi Krad/s).

In Figure 2, we see plots of the magnitude, phase and group delay of the frequency response H(f) of the system that is processing x(t). From the magnitude response, we can clearly see that we are dealing with an "allpass" system (i.e., a system for which the magnitude response is one for all frequencies). Note that the plot of the phase of H(f) appears to have discontinuities but this is simply the "wrap-around" effect due to the fact that the phase is measured modulo
2\pi. Also, note that the phase delay at f_c = 0.8KHz is about pi/2 rad and the group delay at f_c = 0.8KHz is about 2ms.

In Figure 3, we see the output that we get after one, two, three, ..., fourteen, fifteen passes through the system with frequency H(f). Each time, the envelop gets delayed by the group delay (i.e., about 2ms); this is tracked by the circle "o" in the figure. Also, each time, the phase of the underlying cosine gets delays by the phase delay (i.e., about \pi/2 rad); the phase delay is less visible in the figure and its tracked by the cross "x".