Given a signal x(n), we can create echoes and reverberations by delaying and scaling the...

Given a signal x(n), we can create echoes and reverberations by delaying and scaling the signal as follows

where D is positive integer and

(a) Explain why the comb filter

can be used as a reverberator (i.e., as a device to produce artificial reverberations). (Hint: Determine and sketch its impulse response.)

(b) The all-pass comb filter

is used in practice to build digital reverberators by cascading three to five such filters and properly choosing the parameters a and D. Compute and plot the impulse response of two such reverberators each obtained by cascading three sections with the following parameters.

(c) The difference between echo and reverberation is that with pure echo there are clear repetitions of the signal, but with reverberations, there are not. How is this reflected in the shape of the impulse response of the reverberator? Which unit in part (b) is a better reverberator?

(d) If the delays

(e) Plot the phase response of units 1 and 2 and continent on them.

(f) Plot h(n) for being nonprime. What do you notice?

More details about this application can be found in a paper by I A. Moorer, "Signal Processing Aspects of Computer Music: A Survey," Proc. IEEE, Vol. 65, No. 8, Aug. 1977, pp. 1108-1137.