n L$-$level uniform quantizer is used for audio quantization, where L is an even number.

The peak$-$to$-$peak value Vpp $=2$Vp is equal to Lq$,$ in which Vp is the peak value and q is the

step size or quantile interval. Consider an audio signal with spectral components limited

to the frequency band $300$ to $3300$ Hz$.$ Assume that a sampling rate of $8000$ samples$/$s will

be used to generate a baseband binary PCM signal and the ratio of peak signal power to

average quantization noise power at the output needs to be $30$ dB$.$

$($a$)$ If we assume that the quantization error, e$,$ is uniformly distributed over a symmetrical

interval $(-$q$/2,$ q$/2),$ find the quantization noise power E$[$e$2].$ Show that the ratio of peak

signal power to average quantization noise power is $222/$ E$[]3$

p V e L$=$

$.$

$($b$)$ What is the minimum number of uniform quantization levels needed? What is the

minimum number of bits per sample needed?

$($c$)$ Calculate the system bandwidth $($as specified by the main spectral lobe of the signal$)$

required for the detection of such a binary PCM signal. The system is assumed to use

raised cosine pulses with roll$-$off factor r$.$