Question

A low-pass analog signal having $4kHz$ bandwidth is sampled at $1.25$ times the Nyquist rate and each samples is quantized to one of $256$ equiprobable levels. If the digitized signal is required to transmit through an ideal $AWGN$ channel using a modulation scheme by maintaining a signal-to-noise ratio of $20dB$ in the channel, then the minimum bandwidth required by the channel for error free transmission will be approximately

• A. $12kHz$
• B. $80kHz$
• C. $40kHz$
• D. $10kHz$

Answer 1

The correct option is A $12kHz$

$R_b=rH\left(X\right)$

$H\left(X\right)=lo{g}_{2}256=8$ bits/symbol

$r=1.25\times 2f_m=2.5f_m$
$=2.5\times 4kHz=10kHz$
$R_b=10\times 8=80kbps$
When $\frac{S}{N}=20dB=100,C=Blo{g}_2(1+\frac{S}{N})$
$=Blo{g}_2(1+100)\ge 8\left({10}^4\right)$
$\because C\ge R_b$ for error free transmission for ideal channel
$B\ge \frac{8\left({10}^4\right)}{lo{g}_2\left(101\right)}Hz\approx 1.2\left({10}^4\right)Hz\approx 12kHz$
$B_{min}=12kHz$