Question

A low pass analog signal with bandwidth of 20 kHz is sampled at nyquist rate and quantized using a 16-level uniform quantizer. The resultant digital signal is transmitted using M-ary PSK with raised cosine pulse (roll-off factor 0.3) shaping of baseband data. If the channel bandwidth available to transmit the M-ary PSK signal is 110 kHz. then the minimum value of M required is________.

• A. 4

Answer 1

The correct option is A 4
$f_m=20kHz$

$f_s=2f_m=40kHz$

$n=lo{g}_2\left(L\right)=lo{g}_2\left(16\right)=4bits/symbol$

Bit rate, $R_b=n{f}_s=160kbps$

Bandwidth of baseband data modelled using raised cosine pulse is,

$(BW{)}_{baseband}=\frac{R_b}{2}(1+\alpha )$

Bandwidth of M-ary PSK signal (for the given modelling of baseband date) is,

$B_T=\frac{2(BW{)}_{baseband}}{lo{g}_{2}M}=\frac{R_b(1+\alpha )}{lo{g}_{2}M}$

For proper signal transmission, $B_T\le channelBW$

So, $\frac{R_b(1+\alpha )}{lo{g}_{2}M}\le 110kHz$

$lo{g}_{2}M\ge \frac{160(1+0.3)}{110}$

$Lo{g}_{2}M\ge 1.89$

$M\ge 2^{1.89}$

$M\ge 3.7$

So, $M_{min}=4$