Question

A baseband message signal $m\left(t\right)=sin{c}^2\left({10}^{4}t\right)$ is sampled at Nyquist rate and quantized using and 8 bit quantizer. The resultant binary bits are baseband modelled using raised cosine pulse with a roll-off factor of unity. The minimum channel bandwidth required to transmit this baseband signal will be ______kHz.

• A. 160

Answer 1

The correct option is A 160
Given that, $m\left(t\right)=sin{c}^2\left({10}^{4}t\right)$

$sinc\left(t\right)\stackrel{CTFT}{⟷}rect\left(f\right)$

$sinc\left({10}^{4}t\right)\stackrel{CTFT}{⟷}\frac{1}{{10}^4}rect\left(\frac{f}{{10}^4}\right)$

$sin{c}^2\left({10}^{4}t\right)\stackrel{CTFT}{⟷}\frac{1}{{10}^4}rect\left(\frac{f}{{10}^4}\right)\ast \frac{1}{{10}^4}rect\left(\frac{f}{{10}^4}\right)$

Maximum frequency of $rect\left(\frac{f}{{10}^4}\right)=\frac{{10}^4}{2}=5kHz$

Maximum frequency of $rect\left(\frac{f}{{10}^4}\right)\ast rect\left(\frac{f}{{10}^4}\right)=2\times 5=10kHz$

Nyquist sampling rate, $f_{\left(min\right)}=2f_{m\left(max\right)}=2\left(10\right)=20kHz$

Bite/sample used in quantizer, n = 8

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

Minimum channel BW required to transmit a signal, which is baseband modelled using a raised cosine filter will be,

$(BW{)}_{min}=\frac{R_b}{2}(1+\alpha );$ $\alpha =roll-offfactor$

For $\alpha =1$ $(BW{)}_{min}=\frac{R_b}{2}(1+1)=R_b=160kHz$