Question

A data is transmitted through a voice-grade telephone channel of bandwidth 3.4 kHz. All 12 samples stored in the terminal are equiprobable and independent. The channel output signal to noise ratio is 15 dB. The channel capacity. C and maximum symbol rate, $f_s$ for error free transmission is :

• A. $C=17.09kbps;f_s=17.09k\text{samples}/sec$
• B. $C=17.09kbps;f_s=4.77k\text{samples}/sec$
• C. $C=12.34kbps;f_s=1.42k\text{samples}/sec$
• D. $C=12.34kbps;f_s=1.11k\text{samples}/sec$

Answer 1

The correct option is B $C=17.09kbps;f_s=4.77k\text{samples}/sec$

$\begin{array}{ccc}\text{Channel, capacity,}C& =B{\log }_2(1+\frac{S}{N})& \\ \frac{S}{N}& ={10}^{1.5}=31.62& \\ \therefore C& =17.09kbits/sec& \\ \text{For error free transmission,}& R_b\le C& \\ \text{Given that samples are code based on probabilities}& & \\ \text{So, that,}{R}_b& =\frac{Bits}{sample}\times \frac{samples}{sec}& \\ R_b& =H\times f_s& \\ \text{12 samples having equal probability}\frac{1}{12}\text{each}& & \\ \Rightarrow H& ={\log }_{2}12& =3.58\frac{Bits}{sample}\\ R_b& =3.58\times f_s& \\ & 3.58\times f_s\le 17.09kbits/sec& \\ f_s& \le 4.77k\text{samples/sec}& \\ (f_s{)}_{max}& =4.77k\text{samples /sec}& \end{array}$