Question

A binary channel with capacity 36 kbit/sec is available for PCM voice transmission. If the signal is band-limited to 3.2 kHz, then the appropriate values of quantizing level and the sampling frequency will be, respectively
1. 32, 3.6 kHz
2. 64, 7.2 kHz
3. 64, 3.6 kHz
4. 32, 7.2 kHz

Answer 1

Correct Answer - Option 4 : 32, 7.2 kHz

Concept:

The transmission rate of a PCM signal is given by:

R_b = nf_s

n = number of bits to encode a sample

f_s = Sampling frequency.

Analysis:

Given the transmission bandwidth capacity of the binary channel is:

B_ch = 36 k bits / sec

The bandwidth of the message signal is f_m = 3.2 kHz.

Sampling frequency must be at least:

f_s ≥ 2 f_m

f_s ≥ 2 × 3.2 kHz

f_s ≥ 6.4 kHz

Let the quantization level be q, we can write:

q = 2ⁿ

n = number of bits required to encode the sample.

Since the channel must support the data rate of the PCM signal, we can write:

R_b ≤ R_ch

nf_s ≤ 36        ---(1)

From the given options, the sampling frequency will be 7.2 kHz, i.e. we can write:

7.2k × n ≤ 36k

n ≤ 5

2ⁿ ≤ 2⁵

q < 32

∴ The appropriate values of the quantizing level and sampling frequencies are q = 32 and f_s = 7.2 kHz.