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

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.