GATE 2021 EC – Question 33
A speech signal, band limited to 4 kHz, is sampled at 1.25 times the Nyquist rate. The speech samples, assumed to be statistically independent and uniformly distributed in the range $-5$ V to $+5$ V, are subsequently quantized in an 8-bit uniform quantizer and then transmitted over a voice-grade AWGN telephone channel. If the ratio of transmitted signal power to channel noise power is 26 dB, the minimum channel bandwidth required to ensure reliable transmission of the signal with arbitrarily small probability of transmission error (rounded off to two decimal places) is ________ kHz.
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Correct answer: 9.21 to 9.31
Explanation
The Nyquist rate is $8$ kHz, so the sampling rate is $1.25\times8=10$ kHz. At 8 bits per sample the bit rate is $80$ kbps. The SNR is $10^{2.6}\approx398.1$. Reliable transmission needs $B\log_2(1+\text{SNR})\ge80$ kbps, so $B\ge\frac{80}{\log_2399.1}=\frac{80}{8.641}\approx9.26$ kHz.