GATE EC 2014 Set 3 — Question 61
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Communications → Source Coding & Baseband Transmission → PCM & DPCM
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Question
In a PCM system, the signal is sampled at the Nyquist rate. The samples are processed by a uniform quantizer with step size 0.75 V. The minimum data rate of the PCM system in bits per second is ________.
Correct answer
199 to 201
Solution
The given signal is .
This can be rewritten as .
Using the identity , we can write:
.The maximum frequency component of the signal is .According to the Nyquist sampling theorem, the minimum sampling rate (Nyquist rate) is .
.The peak amplitude of the signal is .
The dynamic range of the signal is .
.The uniform quantizer has a step size .The number of quantization levels required is given by .
.Since the number of quantization levels must be an integer and typically a power of 2 for bits, we choose the smallest power of 2 that is greater than or equal to .
If is the number of bits per sample, then .
We need .
For , (too small).
For , (sufficient).
So, the minimum number of bits per sample required is bits/sample.The minimum data rate of the PCM system is given by .
.The final answer is .
This can be rewritten as .
Using the identity , we can write:
.The maximum frequency component of the signal is .According to the Nyquist sampling theorem, the minimum sampling rate (Nyquist rate) is .
.The peak amplitude of the signal is .
The dynamic range of the signal is .
.The uniform quantizer has a step size .The number of quantization levels required is given by .
.Since the number of quantization levels must be an integer and typically a power of 2 for bits, we choose the smallest power of 2 that is greater than or equal to .
If is the number of bits per sample, then .
We need .
For , (too small).
For , (sufficient).
So, the minimum number of bits per sample required is bits/sample.The minimum data rate of the PCM system is given by .
.The final answer is .
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