GATE EC 2025 Set 1 — Question 15
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Communications → Information Theory → AWGN Channel Capacity
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Question
Consider an additive white Gaussian noise (AWGN) channel with bandwidth and noise power spectral density . Let denote the average transmit power constraint. Which one of the following plots illustrates the dependence of the channel capacity on the bandwidth (keeping and fixed)?
Correct answer
(A) [figure]
Solution
The channel capacity for an Additive White Gaussian Noise (AWGN) channel is given by the Shannon-Hartley theorem:
We need to analyze the behavior of as a function of bandwidth , keeping and fixed.
.
As , . So, .
We know that for small , .
So, .
This shows that as approaches infinity, the channel capacity approaches a constant value, .Combining these observations, the channel capacity initially increases with and then saturates, approaching a constant value as becomes very large. This behavior is characteristic of a function that grows and then flattens out.Let's examine the options:
We need to analyze the behavior of as a function of bandwidth , keeping and fixed.
1.As :
. This is an indeterminate form. However, as approaches 0, the term becomes very large, so ..
As , . So, .
2.As :
Let . As , .We know that for small , .
So, .
This shows that as approaches infinity, the channel capacity approaches a constant value, .Combining these observations, the channel capacity initially increases with and then saturates, approaching a constant value as becomes very large. This behavior is characteristic of a function that grows and then flattens out.Let's examine the options:
- Option (A) shows increasing with and then flattening out, which matches our analysis.
- Option (B) shows increasing linearly with , which is incorrect.
- Options (C) and (D) show increasing and then decreasing, which is also incorrect.
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