Question

Consider an additive white Gaussian noise (AWGN) channel with bandwidth \( W \) and noise power spectral density \( \frac{N_0}{2} \). Let \( P_{{av}} \) denote the average transmit power constraint. Which one of the following plots illustrates the dependence of the channel capacity \( C \) on the bandwidth \( W \) (keeping \( P_{{av}} \) and \( N_0 \) fixed)?

• A.
• B.
• C.
• D.

Hint:

The channel capacity of an AWGN channel increases with bandwidth, but the rate of increase slows down as bandwidth increases. This is captured by the Shannon-Hartley theorem.

Answer 1

The Correct Option is A

The channel capacity \( C \) of an AWGN channel is given by the Shannon-Hartley theorem: \[ C = W \log_2 \left( 1 + \frac{P_{{av}}}{N_0 W} \right). \] This equation shows that the channel capacity increases with the bandwidth \( W \), but the increase is not linear. For smaller values of \( W \), the channel capacity increases rapidly, but as \( W \) gets larger, the rate of increase slows down and approaches a limit. This behavior is reflected in Option (A), which shows a curve where capacity increases rapidly at first but gradually levels off as bandwidth increases.

Analyze the relationship.
Option (A) correctly represents this characteristic behavior: a rapid increase in capacity at small values of \( W \) followed by a flattening as \( W \) grows.
Option (B) suggests a linear increase, which is not consistent with the Shannon-Hartley theorem.
Option (C) suggests a logarithmic growth, which is not accurate for the full range of bandwidth values in this context.
Option (D) suggests an oscillatory pattern, which is not valid for this type of channel capacity.

Thus, the correct answer is (A).