GATE ECE 2025 | Question: 5

Question

Consider an additive white Gaussian noise (AWGN) channel with bandwidth $W$ and noise power spectral density $\frac{N_{0}}{2}$. Let $P_{a v}$ 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 $P_{a v}$ and $N_{0}$ fixed)?

• A.
• B.
• C.
• D.

 

Answer 1

A

Answer 2

 

The Shannon-Hartley theorem for channel capacity in an AWGN channel is given by:

 

$$C = W \log_2 \left( 1 + \frac{P_{av}}{N_0 W} \right)$$

• At low bandwidth ($W \to 0$): The capacity starts at 0.

• At high bandwidth ($W \to \infty$): As $W$ increases, the noise power ($N_0 W$) increases, which decreases the Signal-to-Noise Ratio (SNR). However, the linear factor $W$ outside the logarithm drives the growth.

  To find the behavior as $W \to \infty$, we can use the limit $\ln(1+x) \approx x$ for small $x$. Let $x = \frac{P_{av}}{N_0 W}$. As $W \to \infty$, $x \to 0$.

   

  $$C \approx W \cdot \frac{1}{\ln 2} \cdot \left( \frac{P_{av}}{N_0 W} \right) = \frac{P_{av}}{N_0 \ln 2} \approx 1.44 \frac{P_{av}}{N_0}$$

  This means the capacity does not increase infinitely. Instead, it saturates and approaches a constant horizontal asymptote determined by the power and noise density.

Conclusion:

The plot must start at 0, increase, and then flatten out (saturate) towards a maximum limit.

• Plot A shows this behavior correctly.

• Plot B suggests infinite linear growth (incorrect).

• Plot C suggests exponential growth (incorrect).

Correct Answer: Plot A