Channel Capacity — Required SNR: 25-Comp-B5 Computer Communications, December 2016, Q2

Question 2: Channel Capacity — Required SNR (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Quantity	Value
Intended capacity $C$	500 Mbps
Bandwidth $B$	25 MHz

Find. The signal-to-noise ratio (as a ratio, and in dB) needed for the Shannon–Hartley capacity to reach $500\text{ Mbps}$.

Approach. Invert the Shannon–Hartley capacity formula $C = B\log_2(1+\text{SNR})$ to solve for SNR given $C$ and $B$, then convert the ratio to dB.

Set up Shannon–Hartley and isolate SNR. $$C = B\log_2(1+\text{SNR}) \;\Rightarrow\; \text{SNR} = 2^{C/B} - 1$$
Substitute the given values. $$\frac{C}{B} = \frac{500\times10^{6}}{25\times10^{6}} = 20$$ $$\text{SNR} = 2^{20} - 1 = 1{,}048{,}576 - 1 = \boxed{1{,}048{,}575}$$
Convert to decibels. $$\text{SNR(dB)} = 10\log_{10}(\text{SNR}+1) = 10\log_{10}(2^{20}) = 10 \times 20\log_{10}(2) = 200 \times 0.30103$$ $$\text{SNR(dB)} = \boxed{60.21\text{ dB}}$$

Quantity	Value
Required SNR (ratio)	1,048,575 ($\approx 2^{20}$)
Required SNR (dB)	≈ 60.21 dB

Topic: Shannon–Hartley channel capacity, inverted to solve for SNR.

Key relations: $C = B\log_2(1+\text{SNR})$; inverted, $\text{SNR} = 2^{C/B}-1$; $\text{SNR(dB)} = 10\log_{10}(\text{SNR})$.

Why this works: Shannon's theorem gives the maximum error-free rate a channel of bandwidth $B$ can support at a given noise level; because $\log_2$ is monotonic and invertible, exactly the same formula can be solved backward to ask "how quiet must the channel be" to reach a target rate, which is the everyday link-budget question a designer actually needs answered.

Common pitfall: forgetting that $C/B=20$ means the SNR must satisfy $2^{20}$, not $20$ itself — a linear reading of "20 dB-like" ratio badly understates the true noise requirement; also reporting $2^{C/B}$ instead of $2^{C/B}-1$ as the SNR.

Source: Stallings, Data and Computer Communications, 10th ed., Ch.3 (Data Transmission) §3.4, Channel Capacity.

Question 2 of 7: Channel Capacity — Required SNR

Question 2: Channel Capacity — Required SNR (20 marks)