Which expression represents the Shannon-Hartley channel capacity for bandwidth B and signal-to-noise ratio S/N?

• A. C = B loge(1 + S/N)
• B. C = B log2(1 + S/N)
• C. C = 2B log2(1 + S/N)
• D. C = B log2(1 + S^2/N)

Answer: (B)

The key idea is that a channel's maximum data rate grows with its bandwidth but only increases logarithmically as the signal quality (S/N) improves. The Shannon–Hartley expression C = B log2(1 + S/N) captures this: capacity C in bits per second, bandwidth B in hertz, and S/N as the signal-to-noise ratio.

Using log base 2 is important because it yields units of bits per second. If you used the natural logarithm, you’d get a result in nats per second and would need a conversion factor to express the rate in bits per second. The 1 inside the logarithm ensures the correct baseline; as S/N becomes large, the capacity grows roughly like B log2(S/N), and when S/N is small, the capacity approaches zero.

The other forms misstate the relationship: using a natural log changes the units, a factor of 2 would double the capacity, and squaring S in the ratio changes the dependence on signal quality. The standard, correct form is the one with log2(1 + S/N).