A voice-grade line has SNR = 30.1 dB (power ratio ≈ 1023:1) and usable spectrum from 300 Hz to 4300 Hz (bandwidth = 4000 Hz). Using Shannon’s capacity formula, what is the maximum achievable data rate on this line?

Introduction / Context:
Channel capacity places an upper bound on error-free data rate for a given bandwidth and signal-to-noise ratio. For telephony-grade channels, Shannon’s theorem provides the theoretical limit irrespective of modulation specifics.

Given Data / Assumptions:

• SNR = 30.1 dB ≈ power ratio 1023:1.
• Frequency span: 300 Hz to 4300 Hz → bandwidth B = 4000 Hz.
• We apply Shannon capacity: C = B * log2(1 + SNR).

Concept / Approach:
Convert SNR from dB to linear: SNR_linear ≈ 10^(30.1/10) ≈ 1023. Then compute C with base-2 logarithm. Notation: use * for multiplication and / for division.

Step-by-Step Solution:

Verification / Alternative check:
If one mistakenly uses 3000 Hz bandwidth, C would be 30000 bps, which is lower and does not reflect the given 4000 Hz span. The provided spectrum explicitly yields 40000 bps.

Why Other Options Are Wrong:

Common Pitfalls:
Mixing Nyquist and Shannon formulas, using dB directly without converting to linear ratio, or assuming 3000 Hz bandwidth by habit.

Final Answer:
40,000 bps.