Maximum error-free data rate for 1 MHz, 10 dB SNR

The maximum error-free data rate is the channel capacity, given by the Shannon–Hartley theorem for an AWGN channel:

$$C = B \log_2\!\left(1 + \frac{S}{N}\right)$$

Given

• Bandwidth $B = 1$ MHz $= 10^6$ Hz
• $(S/N)_{dB} = 10$ dB

Step 1: Convert SNR to a ratio

$$\frac{S}{N} = 10^{(S/N)_{dB}/10} = 10^{10/10} = 10$$

Step 2: Capacity

$$\begin{aligned} C &= 10^6 \log_2 (1 + 10) = 10^6 \log_2 11 \\ &= 10^6 \times \frac{\ln 11}{\ln 2} = 10^6 \times \frac{2.3979}{0.6931} \\ &= 10^6 \times 3.4594 \\ &= 3.459 \times 10^6\ \text{b/s} \end{aligned}$$

Interpretation

• The spectral efficiency is $C/B = 3.46$ b/s/Hz.
• Any rate up to about 3.46 Mb/s can, in theory, be sent with an arbitrarily small error probability using suitable (possibly very long) channel codes. Rates above this cannot be sent error-free.
• At the Nyquist rate of $2B = 2$ Msymbols/s, this rate needs $3.46/2 \approx 1.73$ bits/symbol. That is more than binary signalling can carry, so a practical scheme needs multilevel signalling, e.g. M-ary with $M \ge 4$, together with strong coding.
• The "10 dB at the input of the channel decoder" means that the SNR is measured where decoding happens, so it is the SNR that sets the capacity.

Answer: Maximum error-free data rate $C \approx 3.46$ Mb/s.