Shannon’s Theorem

Shannon Hartley Theorem This is a measure of the capacity on a channel; it is impossible to transmit information at a faster rate without error. C = capacity (in bit/s) B = bandwidth of channel (Hz) S = signal power (in W) N = noise power (in W) It is more usual to use SNR (in dB) instead of power ratio. If (as with terrestrial and commercial communications systems) S/N >> 1, then rewriting in terms of log10.

More accurate approximation S/N >1 Write S/N as x and rewriting Shannon 𝐶=𝐵 𝑙𝑜𝑔 2 𝑥 1/𝑥+1 1/𝑥+1 or 𝐶=𝐵 𝑙𝑜𝑔 2 𝑥 +𝐵 𝑙𝑜𝑔 2 1/𝑥+1 Since 1 𝑥 <1 a Pade expression can be used ln 1+𝑦 =y(6+y)/(6+4y) where 𝑦<1 𝐶=𝐵 𝑙𝑜𝑔 2 𝑥 + 𝐵 ln 2 ln 1+ 1 𝑥 = 𝐵 𝑙𝑜𝑔 2 𝑥 + 𝐵 ln 2 ( 1 𝑥 )(6+1/x)/(6+4/x) Rearranging: 𝐶= 𝐵×10 𝑙𝑜𝑔 10 𝑥 10 𝑙𝑜𝑔 10 (2) + 𝐵 ln 2 6x+1 𝑥(6x+4) =B× SNR 3.01 + 𝐵 ln 2 6x+1 x(6x+4)

S/N<1 Using a similar approach: 𝐶=𝐵 𝑙𝑜𝑔 2 1+𝑥 = 𝐵 ln 2 𝑥(6+x)/(6+4x)

Overall effect

AWGN Strictly Shannon Hartley is for AWGN (Additive White Gaussian Noise) Linear addition of white noise with a constant spectral density (expressed as watts per hertz of bandwidth) and a Gaussian distribution of amplitude. Does not account for fading, frequency selectivity, interference, nonlinearity or dispersion.

Shannon and AWGN (energy/bit) * (bit/s) gives (energy/s), i.e. power For AWGN N=N0B Also S=EbC (Eb is energy/bit) So we can rewrite We often use C/B as a measure of how good the transmission is – bit/s/Hz – spectral efficiency

Usable region Note – minimum value of Eb/No is -1.6dB – this is the Shannon limit for communication to take place. The value for C/W does continue to climb slowly

Multi-level transmission 1 1111 11 10 01 00 0000 2 bits/cycle 4 bits/cycle 8 bits/cycle 1 1 0 0 1 0 0 Not necessarily done like this – but modulation aims to get the maximum no of bits/cycle

Error margin less with multilevel Noise changes level but does not cause error Same noise now can cause an error depending on the sampling