SHANNONS THEOREM Shannon Hartley Theorem This is a

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 d. B) instead of power ratio. If (as with terrestrial and commercial communications systems) S/N >> 1, then rewriting in terms of log 10. 2

More accurate approximation S/N >1 q 3

S/N<1 q 4

Overall effect 5

AWGN q q q 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. 6

Shannon and AWGN For AWGN N=N 0 B Also S=Eb. C (Eb is energy/bit) So we can rewrite (energy/bit) * (bit/s) gives (energy/s), i. e. power We often use C/B as a measure of how good the transmission is – bit/s/Hz – spectral efficiency 7

Usable region Note – minimum value of Eb/No is 1. 6 d. B – this is the Shannon limit for communication to take place. The value for C/W does continue to climb slowly 8

Multi-level transmission 1 1111 11 10 01 0 2 bits/cycle 1 1 1 0 0 00 4 bits/cycle 0000 8 bits/cycle Not necessarily done like this – but modulation aims to get the maximum no of bits/cycle 9

Error margin less with multilevel Noise changes level but does not cause error Same noise now can cause an error depending on the sampling 10