1. Shannon’s Theorem

2. 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.

3. 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 ln 𝑥 = 𝐵 𝑙𝑜𝑔 2 𝑥 + 𝐵 ln ( 1 𝑥 )(6+1/x)/(6+4/x)
Rearranging:
𝐶= 𝐵×10 𝑙𝑜𝑔 10 𝑥 10 𝑙𝑜𝑔 10 (2) + 𝐵 ln x+1 𝑥(6x+4) =B× SNR 𝐵 ln x+1 x(6x+4)

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

5. Overall effect

6. 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.

7. 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

8. 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

9. Multi-level transmission1
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

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