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# Shannon-Hartley Channel Capacity Theorem

In Satellite, we talk about the Shannon Limit or Shannon Bound. "Shannon" refers to Claude Shanon who is credited with being the father of the modern information age.

In this example, we are referring to the Shannon-Hartley theorem which established the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise.

The Shannon-Hartley theorem establishes Claude Shannon’s channel capacity for a communication link which is a bound on the maximum amount of error-free information per time unit that can be transmitted within a specified bandwidth in the presence of noise interference, assuming that this signal power is bounded and that the Gaussian noise process is characterized by a known power or power spectral density.

That’s a mouthful, so in simpler terms, the Shannon limit is the theoretical limit to how much you throughput you can get from a wireless channel given a specified bandwidth and signal to noise ratio.

## Examples

1. At a Signal to Noise Ratio of 0 where Signal Power = Noise Power, the channel capacity in bits per second equals the bandwidht in Hertz. It is possible to transmit signals below the noise level, but the error rates will grow rapidly.

2. If the SNR is 20 dB, and the bandwidth available is 4 kHz, which is appropriate for telephone communications, then C = 4000 log2(1 + 100) = 4000 log2 (101) = 26.63 kbit/s. Note that the value of S/N = 100 is equivalent to the SNR of 20 dB.

3. If the requirement is to transmit at 5 mbit/s, and a bandwidth of 1 MHz is used, then the minimum S/N required is given by 5000 = 1000 log2(1+S/N) so C/B = 5 then S/N = 25 −1 = 31, corresponding to an SNR of 14.91 dB (10 x log10(31)).

4. Channel capacity is proportional to the bandwidth of the channel and to the logarithm of SNR. Thus channel capacity can be increased by either increasing the channel's bandwidth given a fixed SNR requirement or, with fixed bandwidth, by using higher-order modulations that need a higher Signal to Noise ratio to operate.

As the modulation rate increases, the spectral efficiency improves, but at the expense of the SNR requirement. Hence, there is an exponential rise in the SNR requirement as one adopts higher order modulations; however, the spectral efficiency improves.