## Shannon's Theorem

Shannon's Theorem gives an upper bound to the capacity of a link, in bits per second (bps), as a function of the available bandwidth and the signal-to-noise ratio of the link.

The Theorem can be stated as:

## C = B * log2(1+ S/N)

where C is the achievable channel capacity, B is the bandwidth of the line, S is the average signal power and N is the average noise power.

The signal-to-noise ratio (S/N) is usually expressed in decibels (dB) given by the formula:

### 10 * log10(S/N)

so for example a signal-to-noise ratio of 1000 is commonly expressed as

### 10 * log10(1000) = 30 dB.

Here is a graph showing the relationship between C/B and S/N (in dB):

## Examples

Here are two examples of the use of Shannon's Theorem.

### Modem

For a typical telephone line with a signal-to-noise ratio of 30dB and an audio bandwidth of 3kHz, we get a maximum data rate of:

### C = 3000 * log2(1001)

which is a little less than 30 kbps.

### Satellite TV Channel

For a satellite TV channel with a signal-to noise ratio of 20 dB and a video bandwidth of 10MHz, we get a maximum data rate of: