Shannon s Capacity Theorem

Shannon's Capacity Theorem

Error-correcting codes can alleviate some errors, but cannot alleviate all the errors introduced by the channel in a digital communications system. Shannon's Capacity Theorem states that error-free transmission is possible as long as the transmitter does not exceed the channel's capacity. N bits constitute a block of information bits that have K bits of error correction tacked on. In terms of the bit error probability (p), the probability of an error can reach 0 as N becomes larger if the following is true:

• The ratio N / (N + K) = R, the rate (R) is kept constant.

• R is less than the channel's capacity (C).

• C = 1 + plog ₂ p + (1 - p)log ₂ (1 - p).

Thus, if a repetition code has a rate of 1/3, and if you used more data bits and repeated them twice, you could transmit them through a channel error-free, if the error probability is less than 2 ^{(1- R )} . The capacity sets a limit on your ability to transmit digital information through a channel. The converse to the Capacity Theorem states that if R > C, the probability of a word error approaches 1 as N becomes greater. The Capacity theorem can also be stated in terms of transmission rates, by dividing the coding rate (and the capacity) by the duration of the bit interval.

Shannon showed that the capacity of an additive white Gaussian noise channel is given by the following:

C = BW log ₂ (1 + S / N)

C Channel capacity in kbps

BW Bandwidth of transmission medium

S Power of signal at the transmitting device

N Power of noise received at the destination

Therefore, the telephone channel with BW = 3 kHz and S / N = 1000, yields a capacity of about 30,000 bits per second (bps). The bandwidth of a twisted pair is 4 kHz, which covers the frequency spectrum for voice. Assuming a signal-to-noise ratio (SNR) of (P0/Pn) equals 1000, (30 dB), the Shannon channel capacity is as follows :

C = 4000 x Log ₂ (1 + 1000) = 40 kbps

The following sections are based on the assumption that the modulation techniques in wired, wireless, and hybrid media differ from each other. This classification is symbolic and is not precisely correct. The sole purpose of this classification is to represent the most common technical solutions for each media.