Maximum Data Rate (channel capacity) for Noiseless and Noisy channels

Maximum Data Rate (channel capacity) for Noiseless and Noisy channels


Data rate governs the speed of data transmission. A very important consideration in data communication is how fast we can send data, in bits per second, over a channel. 

Data rate depends upon 3 factors:

-> The bandwidth available

-> Number of levels in digital signal

->The quality of the channel – level of noise


Two theoretical formulas were developed to calculate the data rate: one by Nyquist for a noiseless channel, another by Shannon for a noisy channel.





1. Noiseless Channel : Nyquist Bit Rate -

Nyquist bit rate was developed by Henry Nyquist who proved that the transmission capacity of even a perfect channel with no noise has a maximum limit.



The theoretical formula for the maximum bit rate is:

BitRate = 2 × Bandwidth × log₂ (L)

In the above equation, bandwidth is the bandwidth of the channel, L is the number of signal levels used to represent data, and BitRate is the bit rate in bits per second.

Bandwidth is a fixed quantity, so it cannot be changed. Hence, the data rate is directly proportional to the number of signal levels.


NOTE:-   Increasing the levels of a signal may reduce the reliability of the system.



Ex:1   Consider a noiseless channel with a bandwidth of 3000 Hz transmitting a signal with two signal levels. What can be the maximum bit rate?

Ans: 1  BitRate = 2 * 3000 * log2(2) = 6000bps


Ex:2  We need to send 265 kbps over a noiseless channel with a bandwidth of 20 kHz. How many signal levels do we need?

Ans: 2  265000 = 2 * 20000 * log2(L)
log2(L) = 6.625

L = 26.625 = 98.7 levels





2. Noisy Channel : Shannon Capacity -

In reality, we cannot have a noiseless channel; the channel is always noisy.  
Claude Shannon extended Nyquist's work for actual channels that are subject to noise. Noise can be of various types like thermal noise, impulse noise, cross-talks etc. Among all the noise types, thermal noise is unavoidable. The random movement of electrons in the channel creates an extraneous signal not present in the original signal, called the thermal noise. The amount of thermal noise is calculated as the ratio of the signal power to noise power, SNR.



Shannon capacity is used to determine the theoretical highest data rate for a noisy channel -

Capacity = bandwidth * Log₂(1 + SNR)

In the above equation, bandwidth is the bandwidth of the channel, SNR is the signal-to-noise ratio, and capacity is the capacity of the channel in bits per second.

Bandwidth is a fixed quantity, so it cannot be changed. Hence, the channel capacity is directly proportional to the power of the signal, as SNR = (Power of signal) / (power of noise).

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. 

Ex 1:  If the bandwidth of a noisy channel is 4 KHz, and the signal to noise ratio is 100, then the maximum bit rate can be computed as:

Ans:  Capacity = 4000 × log2( 1+100 ) = 26,633 bps = 26.63 kbps





Ex 2:  The SNR is often given in decibels. Assume that SNR(dB) is 36 and the channel bandwidth is 2 MHz. Calculate the theoretical channel capacity.

Ans:  SNR(dB) = 10 * log10(SNR)
SNR = 10(SNR(dB)/10)
SNR = 103.6 = 3981

Hence, C = 2 * 106 * log2(3982) = 24 MHz