# Noise

In any communication system, the received signal will consist of the transmitted signal, attenuated as it has propagated along the transmission media and suffering from some distortion due to the characteristics of the system. In addition, unwanted signals (or *noise*) may occur between the transmitter and the receiver which are *added* to the transmitted signal. Noise is the main factor that limits the performance of a communications system.

The effect of noise on a digital signal

There are four categories of noise:

*Thermal (Gaussian) noise*- this is due to the thermal agitation of electrons in a conductor, is present in all electronic devices and transmission lines, and is a function of temperature. It is distributed uniformly across the frequency spectrum, and is often referred to as white noise. It cannot be eliminated, and limits overall system performance.*Intermodulation noise*- this can occur if signals at different frequencies share the same transmission line. It results in signals that are the sum or difference of the original signals, and occurs when there is some non-linearity in the communication system (which may be caused by component malfunction or excessive signal strength).*Crosstalk*- this is the phenomenon that allows you to hear someone else's conversation whilst using the telephone, and occurs due to electrical coupling between two or more transmission paths (such as adjacent twisted-pair cables).*Impulse noise*- this consists of random pulses (or spikes) of noise, usually of short duration and relatively high amplitude. Causes include external electromagnetic disturbances such as lightning, vehicle ignition systems, heavy-duty electrical equipment, and faults in the communications system itself. It is usually only a minor annoyance for analogue systems such as a telephone link, but is the primary cause of errors in digital communication.

## Shannon Limit

In 1924 Harry Nyquist derived an equation expressing the maximum data rate for a noiseless channel. Nyquist proved that if an arbitrary signal is run through a low-pass filter of a given bandwidth (*H*), the filtered signal could be completely reconstructed by line samples taken at a rate equivalent to twice the bandwidth. Sampling the line more frequently is pointless, because the higher frequency components that such sampling could recover have already been filtered out. If the signal consists of *V* discrete levels, Nyquist's theorem states:

**Maximum data rate = 2 H log_{2} V bits per second**

In 1948 Claude Shannon took this work further and extended it to the case of a channel subject to random (thermal) noise. According to Nyquists, a noiseless 3 KHz channel cannot transmit binary (i.e. two-level) signals at a rate exceeding 6000 bits per second. If random noise is introduced, the situation deteriorates rapidly. The amount of thermal noise present in a signal is expressed as the ratio of signal power (*S*) to noise power (*N*), and is called the *signal-to-noise ratio* (*SNR*). The ratio will become smaller as the signal propagates through the transmission medium due to attenuation of the transmitted signal. The SNR is not usually usually expressed as a ratio. Instead, the value *10 log _{10} S/N* is used. The unit thus derived is known as a

*decibel*(dB). A signal-to-noise ratio of 10 would be expressed as 10 dB; a ratio of 100 as 20 dB; a ratio of 1000 as 30 dB and so on. Shannon found that the maximum data rate of a noisy channel with a bandwidth of

*H*Hz, and a signal-to-noise ratio

*S/N*is given by:

**Maximum data rate = H log_{2} (1+S/N) bits per second**

As an example, a channel of 3000-Hz bandwidth, and a signal to thermal noise ratio of 30 dB (typical parameters for an analogue telephone line) can never transmit much more than 30,000 bps, no matter how many signal levels are used, and no matter how frequently samples are taken. Shannon's result can be applied to any channel subject to *Gaussian* (thermal) noise. It should also be noted that this limitation is an upper bound, and real systems will rarely achieve it.