Noise and interference

We have mentioned that communications are, to varying degrees, subject to interference and noise. It's time tobe more precise about what these quantities are and how they differ.

Interference represents man-made signals. Telephone lines are subject to power-line interference (in the UnitedStates a distorted 60 Hz sinusoid). Cellular telephone channels are subject to adjacent-cell phone conversations usingthe same signal frequency. The problem with such interference is that it occupies the same frequency band as the desiredcommunication signal, and has a similar structure.

We use the notation $i(t)$ to represent interference. Because interference has man-madestructure, we can write an explicit expression for it that may contain some unknown aspects (how large it is, for example).

Noise signals have little structure and arise from both human and natural sources. Satellite channels are subjectto deep space noise arising from electromagnetic radiation pervasive in the galaxy. Thermal noise plagues all electronic circuits that contain resistors. Thus, in receiving small amplitude signals, receiveramplifiers will most certainly add noise as they boost the signal's amplitude. All channels are subject to noise, and weneed a way of describing such signals despite the fact we can't write a formula for the noise signal like we can forinterference. The most widely used noise model is white noise . It is defined entirely by its frequency-domain characteristics.

• White noise has constant power at all frequencies.
• At each frequency, the phase of the noise spectrum is totally uncertain: It can be any value in between $0$ and $2\pi$ , and its value at any frequency is unrelated to the phase atany other frequency.
• When noise signals arising from two different sources add, the resultant noise signal has a power equal to the sum of thecomponent powers.

Because of the emphasis here on frequency-domain power, we are led to define the power spectrum . Because of Parseval's Theorem , we define the power spectrum $P_s(f)$ of a non-noise signal $s(t)$ to be the magnitude-squared of its Fourier transform.

When we pass a signal through a linear, time-invariant system, theoutput's spectrum equals the product of the system's frequency response and the input's spectrum. Thus, the powerspectrum of the system's output is given by