### SIGNAL FILTERING AND FREQUENCY DOMAIN NOISE BASIC INFORMATION

Effect of Filtering on Signals
As we have seen, many signals have a bandwidth that is theoretically infinite. Limiting the frequency response of a channel removes some of the frequency components and causes the time-domain representation to be distorted.

An uneven frequency response will emphasize some components at the expense of others, again causing distortion. Nonlinear phase shift will also affect the time-domain representation.

For instance, shifting the phase angles of some of the frequency components in the square-wave representation changed the signal to something other than a square wave. However, while an infinite bandwidth may theoretically be required, for practical purposes quite a good representation of a square wave can be obtained with a band-limited signal.

In general, the wider the bandwidth, the better, but acceptable results can be obtained with a band-limited signal. This is welcome news, because practical systems always have finite bandwidth.

Noise in the Frequency Domain
Noise power is proportional to bandwidth. That implies that there is equal noise power in each hertz of bandwidth. Sometimes this kind of noise is called white noise, since it contains all frequencies just as white light contains all colors.

In fact, we can talk about a noise power density in watts per hertz of bandwidth. The equation for this is very simply derived. We start with Equation (1.3):

PN = kTB

This gives the total noise power in bandwidth, B. To find the power per hertz, we just divide by the bandwidth to get an even simpler equation:

N0 = kT (1.10)

where

N0 = noise power density in watts per hertz
k = Boltzmann’s constant, 1.38 × 10−23 joules/kelvin (J/K)
T = temperature in kelvins