# Additive White Gaussian Noise

A basic and generally accepted model for thermal noise in communication channels, is the set of assumptions that
• the noise is additive, i.e., the received signal equals the transmit signal plus some noise, where the noise is statisticaly independent of the signal.
• the noise is white, i.e, the power spectral density is flat, so the autocorrelation of the noise in time domain is zero for any non-zero time offset.
• the noise samples have a Gaussian distribution.
Mostly it is also assumed that the channel is Linear and Time Invariant. The most basic results further asume that it is also frequency non-selective.

## Optimal signal detection in AWGN LTI channel

The theory for signal transmission over AWGN LTI channels is very well developed and covered in many excellent text books. Many fundamental theorems in signal detection theory have been deveoped during World War II, to improve and automate the radar detection of enemy airplanes and ships. The theory of the matched filter receiver is of particular interest. The concept was introduced by D.O. North with the RCA labs in Princeton, in 1943.

 Figure: possible implementation of a matched filter receiver. The signal is multiplied by a locally stored reference copy and integrated over time (correlation).

The matched filter correlates the incoming signal with a locally stored reference copy of the transmit waveform. The matched filter maximizes the signal-to-noise ratio for a known signal. It can be shown to be the optimal detector if

• the channel produces Additive White Gaussian Noise (AWGN),
• the channel is linear and time-invariant (LTI), and
• an exact time reference is available, the signal amplitude as a function of time is precisely known.
 3 minute video clip from cource on radio communication, covering the matched filter princle (mov)

### Whitened Matched Filter

 non-white noise If the noise is non-white, the matched filter can still be applied. In this case, one can pre-filter the incoming signal, to make the noise component white. This is called a whitening filter. Evidently, this also filters the wanted signal. Therefore, the filtered incoming signal is not "matched to" (i.e., correlated with) the reference transmit signal, but with a reference signal that is fed through the a filter that is identical to the whitening filter. Thus, two copies of the whitening filter are needed. It can be shown that one can build a detector that mathematically is equivalent, but only uses one filter.

### Matched Filter for Wireless Channels

 dispersivechannel If the channel is dispersive, the matched filter concept can still be used, but one must multiply the incoming signal with a locally generated copy of the expected waveform after transmission over the channel. That is, the receiver must estimate the channel impulse response and apply this to the reference signal waveform. The incoming signal is correlated with a reference waveform, which is dispersed in the same manner as the channel disperses the radio signal.
A complication is that such dispersion causes intersymbol interference. Theorectically, it is no longer optimum to detect the received symbols one by one. The "maximum likelihood" (ML) receiver correlates the incoming sequence with dispersed sequences of potentially transmitted waveforms, containing multiple successive bits.

In the special case that the received signal is a direct sequence CDMA signal, this leads to the concept of a rake receiver.