# Optimum Filter Signal Detection for Non AWGN Channels

I have been reading this question and it confirms that the matched filter is the maximum-likelihood receiver in the presence of additive white Gaussian noise. So in the AWGN channel it maximizes the signal-to-noise ratio at the decision instant at the end of each symbol.

However,according to Wikipedia

AWGN channel is not a good model for most terrestrial links because of multipath, terrain blocking, interference, etc.

I would like to know:

1. If the transmission channel could not be considered as affected by AWGN. Which filter would yield the optimum SNR? I'm particularly interested in the noise present in sound channel (sound channel through air not underwater).
2. Is it possible to "adapt" matched filter to non AWGN channels?
• you typically do your best to reverse your channel so that it looks AWGN. – Marcus Müller Mar 7 '18 at 17:17
• What kind of sound channel – Stanley Pawlukiewicz Mar 7 '18 at 17:29
• @MarcusMüller I'm sorry what do you mean reverse? – VMMF Mar 7 '18 at 17:30
• @StanleyPawlukiewicz sound channel through air not underwater – VMMF Mar 7 '18 at 17:31
• Note that there is always AWGN, since at a minimum there is thermal noise in the receiver. The matched filter is still used in other channels, but you add an equalizer to try to revert the distortion introduced by the channel (as @MarcusMüller mentioned). – MBaz Mar 7 '18 at 17:49

There are two different phrases that you are mixing up. Additive White Gaussian noise is something that is present in mathematical models of communication systems, even those classified as communication systems operating over acoustic channels or fading channels etc because this noise is not actually something that the channel adds, but rather something that models the thermal noise in the receiver, that is, the electronic components that actually do the stuff that dsp.SE folks write equations about. The phrase Additive White Gaussian noise channel describes a channel in which the only impairment is that the received signal gets white Gaussian noise added to it: what is transmitted is exactly what is received with the exception that white Gaussian noise has been added to the transmitted signal; no filtering of any kind, no multipath, no fading, no reverberation, no psychoacoustic addition of harmonic overtones where none existed before, no intersymbol interference, no jamming signals, no adjacent-channel interference, no multiple-access interference, nothing. All mathematical models of channels should include Gaussian noise as a stand-in for thermal noise, though in several instances, the thermal noise has little effect on the performance because it is so weak compared to the other channel impairments that it can be ignored for simplicity of analysis and exposition.

Turning to the questions asked, Yes, it is possible to design matched filters for channels that are not described as AWGN channels but the criteria used can be different and the analyses can be different too. As described in this answer a matched filter can be viewed as an LTI system designed to maximize the output signal value at a specific time instant but that might not the optimum thing to do for other noise models or other channel impairments. For such channels, a nonlinear filter which produces a smaller output but provides better noise suppression or better removal of channel impairments might be preferable to the canonical matched filter that works perfectly with AWGN but is suspect in other cases.

• Yes! The answer you mentioned is exactly the one I referenced in my question... which is yours by the way :-) . So from your answer I understand: 1) AWGN should always be considered even if the channel is ideal because there's always thermal noise in the electronics. 2) An AWGN channel adds only white Gaussian noise itself, independently of the AWGN also added by the electronics. 3) Either for 1) or 2) the matched filter is the optimum filter. 4) What do you mean nonlinear filter? Nonlinear-phase? – VMMF Mar 8 '18 at 20:34
• @VMMF In your 2), the actual channel adds no noise (or very little noise) compared to the thermal noise. in the receiver. For example, the deep-space channel adds thermal noise with noise temperature of $3\deg K$ compared to receiver thermal noise of close to $300\deg K$ noise temperature. In your 4), I mean a nonlinear circuit such as a decision-feedback equalizer or a square-law detector, a truly nonlinear system rather than a nonlinear phase but nonetheless LTI system. – Dilip Sarwate Mar 8 '18 at 22:15

AWGN is a specific noise model that is to varying physical circumstances, of varying utility.

Not AWGN is not a specific noise model, but you can look at it as detection not dependent on Gaussian assumptions. There is no one kind of optimal detector for every kind of non Gaussian noise.

A good book is