Signal-to-Noise Ratio

Introduction

Signal-to-noise ratio (SNR) is a figure-of-merit, widely used in many areas for different purposes:

as measure of quality for a signal
as measure of quality for devices: sound cards, measurement instruments etc.
in audio context
for baseband signals
for radio frequency signals
...

SNR describes loss of information. The underlying theory is discussed in many communications textbooks, for example Proakis.
Unfortunately, the textbook approach isn't directly applicable to most engineering problems.
Still, it provides key concepts that remain the same, no matter whether one is dealing with audio or radio frequency, quality of signals or equipment.

The purpose of this page is to shed some light on the underlying idea and give hints for an unambiguous SNR definition.

matched filtering

A key result from communications theory is the so-called "matched filter".
To make a long story short, the signal may be more “valuable” at some frequencies than at others:
A matched filter is "matched" to the signal, because it attenuates frequency regions with lower signal energy.
Frequency bands containing no signal energy at all are rejected completely.

For example, one might encounter an audio recording with a lot of high frequency tape hiss.
The logical thing to do is to turn down the treble button on the stereo: Even though some signal information is lost, the overall signal-to-noise ratio improves.
A matched filter works in a similar way.

Informally speaking:

“A matched filter hurts the noise more than it hurts the signal”

Now here's the catch: If it is possible to attenuate the noise more than the signal, doesn't this actually improve the SNR? Isn't that unphysical?
The answer is "yes, but...":
A matched filter is the best possible receiver one can build. At its output, it will give the highest possible SNR.
It is not unphysical, because any other receiver is “suboptimum” and has potential for SNR improvement by better filtering.

SNR and filter

The matched filter results in an optimum receiver that gives the best possible SNR. It is however not common or practical in many fields, for example for audio applications.
But if no filter at all is used, the resulting SNR depends on the bandwidth of the measurement instruments.
This approach should be avoided, or at least well understood.

A proper SNR definition should include a filter, designed for the following two aspects:

An appropriate amplitude response within the signal bandwidth (weighting), or flat
sufficient rejection of signal energy outside of the signal bandwidth

Even if no weighting for the inband response is desired, it is necessary to remove out-of-band noise.
For example, analog-to-digital converters may produce DC offset and 1/f noise at lowest frequencies, and increased noise at highest frequencies due to noise shaping in sigma-delta ADC.

SNR for communications signals

When dealing with baseband or radio frequency signals, accuracy of SNR measurements becomes critical. Bit error rates (BER) are very sensitive with regard to SNR, and one dB difference may change BER by a factor of 10.

Many modern communications systems use signals with a rectangular spectrum, for example OFDM, SC-FDMA or to some extent CDMA/WCDMA.
In those cases, one can read a good estimate of SNR by looking at the spectrum:

© Markus Nentwig 2007-2008
The content of this page is provided without any warranty and may not be reproduced without permission.

Comments? Questions?

Please send me a mail! mnentwig@elisanet.fi