# Signal-noise ratio

[Martin Taylor 970422 10:10]

Bruce Abbott (970421.1515 EST)]

>Richard Kennaway (970421.1610 BST) --

>>Bruce Abbott (970419.1740 EST):

>>Because you are more familiar with concepts like the S/N ratio than I, I
>>wonder if you could tell me what relationship exists between the S/N and the
>>proportions of variance accounted and unaccounted for. I have a feeling
>>that a correlation could be reexpressed as a signal-to-noise ratio, although
>>information about the sign of the correlation would vanish.

>SN power ratio in dB (call it SNdB) is 10*log-base-10(Var R/Var N).
>
>Therefore c-squared = 1/(1 + 1/SN), where SN = 10^(SNdB/10)
>
>Equivalently, SNdB = -10*log-base-10( 1/c^2 - 1 ).

There are a couple more relationships that could be useful. If a signal channel
has signal-noise power ratio P/N (Var R/Var N in the above) and bandwidth W,
it is capable of passing W*log2(1 + P/N) bits of information per second. In
T seconds, then, a signal of power P in noise of power N could pass
B = W*T*log2(1 + P/N) bits.

For the mathematically ideal observer, the index of discriminability (the
ability to distinguish signals into two distinct categories) called d'
(d-prime) was long ago devised. It is equal to sqrt(2E/No) where E is the
energy of the signal (conventionally to be distinguished from a signal of
zero energy) and No is the noise power per unit bandwidth. Crudely, a
signal can just be heard as being there if d' = 1 (so-called "threshold").
At d' = 2, it is quite distinct, and at d' = 3 it's "loud and clear."

The value of d' relates to the information conveyable by the signal. For
the mathematically ideal observer:
d'^2 = 2B*ln(2)

(Taylor, Lindsay and Forbes, Acta Psychologica, 27, 1967, 223-229)

A "mathematically ideal" observer is one that has the best possible
performance under the given conditions. It is not necessarily true that
anyone knows how to make a practical system that achieves this performance
in any specified circumstance. It's a limiting value. But people well
trained in listening (I speak of auditory psychophysics) typically are
able to act as if they were ideal observers of a signal 3 or 4 dB louder
than the ideal observer would need for the same performance. It's hard
to do much better, and this same 3-4 dB applies over a wide variety of
conditions and signal types.

Martin