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From: dvsarwate on 7 Jul 2010 16:34
Let us ignore ISI, receiver imperfections,
Doppler shifts, timing errors etc for the sake
of simplicity and suppose that the signal at
the receiver input is x(t) and has energy E.
The instantaneous power at time t is [x(t)]^2
and varies with t. The maximum value of
the instantaneous power is relevant for
circuit design but is often not of interest
to communications engineers who are
wanting to calculate error probabilities.

The received signal plus noise is passed
through a matched filter. The signal
output is the autocorrelation function R(t)
for x(t) or possibly a delayed version of the
R(t) for reasons of causality and realizability
of the matched filter etc. The ***sampling
instant*** is the location of the peak of
R(t), say R(0), where the signal sample
value is E. The error probability depends
*only* on the value of R(t) at t = 0, and *not*
upon the values of R(t) at other time instants.
Thus, a major performance measure is
*not dependent* on the off-peak values of
R(t). In other words, the energy or the
average power in the matched filter output
(both of which are dependent on the values
of R(t) for ALL time instants t) do not affect
the error probability which is determined by
the maximum value of R(t). Are the energy
or the average power of the matched filter
quantities that are totally irrelevant? Of
course not, the energy consumption of the
receiver depends on the energy in R(t),
since this voltage appears across the output
impedance of the matched filter, etc.

In summary, the energy or average signal
power of the matched filter output do not
affect the error probability and do not occur
in many definitions of SNR. But, to parody
Rudyard Kipling,

"There are nine and sixty ways
Of constructing definitions of SNR
And every single one of them is right!"

and thus SNR, defined as the ratio of
equivalent signal bandwidth to equivalent
noise bandwidth, as measured at the output
of the matched filter is undoubtedly useful
for something, but it might not give the right
answer if you plug it into an expression such
as Q(sqrt(SNR)) for the error probability.

--Dilip Sarwate


From: Steve Pope on 7 Jul 2010 16:51
dvsarwate <dvsarwate(a)gmail.com> wrote:

>Let us ignore ISI, receiver imperfections,
>Doppler shifts, timing errors etc for the sake
>of simplicity and suppose that the signal at
>the receiver input is x(t) and has energy E.
>The instantaneous power at time t is [x(t)]^2
>and varies with t. The maximum value of
>the instantaneous power is relevant for
>circuit design but is often not of interest
>to communications engineers who are
>wanting to calculate error probabilities.

>The received signal plus noise is passed
>through a matched filter. The signal
>output is the autocorrelation function R(t)
>for x(t) or possibly a delayed version of the
>R(t) for reasons of causality and realizability
>of the matched filter etc. The ***sampling
>instant*** is the location of the peak of
>R(t), say R(0), where the signal sample
>value is E. The error probability depends
>*only* on the value of R(t) at t = 0, and *not*
>upon the values of R(t) at other time instants.
>Thus, a major performance measure is
>*not dependent* on the off-peak values of
>R(t). In other words, the energy or the
>average power in the matched filter output
>(both of which are dependent on the values
>of R(t) for ALL time instants t) do not affect
>the error probability which is determined by
>the maximum value of R(t). Are the energy
>or the average power of the matched filter
>quantities that are totally irrelevant? Of
>course not, the energy consumption of the
>receiver depends on the energy in R(t),
>since this voltage appears across the output
>impedance of the matched filter, etc.

>In summary, the energy or average signal
>power of the matched filter output do not
>affect the error probability and do not occur
>in many definitions of SNR. [snip]

>and thus SNR, defined as the ratio of
>equivalent signal bandwidth to equivalent
>noise bandwidth, as measured at the output
>of the matched filter is undoubtedly useful
>for something, but it might not give the right
>answer if you plug it into an expression such
>as Q(sqrt(SNR)) for the error probability.

Thank you for writing this.

I, personally, in the above scenario, would
consider the SNR at the input of the demodulator
to be the SNR of the appropriately sampled signal
following the matched filter. So, yes, it is
not the same as any SNR you could define on the
un-sampled signal, but it is still dimensionally and
conceptually an SNR, being a power ratio equal to a signal
variance divided by a noise variance, and you
can usefully plug it into expressions like Q(sqrt(SNR)).


S.
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