Processing Gain
A basic property of
noise signals is that they add
non-coherently. This means they sum on a
power basis instead of an
amplitude basis. Thus, for
example, if you add two separate realizations of a random process
together, the total energy rises by approximately 3
dB
(

). In contrast to this,
sinusoids and other
deterministic signals can add
coherently. For example, at the midpoint between two
loudspeakers putting out identical signals, a
sinusoidal signal is
dB louder than the signal out of each loudspeaker alone (
cf.
dB
for uncorrelated
noise).

Coherent addition of
sinusoids and noncoherent addition of noise can
be used to obtain
any desired signal to noise ratio in a
spectrum analysis of sinusoids in noise. Specifically, for each
doubling of the
periodogram block size in
Welch's method, the signal
to noise ratio (
SNR) increases by 3 dB (6 dB spectral amplitude
increase for all sinusoids, minus 3 dB increase for the noise
spectrum).
Consider a single
complex sinusoid in
white noise as
introduced in (
5.32):
 |
(7.39) |
where

is the complex amplitude. Then the length
DFT of the first block of

is
For simplicity, let

for some

. That is, suppose for
now that

is one of the DFT frequencies

,

.
Then
for

. Squaring the absolute value gives
re |
(7.40) |
Since

is zero mean, so is

for all

.
Therefore, the average over many length-

blocks will converge
to
 |
(7.41) |
where

denotes
time averaging which, for stationary
stochastic processes, is equivalent to taking the
expected value
(§
C.1.6).
The final term can be expanded as
since

because

is white noise.
In conclusion, we have derived that the average squared-magnitude DFT
of

samples of a sinusoid in white noise is given by
 |
(7.42) |
where

is the amplitude of the complex sinusoid, and

is the variance (mean square) of the noise. We see that the signal
to noise ratio is
zero in every bin but the

th, and in that
bin it is
 |
(7.43) |
In the time domain, the mean square for the signal is

while the
mean square for the noise is

. Thus, the DFT gives a
factor of
processing gain in the bin where the sinusoid
falls. Each doubling of the DFT length adds 3 dB to the within-bin
SNR. (Remember that we use

for
power ratios.)
Another way of viewing processing gain is to consider that the DFT
performs a
change of coordinates on the observations

such
that all of the
signal energy ``piles up'' in one coordinate

, while the noise energy remains uniformly distributed
among all of the coordinates.
A practical implication of the above discussion is that it is
meaningless to quote signal-to-noise ratio in the
frequency domain
without reporting the relevant
bandwidth. In the above example, the
SNR could be reported as

in band

.
The above analysis also makes clear the effect of
bandpass
filtering on the signal-to-noise ratio (SNR). For example, consider a
dc level

in white noise with variance

. Then the SNR
(mean-square level ratio) in the time domain is

.
Low-pass filtering at

cuts the noise energy in half but
leaves the dc component unaffected, thereby increasing the SNR by

dB. Each halving of the lowpass cut-off
frequency adds another 3 dB to the SNR. Since the signal is a dc
component (zero bandwidth), this process can be repeated indefinitely
to achieve any desired SNR. The narrower the
lowpass filter, the
higher the SNR. Similarly, for sinusoids, the narrower the
bandpass
filter centered on the sinusoid's frequency, the higher the SNR.
Next Section: The Panning ProblemPrevious Section: Filtered White Noise