## 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 Problem

**Previous Section:**

Filtered White Noise