A Quadrature Signals Tutorial: Complex, But Not Complicated

Understanding the 'Phasing Method' of Single Sideband Demodulation

Complex Digital Signal Processing in Telecommunications

Introduction to Sound Processing

Introduction of C Programming for DSP Applications

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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.

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 (4.9):

For simplicity, let for some . That is, suppose for now that is one of the DFT frequencies , . Then

for . Squaring the absolute value gives

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

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 *band-pass
filtering* on signal to noise ratio. 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 low-pass 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.

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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