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Figure 2 shows some common signals and their
autocorrelations.
Figure 3 shows the cosine transforms of
the autocorrelations.
Cosine transform takes us from time to frequency and it also takes
us from frequency to time.
Thus, transform pairs in Figure 3
are sometimes more comprehensible
if you interchange time and frequency.
The various signals are given names in the figures,
and a description of each follows:
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autocor
Figure 2.
Common signals and one side of their autocorrelations.
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spectra
Figure 3.
Autocorrelations and their cosine transforms,
i.e., the (energy) spectra of the common signals.
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- cos
- The theoretical spectrum of a sinusoid is an impulse,
but the sinusoid was truncated (multiplied by a rectangle function).
The autocorrelation is a sinusoid under a triangle,
and its spectrum is a broadened impulse
(which can be shown to be a narrow sinc-squared function).
- sinc
- The sinc function is
.
Its autocorrelation is another sinc function, and its spectrum
is a rectangle function.
Here the rectangle is corrupted slightly by
``Gibbs sidelobes,''
which result from the time truncation of the original sinc.
- wide box
- A wide
rectangle function
has a wide triangle function for
an autocorrelation and a narrow sinc-squared spectrum.
- narrow box
- A narrow rectangle has a wide sinc-squared spectrum.
- twin
- Two pulses.
- 2 boxes
- Two separated narrow boxes have the spectrum of one of them,
but this spectrum is modulated (multiplied) by a sinusoidal function
of frequency, where the modulation frequency measures the
time separation of the narrow boxes.
(An oscillation seen in the frequency domain
is sometimes called a ``quefrency.'')
- comb
- Fine-toothed-comb
functions are like rectangle functions with a lower Nyquist frequency.
Coarse-toothed-comb functions have a spectrum which is a fine-toothed comb.
- exponential
- The autocorrelation of a transient exponential function
is a double-sided exponential function.
The spectrum (energy) is a Cauchy function,
.
The curious thing about the
Cauchy function
is that the amplitude spectrum
diminishes inversely with frequency to the first power;
hence, over an infinite frequency axis, the function has infinite integral.
The sharp edge at the onset of the transient exponential
has much high-frequency energy.
- Gauss
- The autocorrelation of a Gaussian
function is another Gaussian,
and the spectrum is also a Gaussian.
- random
- Random
numbers have an autocorrelation that is an impulse
surrounded by some short grass.
The spectrum is positive random numbers.
- smoothed random
- Smoothed random numbers are much the same as random numbers,
but their spectral bandwidth is limited.
Next: SETTING UP THE FAST
Up: CORRELATION AND SPECTRA
Previous: Two ways to compute
2013-01-06