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Common signals

Figure 11 shows some common signals and their autocorrelations. Figure 12 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 12 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:

Figure 11
Common signals and one side of their autocorrelations.

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Figure 12
Autocorrelations and their cosine transforms, i.e., the (energy) spectra of the common signals.

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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).
The sinc function is $\sin(\omega_0 t)/(\omega_0 t)$.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.
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.'')
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.
The autocorrelation of a transient exponential function is a double-sided exponential function.

The spectrum (energy) is a Cauchy function, $1/(\omega^2+\omega_0^2)$.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.

The autocorrelation of a Gaussian function is another Gaussian, and the spectrum is also a Gaussian.
Random numbers have an autocorrelation that is an impulse surrounded by some short grass. The spectrum is positive random numbers. For more about random signals, see chapter [*].
smoothed random
Smoothed random numbers are much the same as random numbers, but their spectral bandwidth is limited.

next up previous print clean
Next: Spectra of complex-valued signals Up: CORRELATION AND SPECTRA Previous: Two ways to compute
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