Figure 2

Figure 3

**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.

12/26/2000