Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

8.11 BANDWIDTH

In Section 8.10, we described a bandlimited white-noise process where its PSD has been modified by a filter such that it is nonzero only in the frequency range . This ideal spectrum can be represented by the rectangle function:

(8.206)

whose autocorrelation is the sinc function in (8.194). Obviously not all PSDs have such a well-defined bandwidth. For example, Figure 8.19 shows the PSD of the exponential autocorrelation function RXX(τ) = exp(−α |τ |) given by

Figure 8.19 PSD for an exponential autocorrelation function. (a) 3-dB bandwidth. (b) Noise-equivalent bandwidth.

(8.207)

where α >0. For this type of PSD, we can define an effective bandwidth. This approach is often taken in filter design where the one-sided bandwidth B (positive frequencies) is defined such that the transfer function satisfies

(8.208)

If |H(f)|2 is expressed in dB, then the cutoff frequencies have a magnitude squared that is 3 dB below the maximum at f = 0. We can apply a similar definition ...