563489578934

436
Random Processes and Spectral Analysis Chap. 6
where N 0 is a positive constant.
The autocorrelation function for the white-noise process is obtained by taking the
inverse Fourier transform of Eq. (6–71). The result is
R x (t) = N 0
(6–72a)
2 d(t)
For example, the thermal-noise process described in Sec. 8–6 can be considered to be a whitenoise
process over the operating band where
N 0 = kT
(6–72b)
Thermal noise also happens to have a Gaussian distribution. Of course, it is also possible to
have white noise with other distributions.
Measurement of PSD
The PSD may be measured by using analog or digital techniques. In either case, the measurement
can only approximate the true PSD, because the measurement is carried out over a finite
time interval instead of the infinite interval specified Eq. (6–42).
Analog Techniques. Analog measurement techniques consist of using either a bank
of parallel narrowband bandpass filters with contiguous bandpass characteristics or using a
single bandpass filter with a center frequency that is tunable. In the case of the filter bank,
the waveform is fed simultaneously into the inputs of all the filters, and the power at the output
of each filter is evaluated. The output powers are divided (scaled) by the effective bandwidth
of the corresponding filter so that an approximation for the PSD is obtained. That is,
the PSD is evaluated for the frequency points corresponding to the center frequencies of the
filters. Spectrum analyzers with this parallel type of analog processing are usually designed
to cover the audio range of the spectrum, where it is economically feasible to build a bank of
bandpass filters.
RF spectrum analyzers are usually built by using a single narrowband IF bandpass filter
that is fed from the output of a mixer (up- or down-converter) circuit. The local oscillator
(LO) of the mixer is swept slowly across an appropriate frequency band so that the RF spectrum
analyzer is equivalent to a tunable narrowband bandpass filter wherein the center
frequency is swept across the desired spectral range. Once again, the PSD is obtained by evaluating
the scaled power output of the narrowband filter as a function of the sweep frequency.
Numerical Computation of the PSD. The PSD is evaluated numerically in spectrum
analyzers that use digital signal processing. One approximation for the PSD is
T (f) = |X T(f)| 2
(6–73a)
T
where the subscript T indicates that the approximation is obtained by viewing x(t) over a T-s
interval. T is called the observation interval or observation length. Of course, Eq. (6–73a) is an
approximation to the true PSD, as defined in Eq. (6–42), because T is finite and because only a
sample of the ensemble is used. In more sophisticated spectrum analyzers, T ( f ) is evaluated
for several x(t) records, and the average value of T ( f ) at each frequency is used to approximate