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Sec. 6–3 DC and RMS Values for Ergodic Random Processes 437
the ensemble average required for a true x ( f ) of Eq. (6–42). In evaluating T ( f ), the DFT is
generally used to approximate X T ( f ). This brings the pitfalls of DFT analysis into play, as
described in Sec. 2–8.
It should be stressed that Eq. (6–73a) is an approximation or estimate of the PSD. This
estimate is called a periodogram, because it was used historically to search for periodicities in
data records that appear as delta functions in the PSD. [Delta functions are relatively easy to
find in the PSD, and thus, the periodicities in x(t) are easily determined.] It is desirable that
the estimate have an ensemble average that gives the true PSD. If this is the case, the estimator
is said to be unbiased. We can easily check Eq. (6–73a) to see if it is unbiased. We have
and, using Eq. (6–50a) for T finite, we obtain
T (f) = c |X T(f)| 2
d
T
T (f) = cR x (t) a t T bd
and then, referring to Table 2–2,
sin pfT
2
T (f) = T x (f) * a (6–73b)
pfT b
Because T (f) Z x (f), the periodogram is a biased estimate. The bias is caused by the
triangular window function ∂ (tT) that arises from the truncation of x(t) to a T-second
interval. Using Sec. A–8, where a = pT, we see that lim T : q [ T (f)] = x (f), so that the
periodogram becomes unbiased as T → q. Consequently, the periodogram is said to be
asymptotically unbiased.
In addition to being unbiased, it is desirable that an estimator be consistent. This means
that the variance of the estimator should become small as T → q. In this regard, it can be
shown that Eq. (6–73a) gives an inconsistent estimate of the PSD when x(t) is Gaussian
[Bendat and Piersol, 1971].
Of course, window functions other than the triangle can be used to obtain different PSD
estimators [Blackman and Tukey, 1958; Jenkins and Watts, 1968]. More modern techniques
assume a mathematical model (i.e., form) for the autocorrelation function and estimate parameters
for this model. The model is then checked to see if it is consistent with the data. Examples are the
moving average (MA), autoregressive (AR), and autoregressive-moving average (ARMA) models
[Kay, 1986; Kay and Marple, 1981; Marple, 1986; Scharf, 1991; Shanmugan and Breipohl, 1988].
Spectrum analyzers that use microprocessor-based circuits often utilize numerical techniques
to evaluate the PSD. These instruments can evaluate the PSD only for relatively
low-frequency waveforms, say, over the audio or ultrasonic frequency range, since real-time
digital signal-processing circuits cannot be built to process signals at RF rates.
6–3 DC AND RMS VALUES FOR ERGODIC RANDOM PROCESSES
In Chapter 2, the DC value, the RMS value, and the average power were defined in terms
of time average operations. For ergodic processes, the time averages are equivalent to ensemble
averages. Thus, the DC value, the RMS value, and the average power (which are all