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Unformatted text preview: Sequences with Rational Power Spectra
Consider the class of zeromean, ﬁnite variance, WSS, random sequences with
rational PSD. These random signals are in general obtained by the processing of
a zeromean, WSS, white, random signal through a LTI system and consequently
the general form for the PSD of these signals is:
2
Pxx (z ) = σo B (z )B ∗
A(z )A∗ 1
z∗
1
z∗ If the PSD of the signal contains just P poles, i.e, there are no zeroes or in other
words B (z ) = K then the PSD takes the form:
Pxx (z ) = 2
Kσo
∗
A(z )A 1
z∗ These processes are termed as autoregressive or AR(p) random signals. It is not
diﬃcult to show that the ACF of the random sequence x[n] in this case takes
the form:
p k  Rxx [k ] = ci ai , ai  < 1
i Note that Gaussian AR random signals are ergodic in the general sense because:
∞ Rxx [k ] < ∞.
k=−∞ If the PSD of the random signal has just q zeroes in it, i.e, A(z ) = 1 then the
PSD takes the form:
1
2
Pxx (z ) = σo B (z )B ∗
.
z∗
These processes are termed as moving average or MA(q ) random signals and
the ACF of the random sequence can be shown to satisfy:
Rxx [k ] = 0, k  > q Signals with both poles and zeroes in their PSD are referred to as autoregressive
moving average or ARMA(p, q ) random signals. 2 AR PROCESS, p = 1, L = 2000, σo = 0.5
SIGNAL 0.5 0 −0.5 SAMPLE REALIZATION OF 1 AR(1) 1.5 −1 −1.5
0 500 1000
TIME SAMPLES 1500 2000 (a) 2
AR PROCESS, p = 1, L = 2000, σo = 0.5 1.2
1 Rxx[k] 0.8
0.6
0.4
0.2
0
−0.2
−20 −15 −10 −5
0
5
10
ACF LAG VARIABLE (k) 15 20 (b) 2 AR PROCESS, p = 1, L = 2000, σo = 0.5
1.4 ACT
THEO 1.2
1 Pxx(ej ω) 0.8
0.6
0.4
0.2
0
0 0.1 0.2
0.3
NORMALIZED FREQUENCY 0.4 0.5 (c) Figure 1: AR(1) example: (a) sample realization of a AR(1) process, (b) sample
ACF of the process, and (c) estimated PSD of the process. ...
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 Fall '11
 GraceGraham
 spectral density, Autoregressive moving average model, Autoregressive model, Random sequence, σo

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