ratpsd - Sequences with Rational Power Spectra Consider the...

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Unformatted text preview: Sequences with Rational Power Spectra Consider the class of zero-mean, finite variance, WSS, random sequences with rational PSD. These random signals are in general obtained by the processing of a zero-mean, 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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ratpsd - Sequences with Rational Power Spectra Consider the...

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