# Evaluate the co-, quadrature, cross-amplitude, phase and coherency spectra.

Assignment + Section9.3.4(which will be useful in some question)

8.2 Define the cross-correlation function of a bivariate stationary process and show that for all

Two first-order moving average processes

are formed from a purely random process , which has mean zero and variance . Find the cross-covariance and cross-correlation functions of the bivariate process and hence show that the cross-spectrum is given by

Evaluate the co-, quadrature, cross-amplitude, phase and coherency spectra.

9.2 Find the impulse response function, the step response function, the frequency response function, the gain and the phase shift for the following linear systems (or filters):

(a)

(b)

(c)

(d)

where in each case t is integer valued. Plot the gain and phase shift for filters (a) and (c). Which of the filters are low-pass and which high-pass?

If filters (a) and (b) are joined in series, find the frequency response function of the combined filter.

9.4 Consider the AR(1) process, given by , where denotes a purely random process with zero mean and constant variance and |a|<1 for stationarity. If we regard the model equation as a linear system with denoting the input and denot ing the output, find the frequency response function of the system. Hence find the power spectrum of , knowing the spectrum of the ‘output’ is constant—see Section 9.3.4 if you get stuck.

Now consider the one-parameter second-order AR process

where the lagged value of X is two steps before. Show that the process is second-order stationary if |a|<1 (as for the AR(1) process), and find theautocovariance and autocorrelation functions.

Show that the power spectral density function of the process is given by

using two different methods: (a) by transforming the autocovariance function; (b) by using the approach of Section 9.3.4.

Suppose now that {} is any stationary process with power spectrum . when Then is the power spectrum of {} as defined by the above AR(1) model?

9.6 Show that the power spectral density function of the ARMA(1, 1) process

is given by for, using the approach of Section 9.3.4. It may help to let (This power spectrum may be shown to be equivalent to the normalized spectrum in Exercise 6.7 after some algebra.) More generally, for the MA process show that the frequency response function of the filteris , so that the spectrum of is given by for . Hence show that the spectrum of the general ARMA process is given by . Check this result on the above ARMA(1, 1) process.

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