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Unformatted text preview: 9.4 Linear Systems with DiscreteTime Random Inputs 315 = ∞ summationdisplay j =−∞ ∞ summationdisplay l =−∞ h [ j ] h [ l ] R XX [ n − j , n + k − l ] If X [ n ] is a widesense stationary discrete sequence, then we obtain R YY [ n , n + k ] = ∞ summationdisplay j =−∞ ∞ summationdisplay l =−∞ h [ j ] h [ l ] R XX [ k + j − l ] = R YY [ k ] In a manner similar to the continuoustime case, it can be shown that the power spectral density of Y [ n ] is given by S YY (Omega1) =  H (Omega1)  2 S XX (Omega1) Example 9.5 The impulse response of a discrete linear timeinvariant system is given by h [ n ] = a n u [ n ] where  α  < 1, and u [ n ] is the unit step sequence defined by u [ n ] = braceleftBig 1 n ≥ n < If the input sequence X [ n ] is a discretetime white noise with power spectral den sity N / 2, find the power spectral density of the output Y [ n ] . Solution The system response is given by H (Omega1) = ∞ summationdisplay n =−∞ h [ n ] e − j Omega1 n = ∞ summationdisplay n = a n e − j Omega1 n = 1 1 − ae − j Omega1 Since S XX (Omega1) = N / 2, we have that S YY (Omega1) =  H (Omega1)  2 S XX (Omega1) = H ∗ (Omega1) H (Omega1) S XX (Omega1) = parenleftbigg 1 1 − ae j Omega1 parenrightbiggparenleftbigg 1 1 − ae − j Omega1 parenrightbigg N 2 = N 2 { 1 − ae − j Omega1 − ae j Omega1 + a 2 } = N 2 { 1 − a [ e − j Omega1 + e j Omega1 ] + a 2 } = N 2 { 1 − 2 a ( [ e − j Omega1 + e j Omega1 ] / 2 ) + a 2 } = N 2 { 1 − 2 a cos (Omega1) + a 2 } trianglesolid 316 Chapter 9 Linear Systems with Random Inputs 9.5 Autoregressive Moving Average Process The autoregressive moving average (ARMA) process is frequently used in time series analysis. It consists of two parts: the moving average process and the au toregressive process. Let { W [ n ] , n ≥ } be a widesense stationary random input sequence with zeromean and variance σ 2 W . In general the W [ n ] are assumed to be uncorrelated. An example of such a sequence is noise. The different processes are defined as follows. 9.5.1 Moving Average Process A moving average process of order q is a process whose current value Y [ n ] de pends linearly on the q past values of the random input process. Thus, given a set of constants β , β 1 ,..., β q , the output process defined by Y [ n ] = β W [ n ] + β 1 W [ n − 1 ] + β 2 W [ n − 2 ] + ··· + β q W [ n − q ] = q summationdisplay k = β k W [ n − k ] n ≥ is called a moving average process of order q , MA( q ). The moving average process is a special case of the purely feedforward system called the finite im pulse response (FIR) system with input X [ n ] that has nonzero mean. The general structure of FIR systems is shown in Figure 9.4, where D indicates a unit delay....
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This note was uploaded on 01/05/2010 for the course STAT 350 taught by Professor Carlton during the Fall '07 term at Cal Poly.
 Fall '07
 Carlton

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