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Chapt 9a - 9 Linear Systems with Random Inputs 9.1 9.2 9.3...

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9 Linear Systems with Random Inputs 9.1 Introduction 305 9.2 Overview of Linear Systems with Deterministic Inputs 305 9.3 Linear Systems with Continuous-Time Random Inputs 307 9.4 Linear Systems with Discrete-Time Random Inputs 313 9.5 Autoregressive Moving Average Process 316 9.6 Chapter Summary 323 9.7 Problems 323 9.1 Introduction In Chapter 8 the concept of random processes was introduced. Different param- eters that are associated with random processes were also discussed. These pa- rameters include autocorrelation function, autocovariance function, crosscorre- lation function, crosscovariance function, and power spectral density. The goal of this chapter is to determine the response or output of a linear system when the input is a random signal instead of a deterministic signal. To set the stage for the discussion, we first begin with a brief review of linear systems with deterministic inputs. Then we examine the response of linear systems to random inputs. 9.2 Overview of Linear Systems with Deterministic Inputs Consider a system with a deterministic input signal x ( t ) and a deterministic re- sponse y ( t ) . The system is usually represented either in terms of its impulse func- 305
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306 Chapter 9 Linear Systems with Random Inputs Figure 9.1 Time-Domain and Frequency-Domain Representation of Linear System tion (or impulse response ) h ( t ) or its system response H ( w ) , which is the Fourier transform of the impulse function. This is illustrated in Figure 9.1. The system response is also called the transfer function of the system. The system is defined to be linear if its response to a sum of inputs x k ( t ) , k = 1 , 2 ,..., K , is equal to the sum of the responses taken separately. Also, the system is said to be a time-invariant system if the form of the impulse response does not depend on the time the impulse is applied. For linear time-invariant systems, the response of the system to an input x ( t ) is the convolution of x ( t ) and h ( t ) . That is, y ( t ) = x ( t ) * h ( t ) = integraldisplay -∞ x ( τ ) h ( t - τ ) d τ where the last equation is called the convolution integral of x ( t ) and h ( t ) . If we define u = t - τ , we see that y ( t ) = integraldisplay -∞ x ( t - u ) h ( u ) du = integraldisplay -∞ h ( u ) x ( t - u ) du = h ( t ) * x ( t ) Thus, the convolution equation can be written in one of two forms: y ( t ) = integraldisplay -∞ x ( τ ) h ( t - τ ) d τ = integraldisplay -∞ h ( τ ) x ( t - τ ) d τ In the frequency-domain, we can compute the Fourier transform of y ( t ) as fol- lows: Y ( w ) = F [ y ( t ) ] = integraldisplay -∞ y ( t ) e - jwt dt = integraldisplay -∞ braceleftbiggintegraldisplay -∞ x ( τ ) h ( t - τ ) d τ bracerightbigg e - jwt dt
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9.3 Linear Systems with Continuous-Time Random Inputs 307 where F [ y ( t ) ] denotes the Fourier transform of y ( t ) . Interchanging the order of integration and since x ( τ ) does not depend on t , we obtain Y ( w ) = integraldisplay -∞ x ( τ ) braceleftbiggintegraldisplay -∞ h ( t - τ ) e - jwt dt bracerightbigg d τ But the inner integral is the Fourier transform of h ( t - τ ) , which is e - jw τ H ( w ) .
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