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# ch09 - 9 9.1 Linear Systems Introduction We consider linear...

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9 Linear Systems 9.1 Introduction We consider linear systems which have stochastic processes as their input and output. Definition 9.1.1: A system is linear if each input λ 1 x 1 ( t )+ λ 2 x 2 ( t ) gives rise to output λ 1 y 1 ( t ) + λ 2 y 2 ( t ) where y 1 ( t ) and y 2 ( t ) are output of x 1 ( t ) and x 2 ( t ). Definition 9.1.2: A linear system is time-invariant if and only if a delay in the input x ( t - τ ) produces the same delay in the output y ( t - τ ). 9.2 Linear Systems in the time domain A time-invariant linear system may generally be written in the form y ( t ) = integraldisplay -∞ h ( u ) x ( t - u ) (9.1) in continuous time and y t = summationdisplay k = -∞ h k x t - k (9.2) in discrete time. Definition 9.2.1: The function h ( u ) or h k is called the impulse response function. Often we assume that h ( u ) = 0 or h k = 0 for u < 0 or k < 0 respectively. 9.2.1 Some examples The impulse response of a moving average filter y t = ( x t - 1 + x t + x t +1 ) / 3 is h k = braceleftBigg 1 3 , k = - 1 , 0 , 1 0 , otherwise.

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ch09 - 9 9.1 Linear Systems Introduction We consider linear...

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