EE200_Weber_12-4

EE200_Weber_12-4 - EE 200 Chapter 5 - Linear Systems A...

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1 EE 200 Chapter 5 - Linear Systems A state machine is define by a five-tuple M = (States, Inputs, Outputs, update, initialState) N-tuple M-tuple K-tuple For a input sequence of M-tuples… The state update equation generates new N-tuples n Integers, n 0, s(n+1) = nextState(s(n), x(n)) The output equation generates output K-tuples n Integers, n 0, y(n) = output(s(n), x(n)) The state update equation and output equation together are the state-space model of the system.
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2 EE 200 Chapter 5 - Linear Systems A system is time invariant if a time shift in the input signal results in a corresponding time shift in the output signal. If x(t) results in y(t), then x(t - t 0 ) results in y(t - t 0 ) A linear function possesses the properties: Homogeneity : f( α u) = α f(u) Additivity : f(u 1 + u 2 ) = f(u 1 ) + f(u 2 ) If the input to a linear function is a weighted sum of signals, then the output is the weighted sum of the responses of the individual signals.
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3 EE 200 Chapter 5 - Linear Systems The state space model of a system is given by the state update equation and the output equation. s(n+1) = nextState(s(n), x(n)) y(n) = output(s(n), x(n)) If the nextState and output functions are both linear, and the initial state is an N-tuple of zeros, then it is a linear system . If a linear system is also time invariant, then it is a linear time-invariant system or LTI .
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4 EE 200 Chapter 5 - Linear Systems For an LTI system, the nextState and output functions can be represented by matrices. s(n+1) = nextState(s(n), x(n)) = As(n) + Bx(n) y(n) = output(s(n), x(n)) = Cs(n) + Dx(n) This is the [A,B,C,D] representation of an LTI system. For N state variables, M inputs, and K outputs A = N × N B = N × M C = K × N D = K × M
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5 EE 200 Chapter 5 - Linear Systems For single-input, single-output, one-dimensional systems, (N=M=K=1), the state response and the output response can be written in a form that shows the response as a function of the input sequence Both responses are the sum of a zero-input response and a zero state response . s ( n ) = a n s (0) + a n " 1 " m bx ( m ) m = 0 n " 1 # y ( n ) = ca n s (0) + ca n " 1 " m bx ( m ) m = 0 n " 1 # $ % ( ) + dx ( n )
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6 EE 200 Chapter 5 - Linear Systems The zero-state output response with an input of a Kronecker delta function give the impulse response of the system. h ( n ) = ca n " 1 " m b # ( m ) m = 0 n " 1 $ % ( ) * + d ( n ) = 0 if n < 0 d if n = 0 ca n " 1 b if n + 1 % , ,
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7 EE 200 Chapter 5 - Linear Systems The expression for the impulse response can be put in the equation for the zero-state output response, y(n), to remove the a, b, c and d terms. The zero-state output response is then given by The impulse response is ALL the information we need in order to determine the response of the system to ANY input, assuming an initial state of zero. This is called a convolution sum and is written as " n # 0 y ( n ) = h ( n $ m ) x ( m ) m = 0 n % = h ( n $ m ) x ( m ) m = % y = h " x = x " h
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This note was uploaded on 12/06/2009 for the course EE 200 taught by Professor Zadeh during the Fall '08 term at USC.

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EE200_Weber_12-4 - EE 200 Chapter 5 - Linear Systems A...

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