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EE200_Weber_10-21

# EE200_Weber_10-21 - EE 200 Linear Time Invariant Systems...

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1 EE 200 Linear Time Invariant Systems Previously we discussed analyzing LTI systems using the state-space method (update function, A,B,C,D matrices, etc.) These methods can precisely define how a system works, but often doesn’t give us much insight as to what the system does. Frequency domain methods can show us more about how the system works. It gives a clearer picture as to what output we can expect to see for a given input. Many real-world systems are close enough to being LTI systems that we choose to model them as LTI in order to take advantage of the frequency domain tools.

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2 EE 200 Time Invariance For system that operate on time-based signals we can divide the systems and the signals they work on into two classes: continuous and discrete. Continuous Time System Continuous-time input signal Continuous-time output signal Discrete Time System Discrete-time input signal Discrete-time output signal
3 EE 200 Time Invariance A time-invariant system has a response that does not change with time. If the input to the system is delayed by amount τ , then the output from the system is only delayed by the same amount. For continuous time systems, D τ delays an input signal by amount τ but otherwise does not change it. t Reals, y(t) = x(t - τ ) D τ t t t 0 t 0 τ

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4 EE 200 Time Invariance A system S is time-invariant if D τ " # \$ Reals, S o D # = D # o S S D τ S " # , t , x S ( D # ( x ))( t ) = D # ( S ( x ))( t ) τ τ τ
5 EE 200 Time Invariance Time invariance also holds for discrete-time systems. The system D M can be defined as an M-sample delay n Integers, y(n) = x(n - M) A discrete-time system is time invariant if " M , x S ( D M ( x )) = D M ( S ( x ))

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6 EE 200 Time Invariance For systems that operate in a spatial domain, a similar property of shift invariance exists.
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