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Unformatted text preview: Lecture 4: Basic System Properties Tie Liu September 6, 2011 1 Causality A system is causal if the output does not anticipate future values of the input, i.e., if the output at any time depends only on values of the input up to that time. • All real-time physical systems are causal, because time only moves forward. Effect occurs after cause. (Imagine if you own a non-causal system whose output depends on tomorrows stock price.) • Causality does not apply to spatially varying signals. (We can move both left and right, up and down.) • Causality does not apply to systems processing recorded data. Example (Causal or non-causal): • y ( t ) = x 2 ( t − 1) • y ( t ) = x ( t + 1) • y [ n ] = x [ − n ] • y [ n ] = 2 n +1 x 3 [ n − 1] Mathematically (in CT), consider a system x ( t ) → y ( t ). Let x 1 ( t ) and x 2 ( t ) be two input signals with corresponding output signals y 1 ( t ) and y 2 ( t ), respectively. Then, the system is causal if and only if x 1 ( t ) = x 2 ( t ) , ∀ t < t = ⇒ y 1 ( t ) = y 2 ( t ) , ∀ t < t 1 2 Time-invariance Informally, a system is time-invariant (TI) if its behavior does not depend on what time it is. Mathematically (in CT), a system x ( t ) → y ( t ) is TI if x ( t ) → y ( t ) =...
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- Fall '09
- Trigraph, LTI system theory, Nonlinear system