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ECSE304_3_2010

# ECSE304_3_2010 - a[1.27a y(0 = x(2 x(2 output at t0 has...

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a) [1.27a] ) 2 ( ) 2 ( ) 0 ( x x y + = Î output at t 0 has dependence on time other than Î 0 t t = not causal and, hence, not memoryless Let ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) ( ) ( ) ( 0 0 2 2 2 0 2 t t x t t x t x t x t y t t x t x + = + = = How, ) 2 ( ) 2 ( ) ( 0 0 0 t t x t t x t t y + + = Î not time-invariant Let ) ( ) ( ) ( 2 1 3 t x t x t x β α + = , and using as input, ) ( 3 t x ) ( ) ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) ( 2 1 2 2 1 1 3 3 3 t y t y t x t x t x t x t x t x t y β α β β α α + = + + + = + = Î linear If an input , i.e. the input is bounded, the output < M t x ) ( M t y 2 ) ( , i.e. the output is also bounded. Î BIBO stable In a similar fashion it is possible to show for the rest: [1.27b] ) ( ) 3 cos( ) ( 0 0 0 t x t t y = Î output at t 0 depends only on input at t 0 (cosine is not an input) Î memoryless and hence causal as well [ ] ) ( ) ( ) ( ) 3 cos( ) ( ) 3 cos( ) ( ) ( ) 3 cos( ) ( 2 1 2 1 2 1 3 t y t y t x t t x t t x t x t t y β α β α α + = + = + = β Î linear 1 ) 3 cos( t and hence for < M t x ) ( , M t y ) ( Î BIBO stable ) ( ) ( 3 cos( ) ( ) ( ) 3 cos( ) ( ) 3 cos( ) ( ) ( ) ( 0 0 0 0 2 2 0 2 t t x t t t t y t t x t t x t t y t t x t x = = = = Î output at t=t 0 does not correspond to a output due to an input time-shifted by t 0 Î not time- invariant [1.27c] τ τ d x t y t = 0 ) ( ) ( 0 Î output at t 0 depends on all values of input from - to t 0 , but not on future values of input Î causal but not memoryless

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[ ] [ ] [ ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 1 2 1 2 1 3 t y t y d x d x d x x t y t t t β α τ τ β τ τ α τ τ β τ α + = + = + = ] Î linear Let , i.e., input is bounded, but corresponding output 1 | ) ( | hence and ) cos( ) ( = t x t t x = ) ( t y Î not BIBO stable = = = = 0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 0 0 2 2 0 2 t t t t t y d x t t y d t x d x t y t t x t x τ τ τ τ τ τ Î not time-invariant [1.27a] Æ Linear, stable [1.27b] Æ Memoryless, linear, causal, stable [1.27c] Æ Linear b) This problem can be solved by two methods: First is to use the reasoning that the ROC of an inter-connect of two systems must include the intersection of the two individual ROC’s, i.e. 2 1 ROC ROC ROC = Hence, if both the ROC 1 and ROC 2 include the ω j axis – as System 1 and System 2 both stable – the intersection must also include the ω j axis making the cascade stable as well.
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