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102_1_Spring09-102-MT-Q1-Sols

# 102_1_Spring09-102-MT-Q1-Sols - Put FIRST Letter of LAST...

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QUETION 1 (40%) (Parts (i), (ii), and (iii) are standing-alone) (i) (10%) Let h ( t, τ ) be the IRF of a L system S . If h ( t, τ ) = h ( t τ ) , −∞ ≤ t, τ ≤ ∞ , then S is TI. True or False? Why? Conversely, if S is TI then h ( t, τ ) = h ( t τ ) , −∞ ≤ t, τ ≤ ∞ . True or False? Why? SOLS. Given that S is L and h ( t, τ ) = h ( t τ ) , −∞ ≤ t, τ ≤ ∞ , Then by BT: x ( t ) [ S ] y ( t ) = Z −∞ h ( t τ ) x ( τ ) d τ From which we find y ( t A ) = Z −∞ h ( t A τ ) x ( τ ) d τ ( ) Then consider x ( t A ) [ S ] = Z −∞ h ( t τ ) x ( τ A ) d τ := z ( t ) (set t τ = σ ) z ( t ) = Z −∞ h ( t A σ ) x ( σ ) d σ , ( ) Then look at ( ) and (
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102_1_Spring09-102-MT-Q1-Sols - Put FIRST Letter of LAST...

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