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# lect3 - EL 625 Lecture 3 1 EL 625 Lecture 3 Principles of...

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EL 625 Lecture 3 1 EL 625 Lecture 3 Principles of time-domain analysis Singularity functions : μ i +1 ( t - τ ) = Z t -∞ μ i ( λ - τ ) - Z - μ i ( t - τ ) μ i +1 ( t - τ ) μ 0 ( t ) = 4 δ ( t ) μ 1 ( t ) = 4 1( t ) Unit impulse or δ -function : - 6 0 t Z t -∞ f ( λ ) δ ( λ - τ ) = 0 for t < τ f ( τ ) for t > τ

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EL 625 Lecture 3 2 δ ( t ) = δ ( - t ) for all t Z t -∞ δ ( λ - τ ) = 0 for t < τ 1 for t > τ = 4 1( t - τ ) Z τ + τ - δ ( λ - τ ) δλ = 1 : The area under the impulse is 1. f ( τ ) = Z -∞ f ( λ ) δ ( τ - λ ) δλ : the sifting property Expressing an arbitrary function as a linear combination of δ -functions : f ( t ) = R -∞ f ( λ ) δ ( t - λ ) δλ Impulse response = 4 response to an impulse = S { δ ( t - λ ) } = 4 h ( t,λ ) y ( t ) = S { u ( t ) } = S ±Z -∞ u ( λ ) δ ( t - λ ) ² = Z -∞ u ( λ ) S { δ ( t - λ ) } = Z -∞ u ( λ ) h ( t,λ ) Expressing the response to an arbitrary function as a linear combination of impulse responses : y ( t ) = R -∞ u ( λ ) h ( t,λ )
EL 625 Lecture 3 3 For a time-invariant system, h ( t,λ ) = h ( t - λ, 0) = 4 h ( t - λ ) y ( t ) = R -∞ u ( λ ) h ( t - λ ) ‘Convolution integral’ Step function, 1( t ) - 1

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## This note was uploaded on 12/06/2010 for the course EL 625 taught by Professor Khorami during the Spring '10 term at NYU Poly.

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lect3 - EL 625 Lecture 3 1 EL 625 Lecture 3 Principles of...

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