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(12 points) LTI Systems and impulse response Consider the following three LTI systems:S1: y(t) = ∫−∞te−3(t−τ)x(τ)dτ S2: y(t) = ∫−∞t−2 x(τ)dτ S3 is characterized by its impulse response: h3(t) = δ(t−3). (a) (4 points) Compute the impulse response h1(t) of S1. (b) (2 points) Define w(t) = S1[x(t)]− S3{S2[x(t)]}. Represent this relationship using a block diagram where x(t) is the input and w(t) is the output. (c) (2 points) Determine the impulse response heq(t) of the above system. (d) (4 points) Determine the response of the overall system to δ(t)+2 δ(t−3).

(12 points) LTI Systems and impulse response Consider the following three LTI systems:S1: y(t) = ∫−∞te−3(t−τ)x(τ)dτ S2: y(t) = ∫−∞t−2 x(τ)dτ S3 is characterized by its impulse response: h3(t) = δ(t−3). (a) (4 points) Compute the impulse response h1(t) of S1. (b) (2 points) Define w(t) = S1[x(t)]− S3{S2[x(t)]}. Represent this relationship using a block diagram where x(t) is the input and w(t) is the output. (c) (2 points) Determine the impulse response heq(t) of the above system. (d) (4 points) Determine the response of the overall system to δ(t)+2 δ(t−3).

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  1. (12 points) LTI Systems and impulse response Consider the following three LTI systems:
  • S 1 : y ( t ) = ∫ − ∞ t e − 3 ( t − τ ) x ( τ ) d τ
  • S 2 : y ( t ) = ∫ − ∞ t − 2 x ( τ ) d τ
  • S 3 is characterized by its impulse response: h 3 ( t ) = δ ( t − 3 ) . (a) (4 points) Compute the impulse response h 1 ( t ) of S 1 . (b) (2 points) Define w ( t ) = S 1 [ x ( t ) ] − S 3 { S 2 [ x ( t ) ] } . Represent this relationship using a block diagram where x ( t ) is the input and w ( t ) is the output. (c) (2 points) Determine the impulse response h e q ( t ) of the above system. (d) (4 points) Determine the response of the overall system to δ ( t ) + 2 δ ( t − 3 ) .

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