The impulse responses of two LTI systems are given below. These two LTI systems are connected in series to form anoverall LTI system ℎ𝑜𝑜(𝑡𝑡).ℎ1(𝑡𝑡) = 4 𝑒𝑒 − 4𝑡𝑡 𝑢𝑢(𝑡𝑡) ℎ 2(𝑡𝑡) = 𝛿𝛿 �𝑡𝑡 − 𝜋𝜋8 �e) (2) Find the system functions 𝐻𝐻1(𝑠𝑠) and 𝐻𝐻2(𝑠𝑠) by integration.f) (2) Find the frequency response of the overall system 𝐻𝐻𝑜𝑜(𝑗𝑗𝑗𝑗)
Question
The impulse responses of two LTI systems are given below. These two LTI systems are connected in series to form anoverall LTI system ℎ𝑜𝑜(𝑡𝑡).ℎ1(𝑡𝑡) = 4 𝑒𝑒 − 4𝑡𝑡 𝑢𝑢(𝑡𝑡) ℎ 2(𝑡𝑡) = 𝛿𝛿 �𝑡𝑡 − 𝜋𝜋8 �e) (2) Find the system functions 𝐻𝐻1(𝑠𝑠) and 𝐻𝐻2(𝑠𝑠) by integration.f) (2) Find the frequency response of the overall system 𝐻𝐻𝑜𝑜(𝑗𝑗𝑗𝑗)
Solution
The given impulse responses are for two Linear Time-Invariant (LTI) systems. The system functions H1(s) and H2(s) can be found by taking the Laplace transform of the impulse responses h1(t) and h2(t).
a) The Laplace transform of h1(t) = 4e^(-4t)u(t) is given by:
H1(s) = ∫ h1(t)e^(-st) dt = ∫ 4e^(-4t)u(t)e^(-st) dt = 4/(s+4) for Re(s) > -4
b) The Laplace transform of h2(t) = δ(t - π/8) is given by:
H2(s) = ∫ h2(t)e^(-st) dt = ∫ δ(t - π/8)e^(-st) dt = e^(-sπ/8)
c) The frequency response of the overall system Hoo(jω) can be found by substituting s = jω in the product of H1(s) and H2(s). This is because when two LTI systems are connected in series, the overall system function is the product of the individual system functions.
Hoo(jω) = H1(jω)H2(jω) = [4/(jω+4)] * e^(-jωπ/8)
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