Consider an LTI system whose response to the input x(t) = [e −t + e −3t ]u(t) is y(t) = (2e −t − 2e −4t )u(t). (a) Determine the frequency response of this system
Question
Consider an LTI system whose response to the input x(t) = [e −t + e −3t ]u(t) is y(t) = (2e −t − 2e −4t )u(t). (a) Determine the frequency response of this system
Solution
To determine the frequency response of a Linear Time-Invariant (LTI) system, we need to find the system's transfer function H(s). The transfer function is the ratio of the Laplace transform of the output Y(s) to the Laplace transform of the input X(s).
Given the input x(t) = [e^(-t) + e^(-3t)]u(t) and the output y(t) = (2e^(-t) - 2e^(-4t))u(t), we first need to find the Laplace transforms of these signals.
The Laplace transform of x(t) is:
X(s) = L{x(t)} = 1/(s+1) + 1/(s+3)
The Laplace transform of y(t) is:
Y(s) = L{y(t)} = 2/(s+1) - 2/(s+4)
Now, we can find the transfer function H(s) as the ratio of Y(s) to X(s):
H(s) = Y(s) / X(s) = [2/(s+1) - 2/(s+4)] / [1/(s+1) + 1/(s+3)]
Solving this equation will give us the transfer function H(s), which describes the system's frequency response.
The frequency response of the system is obtained
Similar Questions
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