Question

# For a continuous time linear time-invariant system, the input-output relation is the following (x(t) the input, y(t) the...

For a continuous time linear time-invariant system, the
input-output relation is the following (x(t) the input, y(t) the
output):

, where h(t) is the impulse response function of the system.
Please explain why a signal like e/“* is always an eigenvector of
this linear map for any w. Also, if ¥(w),X(w),and H(w) are the
Fourier transforms of y(t),x(t),and h(t), respectively. Please
derive in detail the relation between Y(w),X(w),and H(w),
which means to reproduce the proof of the basic convolution
property of Fourier transform.

4. For a continuous time linear time-invariant system, the input-output relation is the following (x(t) the input, y(t) the output) y(t)h(t-T)x (r) d , where h(t) is the impulse response function of the system Please explain why a signal like eir is always an eigenvector of this linear map for any o. Also, if Y(a),X(w), and H(w) are the Fourier transforms of y(t), x(t), and h(t), respectively. Please derive in detail the relation between Y(w), X(w) , and H(w) which means to reproduce the proof of the basic convolution property of Fourier transform

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