Solution Found!
Consider a continuous-time LTI system with frequency response H(jw) = IH(jw)l eJ
Chapter 6, Problem 6.1(choose chapter or problem)
Consider a continuous-time LTI system with frequency response \(H(j \omega)=|H(j \omega)| e^{j \sphericalangle H(j \omega)}\) and real impulse response h(t). Suppose that we apply an input \(x(t)=\cos(\omega_0t+\phi_0)\) to this system. The resulting output can be shown to be of the form
\(y(t)=Ax(t-t_0)\),
where A is a nonnegative real number representing an amplitude-scaling factor and \(t_0\) is a time delay.
(a) Express A in terms of \(|H(j \omega_0)|\).
(b) Express \(t_0\) in terms of \(\sphericalangle H\left(j \omega_{0}\right)\).
Questions & Answers
QUESTION:
Consider a continuous-time LTI system with frequency response \(H(j \omega)=|H(j \omega)| e^{j \sphericalangle H(j \omega)}\) and real impulse response h(t). Suppose that we apply an input \(x(t)=\cos(\omega_0t+\phi_0)\) to this system. The resulting output can be shown to be of the form
\(y(t)=Ax(t-t_0)\),
where A is a nonnegative real number representing an amplitude-scaling factor and \(t_0\) is a time delay.
(a) Express A in terms of \(|H(j \omega_0)|\).
(b) Express \(t_0\) in terms of \(\sphericalangle H\left(j \omega_{0}\right)\).
ANSWER:Step 1 of 4
The following are given from the question.
The frequency response of the system, \(H(j \omega)=|H(j \omega)| e^{j \sphericalangle H(j \omega)}\)
The real impulse response of the system is h(t).
The input, \(x(t)=\cos \left(\omega_{0} t+\phi_{0}\right)\).
The output of the system form, \(y(t)=A x\left(t-t_{0}\right)\)