A fundamental property of linear time-invariant (LTI) systems is that whenever t

Question

A fundamental property of linear time-invariant (LTI) systems is that whenever the input of the system is a sinusoid of a certain frequency, the output will also be a sinusoid of the same frequency but with an amplitude and phase determined by the system. For the following systems, let the input x(t) = cos(t), 1 < t < 1, and ?nd the output y(t) to determine if the system is LTI. (a) y(t) = |x(t)|^2 (b) y(t) = 0.5[x(t) -x(t - 1)] (c) y(t) = x(t)1(t) (d) y(t)= 0.5* int(x(t)dt) (limits are t-1 to t)

Explanation / Answer

A straightforward way to show that LTI systems have this property starts by considering complex exponentials. A complex exponential is a signal e ? [Time? Complex] where for all t ? Time, e(t) = exp(j? t) = cos(?t) + j sin(?t). Complex exponential functions have an interesting property that will prove useful to us: For all t and t ? Time, e(t - t ) = exp(j?(t - t )) = exp(-j?t ) exp(j?t). This represents the function Dt

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.