Question

The continuous time convolution integral y(t) = cos πt [u (t+1) – u (t-1) * u(t)] is __________

(a) \(\frac{sin⁡πt}{π}\) [u (t+1) – u(t-1)]

(b) \(\frac{sin⁡πt}{π}\) u(t-1)

(c) \(\frac{sin⁡πt}{π}\) u(t+1)

(d) \(\frac{sin⁡πt}{π}\) u(t)

The question was posed to me in unit test.

This is a very interesting question from Properties of Systems topic in chapter Signals and Systems Basics of Signals and Systems

Answer 1

Correct choice is (a) \(\frac{sin⁡πt}{π}\) [u (t+1) – u(t-1)]

Best explanation: For t<-1, y (t) = 0

For t<1, y (t) = \(\int_{-1}^t cos⁡πt \,dt = \frac{sin⁡πt}{π}\)

For t>1, y (t) = \(\int_{-1}^t\) cos⁡ πt dt = 0

∴ y (t) = \(\frac{sin⁡πt}{π}\) [u (t+1) – u(t-1)].