Question

Suppose signal \( y(t) \) is obtained by the time-reversal of signal \( x(t) \), i.e., \( y(t) = x(-t) \),\(-\inftyLt;tLt;\infty\) Which of the following options is always true for the convolution of \( x(t) \) and \( y(t) \)?

• A. It is an even signal
• B. It is an odd signal
• C. It is a causal signal
• D. It is an anti-causal signal

Hint:

Convolution of a signal with its time-reversed version always produces an even signal.

Answer 1

The Correct Option is A

Step 1: Understanding convolution with time-reversed signals. (i) Time Scaling Property:
The time scaling property states that: \[ x(at) * y(at) = \frac{1}{|a|} x(t) * y\left(\frac{t}{a}\right) \] Substituting \(a = -1\): \[ x(-t) * y(-t) = x(t) * y(t) \] (ii) Commutative Property:
The commutative property is expressed as: \[ x(t) * y(t) = y(t) * x(t) \] From equation (1), we have: \[ z(t) = y(t) * x(t) = x(-t) * y(-t) \tag{2} \] Substitute \(t = -t\): \[ z(-t) = y(-t) * x(-t) \] From the commutative property, we get: \[ z(-t) = x(-t) * y(-t) \tag{3} \] From equations (2) and (3): \[ z(t) = z(-t) \] Step 2: Applying properties of convolution. - Convolution of \( x(t) \) with its time-reversed version \( x(-t) \) results in a symmetric output. - The symmetry implies \( z(t) \) is an even signal: \[ z(t) = z(-t). \]
Hence, the correct option is (1) It is an even signal.