Question

The convolution of \(x(-t)\) with impulse function \(\delta(-t - t_0)\) is equal to

• A. \( x(t+t_0) \)
• B. \( x(t-t_0) \)
• C. \( x(-t+t_0) \)
• D. \( x(-t-t_0) \)

Hint:

Remember two key properties of the delta function: 1. Scaling: \( \delta(at+b) = \frac{1}{|a|}\delta(t + b/a) \). 2. Sifting (Convolution): \( f(t) \delta(t-T) = f(t-T) \). First, simplify the delta function, then apply the sifting property.

Answer 1

The Correct Option is D

Step 1: Simplify the impulse function using the scaling property.The Dirac delta function, \( \delta(t) \), satisfies \( \delta(at) = \frac{1}{|a|}\delta(t) \).Applying this to \( \delta(-t-t_0) \):\[ \delta(-t-t_0) = \delta(-(t+t_0)) \]With \(a = -1\), we get:\[ \delta(-(t+t_0)) = \frac{1}{|-1|}\delta(t+t_0) = \delta(t+t_0) \]
Step 2: Convolve the simplified impulse.Now we convolve \( x(-t) \) with \( \delta(t+t_0) \).Using the sifting property \( f(t) \delta(t-T) = f(t-T) \), where \( f(t) = x(-t) \) and \( T = -t_0 \):Substitute \(t\) with \(t - (-t_0) = t+t_0\) in \(x(-t)\).\[ x(-t) \delta(t+t_0) = x(-(t+t_0)) \]\[ = x(-t - t_0) \]