Question

Convolution of x(t+5) with impulse function delta(t-7) is equal to

Answer 1

Convolution of x(t+5) with impulse function  $\begin{equation}\delta\end$ (t-7) is x(t-2).

Given  

To find the convolution of x(t+5) with impulse function  $\begin{equation}\delta\end$ (t-7).

Convolution has two functions f and g, which results to give third function.

Formula for convolution :

                (f*g) (t) = $\begin{equation}\int_{-\infty}^{\infty} f(\tau) g(t-\tau) d \tau\end$      

Hence,

               = $\begin{equation}\int \delta(\tau-\tau) \times(t-\tau+5) d \tau\end$        

Applying,  $\begin{equation}\tau\end$ = 7 in the above equation, we get                

               = x(t-$\begin{equation}\tau\end$+5)     ( where, $\begin{equation}\tau\end$ = 7 )

               = x(t-7+5)                                

               = x(t-2)

Therefore, convolution of x(t+5) with impulse function  $\begin{equation}\delta\end$ (t-7) is x(t-2).

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