convolution

[logistic_guy]

Let

\(\displaystyle x[n] = \delta[n] + 2\delta[n - 1] - \delta[n - 3] \ \) and \(\displaystyle \ h[n] = 2\delta[n + 1] + 2\delta[n - 1]\).

Compute and plot each of the following convolutions:

\(\displaystyle \bold{(a)} \ y_1[n] = x[n] \ * \ h[n]\)
\(\displaystyle \bold{(b)} \ y_2[n] = x[n + 2] \ * \ h[n]\)
\(\displaystyle \bold{(c)} \ y_3[n] = x[n] \ * \ h[n + 2]\)

 

[logistic_guy]

\(\displaystyle \bold{(a)} \ y_1[n] = x[n] \ * \ h[n]\)

\(\displaystyle y_1[n] = x[n] \ * \ h[n] = \sum_{k=-\infty}^{\infty}x[k]h[n - k] = \sum_{k=-\infty}^{\infty}x[k]\bigg(2\delta[n + 1 - k] + 2\delta[n - 1 - k]\bigg)\)


\(\displaystyle = 2x[n+1] + 2x[n - 1]\)


\(\displaystyle = 2\delta[n + 1] + 4\delta[n - 1 + 1] - 2\delta[n - 3 + 1] + 2\delta[n - 1] + 4\delta[n - 1 - 1] - 2\delta[n - 3 - 1]\)


\(\displaystyle = 2\delta[n + 1] + 4\delta[n] - 2\delta[n - 2] + 2\delta[n - 1] + 4\delta[n - 2] - 2\delta[n - 4]\)

Or

\(\displaystyle y_1[n] = \textcolor{blue}{2\delta[n + 1] + 4\delta[n] + 2\delta[n - 1] + 2\delta[n - 2] - 2\delta[n - 4]}\)

 

[logistic_guy]

\(\displaystyle \bold{(b)} \ y_2[n] = x[n + 2] \ * \ h[n]\)

\(\displaystyle y_2[n] = x[n + 2] \ * \ h[n] = \sum_{k=-\infty}^{\infty}x[n + 2 - k]h[k] = \sum_{k=-\infty}^{\infty}\bigg(\delta[n + 2 - k] + 2\delta[n - 1 + 2 - k] - \delta[n - 3 + 2 - k]\bigg)h[k]\)


\(\displaystyle = \sum_{k=-\infty}^{\infty}\bigg(\delta[n + 2 - k] + 2\delta[n + 1 - k] - \delta[n - 1 - k]\bigg)h[k]\)


\(\displaystyle = h[n + 2] + 2h[n + 1] - h[n - 1]\)


\(\displaystyle =2\delta[n + 1 + 2] + 2\delta[n - 1 + 2] + 4\delta[n + 1 + 1] + 4\delta[n - 1 + 1] - 2\delta[n + 1 - 1] - 2\delta[n - 1 - 1]\)


\(\displaystyle =2\delta[n + 3] + 2\delta[n + 1] + 4\delta[n + 2] + 4\delta[n] - 2\delta[n] - 2\delta[n - 2]\)


\(\displaystyle =2\delta[n + 3] + 2\delta[n + 1] + 4\delta[n + 2] + 2\delta[n] - 2\delta[n - 2]\)


\(\displaystyle =\textcolor{blue}{2\delta[n + 3] + 4\delta[n + 2] + 2\delta[n + 1] + 2\delta[n] - 2\delta[n - 2]}\)

 

[logistic_guy]

\(\displaystyle \bold{(c)} \ y_3[n] = x[n] \ * \ h[n + 2]\)

\(\displaystyle y_3[n] = x[n] \ * \ h[n + 2] = \sum_{k=-\infty}^{\infty}x[n - k]h[k + 2] = \sum_{k=-\infty}^{\infty}\bigg(\delta[n - k] + 2\delta[n - 1 - k] - \delta[n - 3 - k]\bigg)h[k + 2]\)


\(\displaystyle = h[n + 2] + 2h[n - 1 + 2] - h[n - 3 + 2]\)


\(\displaystyle = h[n + 2] + 2h[n + 1] - h[n - 1]\)


\(\displaystyle = 2\delta[n + 1 + 2] + 2\delta[n - 1 + 2] + 4\delta[n + 1 + 1] + 4\delta[n - 1 + 1] - 2\delta[n + 1 - 1] - 2\delta[n - 1 - 1]\)


\(\displaystyle = 2\delta[n + 3] + 2\delta[n + 1] + 4\delta[n + 2] + 4\delta[n] - 2\delta[n] - 2\delta[n - 2]\)


\(\displaystyle = \textcolor{blue}{2\delta[n + 3] + 4\delta[n + 2] + 2\delta[n + 1] + 2\delta[n] - 2\delta[n - 2]}\)