Question

Calculate an plot convolution of the following signals:

×[n]={ 1,1≤n≤3
0,othervise
​
;h[n]={ −1,−3≤n≤−1
0,0 herwise
​ y[n]=h[n]∗x[n]= ?

Answer 1

To calculate the convolution of the given signals x[n] and h[n], we use the convolution sum formula by flipping h[n], shifting it, and summing the product with x[n] to get y[n]. We calculate for the range where x[n] and h[n] overlap and then plot the results.

Step-by-step explanation:

To calculate and plot the convolution of the given signals x[n] and h[n], we will use the convolution sum formula. The signals defined are:

• x[n] = 1 for 1 ≤ n ≤ 3, and x[n] = 0 otherwise.
• h[n] = -1 for -3 ≤ n ≤ -1, and h[n] = 0 otherwise.

The convolution y[n] = x[n] * h[n] involves flipping h[n], shifting it by n, and summing the product with x[n] for all n.

Step-by-step calculation of the convolution:

1. Flip h[n] to get h[-k].
2. Shift h[-k] by n to get h[n-k].
3. Multiply x[k] and h[n-k] for all n and k within the defined ranges.
4. Sum the products over k to obtain y[n] for each n.

The convolution y[n] = x[n] * h[n] can be represented as:

• y[n] = ∑_{k=1}^{3} x[k] ⋅ h[n-k], where the summation is performed over the range where x[k] and h[n-k] are both non-zero.

Since x[n] is non-zero for n from 1 to 3, and h[n] is non-zero for n from -3 to -1, their overlap occurs for n from -2 to 2. Therefore, the non-zero values of y[n] will be calculated for these values of n. after calculating y[n] for all n, we can plot the results to visually represent the convolution of x[n] and h[n].