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[SOWHAT0819]

compute for the convolution of the following
a.
x = {2,-4,2} arrow point at left 2
h = {7,1,4,-4,-4} arrow point at 7


b.
x = {6,-6,-2,3} arrow point at positive 6
h = {1,0,-1,-1,} arrow point at 0


c.
x = {1,1,2,1,1} arrow point at positive 2
h = {5,-9,-1,4} arrow point at 4

 

[🤖]

a. To compute the convolution of x and h, we can use the formula for discrete convolution:

y = x * h = Σ x(k) * h(n-k)

Given:
x = {2, -4, 2}
h = {7, 1, 4, -4, -4}

Let's find y step by step:
n = 0: y(0) = 2 * 7 = 14
n = 1: y(1) = 2 * 1 + (-4) * 7 = 2 - 28 = -26
n = 2: y(2) = 2 * 4 + (-4) * 1 + 2 * 7 = 8 - 4 + 14 = 18
n = 3: y(3) = (-4) * 4 + 2 * 1 = -16 + 2 = -14
n = 4: y(4) = (-4) * 4 = -16

Therefore, the convolution of x and h is y = {14, -26, 18, -14, -16}.

b. Given:
x = {6, -6, -2, 3}
h = {1, 0, -1, -1}

Let's find y step by step:
n = 0: y(0) = 6 * 1 = 6
n = 1: y(1) = 6 * 0 + (-6) * 1 = 0 - 6 = -6
n = 2: y(2) = 6 * (-1) + (-6) * 0 + (-2) * 1 = -6 - 0 - 2 = -8
n = 3: y(3) = (-6) * (-1) + (-2) * 0 + 3 * 1 = 6 + 0 + 3 = 9

Therefore, the convolution of x and h is y = {6, -6, -8, 9}.

c. Given:
x = {1, 1, 2, 1, 1}
h = {5, -9, -1, 4}

Let's find y step by step:
n = 2: y(2) = 1 * 5 + 1 * (-9) = 5 - 9 = -4
n = 3: y(3) = 1 * (-9) + 2 * 5 + 1 * (-1) = -9 + 10 - 1 = 0
n = 4: y(4) = 2 * (-1) + 1 * 5 + 1 * (-9) = -2 + 5 - 9 = -6
n = 5: y(5) = 1 * 4 + 2 * (-1) + 1 * 5 = 4 - 2 + 5 = 7
n = 6: y(6) = 1 * 4 + 1 * (-1) = 4 - 1 = 3

Therefore, the convolution of x and h is y = {0, -4, 0, -6, 7, 3}.