Question

Determine the convolution sum of two sequences x(n) = {1,1,1,1} and h(n)= {2,2,2,2}

Answer 1

Explanation:

Circular convolution using circular convolution:

x1(n) = {1, 2, 3, 4}

and x2 (n) = {1, 2, 1, 2}

L=4, M=4

Length of y(n) = L+M-1=4+4-1=7

∴,x1(n) = {1, 2, 3, 4, 0, 0, 0}

& x2(n) = {1, 2, 1, 2, 0, 0, 0}

For y(0),

y(0)= 1×1=1

For y(1),

∴, y(1)= 2×1+1×2=4

For y(2),

∴ , y(2)= 1×1+2×2+3×1=8

For y(3),

y(3)=1×2+2×1+3×2+4×1=14

For y(4),

∴, y(4)= 4×2+3×1+2×2=15

For y(5),

∴, y(5) = 4×1+3×2=10

For y(6),

∴, y(6) = 4×2=8

∴ ,y(n) = {1, 4, 8, 14, 15, 10, 8}

Result: y(n) = {2, 4, 8, 14, 15, 10, 8}

Linear using circular convolution:

For y(0),

∴ , y(0)= 1+4+3+8=16

For y(1),

∴ , y(1)= 2+2+6+4=14

For y(2),

∴, y(2)= 1+4+3+8=16

For y(3),

∴, y(3)= 2+2+6+4=14

y(n) = {16, 14, 16, 14}

Result: y(n) = {14, 16, 14, 16}

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Answer 2

Answer:-

Thanks for A2A.

As per my knowledge, i will try to answer your question.

First let me try to explain linear convolution.

Linear convolution is the basic operation to calculate the output for any linear time invariant system given its input and its impulse response

It is same as normal multiplication.

Let x[n]={1,2,3}x[n]={1,2,3} of length 3

x[n]={−1,2,2}x[n]={−1,2,2} of length 3 be two sequences

then, linear convolution of two sequences is obtained as follows

Now, add the numbers crosswise as shown.

y[n]={−1,0,3,10,6}y[n]={−1,0,3,10,6}

Now observe the samples in the resultant sequence y(n)y(n).

There are 5 samples, is nothing but 3 + 3 -1 .

we can infer

If x(n)x(n) of length n1n1 & x(n)x(n) of length n2n2 are

two sequences, then linear convolution of two

sequences will have a length

of m=n1+n2−1m=n1+n2−1

Hope this helps you.

Thankyou☘️