5. (Discrete Convolution using Circular Convolution) In lecture, we said that th

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5. (Discrete Convolution using Circular Convolution) In lecture, we said that the convolution of a length-N1 discrete sequence anl (potentially non-zero for OS n S N1-1) with a length-N2 discrete sequence hinl (potentially non-zero for OS n S N2-1) could be computed using zero- padding and circular convolution, which could ultimately be implemented using DFTs/FFTs. (For all parts of this problem, it may be helpful to draw a graphical sketch of the convolution sum to help you see what's going on more clearly (a) Assume that N1 S N2 without loss of generality (we can always take N1 S N2 because of the commutativity of convolution), and that N N1 N2 1 and TIn and h n are zero-padded length-N versions of zlal and h ml, respectively. Show that the zero-padded length-N circular convolution N-1 ycln

Explanation / Answer

Methods of Circular Convolution

Generally, there are two methods, which are adopted to perform circular convolution and they are

Concentric Circle Method

Let x1(n)x1(n) and x2(n)x2(n) be two given sequences. The steps followed for circular convolution of x1(n)x1(n) and x2(n)x2(n) are

Take two concentric circles. Plot N samples of x1(n)x1(n) on the circumference of the outer circle (maintaining equal distance successive points) in anti-clockwise direction.

For plotting x2(n)x2(n), plot N samples of x2(n)x2(n) in clockwise direction on the inner circle, starting sample placed at the same point as 0th sample of x1(n)x1(n)

Multiply corresponding samples on the two circles and add them to get output.

Rotate the inner circle anti-clockwise with one sample at a time.

Matrix Multiplication Method

Matrix method represents the two given sequence x1(n)x1(n) and x2(n)x2(n) in matrix form.

One of the given sequences is repeated via circular shift of one sample at a time to form a N X N matrix.

The other sequence is represented as column matrix.

The multiplication of two matrices give the result of circular convolution.

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