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Connexions module: m12054 1

Circular Convolution and

Zeropadding ∗

Richard Baraniuk

This work is produced by The Connexions Project and licensed under the

Creative Commons Attribution License †

Abstract

Introduction to circular convolution with zeropadding.

Circular convolution has a "wraparound eect." Given Figure 1(a) we desire to smooth circular convolu- tion with h (Figure 1(b)) then we get Figure 1(c).

(a)

(b)

(c)

Figure 1: (a) The horizontal axis is the number of cell phones purchased per month. (c) The plot is clearly smoother! The wraparound eect can be clearly seen near 1900.

1

Idea

To eliminate wraparound, zeropad x and h with zeros and then do the circular convolution (Figure 2).

Figure 2

note: How much to zeropad to guarantee to wraparound?

∗Version 1.3: Jan 18, 2005 2:14 pm US/Central †http://creativecommons.org/licenses/by/1.0

http://cnx.org/content/m12054/1.3/

Connexions module: m12054 2

2 General Case

Where x ∈ RN and h ∈ RN , we must zeropad out to length 2N − 1. That is, we embed x and h into vectors xz ∈ R2N−1 and hz ∈ R2N−1.

note:

yz = hz~2N−1xz

y = h~Nx

Therefore:

yz 6= y

3 Special Case

Often most elements of h equal zero.

Example 1

4 point smoother for 1024 point signals (Figure 3).

Figure 3

Denition 1: support

The support of a signal h is the length of the nonzero portion (Figure 4). Example

Figure 4: Support = 6.

Now if x ∈ RN and h ∈ RN and support (h) = L, we must zeropad only to length N + L − 1 to avoid wraparound eects.

note: Does it matter where you zeropad?

http://cnx.org/content/m12054/1.3/