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Circular Convolution with Zeropadding: Eliminating Wraparound Effects, Study notes of Signals and Systems Theory

An introduction to circular convolution and explains the use of zeropadding to eliminate the wraparound effect. Figures and examples to illustrate the concepts. It also discusses the general and special cases of circular convolution with zeropadding.

Typology: Study notes

2011/2012

Uploaded on 10/17/2012

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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 eect." 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 eect 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/
pf2

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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.

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.

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 2 N − 1. That is, we embed x and h into vectors xz ∈ R^2 N^ −^1 and hz ∈ R^2 N^ −^1.

note: yz = hz ~ 2 N − 1 xz

y = h~N x 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/