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Connexions module: m12054 1
Circular Convolution and
This work is produced by The Connexions Project and licensed under the
Creative Commons Attribution License †
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).
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.
To eliminate wraparound, zeropad x and h with zeros and then do the circular convolution (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
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.
yz = hz~2N−1xz
y = h~Nx
yz 6= y
3 Special Case
Often most elements of h equal zero.
4 point smoother for 1024 point signals (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?