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Circular Convolution
Richard Baraniuk
This work is produced by The Connexions Project and licensed under the
Creative Commons Attribution License ā
Abstract
Introduction to circular convolution.
- DSP-speak for the operation
y = Hx (1)
when H is a circulant matrix corresponding to an LSI system (Figure 1).
- Write out matrix multiply
y = Hx
where the Ā· is in the n th row.
y [n] =
N ā 1 ā
k=
Hn,kx [k] (3)
where H is a circulant matrix and Hn,k = h [(n ā k) modN ]
Figure 1: H is LSI.
y [n] =
N ā 1 ā
k=
h [(n ā k) modN ] x [k] (4)
ā Version 1.1: Jun 25, 2004 9:39 am GMT- ā http://creativecommons.org/licenses/by/1.
y = h~N x (5)
y [n] = h~N x [n] (6)
1 Notation
Since impulse response h completely describes H, we often write: Figure 2 and Figure 3.
Figure 2
Figure 3
2 Inner Product Interpretation of Circular Convolution
De ne the time reversal matrixR as a matrix that reverses the time axis of a column vector (Figure 4).
Rh [k] = h [(āk) modN ] (7)
h [0modN ]
h [ā 1 modN ]
h [ā 2 modN ]
h [ā 3 modN ]
h [0modN ]
h [3modN ]
h [2modN ]
h [1modN ]
h [0modN ]
h [1modN ]
h [2modN ]
h [3modN ]
note: Given circulant
H =
zeroth column
h
c 0
zeroth row
(Rh
c 0 )
T
T
So circular convolution can be written as this (list, p. 1)
y [n] = inner product of row n of H (turned into a column) =< x, row n of H tipped into a column >
but row n of H tipped into a column vector is
h
r n
T = Cnh 0
T
= CnRh
c 0
which is the circular shift of the zeroth row and where h 0
T = Rh
c 0 and is the time reversed column.
& so...
y [n] =< x, CnRh
c 0 >^ (10)
for R
N ; put a * in second entry for C
N .
3 The Ring of Doom
modN operations are natural on a circle! Since they are naturally N -periodic (Figure 5).
(a)
(b)
Figure 5: x = (3, 2 , 1 , 0)
T
. (a) 0 ā¤ n ā¤ N ā 1. (b) āā < n < ā.
We can put x on a circle/wheel (Figure 6).
Figure 6: Time runs counter-clockwise.
To do a circular shift by m, Cmx: just spin the wheel counter-clockwise m units and read o the new
signal.
Example 1
m = 2, C 2 (Figure 7).
(a)
(b)
Figure 8: Read o in time reversed order then Rx = (x [0] , x [3] , x [2] , x [1])
T .
4 "How to do" Cyclic Convolution
Cyclic convolution works modN is equivalent to "on the wheel," where the cylinder analogy is powerful.
y [n] =
N ā 1 ā
m=
x [m] h [(n ā m) modN ] (11)
4.1 Step 1
Plot x [m] (Figure 9).
Figure 9
4.2 Step 2
Plot h [m] (backwards on cylinder) (Figure 10).
Figure 10
4.3 Step 3
Spin h [ām]n steps to implement h [(n ā m) modN ]anti-clockwise.
