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Circular convolution, college study notes - Introduction to circular convolution., Study notes of Signals and Systems Theory

College Notes. Introduction to circular convolution. Circular Convolution, Connexions Web site. http://cnx.org/content/m12053/1.1/, Jun 25, 2004. Circular, Convolution, Richard, Baraniuk, Criculant, Matrix, Lsi system, Inner, Product, Interpretation, Ring, Doom, Circle, Anti-clockwise.

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Connexions module: m12053 1
Circular Convolution
āˆ—
Richard Baraniuk
This work is produced by The Connexions Project and licensed under the
Creative Commons Attribution License
†
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







Ā·







=










Ā· Ā· Ā· Ā· Ā· Ā· Ā· Ā· Ā· Ā· Ā· Ā· →






















.
.
.
.
.
.
.
.
.
.
.
.
↓












(2)
where the
Ā·
is in the
nth
row.
y[n] =
Nāˆ’1
X
k=0
Hn,kx[k]
(3)
where
H
is a circulant matrix and
Hn,k =h[(nāˆ’k)modN]
Figure 1:
H
is LSI.
y[n] =
Nāˆ’1
X
k=0
h[(nāˆ’k)modN]x[k]
(4)
āˆ—
Version 1.1: Jun 25, 2004 9:39 am GMT-5
†
http://creativecommons.org/licenses/by/1.0
http://cnx.org/content/m12053/1.1/
pf3
pf4
pf5
pf8
pf9

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Circular Convolution

Richard Baraniuk

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

Creative Commons Attribution License †

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 āˆ’ 1 āˆ‘

k=

Hn,kx [k] (3)

where H is a circulant matrix and Hn,k = h [(n āˆ’ k) modN ]

Figure 1: H is LSI.

y [n] =

N āˆ’ 1 āˆ‘

k=

h [(n āˆ’ k) modN ] x [k] (4)

āˆ— Version 1.1: Jun 25, 2004 9:39 am GMT- † 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

Dene the time reversal matrixR as a matrix that reverses the time axis of a column vector (Figure 4).

Rh [k] = h [(āˆ’k) modN ] (7)

h [0modN ]

h [āˆ’ 1 modN ]

h [āˆ’ 2 modN ]

h [āˆ’ 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 ≤ n ≤ N āˆ’ 1. (b) āˆ’āˆž < n < āˆž.

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 āˆ’ 1 āˆ‘

m=

x [m] h [(n āˆ’ 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 [āˆ’m]n steps to implement h [(n āˆ’ m) modN ]anti-clockwise.