Skip to content Skip to navigation

Connexions

You are here: Home » Content » Circular Convolution

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

      What are tags? tag icon

      Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

    • External bookmarks
  • E-mail the author
  • Rate this module (How does the rating system work?)

    Rating system

    Ratings

    Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

    How to rate a module

    Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

    		(0 ratings)

Recently Viewed

This feature requires Javascript to be enabled.

Circular Convolution

Module by: Richard Baraniuk

Summary: Introduction to circular convolution.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

  • DSP-speak for the operation
    y=ℋx y ℋ x (1)
    when ℋℋ is a circulant matrix corresponding to an LSI system (Figure 1).
  • Write out matrix multiply y=ℋx y ℋ x
    ·=⋯⋯⋯⋯→⋮⋮⋮⋮↓ · ⋯ ⋯ ⋯ ⋯ → ⋮ ⋮ ⋮ ⋮ ↓ (2)
    where the · is in the nth n th row.
    yn=∑k=0N−1ℋnkxk y n k 0 N 1 ℋ n k x k (3)
    where ℋ ℋ is a circulant matrix and ℋnk=hn−kmodN ℋ n k h n k N
Figure 1: ℋ ℋ is LSI.
Figure 1 (matmult.png)
yn=∑k=0N−1hn−kmodNxk y n k 0 N 1 h n k N x k (4)
y= h ⊛ N x y h ⊛ N x (5)
yn= h ⊛ N x n y n h ⊛ N x n (6)

Notation

Since impulse response hh completely describes ℋℋ, we often write: Figure 2 and Figure 3.

Figure 2
Figure 2 (matmult2.png)
Figure 3
Figure 3 (matmult3.png)

Inner Product Interpretation of Circular Convolution

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

ℛhk=h-kmodN ℛ h k h k N (7)
Figure 4: N=4 N 4 .
(a) hh.
Figure 4(a) (trm1.png)
(b) flip h h.
Figure 4(b) (trm2.png)
(c) flip hmodN flip h N i.e.: to periodize with a period of NN.
Figure 4(c) (trm3.png)
(d) ℛh ℛ h .
Figure 4(d) (trm4.png)
h0modNh-1modNh-2modNh-3modN=h0modNh3modNh2modNh1modN=1000000100100100h0modNh1modNh2modNh3modN h 0 N h -1 N h -2 N h -3 N h 0 N h 3 N h 2 N h 1 N 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 h 0 N h 1 N h 2 N h 3 N (8)

Note:

Given circulant ℋ=132213321 ℋ 1 3 2 2 1 3 3 2 1 zeroth column h0c=123 h 0 c 1 2 3 zeroth row ℛh0cT=132T=132 ℛ h 0 c 1 3 2 1 3 2
So circular convolution can be written as this yn= inner product of row n of ℋ (turned into a column) =<x, row n of ℋ tipped into a column > y n inner product of row n of ℋ (turned into a column) x row n of ℋ tipped into a column but row nn of ℋ ℋ tipped into a column vector is
hnrT= C n h0T= C n ℛh0c h n r C n h 0 C n ℛ h 0 c (9)
which is the circular shift of the zeroth row and where h0T=ℛh0c h 0 ℛ h 0 c and is the time reversed column.

& so...

yn=<x, C n ℛh0c> y n x C n ℛ h 0 c (10)
for ℝN N ; put a * in second entry for ℂN N .

The Ring of Doom

modN N operations are natural on a circle! Since they are naturally N N-periodic (Figure 5).

Figure 5: x=3210T x 3 2 1 0 .
(a) 0≤n≤N−1 0 n N 1 .
Figure 5(a) (x.png)
(b) -∞<n<∞ n .
Figure 5(b) (x2.png)
We can put xx on a circle/wheel (Figure 6).
Figure 6: Time runs counter-clockwise.
Figure 6 (circ1.png)

To do a circular shift by m m, C m x C m x : just spin the wheel counter-clockwise mm units and read off the new signal.

Example 1

m=2 m 2 , C 2 C 2 (Figure 7).

Figure 7: Spin two spots counter-clockwise then C 2 x=x2x3x0x1T C 2 x x 2 x 3 x 0 x 1 .
(a)
Figure 7(a) (circ1.png)
(b)
Figure 7(b) (circ2.png)

Time reversal, ℛx ℛ x : just read off wheel in clockwise direction (Figure 8).

Example 2

Figure 8: Read off in time reversed order then ℛx=x0x3x2x1T ℛ x x 0 x 3 x 2 x 1 .
(a)
Figure 8(a) (circ3.png)
(b)
Figure 8(b) (circ4.png)

"How to do" Cyclic Convolution

Cyclic convolution works modN N is equivalent to "on the wheel," where the cylinder analogy is powerful.

yn=∑m=0N−1xmhn−mmodN y n m 0 N 1 x m h n m N (11)

Step 1

Plot xm x m (Figure 9).

Figure 9
Figure 9 (circ5.png)

Step 2

Plot hm h m (backwards on cylinder) (Figure 10).

Figure 10
Figure 10 (circ6.png)

Step 3

Spin h-m h m nn steps to implement hn−mmodN h n m N anti-clockwise.

Step 4

Multiply pointwise xm x m wheel and hn−mmodN h n m N wheel. The sum equals yn y n .

Step 5

Repeat steps 3 and 4 for all n n.

Comments, questions, feedback, criticisms?

Send feedback