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Circular Convolution and the DFT

Module by: Justin Romberg

Summary: This module describes the circular convolution algorithm and an alternative algorithm

Introduction

You should be familiar with Discrete-Time Convolution, which tells us that given two discrete-time signals xn x n , the system's input, and hn h n , the system's response, we define the output of the system as
yn=xn*hn=∑k=-∞∞xkhn-k y n x n h n k x k h n k (1)
When we are given two DFTs (finite-length sequences usually of length NN), we cannot just multiply them together as we do in the above convolution formula, often referred to as linear convolution. Because the DFTs are periodic, they have nonzero values for n≥N n N and thus the multiplication of these two DFTs will be nonzero for n≥N n N . We need to define a new type of convolution operation that will result in our convolved signal being zero outside of the range n=01…N-1 n 0 1 … N 1 . This idea led to the development of circular convolution, also called cyclic or periodic convolution.

Circular Convolution Formula

What happens when we multiply two DFT's together, where Yk Y k is the DFT of yn y n ?
Yk=FkHk Y k F k H k (2)
when 0≤k≤N-1 0 k N 1
Using the DFT synthesis formula for yn y n
yn=1N∑k=0N-1FkHkⅇj2πNkn y n 1 N k 0 N 1 F k H k j 2 N k n (3)
And then applying the analysis formula Fk=∑m=0N-1fmⅇ-j2πNkn F k m 0 N 1 f m j 2 N k n
yn=1N∑k=0N-1∑m=0N-1fmⅇ-j2πNknHkⅇj2πNkn=∑m=0N-1fm1N∑k=0N-1Hkⅇj2πNkn-m y n 1 N k 0 N 1 m 0 N 1 f m j 2 N k n H k j 2 N k n m 0 N 1 f m 1 N k 0 N 1 H k j 2 N k n m (4)
where we can reduce the second summation found in the above equation into h ( ( n − m ) ) N =1N∑k=0N-1Hkⅇj2πNkn-m h ( ( n − m ) ) N 1 N k 0 N 1 H k j 2 N k n m yn=∑m=0N-1fmh ( ( n − m ) ) N y n m 0 N 1 f m h ( ( n − m ) ) N which equals circular convolution! When we have 0≤n≤N-1 0 n N 1 in the above, then we get:
yn≡fn⊛hn y n ⊛ f n h n (5)
note: The notation ⊛ ⊛ represents cyclic convolution "mod N".

Steps for Cyclic Convolution

Steps for cyclic convolution are the same as the usual convo, except all index calculations are done "mod N" = "on the wheel"
    Steps for Cyclic Convolution
  • Step 1: "Plot" fm f m and h ( ( − m ) ) N h ( ( − m ) ) N
cconv_s1.pngcconv_s2.png
Subfigure 1.1
Subfigure 1.2
Figure 1: Step 1
  • Step 2: "Spin" h ( ( − m ) ) N h ( ( − m ) ) N n n notches ACW (counter-clockwise) to get h ( ( n − m ) ) N h ( ( n − m ) ) N (i.e. Simply rotate the sequence, hn h n , clockwise by nn steps).
cconv_s3.png
Figure 2: Step 2
  • Step 3: Pointwise multiply the fm f m wheel and the h ( ( n − m ) ) N h ( ( n − m ) ) N wheel. sum=yn sum y n
  • Step 4: Repeat for all 0≤n≤N-1 0 n N 1
Example 1: Convolve (n = 4) 
cconv_p1.pngcconv_p2.png
Subfigure 3.1
Subfigure 3.2
Figure 3: Two discrete-time signals to be convolved.
  • h ( ( − m ) ) N h ( ( − m ) ) N
cconv_p3.png
Figure 4
Multiply fm f m and sum sum to yield: y0=3 y 0 3
  • h ( ( 1 − m ) ) N h ( ( 1 − m ) ) N
cconv_p4.png
Figure 5
Multiply fm f m and sum sum to yield: y1=5 y 1 5
  • h ( ( 2 − m ) ) N h ( ( 2 − m ) ) N
cconv_p5.png
Figure 6
Multiply fm f m and sum sum to yield: y2=3 y 2 3
  • h ( ( 3 − m ) ) N h ( ( 3 − m ) ) N
cconv_p6.png
Figure 7
Multiply fm f m and sum sum to yield: y3=1 y 3 1
Example 2 
The following demonstration allows you to explore this algorithm for circular convolution. See here for instructions on how to use the demo.

Alternative Algorithm

    Alternative Circular Convolution Algorithm
  • Step 1: Calculate the DFT of fn f n which yields Fk F k and calculate the DFT of hn h n which yields Hk H k .
  • Step 2: Pointwise multiply Yk=FkHk Y k F k H k
  • Step 3: Inverse DFT Yk Y k which yields yn y n
Seems like a roundabout way of doing things, but it turns out that there are extremely fast ways to calculate the DFT of a sequence.
To circularily convolve 2 2 N N-point sequences: yn=∑m=0N-1fmh ( ( n − m ) ) N y n m 0 N 1 f m h ( ( n − m ) ) N For each n n : N N multiples, N-1 N 1 additions
N N points implies N2 N 2 multiplications, NN-1 N N 1 additions implies ON2 O N 2 complexity.

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