Circular Convolution and the DFT 2.8 2002/08/05 2006/07/19 16:36:10.498 GMT-5 Justin Romberg jrom@rice.edu Prashant Singh prash@ece.rice.edu Mariyah Poonawala mariyah@rice.edu Matthew Hutchinson mhutch@rice.edu circular circular convolution convolution convolutions convolve dft discrete fourier transform fourier transform system 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 x n , the system's input, and h n , the system's response, we define the output of the system as y n x n h n k x k h n k When we are given two DFTs (finite-length sequences usually of length N), 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 and thus the multiplication of these two DFTs will be nonzero for 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 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 Y k is the DFT of y n ? Y k F k H k when 0 k N 1 Using the DFT synthesis formula for y n y n 1 N k 0 N 1 F k H k j 2 N k n And then applying the analysis formula F k m 0 N 1 f m j 2 N k n 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 where we can reduce the second summation found in the above equation into h ( ( n − m ) ) N 1 N k 0 N 1 H k j 2 N k n m y n m 0 N 1 f m h ( ( n − m ) ) N which equals circular convolution! When we have 0 n N 1 in the above, then we get: y n ⊛ f n h n 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" f m and h ( ( − m ) ) N

Step 1

Step 2: "Spin" h ( ( − m ) ) N n notches ACW (counter-clockwise) to get h ( ( n − m ) ) N (i.e. Simply rotate the sequence, h n , clockwise by n steps).

Step 2

Step 3: Pointwise multiply the f m wheel and the h ( ( n − m ) ) N wheel. sum y n Step 4: Repeat for all 0 n N 1 Convolve (n = 4)

Two discrete-time signals to be convolved.

h ( ( − m ) ) N

Multiply f m and sum to yield: y 0 3 h ( ( 1 − m ) ) N

Multiply f m and sum to yield: y 1 5 h ( ( 2 − m ) ) N

Multiply f m and sum to yield: y 2 3 h ( ( 3 − m ) ) N

Multiply f m and sum to yield: y 3 1 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 f n which yields F k and calculate the DFT of h n which yields H k . Step 2: Pointwise multiply Y k F k H k Step 3: Inverse DFT Y k which yields 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 N -point sequences: y n m 0 N 1 f m h ( ( n − m ) ) N For each n : N multiples, N 1 additions N points implies N 2 multiplications, N N 1 additions implies O N 2 complexity.