Discrete Convolution by FFT

4.7.2 Discrete Convolution by FFT

The most important operation in signal processing is computing the discrete version of the convolution operator. This awkward operation in the time domain becomes very simple in the frequency domain.

Definition 4.7.3.

The convolution of two sequences f i and g i , i = 0 : N − 1, is conv (f, g) = (h 0 ,h 1 ,...,h N −1 ) T , where

where the sequences are extended to have period N by setting f i =f i +jN ,g i =g i +jN for all integers i, j .

The discrete convolution can be used to approximate the convolution defined for continuous functions in Definition 4.6.5 in a similar way as the Fourier transform was approximated using sampled values in Sec. 4.6.6.

We can write the sum in (4.7.15) as a matrix-vector multiplication h = Gf , where G is a Toeplitz matrix. Writing out components we have

 h 0   g 0 g N −1 g N −2 ···g 1   f 0   h 1   g 1 g 0 g N −1 ···g 2   f 1 

   =  g 2 g 1 g 0 ···g 3   f 2  

g N −1 g N −2 g N −3 ···g 0 f N −1 Note that each column in G is a cyclic down-shifted version of the previous column. Such

h N −1

a matrix is called a circulant matrix. We have

g] =g 0 I +g 1 C N +···+g N −1 C N −1 , (4.7.16) and C N is the circulant permutation matrix

G =[gC N N g ···C N −1

The following result is easily verified.

Lemma 4.7.4.

The eigenvalues of the circulant matrix C N in (

4.7.17) are

ω j =e −2πj/N , j = 0 : N − 1,

510 Chapter 4. Interpolation and Approximation where ω is an Nth root of unity, i.e., ω N = 1. The columns of the DFT matrix F N ,

,...,ω j N = (1, ω j j −1 ) T , j = 0 : N − 1,

are eigenvectors. Since the matrix G in (4.7.16) is a polynomial in C N it has the same set of eigenvectors,

and thus G is diagonalized by the DFT matrix F N ,

G =F N XF N −1 , X = diag (λ 1 ,...,λ n ), (4.7.18) where the eigenvalues of G are λ i

)g, j = 0 : N − 1, which is the FFT of the first column in G. Hence X = diag (F N g) , where diag (x) denotes

= (1, ω N j ,...,ω −1

a diagonal matrix with diagonal elements equal to the elements in the vector x.

Theorem 4.7.5.

Let f i and g i , i = 0 : N − 1, be two sequences with DFTs equal to F N f and F N g. Then the DFT of the convolution of f and g is F N f. ∗F N g (. ∗ denotes the elementwise product).

Proof.

From G = F −1 N diag (F N g)F N it follows that

h = Gf = F N −1 diag (F N g)F N f =F N −1 ((F N g). ∗ (F N f )). (4.7.19) This shows that using the FFT algorithm the discrete convolution can be computed

in O(N log 2 N) operations as follows: First the two FFTs of f and g are computed and multiplied (pointwise) together. Then the inverse DFT of this product is computed. This is one of the most useful properties of the FFT!

Using the Gentleman–Sande algorithm F N =P N A T for the forward DFT and the Cooley–Tukey algorithm F N = AP N for the inverse DFT,

F −1

F N H = AP ¯ N ,

we get from (4.7.19)

A((A ¯ T f ). = T ∗ (A g)). (4.7.20) N

h = AP ¯ N ((P N A f ). ∗ (P N A T g))

This shows that h can be computed without the bit-reversal permutation P N which typically can save 10–30 percent of the overall computation time.