Skip to content Skip to navigation

OpenStax_CNX

You are here: Home » Content » A Bilinear Form for the DFT

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 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.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Rice Digital Scholarship

    This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "Automatic Generation of Prime Length FFT Programs"

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

  • Featured Content display tagshide tags

    This module is included inLens: Connexions Featured Content
    By: ConnexionsAs a part of collection: "Automatic Generation of Prime Length FFT Programs"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • UniqU content

    This module is included inLens: UniqU's lens
    By: UniqU, LLCAs a part of collection: "Automatic Generation of Prime Length FFT Programs"

    Click the "UniqU content" link to see all content selected in this lens.

  • Lens for Engineering

    This module is included inLens: Lens for Engineering
    By: Sidney Burrus

    Click the "Lens for Engineering" link to see all content selected in this lens.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

A Bilinear Form for the DFT

Module by: Ivan Selesnick, C. Sidney Burrus. E-mail the authors

Summary: This collection of modules is from a Rice University, ECE Department Technical Report written around September 1994. It grew out of the doctoral and post doctoral research of Ivan Selesnick working with Prof. C. Sidney Burrus at Rice. Earlier reports on this work were published in the ICASSP and ISCAS conference proceedings in 1992-94 and a fairly complete report was published in the IEEE Transaction on Signal Processing in January 1996.

A Bilinear Form for the DFT

A bilinear form for a prime length DFT can be obtained by making minor changes to a bilinear form for circular convolution. This relies on Rader's observation that a prime pp point DFT can be computed by computing a p-1p-1 point circular convolution and by performing some extra additions [2]. It turns out that when the Winograd or the split nesting convolution algorithm is used, only two extra additions are required. After briefly reviewing Rader's conversion of a prime length DFT in to a circular convolution, we will discuss a bilinear form for the DFT.

Rader's Permutation

To explain Rader's conversion of a prime pp point DFT into a p-1p-1 point circular convolution [1] we recall the definition of the DFT

y ( k ) = n = 0 p - 1 x ( n ) W k n y ( k ) = n = 0 p - 1 x ( n ) W k n
(1)

with W = exp - j 2 π / p W = exp - j 2 π / p . Also recall that a primitive root of p p is an integer r r such that r m p r m p maps the integers m = 0 , , p - 2 m = 0 , , p - 2 to the integers 1 , , p - 1 1 , , p - 1 . Letting n = r - m n = r - m and k = r l k = r l , where r - m r - m is the inverse of r m r m modulo p p, the DFT becomes

y ( r l ) = x ( 0 ) + m = 0 p - 2 x ( r - m ) W r l r - m y ( r l ) = x ( 0 ) + m = 0 p - 2 x ( r - m ) W r l r - m
(2)

for l=0,,p-2l=0,,p-2. The `DC' term fis given by y(0)=n=0p-1x(n).y(0)=n=0p-1x(n). By defining new functions

x ( m ) = x ( r - m ) , y ( m ) = y ( r m ) , W ( m ) = W r m x ( m ) = x ( r - m ) , y ( m ) = y ( r m ) , W ( m ) = W r m
(3)

which are simply permuted versions of the original sequences, the DFT becomes

y ( l ) = x ( 0 ) + m = 0 p - 2 x ( m ) W ( l - m ) y ( l ) = x ( 0 ) + m = 0 p - 2 x ( m ) W ( l - m )
(4)

for l=0,,p-2l=0,,p-2. This equation describes circular convolution and therefore any circular convolution algorithm can be used to compute a prime length DFT. It is only necessary to (i) permute the input, the roots of unity and the output, (ii) add x(0)x(0) to each term in Equation 4 and (iii) compute the DC term.

To describe a bilinear form for the DFT we first define a permutation matrix QQ for the permutation above. If pp is a prime and rr is a primitive root of pp, then let QrQr be the permutation matrix defined by

Q e r k p - 1 = e k Q e r k p - 1 = e k
(5)

for 0kp-20kp-2 where ekek is the kthkth standard basis vector. Let the w˜w˜ be a p-1p-1 point vector of the roots of unity:

w ˜ = ( W 1 , , W p - 1 ) t . w ˜ = ( W 1 , , W p - 1 ) t .
(6)

If ss is the inverse of rr modulo pp (that is, rs=1rs=1 modulo pp) and x˜=(x(1),,x(p-1))tx˜=(x(1),,x(p-1))t, then the circular convolution of Equation 4 can be computed with the bilinear form of (Reference):

Q s t J P t R t B t C t R - t P J Q s w ˜ * A R P Q r x ˜ . Q s t J P t R t B t C t R - t P J Q s w ˜ * A R P Q r x ˜ .
(7)

This bilinear form does not compute y(0)y(0), the DC term. Furthermore, it is still necessary to add the x(0)x(0) term to each of the elements of Equation 7 to obtain y(1),,y(p-1)y(1),,y(p-1).

Calculation of the DC term

The computation of y(0)y(0) turns out to be very simple when the bilinear form Equation 7 is used to compute the circular convolution in Equation 4. The first element of ARPQrx˜ARPQrx˜ in Equation 7 is the residue modulo the polynomial s-1s-1, that is, the first element of this vector is the sum of the elements of x˜x˜. (The first row of the matrix, RR, representing the reduction operation is a row of 1's, and the matrices PP and QrQr are permutation matrices.) Therefore, the DC term can be computed by adding the first element of ARPQrx˜ARPQrx˜ to x(0)x(0). Hence, when the Winograd or split nesting algorithm is used to perform the circular convolution of Equation 7, the computation of the DC term requires only one extra complex addition for complex data.

The addition x(0)x(0) to each of the elements of Equation 7 also requires only one complex addition. By adding x(0)x(0) to the first element of CtR-tPJQsw˜*ARPQrx˜CtR-tPJQsw˜*ARPQrx˜ in Equation 7 and applying QstJPtRtQstJPtRt to the result, x(0)x(0) is added to each element. (Again, this is because the first column of RtRt is a column of 1's, and the matrices QstQst, JJ and PtPt are permutation matrices.)

Although the DFT can be computed by making these two extra additions, this organization of additions does not yield a bilinear form. However, by making a minor modification, a bilinear form can be retrieved. The method described above can be illustrated in Figure 1 with u=CtR-tPJQsw˜u=CtR-tPJQsw˜.

Figure 1: The flow graph for the computation of the DFT.
Figure 1 (fig1.png)

Clearly, the structure highlighted in the dashed box can be replaced by the structure in Figure 2.

Figure 2: The flow graph for the bilinear form.
Figure 2 (figB1.png)

By substituting the second structure for the first, a bilinear form is obtained. The resulting bilinear form for a prime length DFT is

y = 1 Q s t J P t R t B t U p t V p 1 C t R - t P J Q s w * U p 1 A R P Q r x y = 1 Q s t J P t R t B t U p t V p 1 C t R - t P J Q s w * U p 1 A R P Q r x
(8)

where w=(W0,,Wp-1)tw=(W0,,Wp-1)t, x=(x(0),,x(p-1))tx=(x(0),,x(p-1))t, and where UpUp is the matrix with the form

U p = 1 1 1 1 1 U p = 1 1 1 1 1
(9)

and VpVp is the matrix with the form

U p = 1 -1 1 1 1 U p = 1 -1 1 1 1
(10)

References

  1. Burrus, C. S. (1988). Efficient Fourier Transform and Convolution Algorithms. In Lim, Jae S. and Oppenheim, Alan V. (Eds.), Advanced Topics in Signal Processing. Prentice Hall.
  2. Rader, C. M. (1968, June). Discrete Fourier Transform When the Number of Data Samples is Prime. Proc. IEEE, 56(6), 1107-1108.

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 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