DFT Beamforming Algorithms for Space-Time-Frequency Applications D. Rodriguez, M. Vargas Solleiro, ‘19. Avilks

Computational Signal Processing Group Electrical and Computer Engineering Department

University of Puerto Rico - Mayagiiez Campus Mayagiiez, Puerto Rico, 0068 I-9042

ABSTRACT This work presents modified variants, in a recursive format, of the Kahaner’s additive fast Fourier transform (FFT) algorithm. The variants are presented in Kronecker products algebra language. The language serves as a tool for the analysis, design., modification and implementation of the FFT variants on re-configurable field programmable gate array (FPGA) computational structures. The target for these computational structures are discrete Fourier transform (DFT) beamforming algorithms for space-time-frequency applications in wireless.

1. INTRODUCTION This work presents modified variants, in a recursive format, of the Kahaner additive fast Fourier transform (FFT) algorithm. The variants are presented in Kronecker products algebra language. The language serves as a tool for the analysis, design, modification and implementation of the FFT variants on re-configurable field programmable gate array (FPGA) computational structures. The target for these computational structures are discrete Fourier transform (DFT) beamforming algorithms for space-time-frequency applications in wireless. When using Kroncker products algebra, a given FFT algorithm can be written as a decomposition of basic factors or mathematical expressions which we term functional primitives. This decomposition action establishes a one-one correspondence between a mathematical formulation of an algorithm and a given hardware computational structure such as an FPGA. Variants of a given mathematical formulation can be obtained using properties of Kronecker products algebra. These variants may satisfy certain design criteria such as pipelining, parallelism, data flow control, etc. In turn, each of these new variants will produce a di.%erent hardware implementation. The efficiency of each algorithm is evaluated when a cost function is imposed on the design criteria. We proceed to describe in detail a Kronecker decomposition for the Kahaner’s FFT algorithm.

2. KRONECKER DECOMPOSITION OF KAHANER’S ALGORITHM In his paper [ 181, D. K. Kahaner describes a procedure for factoring the Fourier matrix FN when N = p ’ , p and y any integers. Kahaner’s factorization method produces, up to matrix factor expansion, what is commonly known as the Cooley- Tukey (C-T) decimation in frequency algorithm. In this section we describe Kahaner’s algorithm in detail, and then present it a Kronecker product formulation This will aid in the understanding of the Kronecker products language used to analyze other FFT algorithms later on

2.1 Kahaner’s Mathematical Formulation Kahaner starts by defining the discrete Fourier transform of N equally spaced data points X, , k = 0,. . . , N - I :

2iTi --rk N-l 2ni -- F

1 N-l B -- r- N c xke N =-cxkark, O<r<N, a=e N k=O N k=O

In matrix form,

F - - [ F 0 F . . . 1 F 1 N-l

where A is the matrix:

Proc. of SPIE Vol. 4045, Digital Wireless Communication II, ed. R. Rao, S. Dianat, M. Zoltowski (Apr, 2000) Copyright SPIE

86

1 1 1 . . . 1 1 a a2 . . . a CNml)

A = 1 a2 a4 ..* a 2(N-1) . e * 0 . . . ..c . . . . 1 aNsl a2CNs1) . . . aCNmliCN-ll

Kahaner proceeds to write a general expression for F, , r = O,l, . . ., N - 1:

1 N-l

F =- r N c xkark ‘k+tiV a

r(k+tM) ,A4 = py-l,r = O,l,...N 4 k=O

Writing r = pm + I, the following expression is obtained,

N-l

F, = Fpm+l = ‘&ark N k=O

-27ti

xk+.&&?talkapmk ,8 = e ’

For each fixed I, the following vector is formed:

Fp+l &2p+l l l l E;i,-l),,, p , l = O,l, l l l 7 P - 1 After some algebraic manipulations, this vector is written as,

1 1 1 1 ap a 2P

0 0 . 0 0 . 0 0

1 a(ha-l)P a2Cim1

000 1

P-1

c X&p

t=o

aC”-‘lP

II

P-l m . .

c x,+,Pa’ . 0 . 0 0 t=o

t=o

= Isx:,l= - o,...,p-1 - N

(2.3)

(2 7) .

(2.8)

By writing these p vectors (l = O,l, . . . , p - 1) in a column, the following result is obtained:

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-(1) F - -

where,

r F 0

F P

- 1

.

.

.

.F(iM-l)p+o -

F 1

F p+l . . 0

-qMl)p+l . 0 .

!

F P-l

F 2p-1 . * 0

F PM-p+p-1 I- -0) i 1 X0

1 =- - - N

-(l) i I Xl - - x -(‘) = A(o)x

I I . . . I D 6?D e . 0

D 2 ( > 8D 2 . . . ((g$

2

. . . . . . . . . 0 0

D P-l ( > 6D P-l ... (oP-l$-l

,D = diag[l,a,a2 ,..., aMa’]

-0) The vector F differs from the vector F by the permutation matrix YT, :

F=n,F(l)= X0 - N

Since the matrix B has the general form of the matrix A , this result is generalized:

F xO -- - N

(2. no)

(2.11)

(2.12)

(2.13)

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where

A 0 1 - -

hP - i-

I DJ ( ‘> 0 D(i) . ~. op-l D(i)

DJ ( ‘> - - ,j = OJ,..., y-l,Dy-’ = [l]

2.2 Kronecker Products Formulation

(2.14)

(2. as>

(2.16)

We proceed to describe this matrix factorization method in kronecker products form. We start by introducing the following definitions: The matrix P, s, of order n = > r l s is called the stride by s permutation matrix, and is defined by

P,, l d= (do,d~,d2~~***,dl,d,~+l,*~*,d~r-l~~+,s-l) > (2.17) for

The diagonal matrix D, s >’ of order s is defined by

D n,s = diag[l, Wn, W,“, . . . , W,S-’ 1 Wn = em”%

(2.18)

(2.19)

The twiddle factor (phase factor) matrix Dr n of order n : > / s / S

Tn,so = c @ Dr$/ > (2.20) 01 j-3 s

If TZ=PS,then

Tn,,.(n)=Tn, = c OD;;, = x@Dir >I* (2.21) > , 01 j<s S 01 j-3

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To arrive to a general form for the Fourier matrix FN , N = py 9 expressed in Kronecker products, we start with an expression for the Fourier matrix Fp and use this expression, and the Cooley-Tukey decimation in frequency algorithm expressed in Kronecker products, to obtain higher order Fourier matrices expressed in Kronecker products form:

F F N= pi = Fp = pp 1 Ah-1) = T -

> P,P &‘-1) = 1

P Ah-‘), y - 1

(2.22) where

A (r-1) - -

1 1 1 0 0 0 1 1 W w2 . . . w(P-1)

1 w2 py4 QO. w 2(P-1) -2m'

3 / =e P

. . . . . . . . . . . .

; wP-l

. .

w %P-1) . . . w(P-l)(P-l)

The Cooley-Tukey decimation in frequency algorithm allows us to write F 2

Fp2 = pp2 p (Ip QFp jTp2 p (F, QlpP)

in the following form:

Using

we get,

> >

dy-2) = T 2 P YP

(F, 0 Ip )

F P2

= P (I Q Fp)A(y-2) P2YP p

Tp2p(4= Cea:,,~~~ = [i7~p,~;,...,~;-l > > > 01 j-3

Using the expression for Fp given above, we obtain

Fp2 = Pp2 p (Ip 0 Pp 1 XI, 63 d~-1+o-2~ > > For the matrix F p3 , we write down again the expression for the Cooley-Tukey decimation in frequency algorithm:

where,

F P3

=pp3,p2(1p @Fp2bp3,p(Fp @Ip2)

TpJ,(r)= COD’, D > P >P’ P2>P =b,w 2,w2 w P-l

01 j-3 p p2P’*‘> p2 I

Using, and the expression given above for F 2 , we get,

F - P (I 8 Pp2 p B,; Q Pp,l)e (Ip2 0 dy-l’xI, 63 dy-2$o-3) P3 - P3>P2 P >

Continuing in the same manner:

F p4 =

Using

Acy-4) = T (F @ rp3) P4>P p

we get, F -

P4 - P

P4,P3 P @ p,3 p2 )(Ip2 @3 pp2 p )(Ip3 @ %,i )a (Zp3 6 dy-1f)o(p2 Q dy-2)lIp Q A(y-3))&-4) > 3

(2.23)

(2.24)

(2.25)

(2.26)

(2.27)

(2.28)

(2.29)

(2.30)

(2.31)

(2.32)

(2.33)

(2.34)

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III general, for a Fourier matrix 6;_,_, ,O 5 k < y , we write:

We, again, set

and the identity

F py-k = P Py-k y-k-1 ,P

A(Y-(y-k)) = A@)T y-k P

@ 1 y-k-l P

T py-k = YP c

WY Py-k JP y-k-1 ’

osk<y; 01 j-a

wpy-‘-’ = e

may be used to write down the elements of D P y-k

>P yskml ,O 2 k < y

The general expression for Fmysk thus becomes: Y

F py-k = P Py-k y-k-1

” l

,P >P y-k-2

x I

P 2 8 Ppy+2

,P y-k-3

(2.35)

(2.36)

(2.37)

(2.38)

(2.34)

3. CONCLUSIONS This has presented modified variants, in a recursive format, of the Kahaner additive fast Fourier transform (FFT) algorithm. The variants are presented in Kronecker products algebra language. The language has been used as a tool for the analysis, design, modification and implementation of the FFT variants on re-configurable field programmable gate array (FPGA) computational structures

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on SelectedAreas in Communications, vol. 17, NO. 11, pp. 1863, 1999. 4. J. S. Thompson, P. M. Grant B. Mulgrew, “Algorithms for Coherent Diversity Combining of Ilrl-ary Orthogonal

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Channels”, Wireless Networks, 1999. 6. J.-J. van de Beek, P. 0. Borjesson, M.-L. Boucheret, D. Landstrom, J. Martmez Arenas, P. odling, C. &tberg, M.

Wahlqvist, S. K. Wilson, “A Time and Frequency Synchronization Scheme for Multiuser OFDM”, IEEE Journal on Selected Areas in Communications, vol. 17, No. 11, pp. 1900, 1999.

7. J. B. Anderson, “Digital Transmission Engineering’, Prentice Hall, 1999. 8. A. Terras, “Fourier Analysis on Finite Groups and Applications”, Cambridge University Press, 1999. 9. R. Tolirnieri, M. An, “Time-Frequency Representations”, Birkhduser Boston, 1998. 10. E. Biglieri, J. Proakis, S. Shamai, “Fading Channels: Information-Theoretic and Communications Aspects”, IEEE

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13. P. J. Ashenden, P. A. Wilsey, Considerations of Object-Oriented and Generic@ Extensions to VHDL for High-Level Modeling”, Proceedings ofFDL ‘98, 1998.

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17. J. Litva, T. D.-Y. Lo, “Digital Beamforming in Wireless Communications”, Artech House, Inc., 1996. I8 Kahaner, D., “Matiix Description of the Fast Fourier Trasform,” IEEE Trans. on Audio Electroacust., Vol. 18, 1970, pp.

442-450. 19. E. Biglieri, M. Luise, Signal Processing in Telecommunications”, Proceedings of the 7th International Thyrrhenian

Workshop on Digital Communications, Viareggio, Italy, 1995. 20. K. Feher, “Wireless Digital Communications: Modulation and Spread Spectrum Applications”, Prentice Hall PTR, 1995. 21. S. U. Pillai, C. S. Burrus, “Array Signal Processing”, Springer-Verlag New YorkInc., 1989.

Figure 1. Multiplicative Channel Model

Figure 2. Tapped-Delay-Line Channel Model

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I. I I

:’ I t_- I ,--m-v- ---_ --_-_

------------------_-_I_______

1: ,,.,.,,,,,,,,,,,.: ,,,,,, :,.’ .’ ~- -___-----------’

.’ :

Figure 4. Binary Digital Comnmnic ;ations Divers ity Model

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