COMPLEX CYCLOTOMIC POLYNOMIALS AND THE RELATED ALGORITHMS FOR CYCLIC CONVOLUTION

Hari Krishna and F. V. Chrys Mendis Department of Electrical Engineering

National University of Singapore 10 Kent Ridge Crescent

E-mail: hkrishnaBsuvm. acs. syr. edu, elemendi@nusvm. bitnet Singapore 05 1 1

Abstract: This paper investigates the factorization properties of cyclotomic polynomials over the Jeld of complex rational numbers. Based on this factorization and the Chinese remainder theorem, we anaIyze the mathematical structure of the associated algorithms for computing the cyclic convolution of data sequences.

I. Introduction

This paper focuses on the analysis of the Chinese remainder theorem (CRT) based algorithms for computing the one-dimensional cyclic convolution (CC). The factorization properties of cyclotomic polynomials (CPs) over the field of complex rational numbers are investigated. Based on this factorization and the CRT, the mathematical structure of the associated algorithms for computing the cyclic convolution (CCA) is analyzed.

This paper is a continuation of an earlier paper [I]. The notation has been left unchanged for obvious reasons. The readers are assumed to be familiar with [ I ] in order to maintain this continuity.

The organization of this paper is as follows. In Section I1 the problem is defined. In Section 111, the factorization of uN - I in terms of cyclotomic polynomials over the field of complex rationals is presented. Section IV is on the properties of the bilinear algorithm for computing the CC. In Section V, the conclusions of the work are presented.

JI. Notation and Previous Results

The number systems that we encounter in this work are as follows: I, ring of integers Z, field of rational numbers CI, ring of complex integers, a+jb E CI for a,b E I CZ, field of complex rational numbers, a+jb ECZ for a,b E Z

1058-6393/93 $03.00 0 1993 IEEE 354

It is seen that I c CI c CZ, and I c Z c CZ. The CRT based CCA derives its structure from the factorization of uN -1, in terms of irreducible polynomials over Z. These irreducible polynomials are called CPs. The R, T and P matrices involved in the CRT reduction and reconstruction are highly structured and can be determined in closed form [I]. The elements of those blocks of R and P matrices that correspond to the CPs C, ( U ) with d having at most two odd prime factors take values in the set U, U={O,l,-l}. This is a highly desirable property to be pursued further and generalized over CZ.

The factorization over CZ is important and useful in many respects. It is expected to result in CCAs having lower multiplicative complexity for processing data sequences defined over I, Z, CI, CZ and the fields of real and complex numbers.

Complex integers (or more generally complex rational numbers) are also known as Gaussian numbers. Factorization of u N - 1 over CZ or equivalently over CI has not received much attention in the digital signal prqcessing literature. We came across two places (p. 168 of [2] and p.235 of [3]) where it has been mentioned. In [2], it is merely stated that factorization over CZ could be used. In [3], it is incorrectly stated that z8 - I factorizes

as ( z - I ) ( z + I ) ( z - j ) ( z + j ) ( z 4 + I ) over CZ. The

factorization z4 + I = ( z 2 + j ) ( z 2 - j ) has been left out. The approach used to obtain complex rectangular transforms in [4,5] is an ad-hoc one with very few results of analykal interest.

111. Complex Cyclotomic Polynomials

The field CZ can be studied as a polynomial extension of the field Z. An element c E CZ, can be expressed as c=n+jb, a,bEZ. Thus elements in CZ are

polynomials of degree up to 1 over Z. The computation

in CZ is perfornied modulo j 2 + 1, which is an irreducible polynomial over Z. In fact, CZ can be interpreted as a quadratic extension of Z. the discriminant of the polynomial being -4. Such a field is represented as

K = Z(&) in [6].

The factorization of U&' - 1 over Z, is given by

U N -1=n f d ( U )

dln

where the roots of c d ( u ) are primitive d-th roots of unity. Thus.

(I .d)= 1

w = exp( y ) (3)

w being a primitive cl-th root of unity. Since C d ( u ) is irreducible over Z, it may be used to generate a field over Z. Let us represent such a field extension of Z by Z ( d ) . Z(d) is also called a cyclotoniic field. Both CZ and Z(d) are root fields of a polynomial over Z and therefore a Galois estension of Z [6]. The question we face is whether Cd (U) factorizes further over CZ.

The key to answer this question comes from the application of Dedekind's "reciprocity theorem" of Galois theory [7]. The statement of this fundamental theorem is as follows: Theoretir 1 (Dedekiiid's "reciprocitv theorem '3. Let f and g be irreducible polynomials with coefficients in K . Let f

decompose into irreducible factors, f = f , f , ...A, over K(b), where b i s a root of g . Similarly let g decompose

as. g = g , g 2 . . . g y over K(a) . where a is a root of ,f:

Then p=yand the g, can be reordered in such a way that

the ratio of deg ,fk and deg gk is the same for all k=1,2 ,..., p. This statement of the theorem is taken verbatim from (7. p.661. There are certain minor notational discrepancies between its statement and the rest of the paper. We hope that this will not confuse the reader.

We apply this theorem with .f=Cd (U). g =U' + 1,

K(b)=CZ, h=j, K ( a ) = Z ( d ' . and a-w. Since degk) =2,

we have two possibilities, p = l or p=2. For p= 1. Cd (U)

is irreducible over CZ and U' + 1 is irreducible over Z ( d ) . For p=2, u2 +1 factorizes into two degree-one

polynomials over Z ( d ) . de&,) = deg(g2) = l . As a

result, for p=2, C d ( u ) also factorizes into two irreducible factors of equal degree, that is,

Cd(U) =cld(U).c2d(U) (4)

deg( r i d (U)) = deg( c2d ( U)) = deg( c d (U)) / 2 (5) Therefore. if p=2. then Z(d) is a splittingfield for

U' + 1 [SI. Also, since CZ is the splitting field for u2 + 1, we must have CZ c Z ( d ) for the case p=2. In the following, we establish necessary and sufficient conditions for CZ c Z ( d ) .

Theorem 2 [6]. The quadratic number field Z ( n ) with

discriminant -4 is the field ZC4) of the 4-th roots of unity.

Specifically, 4- 1 is the primitive 4-th root of unity. If d = d,d2 ... d , is a decomposition into pairwise

relatively coprime factors. then w (the primitive d-th root of unity) can be uniquely represented as a product of primitive d,-th primitive roots of unity 0, . that is,

w = w,w2 ... 0, corresponding to the decomposition of the residue class ring mod d into direct sum of the residue class rings mod d, based on the CRT for integers. This leads to

that is, Z ( d ) is the compositum of the fields Z'di) [6]. Theorem 2 implies that Z C d ) is a splitting field for U' + 1 if and only if Z") c Z ( d ) . Combining this statement with (S), we get the condition

for Z(4) c Z(d ) . Let us analyze this case further. Taking complex conjugates on both sides of (4), we get

(6)

(7) Z(d) = Z(d,)Z(d,) z'd,' ...

41d (8)

f d ( U ) =cl;(U).c2;(U) (9) Since the coefficients of c l d (U) and C2d (U) are complex

(or else C d ( u ) is reducible over Z), (4) and (9) are valid if and only if

(10) c'ld(U) = ('2i(U) and vice versa. The above analysis can be summarized in the form of the following theorem. Theorem 3 . The CP, C, (U). which is irreducible over Z factorizes into two irreducible factors of equal degree over CZ if and only if 4 I d. The factor polynomials are

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complex conjugates of each other. If d is not divisible by 4, then C, (U) is irreducible over CZ.

The polynomials C1, (U) and C2 (U) will be termed as the complex CPs (CCPs). The focus of this work is on the analysis of the factors C1, (u)and C2, (U) and the resulting CCA for the case 4 I d. Some examples of CCPs are as follows:

Cd(U)=Ud12 +1=(Ud'4+j)(Ud'4 - j ) , d = 2 " , n122,

C , 2 ( ~ ) = ~ 4 - ~ 2 + 1 = ( u 2 + j ~ - 1 ) ( ~ 2 - j ~ - 1 ) , and

c ~ , , ( ~ ) = U* --6 +u4 - U 2 + 1 = (U4 -jU3 - U z + j U + l ) .

(U4 +jU3 - U 2 - j u + l ) Under the condition that 41d, a method to obtain C1, (U) and C2d (U) from C,, (U) is as follows. Case 1. d=4M, (2Sz./)=l. In this case, an integer Id,

(i,d)=l can be expressed as i=(4k+Mi) mod d, klM, fs4, (kSz./)=l and (1,4)=1. Clearly, 1=1 or 1=3. We partition the values of i , (J,d)=l, into two sets corresponding to 1=3

and I=1. Thus, cd(U) as expressed in (2) can be written as:

(k.M)=l (k,M)=l

Since m=exp( j2 l r /d ) , mM =exp(j l r /2)= j and

m4 = exp(j2 lr/ M) , and (1 1) simplifies to:

1

(k.M)=l J

= Cl,(U).C2,(U) (13) Here, $(M) is the Euler's totient function of M. Noting that m4 is an M-th primitive root of unity, (13) leads to

and c~ , (u ) = c ~ , ~ ( u ) = (-j)H')CM(ju) (14)

and >

C2,(U) = ( j ) ~ M ) c M ( ( - j U Z " ' ) (18) The expressions in (17) and (18) follow directly from (14), (15) and (16).

Since the coefficients of C,(U) are integers, the

coefficients of c l d (U) and C2, (U) are complex integers. In addition, their values alternate as either integers or

purely imaginary integers. Let U ' = { 0, +I, + j } . Summarizing this discussion, we have the following lemma. Lemma I. The coefficients of the CCPs, cld(U) and

C ~ , ( U ) take values in CI. These values alternate as either integers or purely imaginary integers. I f M has at most two odd prime factors, then the coefficients of the CCPs take values in the set U' .

At this point, the factorization of uN - 1 in terms of its irreducible factor polynomials over CZ is obvious. The CP c d (U) factorizes further over CZ if and only if

4Id. I ~ ~ I N , that is, N has the form N=2 'n , ,222. (2,n)=I, then it can be easily shown that the fraction of CPs that constitute -1 and further factorize over CZ is given by (c-l)/(c+l). This includes the highest degree CP, that is C, (U), as well. Thus, of all the CPs that constitute u N - I, either all are irreducible over CZ or at least one-third (c22) factorize further. In digital signal processing applications, the condition that 4 IN may not be a restrictive condition.

IV. Structure of Cyclic Convolution Algorithms

Throughout this section, it is assumed that the condition 4 IN is satisfied for the CCPs to exist. Given the cyclotomic factorization of - 1 over CZ, we now turn to the CRT-P reduction and reconstruction matrices R, T and P. Each of these matrices has a block structure with D blocks, D being the number of complex cyclotomic

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factors of uN - 1. Here, D=2cl(c+l) times the number of factors of N , N = 2'n, ( 2 , n ) = 1 .

The elements in the d-th block of the R, T and P matrices depend only on the corresponding CP, c d (U), for d not divisible by 4. They have already been studied in [I] . Therefore, in the following, we focus only on the

of the R, T and P matrices, 4 I d. These blocks will be labeled as the d, I-th and d,2-th blocks respectively.

CCPS, c1d (U) and c2d (U) and the corresponding blocks

Let @ (d) = deg( c1d (U)) = deg( c2d (U)) = $b( d)/2 .

The reconstruction polynomials tld (U) and t2, (U) are obtained as [ I , eqn. (15)]:

f ld(U) Substituting for cld(U) from (17) and simplifying, we get

t ~ , (U) = 2m-2 (- j)",) t , ( j F 2 ) (19)

I, (U) being the reconstruction polynomial for the CP

C , ( U ) . Replacing j by -j in (19), we get t2d(U). The following lemma is obtained from (19) and the analysis for t , (U) in [9]. Lemma 2. The coefficients of the reconstruction polynomials tld (U) andt2, (U) are alternately either integers or purely imaginary integers. If d has at most two odd prime factors, then the largest absolute value of any non-zero coefficient of the polynomials tld(U) andt2,(u) is # ( d ) , (ignoring N in the denominator).

The d,l-th and d,2-th blocks of the T matrix have

dimensions (3@( d) - 2) x (2 @ (d) - 1). Their structure is

analogous to the structure of Td corresponlng to td(U) in eqn.(53) of [ I] . Now, let us analyze the RPs associated with C1, (U) and c2d (U). By definition,

f 1, (U) = ( U N - l)/Cld (U)

For d = 2 " M , ( 2 M = l , m22, N must have the form

N = 2"n, Mln, (2,n)=I and c-z. Thus P Id(u) can be expressed as:

Setting juZw2 = t in (20), we get:

d = 2m.M N = n.2'

MIN (M,2) = (n,2) = I c 2 m 2 2

I d = M

I MIN Following the same derivation as in Appendix A of [l] , we-have:

1 d = M

It = ju'

The expression for f 2, (U) is obtained by replacing j by -j in the RHS of (21). It is clear from [ 1, eqns. (58), (61), (62), (63)] and (21) above, that the coefficients of the RPs, Pld(U) andP2,(u) take values in the set U' if d has at most two odd prime factors. In these cases, the number of non-zero coefficients can be determined in closed form. The structure of the d, I-th and d,2-th blocks of the P matrix is analogous to the structure of the fd

corresponding to f d ( U) in (64) of [I] . The dimension of

these blocks is N x(3@(d)-2) . Summarizing this analysis, we get the following lemma. Lemma 3. The Coefficients of the RPs, Pld(u) and

P2d (U), are alternately either integers or purely imaginaly integers. If d has at most two odd prime factors, then the coefficients take values in the set U ' .

In the following, we analyze the structure of the CRT-P reduction matrix, R. The computation of X 1, (U)

is performed as: (U) = x(U) mod c l d (U)

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where deg(X(u))=N-1 and deg Cl,(u) = &(d) = &d)/2. Following (65)-(67) of [ I ] , we conclude that the i-th column of the block RI , of the reduction matris is obtained by listing the elements of the polynomial

r13,,(u)=u' modCI,(u), i = O , I ,..., N - 1

Once again, Cl,(u) consists of all the primitive d-th

roots of unity. Therefore, U, = I modC1, (U), while

U' $ 1 mod CI, (U) for i<d. Thus, RI, is periodic with the (d+i)-th column the same as the i-th column. Also,

(22)

rl t . , (u) = U', i < @ ( d ) (23)

For @ ( d ) < i < d , w e g e t :

= U' mod C, ( juZm-* ) Writing i = I , 2m-2 + i , , i , < 2m-2. i , < 4 M , we get:

Replacingj by -j, we get an expression for r 2 , , , ( ~ ) . The

polynomial q,,,,, ( t ) is periodic as y, ,+M,M(t) = r',,,,, ( t ) . This analysis leads to the following lemma. ternrna 4. The coefficients of the polynomials

U' modCI,(u), {=@,I, ..., d-I are obtained from

U' modC,(u),r=O,I .... 41-1, d = 2 " h f . (2iM)=I. They occur alternately as either integers or purely imaginary integers and take values in the set U' if d has at most two odd prime factors.

We conclude this section by restating that the coefficients of the CCPs and the elements of the CRT-P matrices R, T and P always alternate between integer and purely imaginary integer values (ignoring N in the denominator for 7). In addition, the coefficients of those CCPs and the elements of those blocks of the R and P matrices for which d has at most two odd prime factors, take values in the set I J ' .

V. Discussion and Conclusions

A preliminary analysis of the computational complexity indicates that complex cyclotomic polynomials lead to a reduction in the number of multiplications for computing the cyclic convolution of complex-valued sequences as compared to the cyclotomic polynomials. For example, a cyclotomic polynomial based cyclic convolution algorithm for N=60 requires 200 MULT. For complex-valued sequences, these multiplications are complex as well. A complex cyclotomic polynomial based cyclic convolution algorithm for N=60 requires 133 MULT. For real-valued sequences, both approaches appear to require the same number of arithmetic operations.

In this paper, we have extended the cyclotomic factorization of uN - I over CZ. ~nteresting~y, the elements of the resulting matrices turn out to be either integers or purely imaginary integers. In many cases of interest, the matrices involved in the computation have simple elements such as 0, 1 , - 1 , j, and -j .

References

[ 1]H. Krishna, "Mathematical Analysis of the Chinese Remainder Theorem based Algorithms for Cyclic Convolution," submitted for publication. [2]R. Tolimieri, M. An and C. Lu, Algorithms for Discrete Fourier Transform and Convolution, Springer- Verlag, 1989. [ 3lD.G. Myers, Digital Signal Processing, Efficient Convolution and Fourier Transform Techniques, Prentice Hall, 1990. I4lV.U. Reddy and N.S. Reddy, "Complex Rectangular Transforms," Proc. International Conference on Acoustics, Speech and Signal Processing, 1979. [5]N.S. Reddy and V.U. Reddy, "Complex Rectangular Transforms for Digital Convolution," IEEE Trans. Acoustics, Speech and Signal Processing, Vol. ASSP-28,

[6]H. Hasse, Number Theoy, Springer-Verlag, 1980. [7]H.M. Edwards, Galois Theory, Springer-Verlag, 1984. [8]L.C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, 1982.

NO. 5 , pp. 592-596, Oct. 1980.

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