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Binary Polynomial Transforms for Nonlinear Signal Processing

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Binary Polynomial Transforms for Nonlinear Signal Processing BINARY POLYNOMIAL TRANSFORMS FOR NONLINEAR SIGNAL PROCESSING Jaakko Astola, Karen Egiazarian, Rugen oktern SOS Agaian Signal Processing Laboratory Electrical Engineering Department Tampere University of Technology University of Texas at San Antonio Tampere, FINLAND San Antonio, Texas, USA ABSTRACT ing computers or systems suitable for simultaneous arith- metical or logical operations. In this paper we introduce parametric binary Rademacher functions of two types. Based on them using different kinds 2. BINARY POLYNOMIAL FUNCTIONS AND of arithmetical and logical operations we generate a set of MATRICES binary polynomial transforms (BPT). Some applications of BPT in nonlinear filtering and in compression of binary 2.1. Rademacher Functions and Matrices images are presented. Let a and b be arbitrary integers. We form the classes r,(t,a, b) and s,(t,a, b) of real periodic functions in the 1. INTRODUCTION following way. Let Spectral analysis is a powerful tool in communication, sig- ro(t.a,b) E so(t,a, b) E 1, (1) nal/ image processing and applied mathematics and there are many reasons behind the usefulness of spectral analysis. First, in many applications it is convenient to transform a a, fortEU~~ol[$,fk++i), problem into another, easier problem; the spectral domain rn+l (t, a, b) = is essentially a steady-state viewpoint; this not only gives b, for t E U~~ol[ $t + 5&l *).2 insight for analysis but also for design. Second, and more (2) important: the spectral approach allows us to treat entire and classes of signals which have similar properties in the spec- tral domain; it has excellent properties such as fast algo- a: for t E LO,&I, n = 0, 1,2, . rithms, high energy compaction property of the transform wl(t,a,b) = b, fort E [h, &) data. 0, otherwise for t E [0, l), However, there are some problems in signal/ image pro- (3) Note that cessing (especially if the input data is binary) where they cannot be applied directly. a, for t E [0,t), rl(t,a,b) = sr(t,a,b) = We may properly ask what is a basic transform for bi- b, for t E [f, 1) nary signal/image processing which plays the same role for it a.5 spectral transform for linear processing. DEFINITION 2.1: The functions r,(t,a, b) arc called Similarly we may ask what is a basic transform for non- Rademacher (a, b)-junctions of I-type. The functions linear signal/image processing which plays the same role for s,,(t, a, b) are called Rademacher (a, b)-functions of II-type. it as spectral transform for linear processing. We will develop a compact representation of signals for later use in digital logic and nonlinear signal analysis. Our Note that when a = 1 and b = -1, the Rademacher approach will be based on introduction of parametric bi- (a, b)-functions of l-type coincide with the Rademacher func- nary Rademacher functions (BRF); on generating a set of tions [5]. Let us consider some properties of the generalized transforms using Rademacher functions and different kinds Rademacher functions defined above. of arithmetical and logical operations. Emphasis is placed PROPERTY 2.1: There are the following representa- on the two classes of transforms, introduced in this paper: tions of the Rademacher (a, b)-functions of the l-type: transforms generated by BRFs using only one operation (arithmetical or logical) and transforms generated by BRFs 1) r+l(t,a,b) = a+ (b- a)c,+l(t), t E [O,l), (4 using two operations (arithmetical or logical). We present the basic properties of these transforms. where c,,+l are the coefficients from This development is motivated also by potential appli- cations in areas such as nonlinear signal/image processing, t = c cn(t)2-* = c&2-n. (5) construction of new architectures of signal/image process- n=l n=l The discrete Rademacher (a, b)-matrices of l-type and II-type have the following properties. 2) r,,(t, a, b) = i [(a - b)sgn [sin(2”nt)] + a + b] , PROPERTY 2.2: ify>O t#$k=O ,..., 2”-‘, if y < 0: l)Rp)(x,a,b)=a+(b-a).xk(x); k=l,..., n, 3) There is the following representation of the where n is a natural number, and x, 0 5 L 5 2” - 1 is Rademacher functions of I l-type: written in the form n-1 s,tl(t,a,b)=[a+(b-a)c,tl(t)l~ci(t). (6) x=Cx,(x).2n-‘-1, j=O,l,..., n-l. (10) :=l 3=0 4) There is the following relation between the Rademacher (a, b)-functions of l-type and II-type: 2) RT(n, a. b) . R(n, a, b) = 2+‘[(a - b)2 . It,,) + (a + 2”-1-l bJ2. J(n)], where 1~~) is the identity matrix of order n and Jc,,) is the r,+l(t,a,b) = c wl(t - s,a,bl. (7) (n x n) matrix whose all entries are equal to 1. In particular, for the classic Rademacher matrices For a natural number n and integers a and b we define the RT(n, 1, -1) . R(n, 1, -1) = 2” . It,). following discrete functions: 3) Let n be a natural number, 1 < k 5 n and suppose that x, 0 5 x 5 2” - 1 is written in the form (10). Then &“)(x,a,b) = rk( s,a,b): (8) k-l k = 0, 1, . n - 1; z = 0, 1, . 2” - 1; n 2 0. Sp)(x,a,b) = [a+(b-a).xk(x)]n[l-xi(x)]; k = l,...,n. i=O 2x + 1 .$“)(x,a, b) = sk(2n+lvav b)g (9) (11) 4) ST(n, a, b).S(n, a, b) = a(a+b).[J( n)+ Cv:2 2’P’i) ] k=O,l,..., n-l; z=O,l,..., 2”-1; n>O. I 1 fn) +b(b - a) . D(,), DEFINITION 2.2: The rectangular (2” x (n + 1)) where J(,) is the (n x n) matrix whose all the entries are matrix R(n,a, b), whose (2, k)th element is RJ,“)(x,a, b), equal to 1, P[z) is the (n x n) matrix whose leftmost (n - x = 0, 1, . ,2”-1, k = 0,. , n, is called Rademacher (a, b) ;) x (n - j) submatrix is Jc,-,), i = 1, . n - 2: and matrix ofl-type (of order n). The rectangular (2” x (n+l)) matrix S(n, a, b), whose (x, k)th element is .S$“)(x, a, b), x = DC,) = diag(2n-1, 2”-‘, . 2,1). 0, 1, . ,2” - 1, k = 0,. , n, is called the Rademacher (a, b) matrix of II-type (of order n). REMARK 2.1: For each pair of integers (n, k), n > 0, 2.2. (a, b; T)-Polynomial Functions of I-Type 1 ,< k 5 n, the kth column of R(n, a, b) defines a func- tion (0, 1, . ,2” - 1) + {a, b}. This set of functions is Let T be an arbitrary associative arithmetical or logical op- called the system of discrete Rademacher (a, b) functions of eration, (mo,ml, . m,-1) be the binary representation I-type. Analogously, for each pair of integers (n, k), n > 0, of an integer m = C::ol m,2n-1-‘, 0 5 m 5 2n - 1. Let 1 5 k 5 n, the kth column of S(n, a, b) defines a function also H @ G be the Kronecker product of the matrix H and (0, 1, . , 2” - 1} -+ {a, b}, the system of which is called the the matrix G, and H@” be the Kronecker nth power of t.he system of discrete Rademacher (a, b) functions of II-type. matrix H, i.e. H@” = p @ H y . @ H,. n times EXAMPLE 2.1: When n = 4, the Rademacher (a, b)- We form a class of functions &)(z, a, b; T) from the matrices of l-type RT(4, a, b) and II-type ST(4,a, b) have system of discrete Rademacher (a, b)-functions of I type (i.e. the following forms, respectively: from {R!f?(x,a,b)}~~~) in the following way. Let 1111111111111111 bbbbbbbb aaaaaaaa d?(x, a,6 7) = d$m, ,...,m,-l (2, a, b; 7) a a a a bbbbaaaabbbb a a bbaabbaabbaabb = [R~)(x,a,b)]mo T... 7 [Rf,“‘l(x,a,b)]m”-l, (12) ( a b a b a bababababab where T is an arithmetical or logical operation. If T is a 1111111111111111 logical operation, then we naturally assume a, b E (0, 1). aaaaaaaa bbbbbbbb a a a a bbbb00000000 DEFINITION 2.3: The functions &)(x,a, b; T), m = a a bb000000000000 0, . ..n - 1, are called (a, b; t)-polynomial functions. If T is ( ab00000000000000 an arithmetical (logical) operation, the (a, b; r)-polynomial where T denotes matrix transposition. functions are called arithmetical (logical). DEFINITION 2.4: The square matrix @(n, a, b; 7) = DEFINITION 2.6: The square matrix H(n, a, b) [&?“)(z, a, b; T)], z, m = 0, 1, . 2”-1, which has as columns =[h?)(z,a,b)], x,r=O,l,..., 2”-1,whichhasascolumns the (a, b; r)-polynomial functions, is called the (a, b; T)- poly- the (a, b)-polynomial functions of II-type, is called the (a, b)- nomial matrix. polynomial matrix of II-type. Let T be multiplication operation. The (a, b, x)-polynomial Note that the classical Haar functions (matrices) are the functions r&’ are formed via Rademacher (a, b)-functions particular case of the (a, b)-polynomial functions (matrices) of II-type where a = 1: b = -1.
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