Efficient Implementations of the Chinese Remainder Theorem for Sign Detection and Residue Decoding

Transcription

1 646 IEEE TRANSACTIONS ON COPUTERS, VOL. c-34, NO. 7, JULY 1985 Efficient Implementations of the Chinese Remainder Theorem for Sign Detection and Residue Decoding THU VAN VU Abstract -Two conversion techniques based on the Chinese remainder theorem are developed for use in residue number systems. The new implementations are fast and simple mainly because adders modulo a large and arbitrary integer are effectively replaced by binary adders and possibly a lookup table of small address space. Although different in form, both techniques share the same principle that an appropriate representation of the summands must be employed in order to evaluate a sum modulo efficiently. The first technique reduces the sum modulo in the conversion formula to a sum modulo 2 through the use of fractional representation, which also exposes the sign bit of numbers. Thus, this technique is particularly useful for sign detection and for any operation requiring a comparison with a binary fraction of. The other technique is preferable for the full conversion from residues to unsigned or 2's complement integers. By expressing the summands in terms of quotients and remainders with respect to a properly chosen divisor, the second technique systematically replaces the sum modulo by two binary sums, one accumulating the quotients modulo a power of 2 and the other accumulating the remainders the ordinary way. A final recombination step is required but is easily implemented with a small lookup table and binary adders. Index Terms -Fractional representation, multioperand modular addition, quotient-remainder representation, residue decoding, residue number system, sign detection. I. INTRODUCTION U NLIKE the fast operations of addition, subtraction, and multiplication, most other operations in residue number systems are found to be slow because of their dependence on the magnitude of residue numbers [1]-[3]. For example, the sign of a residue number is determined by comparing the number with /2 where is the range of the number system. agnitude information is also indispensable in a redundant residue number system because a possible first step toward correcting errors is to compare projections of a number with the ratio T/R where T is the total range and R is the redundant range. Since the magnitude is not explicitly available in the residue representation of a number, some form of conversion from residues to digits in a weighted number system is necessary. Currently known techniques for residue decoding are based either on the Chinese remainder theorem or on a mixed radix conversion [4]-[13]. Although the Chinese remainder theorem provides a direct conversion formula which is simple and potentially fast, its imple- anuscript received December 22, 1983; revised June 1, The author is with the Government Aerospace Systems Division, Harris Corporation, elbourne, FL mentation into high-speed digital systems has been hindered by the lack of adders modulo a large and arbitrary integer. On the contrary, conversion to mixed radix digits can be easily realized with lookup tables of modest size, but the process is notoriously slow due to the propagation of carries, a computational bottleneck characteristic of weighted number systems. In an effort to speed up some of the difficult operations in residue number systems, the major obstacle in the implementation of the Chinese remainder theorem has been examined and successfully removed. This paper presents two new hardware realizations of the theorem which consist strictly of binary adders and lookup tables of small address space. The elimination of modular adders is possible because with an appropriate representation of the summands different than integers, addition modulo a large and arbitrary integer can be transformed equivalently into simple addition modulo a power of 2. Thus, this new development, coupled with the use of fast techniques for multioperand binary addition, should make the theorem a faster alternative for conversion purposes than techniques relying on mixed radix digits. Section II begins the discussion with a review of past implementations of the theorem, emphasizing achievements and limitations which motivate the development of the new implementations. Section III shows how afractional representation can be used to accelerate sign detection and other operations based on magnitude comparison. A detailed error analysis and examples are also included. In Section IV, conversion to binary numbers, signed or unsigned, is demonstrated to be most efficient with the use of a quotientremainder representation. II. THE CHINESE REAINDER THEORE In this theorem [1] an integer X is converted from its residue number representation by the weighted sum X =I (WiXi)m) (1) where mi's are the system moduli, = mim2 a * * m, is the system range, wi's are constants such that ((/mi)wi)mi = 1, is -the residue of X with respect to mi. As only and each xi integers in the range [0, ) are considered, the weighted sum is evaluated modulo ; that is, X is the nonnegative remainder obtained when the sum is divided by. The Chinese remainder theorem may be used as the basis for a fast and simple residue number decoder (Fig. 1). Since /85/ $01.00 C 1985 IEEE

2 VAN VU: IPLEENTATIONS OF THE CHINESE REAINDER THEORE xi x2 X3 xni n steps. Then the raw result is compared with a series of reference points, and a proper correction-term is generated to be RO RO applied to the sum. Normally, for a sum of n operands, all of which are less than, n - 1 reference points are required. sn-i sn By shifting the sum and scrutinizing overflow bits from binary adders employed in accumulating the sum, Fraser and Bryg [7] managed to cut the number of comparisons down to exactly one. Their technique worked correctly for the example given in [7]. Later, Cheng and Huang [8] formalized the technique and revealed, however, that a certain inequality must be satisfied by in order to keep the number of comparisons at one. It was found that as the n'umber of moduli x increases, must approach a power of 2, a restriction which Fig. 1. Residue decoding by the Chinese remainder theorem. also complicates the selection of mi's. The new conversion techniques described in this paper are xi is usually small, the values motivated by both accomplishments and shortcomings of past implementations of the theorem. First, it is ideal to have the Si =- (WiXi)m (2) bulk of the computation (i.e., the summation of n numbers) 'mi carried out by binary adders. Although intermediate coras a function of xi. rection steps are allowed, they must be in a very simple form can be stored practically in a lookup table For example, 5 bit residues occupy only 332 locations with a such as dropping the most significant bits. This is to make word length equal to Flog2 ]. However, Ithe sum modulo possible the use of standard binary hardware and fast techcauses a major implementation problem be.cause is gener- niques for multioperand binary addition to fully exploit the ally a very large and arbitrary integer. The common practice parallel conversion algorithm provided by the theorem. It is of substituting lookup tables for modular adders is obviously also desired to complete any final correction step at a minimal out of the question. On the other hand,,a modular adder cost using as much simple hardware as possible. Last and designed from a conventional binary ad( der would require most importantly, there must not be any severe restrictions on additional logic to detect overflow and correct the over- the definition of the system moduli. flowed sum [4]. Such an adder must be custom designed and It turns out that these objectives can be achieved by an is more expensive to operate than an ordi: nary binary adder. appropriate representation of si's other than the usual integer Hence, clever ways for handling the sum rmodulo must be representation. Since the summands are intended to be found if an efficient residue number decoder based on the fetched from lookup tables, representing them in a different theorem is to be built. form does not require any special real-time processing. The Two approaches have been taken by ivarious authors to true requirements for a candidate number representation are compute the sum modulo in (1). In the first approach that addition modulo be transformed into the much desired overflow is detected and a correction mlade as soon as a addition modulo a power of 2, and that the adjustment of the partial sum is formed. The total sum is c)btained either se- final sum back to a standard form be easy. In the following, quentially by a modulo accumulator or iin parallel by a tree two alternatives for representing the summands are proposed array of modulo adders. Thus, the basic ] nrohlet-n oroblem of mecha-v to efficiently implement the Chinese remainder theorem. nizing addition of two numbers modulo is encountered here. Taylor [13] assumed that = 2' - 2b (a, b integers) because for this particular value he could implement the de- programmable It has been mentioned that some residue number operations III. IPLEENTATION BY A FRACTIONAL REPRESENTATION tection and correction steps with a simlple logic array. Nevertheless, compared with a binary adder, his depend totally on the outcome of a comparison with a constant fraction of. To perform these operations by way of proposed adder is still expensive, and the riestriction imposed on severely limits the selection of th.e system moduli, the Chinese remainder theorem, one might first compute (1), which must be relatively prime in pairs. T 4aking use of only then compare the result with the constant. This solution is binary adders, Jenkins [12] computed an ordinary sum but disappointingly expensive because it requires the explicit biased the sum so that the detection step became completely evaluation of a sum modulo and a separate comparison. trivial. With the bias included, overflow in addition modulo However, if the summands are changed to reflect the constant, the sum modulo may become easier to deal with, coincides with overflow in binary addit on, which is indithe correction step and the comparison step may be completely eliminated. As cated by the most significant carry bit, but may still be required in the form of another addition. So when an example, the implementation of sign detection is now discussed in detail. a sum of n operands is carried out, the aidditional delay is multiplied either by n - 1 for sequential summation or by A standard residue number system is defined exclusively Flog2 ni for parallel summation. for positive integers in the range [0, ). To accommodate In the second approach, the sum is allowed to grow without negative integers, an implicit signed number system may be the interference of any intermediate detecttion and correction considered in which the range is split evenly into a positive 647

3 648 half and a negative half. Thus, integers X in the interval [0, /2) are interpreted as positive X; integers X in the interval [/2, ) are interpreted as negative X -. The objective of sign detection is to construct a sign function S(X) such that S S(X) = ifo X K 2 ' if- X <. 2 Note that when is even, the positive and negative halves are [0, (/2) - 1] and [/2, - 1], respectively. For odd they are [0, ( - 1)/2] and [( + 1)/2, - 1], respectively. In both cases the real number /2 serves as the delimiter for positive and negative integers. Hence, if X is scaled down by /2 to yield X, - (2/)X, this number is a fractional number lying in the range [0, 2) and is such that ( ) If O CXs < I, Therefore, if Xs is represented in a binary form where the binary point is placed after the leftmost bit, the value of this leftmost bit is no other than the sign function S(X,) = S(X). Furthermore, with this fractional representation the Chinese remainder theorem apears to be better suited for computing the scaled value Xs than for computing X itself. To see how addition modulo is eliminated, the expression given in (1) is first rewritten as X = - (waxi)m - p (5) i=l1 for some nonnegative integer p. Then, by dividing both sides of (5) by /2, a more manageable expression is obtained as X, = m: 2 (WiXi)mi -2p n ^} i=l Now, X, is computed simply by taking the sum in (6) as an ordinary sum of fractional numbers and by throwing away any multiples of 2 which show up in the integer part of the result. This can be trivially accomplished with binary adders by assigning only one bit to the integer part. The final sum is then in a form consistent with the representation of X, discussed previously where the sign bit is explicitly present at the leftmost position. The idea of using fractional numbers to simplify addition modulo in (1) has been recently proposed by Soderstrand et al. [ 11] as a technique for residue-to-analog conversion. However, their discussion was based solely on computer simulation of a very specific residue number system. Thus, some claims made with respect to accuracy are not valid when applied to more general systems. Accuracy is a cause for concern here because for most values of mi, the fractional summands Ui = - (WiXi)m mi cannot be represented exactly in a finite number of bits. (3) (6) (7) So IEEE TRANSACTIONS ON COPUTERS, VOL. c-34, NO. 7, JULY 1985 the number of bits sufficient to yield a correct sign function remains to be determined. Let t + 1 be the number of bits needed where the integer part takes 1 bit, and the fraction part occupies t bits. In the place of ui, let the truncated values u& = 1-2tui-12-t be stored in lookup tables. These values are contaminated with truncation errors so that the sum Ui= uii + ei, 0 < ei < 2-' n n n E ui =, ui +, ei = 2p + X, + e i=- i=1 i=l accumulates an error e which is a sum of ei's. This nonnegative quantity must be controlled in order to distinguish the sign of two distinct numbers, in particular the sign of those numbers close to the boundary of the positive and negative intervals. First, it is observed that the numbers X, are equally spaced in the interval [0, 2) by increments of 2/. In addition, the gap between the largest positive number and 1, the delimiter, is 2/ for even and 1 / for odd. Accordingly, the error in (9) must satisfy the upper bounds 2 e for even, < 1 e < (8) (9) for odd. (10) Since the integer part remains, fixed for these bounds, the computed value of X, is given by Xs = i -2p (11) [i.e., the same multiple of 2 is discarded as in (6)] and satisfies A Xs = Xs, + e,9 S(Xs) = S(Xs) = S(X). (12) If specific moduli were given, the accumulated error would be known exactly for every number in the system since there are only a finite number of moduli as well as a finite number of residue combinations. It would then not be difficult to calculate the maximum error and deduce from there the number of bits. In the absence of such specific information, a known upper bound of the error is used instead. It is obvious from (8) that e < n2-'. Hence, the upper bounds (10) are certainly satisfied when n2t c 2 n2-t or equivalently when t 2[log22nl- 1 for even, for odd (13) for even, t : Flog2 n] for odd. (14) Thus, compared with the usual integer representation, the

4 VAN VU: IPLEENTATIONS OF THE CHINESE REAINDER THEORE fractional representation requires roughly Flog2 nl extra bits (a fact omitted in [11]), but the simplicity and speed gained in computing the sum outweigh the minor cost of these bits. From residues to sign bit, it takes only one table lookup cycle and the fastest possible time to add n binary numbers (see, for example, [14] and [15] for a discussion of fast multioperand addition). Although the stored values ui are truncated up in the previous analysis, they may very well be truncated down or rounded. Truncating down generates negative errors with a lower bound - 2-T; rounding results in both negative and positive errors bounded in absolute value by 2-t. Because of negative errors, the sign may be incorrect for X = 0 or, when is even, for X = /2. However, this does not turn out to be a problem since these special boundary values are always correctly represented in fractional form. The same analysis for determining t should apply with little change. To illustrate the use of fractional representation for sign detection, a numerical example using the same system of moduli as in [11] is presented next. In this system, {11, 13, 15, 16}, residues are 4 bit words, and X is a 16 bit word since = Also, w1 = 8, w2 = 1, W3 = 2, and W4= 1. The sum in (6) is reduced to a sum of two terms 4 EUi- (U1 + U2) + (U3 + U4) = V1I + V2 to take advantage of lookup tables with 8 input bits. Thus, X, can be represented as a 17 bit word. To detect the sign of X = 1 = (1,1,1,1), the stored values of VI = --(8 x 1)I + -(1 x 1) _230 = , ~143 V= -(2 x 1)5 + -(1 x 1) _94_... = = are fetched and added: VI: loo V2: A~~~~~~~~~~~~~~~~~~~~ X,: As indicated by the leftmost bit in the sum, S(1) = 0. Likewise, for X =-1 = (10,12,14,15) vi = -(8 x 10),, + -(1 x 12) _ = , =2.-(2 X 14)15 + (1X 15) = Y406 = O O so that VI: V2: X,: Thus, S(- 1) = 1. Note, however, that if v, and V2 were rounded and if only a total of 11 bits were retained in the last calculation, the resulting sign would not be correct. The sum in this case would be leading to a wrong sign. Clearly, this example does not support the claim in [I1] that as few as 11 bits is sufficient for sign detection. IV. IPLEENTATION BY A QUOTIENT-REAINDER REPRESENTATION A residue-to-decimal converter can be obtained by extending the sign detector with a binary multiplier because. X = Xs(/2). Better still, the now available sign bit also indicates whether should be subtracted from the result in order to convert to a conventional signed number system. Computationally, the signed value of X is given by W = 2- (X (15) as a result of (10) and (12). The amount to be subtracted is changed to /2 if the integer part of X, is discarded as in IA A P W = LXS S(XS)) -~ () (16) Because a multiplication is involved in the final step, this type of fractional converter is not very useful unless residues are converted to analog signals as is done in [11]. There, the multiplication is conveniently incorporated into a D/A converter. For normal residue-to-binary conversion, the following technique is suggested which requires only one or two extra additions after the usual table lookups and the summing of n numbers have been completed. The key to the solution is a quotient-remainder representation for the summands. Recall from (2) that the summands s, are such that 0 ' si <. Now, if a particular modulus mj is singled out to be used in expressing si as Si = qi + ri mj (17) where qi and ri are, respectively, the quotient and remainder of si with respect to /mj such that 0 C qi <mk, 649 0, :C r. <- i (18) then the conversion expression in the Chinese remainder theorem becomes

5 650 IEEE TRANSACTIONS ON COPUTERS, VOL. c-34, NO. 7, JULY 1985 X = (ESi) i=l ( qi -r + i=l i ri) ( I qi) + ( ri) (my i=l )i (=l V (19) This leads to an efficient procedure for computing X without adders modulo as follows. Let and q = qij i= I i (20) r = ri) i=1 (21) Computing the sum in (20) is not a problem because the modulus mj is small; lookup tables are commonly used. However, q is obtained most easily when mj = 2' or 2k - 1, for some k, as these values correspond to k bit binary addition with discarded carry or end-around carry. Since it is not unusual to have a power of 2 included in most practical systems of moduli, mi = 2k is assumed from now on. The final sum in (20) will be a nonnegative integer strictly less than 2k, which in turn is multiplied by /2k via a table lookup. As for the sum in (21), the following inequality is true: n 0. rj < n < (22) =1 if n c m= i1 2k. Again, this condition is easily satisfied by most practical residue number systems. For example, when 2k is selected as the largest modulus, the number of relatively prime moduli which might be considered to complete the system is much less than 2k, even less than 2k-1. Thus, as a result of (22) the sum r is computed simply as an ordinary sum. Finally, let Z =-q(/2k) + r. Then from (19), X = (Z) or Fig. 2. Implementation using a quotient-remainder representation. resentation is summarized in Fig. 2 and in the following steps. 1) Use the residues xi to look up the quotients qi and the remainders ri of si with respect to /2k. 2) Compute in parallel the binary sums q = ( qi) mod 2k, r n = i=.] Iri. XZ if <, IZ - if Z ':-. The comparison of Z with can be eliminated by adding in advance a shift [7], [8] (or bias [12]) to Z. ore specifically, let c = Flog2 I and let the positive shift 2c - be included in the lookup table where q gets multiplied by /2k. That is, let Y = q(/2k)'+ (2C - ) be fetched from the table, given the input q. Then, when the shifted sum Z - Y + r = Z + (2c - ) is formed modulo 2c, a most significant carry bit b is also generated. If Z., this bit is 1 because Z + (2c - ). 2c; it is 0 otherwise. Thus, based simply on the bit value of b, the correction by and the reverse shift are applied accordingly to yield X = Z = Z- (2C - ), or X = Z - = Z. The new implementation using a quotient-remainder rep- 3) Use the sum q to look up Y = q(/2k) + (2C- ). 4) Form the binary sum Z = (Y + r) mod 2c, and retain the most significant carry bit b. 5) OutputX in [0,) as (23) (Z + ) mod 2C if b = 0, z (24) if b = 1. It can be seen that representing qi and rg takes the same number of bits as representing si, but in the (qi, ri) format, the summands can be rapidly accumulated by binary adders. Again, column-compression or carry-save summing techniques should be considered for fast-conversion time. As the word length of the sum q is much shorter than that of the sum r (k versus Flog2 n7 - k), step 3) may take place even before the sum r is completed. The remaining steps 4) and 5) are simple binary addition. The number X can be returned in 2's complement binary form by a very simple modification to the decoder. As mentioned in Section III, the signed value is W = X ifx < /2

6 651 VAN VU: IPLEENTATIONS OF THE CHINESE REAINDER THEORE NORAL POSITION: x CIRCULAR SHIFT: V = <X+ 2 > LINEAR SHIFT: W =V _ I L< I I, I exactly to a summand in the theorem, but is expressed in a mixed radix form. Thus, the sum modulo is transformed into a sum of mixed radix numbers. Nevertheless, mixed radix representation is seemingly not a good choice because the generation and propagation of carries is more cumbersome in mixed radix arithmetic than in binary arithmetic. + I REFERENCES 2 Fig. 3. Shifts used in conversion to signed integers. or W = X - if X 2 /2. Instead of the usual comparison and subtraction step following a conversion cycle, a cheaper alternative is taken as follows. Let V = (X + /2)m. This creates the effect of a circular shift, which pushes negative values into the lower half of the range [0, ) and positive values into the upper half. Then W is simply V - /2 because the shift, linear this time, sends values back to their correct position in the interval [-/2, /2). Thus, no comparison is necessary (Fig. 3). The circular shift is accomplished by changing the tables in step 1) so that /2 is added modulo mi to each residue xi. Note that when mj = 2' as assumed, this amounts to adding a zero for moduli with i f j. So only one table needs to be changed: the one which returns qj (rj is already zero!). The linear shift is easily included in step 5) as (Z + W =_ ) mod 2C if b = 0, ) if b = 1. mod 2C V. SUARY New techniques for an efficient implementation of the Chinese remainder theorem have been presented. By reformulating the conversion expression given in the theorem, the new techniques make it possible to evaluate the sum modulo easily with binary adders and lookup tables. As the obstacle posed by adders modulo a very large and arbitrary integer no longer exists, there is no doubt that this development offers a faster alternative to the traditionally slow mixed radix conversion method. Some difficult operations in residue number systems have already benefited from the new discovery. Sign detection and conversion to unsigned or signed binary numbers have been treated here. In [161 a versatile fractional representation is demonstrated to provide complete information for the simultaneous execution of error detection and residue-interacting operations such as sign detection, magnitude comparison, overflow detection, scaling, nonlinear mapping, and residue-to-analog conversion. In light of the various representations for the summands adopted in this paper, it is interesting to note that the parallel mixed radix conversion algorithm proposed by Huang [10] is, in effect, a disguise of the Chinese remainder theorem. Each orthogonal projection in that algorithm corresponds [1] N. S. Szabo and R. l. Tanaka, Residue Arithmetic and its Applications to Computer Technology. New York: cgraw-hill, [2] D. K. Banerji and J. A. Brzozowski, "Sign detection in residue number systems," IEEE Trans. Comput., vol. C-18, pp , Apr [3] F. Barsi and P. aestrini, "Error correcting properties of redundant residue number systems," IEEE Trans. Comput., vol. C-22, pp , ar [4] W. K. Jenkins and B. J. Leon, "The use of residue number systems in the design of finite impulse response digital filters," IEEE Trans. Circuits Syst., vol. CAS-24, pp , Apr [5] W. K. Jenkins, "Techniques for residue-to-analog conversion for residueencoded digital filters," IEEE Trans. Circuits Syst., vol. CAS-25, pp , July [6] A. Baraniecka and G. A. Jullien, "On decoding techniques for residue number system realizations of digital signal processing hardware," IEEE Trans. Circuits Syst., vol. CAS-25, pp , Nov [7] D. F. Fraser and N. J. Bryg, "An adaptive digital signal processor based on the residue number system," in Proc. AIAA 2nd Comput. Aerospace Conf., Los Angeles, CA, Oct , [8] V. S. Cheng and C. H. Huang, "On the decoding of residue numbers," in Proc. Int. Symp. ini-icrocomput. Contr. easurement, San Francisco, CA, ay 20-22, [9] F. J. Taylor and A. S. Ramnarayanan, "An efficient residue-to-decimal converter," IEEE Trans. Circuits Syst., vol. CAS-28, pp , Dec [10] C. H. Huang, "A fully parallel mixed-radix conversion algorithm for residue number applications," IEEE Trans. Comput., vol. C-32, pp , Apr [11]. A. Soderstrand, C. Vernia, and J. H. Chang, "An improved residue number system digital-to-analog converter," IEEE Trans. Circuits Syst., vol. CAS-30, pp , Dec [12] W. K. Jenkins, "A technique for the efficient generation of projections for error correcting residue codes," IEEE Trans. Circuits Syst., vol. CAS-31, pp , Feb [13] F. J. Taylor, "Residue arithmetic: A tutorial with examples," IEEE Comput. ag., vol. 17, pp , al [14] I. T. Ho and T. C. Chen, "ultiple addition by residue threshold functions and their representation by array logic," IEEE Trans. Comput., vol. C-22, pp , Aug [15] D. E. Atkins and S. C. Ong, "Time-component complexity of two approaches to multioperand binary addition," IEEE Trans. Comput., vol. C-28, pp , Dec [16] T. Van Vu, "The use of residue arithmetic for fault detection in a digital flight control system," in Proc. Nat. Aerospace Electron. Conf., Dayton, OH, ay 21-25, Thu Van Vu completed his undergraduate studies at anhattan College, Bronx, NY, in He received the.s. degree in mathematics from Rutgers University, New Brunswick, NJ, in 1979 and the Ph.D. degree in computer science from the University of Illinois, Urbana-Champaign, in His interests have been in computational mathematics and mathematical software for differential equations. He is currently at Harris Corporation, elbourne, FL, developing computation algorithms and architecture for high-speed and fault-tolerant digital systems.