Efficient canonic signed digit recoding - Universidad de Cantabria

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G.A. Ruiz, M. Granda / Microelectronics Journal 42 (2011) 1090–1097 1091Okeya et al. [33] proposed an algorithm to carry out this conversionbut it requires additional memory to store some precomputedelements. Indeed, efficient MSB-to-LSB algorithms yield a MSDrepresentation [34,35] but not a CSD one because of havingconsecutive nonzero digits.This paper presents novel, efficient standard cell-based implementationsto convert a two’s complement binary number into itsCSD representation based on two signals, H and K, functionallyequivalent to two carries. These signals were already defined inthe implementation of 3X terms for radix-8 encoding [36], buthere they are used for the CSD recoding in a more resourcefulway. Implementations in a 130 nm standard cell CMOS technologyof previously published architectures have been comparedwith the proposed scheme demonstrating that the circuits arehighly efficient in area and speed. The remainder of this paper isorganized as follows. In Section 2, a brief introduction anddefinitions related to CSD recoding are presented. New compactalgebraic expressions for optimal CSD recoding in terms of signalsH and K and their application in the efficient implementation ofCSD recoders for different configurations are described in Section3. Simulations and comparisons are listed in Section 4. Finally, theconclusions are stated in Section 5.2. CSD recodingThe CSD representation of an integer number is a signed andunique digit representation that contains no adjacent nonzerodigits. Given an n-digit binary unsigned number X¼{x 0 , x 1 , y,x n 1 } expressed asX ¼ Xn 1x i U2 i , x i Af0,1g ð1Þi ¼ 0then the (nþ1)-digit CSD representation Y¼{y 0 , y 1 ,y, y n }ofX isgiven byY ¼ Xn 1x i U2 i ¼ Xny i U2 i , y i Af1,0,1g ð2Þi ¼ 0i ¼ 0The condition that all nonzero digits in a CSD number areseparated by zeros implies thaty i þ 1 Uy i ¼ 0, 0rirn 1 ð3ÞFrom this property, the probability that a CSD n-digit has anonzero value [13,26] is given byPð9y i 9 ¼ 1Þ¼ 1 3 þ 1 9n 1 1 n ð4Þ2As n becomes large, this probability tends to 1/3 while thisprobability becomes 1/2 in a binary code. Using this property, thenumber of additions/subtractions is reduced to a minimum inmultipliers [14–17] and, as a result, an overall speed-up can beachieved.The adoption of a ternary number system adds some flexibilityto the CSD representation, since it allows the number of nonzerodigits to be minimized, but it requires that each digit y i must beencoded over two bits {y s i , yd i}. Table 1 shows the two mostfrequently used encodings in practice [1]. Encoding 1 can beviewed as a two’s representation. However, encoding 2 is preferablesince it satisfies the following relationy i ¼ y d iy s ið5Þwhere y s irepresents the sign bit and y d ithe data bit. This encodingalso allows an additional valid representation of 0 when y s i¼1 andy d i¼1, which is useful in some arithmetic implementations. In thewhole paper, this encoding is used.Table 1Two most encodings used in the binary representation of a CSD digit (1¯ stands for1). In this paper, encoding 2 is used.Encoding 1 Encoding 2y i y s i yd i y s i yd i0 00 001 01 01¯1 11 10Hashemian [27] presented the binary coded CSD (BCSD)number to avoid extra data word representation. The BCSDrecoding is based on a simple binary representation B¼{b 0 ,b 1 ,y,b n 1 } of a CSD code, which uses the same number of bitsas the original two’s complement representation. It takes advantageof the CSD property, in which no two adjacent digits can bothbe nonzero, to assign the next position of each nonzero CSD digitas sign bit while maintaining the same size data word. Thus, if thebit i in a CSD code is nonzero, y i a0 (which means that y i þ 1 ¼0),then the bit i is nonzero in the BCSD code, b i a0, and thefollowing bit b i þ 1 acts as a sign bit: b i þ 1 ¼0 means y i is positiveand b i þ1 ¼1 means y i is negative. As a result, y i ¼0 is encoded asb i ¼0, y i ¼1 as b i þ 1 b i ¼01 and y i ¼1¯ as b i þ 1 b i ¼11. A simpleconversion between a CSD coding and BCSD coding isb i ¼ y d i þys i 1b i þ 1 ¼ y d i þ 1 þys i0 0 1 0 1 0 1 1 10 0 1 0 1 1 0 0 10 0 1 1 0 1 0 0 10 1 0 1 0 1 0 0 1Fig. 1. Conversion process from binary to CSD code.Since this conversion can be obtained by a simple operation, inthis paper the two-bit encoding representation is used to make itsreading and understanding easier, and, without loss of generality,it can be straightforwardly extended to other representations.The conversion from a binary representation to CSD representationis mostly based on the following identity2 i þ j 1 þ2 i þ j 2 þ...þ2 i ¼ 2 i þ j 2 i ð7ÞThis means that a string of 1s can be replaced by a 1, followedby 0s, followed by a 1¯ . Isolated 1s are left unchanged, but isolated0s are re-examined in such a way that, after applying Eq. (7), pairsof type 11¯ are changed to 01. For example, the binary number(001010111) 2 is equivalent in a CSD representation to 0101¯ 01¯ 001¯ ;the encoding process is graphically shown in Fig. 1. Traditionally,this encoding is performed from LSB to MSB using two adjacentbits and a carry signal according to the recoding algorithm shownin Table 2 [25,26]. Here, the carry-out c i ¼1 if and only if there aretwo of three 1s among the three inputs x iþ1 , x i and c i 1 ,thatisc i ¼ x i x i þ 1 þðx i þx i þ 1 Þc i 1 , being c 1 ¼ 0 ð8ÞThe variable D¼{d 0 , d 1 ,y,d n 1 }, which flags all nonzero digitsin the CSD representation, is defined asd i ¼ x i c i 1 ð9Þð6Þ
1092G.A. Ruiz, M. Granda / Microelectronics Journal 42 (2011) 1090–1097Since y i takes one of the three values {0, 1, 1¯ }, two bits {y s i , yd i }are necessary to encode it, which are defined from Table 2 as(y d i¼ x i þ 1 ðx i c i 1 Þ¼x i þ 1 d iy s i ¼ x ð10Þi þ 1ðx i c i 1 Þ¼x i þ 1 d iFor the sake of clarity, the following example describes thedifferent signals used in the CSD recoding:X ¼ 001010111Carry ¼ 011111110D ¼ 010101001Y ¼ 010101001In the case of a n-bit two’s complement binary number X, theCSD representation is given byY ¼ x n 1 U2 n 1 þ Xn 2x i U2 i ¼ Xn 1y i U2 ið11Þi ¼ 0i ¼ 0here, only n CSD digits are necessary as the value of the binarynumberislimitedto[ 2 n ,2 n 1 ]. Negative integers in CSD can beobtained trivially from their positive counterpart by changing thesigns of all nonzero digits. For example, the CSD code 0101represents the decimal number 3, while 0101 represents thenumber 3. Therefore, the conversion of a negative n-digit two’scomplement binary number X into its CSD representation can beperformed from the well-known property X ¼ X þ1. Then reformulatingthe former equations for the case of a binary number X ina two’s complement representation, the conversion into its CSDrepresentation is given byt i ¼ x i x n 1 , ion 1 ð12Þc 0 i ¼ t i þ 1t i þðt i þ 1 þt i Þc 0 i 1 , being c0 1 ¼ x n 1 ð13Þwhere x n 1 represents the sign of X. From (13), c’ i propagates c i orits complement depending on sign of the X because of all inputs arecomputed according to (12). Therefore, another way to express c’ i isc 0 i ¼ x n 1 c i ð14ÞTable 2CSD coding.Applying these definitions, we getd i ¼ t i c 0 i 1 ¼ðx i x n 1 Þðx n 1 c i 1 Þ¼x i c i 1 ð15ÞThis means that the definition of D is independent of sign X. Ina similar way, the expression of Y is the same as that in (10) as(y d i¼ t i þ 1 d i ¼ t i þ 1 ðt i c 0 i 1 Þ¼ðx i þ 1 x n 1 Þððx i x n 1 Þc 0 i 1 Þ¼x i þ 1d iy s i ¼ t i þ 1d i ¼ t i þ 1 ðt i c 0 i 1 Þ¼ðx i þ 1 x n 1 Þððx i x n 1 Þc 0 i 1 Þ¼x i þ 1d ið16ÞFig. 2 shows the circuit to convert a n¼6 digit binary numberinto its CSD representation according to Eqs. (8)–(10), valid forboth unsigned and two’s complement binary numbers. The onlydifference arises in the last CSD digit. By introducing an extra signextension, x n ¼0 for unsigned numbers and x n ¼x n 1 for two’scomplement numbers, the last section changes depending on thesign of X in the following general expression:8y d n 1 ¼ d n 1>< y s n 1 ¼ 0For an unsigned number Xy d n ¼ d ð17Þn ¼ c n 1 ¼ x n c n 2>: y s n ¼ 0For a signed number X yd n 1 ¼ x nd i ¼ x n 1 c n 2y s n 1 ¼ x nd i ¼ x n 1 c n 2(ð18ÞFor unsigned X, nþ1 CSD digits are necessary to represent thatbinary number. However, as it is a positive number the two MSBdigits of CSD are also positives and, indeed, only their positivedata parts are necessary as shown in Eq. (17); the negative part isalways zero. For signed numbers, only n CSD digits are used andthe last digit is computed according to Eq. (18). As can be seen inFig. 2, the critical path of the circuit is fixed by the propagation ofthe carry signal, in a similar way to a conventional ripple-carryadder. There is a clear similarity between the carry definition inconventional adders and the definition of that carry in Eq. (8). Thissuggests that implementation of fast CSD converters should bebased on fast well-known carry look-ahead structures used inaddition.x i þ1 x i c i 1 y i c i Comments0 0 0 0 0 String of 0s0 0 1 1 0 End of 1s0 1 0 1 0 A single 10 1 1 0 1 String of 1s1 0 0 0 0 String of 0s1 0 1 1¯ 1 A single 01 1 0 1¯ 1 Beginning of 1s1 1 1 0 1 String of 1s3. New CSD recodingIn the definition of carry in Eq. (8), two adjoining carries sharethe same input variable. This means that the same input x i is usedin the generation of both carries c i 1 and c i . This characteristicallows the algebraic expressions of carry generation to be simplifiedin order to obtain efficient circuits.Fig. 2. Schematic circuit for the conversion of a binary number into its CSD representation (n¼6).
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