SCALED RESIDUE REVERSE CONVERTER FOR SIGNED NUMBERS WITH LOW COST MODULI BASE

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No 98 Electrical Engineering 2019 DOI 10.21008/j.1897-0737.2019.98.0009

___________________________________________________

* State University of Applied Sciences in Elbląg

** Gdańsk University of Technology Maciej CZYŻAK^*, Robert SMYK^**

SCALED RESIDUE REVERSE CONVERTER FOR SIGNED NUMBERS WITH LOW COST MODULI BASE

The paper presents a realization of the scaled residue reverse converter for the low cost moduli base {2ⁿ−1, 2 , 2ⁿ ⁿ+1}. The moduli of this type allow for the memoryless reverse conversion using the Chinese Remainder Theorem because the orthogonal projections can be obtained by shifts and additions. Moreover, the modulo reduction of the sum of projections and sign detection algorithms are shown. Also the converter architecture is presented.

KEYWORDS: residue number system, residue reverse conversion, scaling, low-cost moduli.

1.INTRODUCTION

The new generation of portable electronic devices, such as cell phones, cam- corders, digital cameras call for the real-time realization of many complex digital signal processing (DSP) algorithms. The other examples can be radar signal processors. Fast processing may be required as well for cryptographic algorithms when they have to be performed in real-time. The second issue can be the de- mand of the low-power operation. The high-speed DSP and low-power budget are mutually contradictory, because high-speed DSP requires high-clock fre- quencies that increases the needed power consumption. One of the remedies can be the reduction of the IC supply voltage. The tool that could make it possible is the residue number system (RNS) [1-4]. RNS has been of interest as the tool for fast or high-speed implementation of digital signal processing algorithms or cryptographic algorithms. The RNS is a non-positional system that decomposes certain arithmetic operations on long integers into equivalent independent operations on small integers. Residue addition, subtraction and multiplication are relatively easy because they are performed independently on the individual digits of residue representations of arguments. Hence the RNS is advantageous for realization of algorithms where these operations dominate. However, because the

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RNS is an integer system with the limited number range, usually after one multi- plication, scaling i.e. division by a constant is needed which was used for trans- formation of algorithm coefficients to integers. The example can be FIR filters or FFT algorithms. Scaling as well as general division and sign detection belong to difficult RNS operations. The RNS implementation of an algorithm gives an architecture with parallel computational channels that allows for the segmentation of the datapath so that the individual channels operate on subwords. Such a structure can tolerate voltage reduction in individual channels that results in quadratic power reduction. The usefulness of the RNS-based architectures has been demonstrated in several works [31-34]. However, RNS-based architectures may have the bottleneck mainly due to each multiplication and the reverse conversion, i.e. conversion from the RNS to the weighed system such as the natural binary system or the two's complement system. Scaling and residue reverse conversion seem to be the most important problems in RNS. Residue reverse conversion has been considered by many authors. The main approaches are based on the mixed-radix number system (MRS) [1], [5-8], CRT [9-17] or CRT II [19], and core function [21, 22]. For general moduli with the small binary size, i.e.

5–6 bit moduli any approach can be taken. For CRT-based methods the algo- rithm requires the computation of orthogonal projections and their addition modulo M, where M is the RNS dynamic range (DR). In this approach orthogonal projections are obtained by memory look-up using the individual residues of the converted number as the memory addresses. In the next step modulo M reduction has to be performed. In this work we consider the scaled conversion.

The aim of scaled reverse conversion is to obtain scaled representation of the residue number in the weighted number system. Scaled residue conversion in comparison with the ordinary residue conversion allows to reduce the width of the data-path by the binary length of the scaling factor. In Section 2 we review the state-of-the-art, and in Section 3 the residue number system and scaling fundamentals are described. In Section 4 scaled conversion is derived and in Section 5 sign detection problem is analyzed and finally in Section 6 the converter architecture is presented.

2.LITERATUREREVIEW

The RNS-based architecture usually requires output processing, which usually encompasses the conversion from the residue number system to a weighted system. The scaled RNS/B conversion can be treated in one aspect as the special case of RNS/B conversion, on the other hand as a form of residue scaling without computing the residue representation of the scaled number. We first review un- scaled RNS/B conversion methods. The conversion based on the MRS is a relatively slow serial process in which first the RNS/MRS conversion is performed and then subsequently the MRS/B conversion. For the moduli with the binary

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length not exceeding the acceptable memory address length, the arithmetic operations can be performed by look-up of precomputed values. Such conversion can be accelerated by the concurrent use of the CRT and MRS. In such an algorithm the CRT orthogonal projections are represented in the MRS [20] with a succes- sive parallel modular additions for the individual moduli that allows to avoid the complex modulo M reduction. The CRT-based conversion is currently the most frequent approach. There are two main drawbacks of using the CRT. The first is the necessity of performing the modulo M operation with M being a large number where the value to be reduced is the sum of orthogonal projections of the RNS number. The second is the computation of orthogonal projections. For rela- tively small moduli it can be done by memory look-up. The modulo M reduction can be performed in various ways. The most direct approach can be a series of comparisons with n−1 multiplicities of M. Fraser and Bryg [9] limited the number of comparisons to one for a certain example and Cheng and Huang [10]

proved for this technique that as the number of moduli increases, M should ap- proach the power of 2. For n operands the sum can be received using a tree of two-operand modulo M adders, but these devices are, in general, much more expensive than binary adders. Elleithy and Bayoumi [16] proposed an effective structure using a range determinator, so that the appropriate value can be sub- tracted using the binary adder. Piestrak [17] suggested an efficient reduction of the sum of orthogonal projections to the interval [0, 2M) using the carry-save representation and a modulo M generator for the number represented by the most significant bits of carry and save vectors. This allows for the final modulo M generation based on two binary adders working in parallel. This conversion technique uses only one log₂M -bit addition in series. However, for the greater number of moduli this approach becomes less effective due to the growth of the ROM used in the structure. This approach can be extended by using a small binary adder to add high-order bits of carry-save adder output sum that permits to use the shorter ROM address. The new conversion technique named the CRT II was proposed by Wang [19], but this approach can be effective if the use of the multipliers with the maximum size of 0.5 log⋅ ₂M -bits is allowable.

Along with the CRT techniques based on integer arithmetic, the techniques using the real arithmetic have been developed. First of them is the technique called the Approximate Chinese Remainder Theorem (ACRT) introduced by Soderstrand and Vernia [13], where the modulo M operation is avoided by using a sum of fractions that represents the orthogonal projections scaled by M. The conversion result is obtained as X/M. This technique was further developed and analyzed in [6],[15],[18]. The technique based on fractions for residue conversion and sign detection was introduced by Vu [14]. The similar approach was taken in [23] for computation of the excess factor in the CRT formula, i.e. the number of times the number to be reduced overflows M. The modulo M reduction requires in this case one binary adder operating in parallel with the calcula-

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tion of the sum of projections and a multiplier of a small number by M, imple- mented as the look-up table.

The second group are moduli akin to the powers of 2, eg. {2ⁿ −1, 2 , 2ⁿ ⁿ+1}

Several solutions of such converters are known [26,27]. The scaled conversion can be simplified as it can be seen for this base in the case of the residue scalers shown in [24,25]. Such moduli simplify the conversion because after transfor- mation of the CRT formula, it turns out that the CRT projections can be computed by simple shifts and additions. Also the modulo M reduction can be more simple than for the general moduli. Moreover, for scaled conversion with the scaling factor equal to 2ⁿ, we can perform instead of the modulo M reduction, the reduction modulo^M^{/ 2}ⁿ ⁼

(

²²ⁿ⁻¹

)

, that makes that we can use the segmentation of the word to be reduced modulo M using the method given by Avizienis [29] and later generalized by Piestrak [30].

3.RESIDUENUMBERSYSTEMANDSCALING FUNDAMENTALS

The residue number system is determined by its base, B={m m₁, ₂,...,m_n} where m , _i i=1, 2,3,...,n, are nonnegative integers termed the moduli. The num- ber range M of the system is M =

∏

ⁿ_i₌₁m_i. If the moduli are pairwise relatively prime, i.e. if gcd(m m_j, _k) 1,= j≠k, j k, =1, 2,.., n, then every integer X from [0,M−1], is represented by the n-tuple

(

x x0, ,...,1 x_l₋1

)

, where _i _m

x = X i, in one-to-one correspondence manner. The residue operations can be defined as

(

x x1, 2,...,x_n

) (

⊗ y y1, 2,...,y_n

) (

= z z1, 2,...,z_n

)

, where

i i i mi

z = x ⊗y , i=1,2,..., n, and ⊗ may denote addition, subtraction or multiplication. As it can be seen from the above formula the operations are performed in small integer rings R m , i=1,2,..., n. The condition of mutual

( )

i

primality assures that the mapping between the ring modulo M and the direct sum of small rings R m , i=1,2,...,n. is isomorphic. This mapping can be per-

( )

i

formed using the CRT or the MRS.

The value of an integer X, with the use of the CRT [1], is given by the formula

1 1

n n

j j M j j

X = Σ ₌ X = Σ ₌X − ⋅r M , (1)

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where X are termed orthogonal projections equal to _j ¹

j j

j j j m

m

X =M ⋅ M⁻ ⋅ X , where _j

j

M =Mm , and ¹ 1

j

j j

M ⋅M⁻ m = . M_j⁻¹ is called the multiplicative inverse of M modulo _j m , and exists if _j ^gcd

(

^M^j^,^m^j

)

⁼¹^.

Scaling is defined as the division of X by an arbitrarily chosen but fixed scaling factor, K and rounding the result to the smaller,

( (

^⋅ , greater,

( (

^⋅ , or the near- est,

[ ]

^⋅ ^integer.

In the RNS the sign is determined in the following way. If we denote the un- signed number from

[

^0,M as N , then for signed numbers denoted as X , we

]

have, if M is even, X =N for N<M/ 2, and X = −N M for N≥M/ 2. If M is odd, X =N for N<(M−1) / 2, and X = −N Mfor N≥(M −1) / 2.

In contrary to ⊗ operations, residue scaling is relatively complex .The scaling result, Y is determined as follows

X X K

Y X

K K

−

 

= = . (2) Then the residues of Y for j=1,...,n, are obtained as

1

j j

j m m

m m

y =Y = X − X K ⋅ K , (3)

where

( (

¹K m_j is multiplication inverse of K modulo m . _j Provided that 1

mi

K exists, the corresponding residue y_j of Y is given by 1

j j

j m j m

m m

y = Y = x − X K ⋅ K j=1,2,..,n. (4) For X <0 we have

M X M X K

Y X M

K K

− − −

 

= = − (5) and

1

j j

j

j m m K

m m

m

y Y M X M X

= = − − − K j=1,2,..,n. (6)

The formula (6) can be also written as

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1

j j

j

j m j m j K

m m

m

y Y m X m X

= = − − − K j=1,2,..,n (7)

In the case of scaled residue conversion of X ={ ,x x₁ ₂,...,x_n} we have to perform (2) for X ≥ 0 or (5) for X < 0. The calculation of | |

mi

Y is not needed.

We can also perform the scaled conversion using the following formula

1 1

/

j j

n n

j j

M K

X X

X M

Y r

K ⁼ K ⁼ K K

 

= = Σ = Σ − (8)

The direct usage of (8) for scaled conversion is difficult for several reasons.

First X_j /K have to be rounded to integers, that introduces certain error.

Moreover, K must be the divisor of M and the excess factor r has to be calcu- lated. Below we shall show that for the base {2ⁿ−1, 2 , 2ⁿ ⁿ +1} the calculation of (8) using K=2ⁿ can be greatly simplified because of the form of K and of

/ 22ⁿ 1

M K= − .

4. SCALED CONVERSION

In this algorithm we assume the three moduli RNS base B={m m m₁, ₂, ₃} with

1 2ⁿ 1

m = − , m₂=2ⁿ, m₃ =2ⁿ+1. In this case for n=3 the CRT in (1) assumes the following form

1 2 3

1 1 1

1 1 1 2 2 2 3 3 3

1 1 1

2 3 1 1 1 3 2 2 1 2 3 3

m m m M

X M M x M M x M M x

m m M x m m M x m m M x

− − −

= + +

. (9)

The multiplicative inverses M⁻_j¹ fulfill the equations

1 1, 1, 2,3

j

j j m

M⁻ ⋅M = j= . (10) We shall prove that

1

1 1

1 2ⁿ

M⁻ m = ⁻ ,

2

1

2 2ⁿ 1

M⁻ m = − ,

3

1 1

3 2ⁿ 1

M⁻ m = ⁻ + .

Theorem 1. For the base B={2ⁿ −1, 2 , 2ⁿ ⁿ+1} the multiplicative inverses have the values

1

1 1

1 2ⁿ

M⁻ m = ⁻ , (11a)

2

1

2 2ⁿ 1

M⁻ m = − , (11b)

3

1 1

3 2ⁿ 1

M⁻ m = ⁻ + . (11c)

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For simplification we assume the denotations as follows

1

1 1

i = M⁻ m ,

2

1

2 2 m

i = M⁻ ,

3

1

3 3 m

i = M⁻ . Proof for (11a)

We have

2

1 1 1 1 1

2 1 2 1 2 1 2 1 2 1

1 1 2 1

2 (2 1) 2 2 2 2 2

1 1 1.

n n n n n

n

n n n n n n n

i i i i i

i i

− − − − −

−

+ = + = ⋅ ⋅ + ⋅

= ⋅ ⋅ + = (12)

Hence we must have 2i₁−2ⁿ+ =1 1, thus i₁=2ⁿ⁻¹. Proof for (11b)

We have

2

22 22 2 2

(2 1)(2 1) n (2 1) n n 1

n n n

i i i

− + = − = − = . (13)

Inserting 2ⁿ we obtain ₂

2ⁿ−i 2_n =1. If 2ⁿ− =i₂ 1 then i₂ =2ⁿ −1. Proof for (11c)

We have

2

32 3 32 3 32

(2ⁿ −1)2ⁿi _n = 2 ⁿi −2ⁿi _n = 2 2ⁿ ⁿi −2ⁿi _n =1.

In order to obtain the terms which are multiplicities of 2ⁿ+1, we replace 2 2ⁿ ⁿ by (2ⁿ+1)(2ⁿ+ −1) 2ⁿ⁺¹−1. This form results from the fact that

2 1

(2ⁿ +1)(2ⁿ+ =1) 2 ⁿ+2ⁿ⁺ +1. We obtain

1

3 3 3 32 1

1 1

3 3 2 1 3 2 1

(2 1)(2 1) 2 2

2 (2 1) 2 .

n

n n

n n n n

n n n

i i i i

i i i

+

+ +

+ + − − −

= − − + = − (14)

We can introduce the additional factor equal to 2(2ⁿ +1)i₃=0

1 1 1 3

3 3 2 1 3 3 2 1 3 2 1

2(2 1) 2 n 2 2 2 n 2 n 1.

n n n n

i ⁺i ⁺ i i ⁺ i i ₊

+ +

+ − = + − = = (15)

If i is the multiplicative inverse, it has to be the solution of the equation ₃ 2i3−2ⁿ− =1 1. Hence i₃=2ⁿ⁻¹−1.

Inserting the values of moduli into (9) and multiplicative inverses we obtain

2

1 1

2

1

3 (2 1)2

2 (2 1)2

(2 1)(2 1)(2 1) .

2 (2 1)(2 1) _n _n

n n n

x

X x

x

−

+ +

= + − + − +

+ − +

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After reduction we have

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2

3 1 2 1 3 2 1 2 1

1 1 2

3 1 2 2 1

3 (2 1)2

3 1 2 1 3 2 3 1 2 1

1 2 3 (2 1)2

2 2 (2 2 2 2 2 1)

(2 2 2 2 )

(2 2 ) (2 2 2 1) (2 2 2 ) .

n n

n n n n n n n

n n n n

n n n n n n n n

x x x

X

x

x x x

+ + + +

− −

−

+ + − −

−

+ + − + + − +

= + + − −

= + + + − + + − −

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Using the RNS property |PX |_PQ=P X| |_Q, and assuming P=m₂=2ⁿ,

1 3 (2ⁿ 1)(2ⁿ 1)

Q=m m = − + , after grouping the individual terms and scaling by 2ⁿ, we receive

2

2 1 1

1 2 3 1 2 3 2 3 3 (2 1)

( 2 )2 ( 2 )2 ( ) / 2 .

2 ⁿ

n n n

n

Y x x x ⁻ x x x ⁻ x x x

= + + + − + − + + − (18)

The calculation of the scaled conversion using the CRT requires the addition of terms and final reduction modulo 2²ⁿ −1. Such reduction is relatively simple and can be performed by segmentation and addition of segments of the reduced word modulo [17].

Example 1. Computation of (18) for n=5 and n=18.

For n=5 we have

For X =1142 andM = 32736, the residues have the values:x₁= 26 ,

2 22

x = ,x₃ =20, respectively. We obtain the scaled value equal to Y = 35. The exact value is 35.625.

For n=18 we have 9007199254609923

X = , M = 18014398509219840, and residues x₁=3,

2 131075

x = ,x₃=3. We obtain the scaled value equal to Y=34359738367. The exact value is 34359738367.50001.

The RNS/B conversion [17] requires multi-operand addition, modulo reduc- tion to 2M range and final modulo M reduction. In the general case modulo

2M reduction requires binary adders, one carry-save adder and two-input mul- tiplexer, all of log M₂  -bit length. Denote word to be reduced as S . In our

case M =(2ⁿ+1)(2ⁿ−1) We have S≤2M and

max( )S =(2ⁿ+1)(2ⁿ− − =1) 1 22ⁿ−2. In this case we can detect the overflow of M using the value of the MSB bit b of S with the weight 2_n ⁿ or all remaining bits. The overflow of M takes place, when b_n=1 or b_p = Πⁿ_i₌⁻₀¹b_i =1, in the former case the modulo reduction can be performed by setting b to 0 and add-_n ing 1 to LSB. In the letter case if b_p =1 then |S|_M=0.

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5. SIGN DETECTION

In the case of M =2ⁿ −1, M is odd and the positive numbers belong to [0, 2ⁿ −1] and negative numbers belong to [2 , 2ⁿ ²ⁿ −2]. In such a case the bit

1

bn₋ determines the sign. For b_n₋₁=0 the number is nonnegative and for b_n₋₁=1 the number is negative.

Below we shall propose a procedure that performs jointly modulo M reduc- tion and sign detection. This will be based on the values of b_n₋₁ and b . _n

Case 1. b_n =0 and b_n₋₁=0

The reduced word has the form 00xx…x. This means that there is no overflow and the number is non-negative and equal to Y = Σⁿ_i₌⁻₀²b_i.

Case 2. b_n =0and b_n₋₁=1

The reduced word has the form 01xx…x. This means that there is no overflow and the number is negative or zero.

a) If b_p = Π_iⁿ₌⁻₀²b_i =1the Y=|S|_M=0

b) If b_p = Π_iⁿ₌⁻₀²b_i =0 then Y<0 and 1xx…x represents Y<0 in two’s- complement diminished by 1, Y = −(2ⁿ− − Σ1 _iⁿ₌⁻₀²b_i).

The reduced word has the form 10xx…x. This means that there is M overflow and the modulo reduction can be performed by zeroing the MSB and adding 1 to the LSB.

a) If b_p = Πⁿ_i₌⁻₀²b_i =1then Y<0 (negative) and Y =2ⁿ−1 b) If b_p = Πⁿ_i₌⁻₀²b_i =0 then Y= Σⁿ_i₌⁻₀²b_i+1

The reduced word has the form 11xx…x. This means that there is M overflow and the number is reduced as in Case 3.

a) If b_p =b₀(1− Πb₁) ⁿ_i₌⁻₂²b_i =1 then Y =|S|_M=0

b) If b_p =b₀(1− Πb₁) ⁿ_i₌⁻₂²b_i <>1 then Y = −(2ⁿ− − Σ1 _iⁿ₌⁻₀¹b_i) 6. CONVERTER ARCHITECTURE

The converter design requires the evaluation of the number range of the expression in (18). For convenience we cite this expression

2

2 1 1

1 2 3 1 2 3 2 3 (2 1)

( 2 )2 ( 2 )2 ( )

2 ⁿ

n n

n

X x x x ⁻ x x x ⁻ x x

−

 

= + + + − + − +

 

  (18)

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In order to compute the maximum value of the internal expression in (18) we shall substitute the maximum values of the individual moduli, for m₁=2ⁿ −1, max( )x1 =2ⁿ −2, for m₂=2ⁿ, max(x₂)=2ⁿ−1, for m₃ =2ⁿ +1, max( )x₃ =2ⁿ.

2

2 1

1

3 (2 1)(2 1)

1 2 1 2 1 1

3 2 1

3 2 1 2

2 1

(2 2 2(2 1) 2 )2

(2 1 2 ) / 2

(2 4)2 (2 4)2 2 2 / 2

| 2 2 2 2 1|

n n

n

n n n n

n n n

n n n n n n

n n n

x

−

− +

+ − + − +

−

+ +

−

− + − + +

+ − + − + + =

+ − + +

= − + − − + + =

= + − + +

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For the modulus with a binary length n=18 we obtain

36

54 27 20 55

2 1

| 2 +2 −2 +3 | ₋<2 (20) So we see that the word to be reduced has the binary length of 55 bits. The length of the reduction modulus is equal to 2n=36 bits. In this way we can divide the output word in two segments, first of 36 bits and the second of 19 bits.

These segments have to be added using 36-bit binary adder and the received sum

(

³⁶

)

2 2 1

S< − has to be reduced modulo 2³⁶−1. In the next stage the obtained residue X has to be reduced modulo 2 and the sign of the number is to be de-³⁶ tected.

The block scheme of the scaled converter architecture is shown in Fig. 1.

Firstly the binary adder BA1 computes the sum S1. Next S₂= −S₁ 2x₂ along with S₃= +S₁ 2x₂ are computed. BA4 calculates the shifted sum of S and ₂ S . ₃ The obtained sum S if divided into two segments to reduce it to the ^{2 2}

(

²ⁿ⁻¹

)

number range with a first one of 36 bits and the second of 19 bits. Such segmentation of the word to be reduced is possible due to periodicity property of series

2^k [30]. The final block realizes the modulo 2²ⁿ−1 reduction and sign detection as presented in Section 5.

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In this bers for th of multipl tection. A

[1] Szabo nology [2] Soders Digita

Fig 1

work we ha he base {2ⁿ

licative inve Also schemati

N.S, Tanaka y, New York, strand M. et a l Signal Proc

1. Schematic bl

7.

ave presented 1,2 ,2ⁿ ⁿ 1}

 

erses are give ic architectur RE

R.J, Residue McGraw-Hill al., Residue N essing, IEEE

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Arithmetic an l, 1967.

Number System Press, NY, 19

onverter archite

RY

e scaled conv ling factor K th the detaile nverter has be CES

nd its Applica m Arithmetic:

986.

ecture

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(Received: 07.02.2019, revised: 15.03.2019)