CN102184159A - Base is {2n-1, 2n+1,2n} and {2n-1, 2n+1,22n+1} remainder system backward conversion device and method - Google Patents

Abstract

The invention discloses a backward conversion (R2B) device and method of a residue number system (RNS) with bases of M1=[2<n-1>,2<n+1>,2<n>] and M2=[2<n-1>,2<n+1>,2<2n+1>]. An integer of the RNS is converted into a binary integer. In the invention, according to the characteristics of M2 and M1, M1 and M2 have one common group of sub residue bases MS=[2<n-1>,2<n+1>], firstly, backward conversion of the MS is completed, and then backward conversions of M12=[2<2n-1>,2<n>] and M22=[2<2n-1>,2<2n+1>] are respectively completed, and results of the backward conversions are conversion results of M1 and M2. In addition, the form of M22 is similar to the form of MS, therefore, M22 is realized by adopting the structure similar to MS, the module design is better reused, and the trouble of repeated design is avoided. The invention plays a positive pole in application of the RNS with a base of M1 or M2 in a digital signal processing system.

Description

Base is { 2 n-1,2 n+ 1,2 nAnd { 2 n-1,2 n+ 1,2 2nBehind the residue number system of+1} to conversion equipment and method

Technical field

The present invention relates to the signal Processing field, specifically, relate to a kind of be used for communicating by letter and signal Processing back to conversion based on residue number system (RNS)---the RNS integer is to the conversion equipment and the method for bigit.

Background technology

Traditional binary numeration system has chain carry, has often limited the arithmetic speed of computing machine.Residue number system (RNS) is a kind of concurrent working mechanism and the system that follows free carry.Because this characteristic, in the past decade residue number system is being subjected to suitable attention aspect arithmetical operation and the signal Processing.At signal processing method based on residue number system, in Fourier transform, FIR wave filter, matrix inversion etc., the back is to conversion must go on foot through as the conversion from the remainder space to binary space, playing the part of important role, because it not only influences the speed of place system, also the hardware complexity to total system has a significant impact.Therefore, at a high speed, the back of low complex degree plays a part positive to transformational structure to the application of residue number system.

Summary of the invention

The purpose of this invention is to provide a kind of base and be respectively M ₁={ 2 ⁿ-1,2 ⁿ+ 1,2 ⁿAnd M ₂={ 2 ⁿ-1,2 ⁿ+ 1,2 ²ⁿThe residue number system of+1} back to conversion equipment and method thereof.

Particularly, one of purpose of the present invention provides a kind of base and is respectively M ₁={ 2 ⁿ-1,2 ⁿ+ 1,2 ⁿAnd M ₂={ 2 ⁿ-1,2 ⁿ+ 1,2 ²ⁿThe residue number system of+1} (RNS) back is to conversion (R2B) device, and it comprises:

The data pretreatment unit, it is based on remainder base M ₁And M ₂Have one group of common sub-remainder base M _S={ 2 ⁿ-1,2 ⁿ+ 1} is with M ₁Be decomposed into M _SAnd M ₁₂={ 2 ²ⁿ-1,2 ⁿ, with M ₂Be decomposed into M _SAnd M ₂₂={ 2 ²ⁿ-1,2 ²ⁿ+ 1};

The back is finished M respectively to converting unit _S, M ₁₂And M ₂₂Back to conversion.

Further, M _Safterwards adopt binary addition, shifting function, LUT etc. to conversion.

Further, above-mentioned M _Safterwards specifically comprise as lower member to conversion equipment:

(Ripple Carry Adder RCA), is used for calculating (x to the ripple carry adder 201 of a n+1 bit ₁-x ₂) value, Xi is meant, for N coprime in twos positive integer m ₁, m ₂..., m _N, make M=m ₁* m ₂* ... * m _N, any one positive integer X in interval [0, M-1], it is about m ₁, m ₂..., m _NRemainder be respectively

I=1,2 ..., N

The data cutout unit is used to intercept the high n bit of the output of this ripple carry adder, thereby obtains

The data shift cells left, it is right to be used for realizing

The operation of n bit moves to left;

The data splicing unit is used for the output result of data interception unit and data shift cells left unit is spliced, thereby obtains $T=x_2+\frac{(x_1-x_2)+\mathrm{\&alpha;}(2^n-1)}{2}\&times;(2^n+1)$ $=x_2+[{\mathrm{MSB}}_1^n(x_1-x_2)+h]\&times;(2^n+1)$ $=x_2+\left[{\mathrm{MSB}}_1^n(x_1-x_2)\right]\&times;(2^n+1)+h\&times;(2^n+1)$ In $\left[{\mathrm{MSB}}_1^n(x_1-x_2)\right]\&times;(2^n+1);$

Data storage cell is used to store h * (2 ⁿ+ 1) LUT, the degree of depth of LUT is 4, width is 2n;

(Carry Save Adder CSA), is used for finishing the carry save adder of a 2n bit $T=x_2+\frac{(x_1-x_2)+\mathrm{\&alpha;}(2^n-1)}{2}\&times;(2^n+1)$ $=x_2+[{\mathrm{MSB}}_1^n(x_1-x_2)+h]\&times;(2^n+1)$ $(=x_2+[{\mathrm{MSB}}_1^n(x_1-x_2)]\&times;(2^n+1)+h\&times;(2^n+1)$ In the operation on last equal sign the right;

The output of the carry save adder of 2n bit is { 2 ⁿ-1,2 ⁿ+ 1}'s is back to transformation result T, and it is made up of two parts: part and SUM and carry CARRY.

Further, the present invention according to Chinese remainder theorem to M ₁₂Carry out the back to conversion.

Further, M ₁₂afterwards further comprise following content to conversion equipment:

A CSA processor, it comprises three input items: the x of n bit ₃Correspond to remainder base 2 ⁿ, the SUM of 2n bit and CARRY are that the back is to transformation result T;

N bit ripple carry adder RCA is used for the result of CSA processor is carried out carry addition;

The data cutout unit is used for the low n bit of the output of interception unit n bit ripple carry adder RCA, and its output is

In Y (n bit), its as final output result's high n bit, participates on the one hand on the other hand

In the computing of T-Y, the computing of T-Y is as final output result's low 2n bit.

Further, M ₂₂Adopt and M _SSimilarly structure realizes.

Another object of the present invention is to, provide a kind of base to be respectively M ₁={ 2 ⁿ-1,2 ⁿ+ 1,2 ⁿAnd M ₃={ 2 ⁿ-1,2 ⁿ+ 1,2 ²ⁿThe residue number system of+1} (RNS) back is to conversion (R2B) method, and it comprises following steps:

The data pre-treatment step, it is based on remainder base M ₁And M ₂Have one group of common sub-remainder base M _S={ 2 ⁿ-1,2 ⁿ+ 1} is with M ₁Be decomposed into M _SAnd M ₁₃={ 2 ²ⁿ-1,2 ⁿ, with M ₂Be decomposed into M _SAnd M ₂₃={ 2 ²ⁿ-1,2 ²ⁿ+ 1}:

The back is finished M respectively to switch process _S, M ₁₂And M ₂₂Back to conversion.

Further, M _Safterwards adopt binary addition, shifting function, LUT etc. to conversion.

Further, M _Safterwards specifically comprise to switch process:

(Ripple Carry Adder RCA) is used for calculating (x with the ripple carry adder 201 of a n+1 bit ₁-x ₂) value, Xi is meant, for N coprime in twos positive integer m ₁, m ₂..., m _N, make M=m ₁* m ₂* ... * m _N, any one positive integer X in interval [0, M-1], it is about m ₁, m ₂..., m _NRemainder be respectively

I=1,2 ..., N

Intercept the high n bit of the output of this ripple carry adder, thereby obtain

Right

The n bit moves to left;

Above-mentioned intercepting and the output result that moves to left are spliced, thereby obtain

$T=x_2+\frac{(x_1-x_2)+\mathrm{\&alpha;}(2^n-1)}{2}\&times;(2^n+1)$

$=x_2+[{\mathrm{MSB}}_1^n(x_1-x_2)+h]\&times;(2^n+1)$

$=x_2+\left[{\mathrm{MSB}}_1^n(x_1-x_2)\right]\&times;(2^n+1)+h\&times;(2^n+1)$ In $\left[{\mathrm{MSB}}_1^n(x_1-x_2)\right]\&times;(2^n+1);$

Storage h * (2 ⁿ+ 1) in LUT, the degree of depth of LUT is 4, and width is 2n;

Realize by CSA $T=x_2+\frac{(x_1-x_2)+\mathrm{\&alpha;}(2^n-1)}{2}\&times;(2^n+1)$ $=x_2+[{\mathrm{MSB}}_1^n(x_1-x_2)+h]\&times;(2^n+1)$ $=x_2+\left[{\mathrm{MSB}}_1^n(x_1-x_2)\right]\&times;(2^n+1)+h\&times;(2^n+1)$ In the operation on last equal sign the right;

The output of CSA is { 2 ⁿ-1,2 ⁿ+ 1}'s is back to transformation result T, and it is made up of two parts: part and SUM and carry CARRY.

Further, according to Chinese remainder theorem to M ₁₂Carry out the back to conversion.

Further, M ₁₂afterwards specifically comprise to switch process:

A CSA processor is imported 3: the x of n bit ₃Correspond to remainder base 2 ⁿ, the part of 2n bit and SUM and carry CARRY are that the back is to transformation result T;

Result to the CSA processor carries out carry addition;

The low n bit of intercepting abovementioned steps output, its output is

In Y (n bit), its as final output result's high n bit, participates on the one hand on the other hand

In the computing of T-Y, the computing of T-Y is as final output result's low 2n bit.

Further, M ₂₂afterwards adopt and M to conversion _SSimilarly step realizes.

Generally speaking, the back of residue number system is that (Chinese Remainder Theory CRT) finishes by Chinese remainder theorem to conversion.CRT is described below:

For N coprime in twos positive integer m ₁, m ₂..., m _N, make M=m ₁* m ₂* ... * m _N, any one positive integer X in interval [0, M-1], it is about m ₁, m ₂..., m _NRemainder be respectively

I=1,2 ..., N, note X (x ₁, x ₂..., x _N), if known x ₁, x ₂..., x _N, then X can calculate by following formula

$X={<\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{A}_i{<a_i{x}_i>}_{m_i}>}_M---\left(1\right)$

A wherein _i=M/m _i, a _iSatisfy

a _iBe called A _iAbout m _iThe mould inverse, be designated as

Below be an inference of Chinese remainder theorem:

Given remainder base { m ₁, m ₂..., m _NAnd one group of remainder (x ₁, x ₂..., x _N), then the binary representation of this group remainder can be calculated by following formula

$X=x_1+{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_1{<\begin{array}{c}{k}_1(x_2-x_1)+k_2{m}_2(x_3-x_2)+...\\ +k_{N-1}{m}_2{\mathrm{<figure-callout\; id="3"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000022.png"\; state="\{\{state\}\}">m</figure-callout>}}_3...m_{N-1}(x_N-x_{N-1})\end{array}>}_{m_2{\mathrm{<figure-callout\; id="3"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000022.png"\; state="\{\{state\}\}">m</figure-callout>}}_3...m_N}---\left(2\right)$

Wherein ${<k_1{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_1>}_{m_2{\mathrm{<figure-callout\; id="3"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000022.png"\; state="\{\{state\}\}">m</figure-callout>}}_3...m_N}=1,$ ${<k_2{m}_1{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_2>}_{{\mathrm{<figure-callout\; id="3"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000022.png"\; state="\{\{state\}\}">m</figure-callout>}}_3...m_N}=1,$ ..., ${<k_{N-1}{m}_1{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_2...m_N>}_{m_N}=1.$

When N=2, make m ₁＜m ₂, get by formula (2)

$X=x_2+{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_2{<k(x_1-x_2)>}_{m_1}---\left(3\right)$

Wherein

x ₁And x ₂Bit wide be respectively n ₁And n ₂Bit.

By formula (3),

X＝x ₂+k(x ₁-x ₂)m ₂+tm ₁m ₂ (4)

T ∈ wherein.Select suitable t can guarantee X ∈ [0, m ₁m ₂-1].Formula (4) both sides are with multiply by (m ₂-m ₁), delivery m then ₁m ₂

${<X(m_2-m_1)>}_{m_1{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_2}={<[x_2+k(x_1-x_2){\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_2+t{m}_1{m}_2]\&CenterDot;(m_2-m_1)>}_{m_1{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_2}$

$=<{\begin{array}{c}{x}_2(m_2-m_1)+k(x_1-x_2)m_2(m_2-m_1)\\ +t{m}_1{m}_2(m_2-m_1)\end{array}>}_{m_1{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_2}$

$=<{\begin{array}{c}{x}_2(m_2-m_1)+k(x_1-x_2)m_2{m}_2\\ -k(x_1-x_2)m_2{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_1+t{m}_1{m}_2(m_2-m_1)\end{array}>}_{m_1{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_2}$

$=<{\begin{array}{c}{x}_2(m_2-m_1)+m_2(x_1-x_2)\\ -k{m}_1{m}_2(x_1-x_2)+t{m}_1{m}_2(m_2-m_1)\end{array}>}_{m_1{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_2}$

$=<x_2(m_2-m_1)+m_2{\left[\begin{array}{c}(x_1-x_2)+\\ m_1(-k(x_1-x_2)+t(m_2-m_1))\end{array}\right]>}_{m_1{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_2}$

$={<x_2(m_2-m_1)+m_2[(x_1-x_2)+\mathrm{\&alpha;}{m}_1]>}_{m_1{m}_2}---\left(5\right)$

Wherein α=-k (x ₁-x ₂)+t (m ₂-m ₁).Same on formula (5) both sides divided by (m ₂-m ₁), then

$X=x_2+\frac{(x_1-x_2)+\mathrm{\&alpha;}{m}_1}{m_2-{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_1}\&times;{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_2.---\left(6\right)$

If m ₂-m ₁=2 ^k(k ∈), then formula (6) can be rewritten as

$X=x_2+\frac{(x_1-x_2)+\mathrm{\&alpha;}{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_1}{2^k}\&times;{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA0000063994580000011.png,HDA0000063994580000013.png"\; state="\{\{state\}\}">m</figure-callout>}}_2$

$=x_2+[{\mathrm{MSB}}_k^{n-1}(x_1-x_2)+h]\&times;m_2---\left(7\right)$

Wherein

And n=max (n ₁, n ₂),

(x is got in expression ₁-x ₂) high n-k bit,

(x is got in expression ₁-x ₂) low k bit.In formula (8), because x ₁, x ₂Be known with k, therefore, can pass through (x ₁-x ₂) sign bit Sign (x ₁-x ₂) and low k position, calculate in advance the value of h.

In conjunction with M ₁And M ₂The characteristics of these two groups of remainder bases, the present invention at first divides into groups { (2 with three mould remainder bases ⁿ-1,2 ⁿ+ 1), 2 ⁿAnd { (2 ⁿ-1,2 ⁿ+ 1), 2 ²ⁿ+ 1} is promptly earlier M _s={ 2 ⁿ-1,2 ⁿ+ 1} handles as a sub-remainder base, then with 2 ⁿ-1 and 2 ⁿ+ 1 multiplies each other obtains a new remainder base, is designated as m _t=(2 ⁿ-1) * (2 ⁿ+ 1)=2 ²ⁿ-1, at last more respectively to M ₁₂={ 2 ²ⁿ-1,2 ⁿAnd M ₂₂={ 2 ²ⁿ-1,2 ²ⁿ+ 1} handles, and can obtain required M ₁And M ₂Back to conversion.

For M _s={ 2 ⁿ-1,2 ⁿ+ 1}, m ₂-m ₁=2, by formula (7) and formula (8),

$T=x_2+\frac{(x_1-x_2)+\mathrm{\&alpha;}(2^n-1)}{2}\&times;(2^n+1)$

$=x_2+[{\mathrm{MSB}}_1^n(x_1-x_2)+h]\&times;(2^n+1)$

$=x_2+\left[{\mathrm{MSB}}_1^n(x_1-x_2)\right]\&times;(2^n+1)+h\&times;(2^n+1)---\left(9\right)$

Wherein

And Sign (x ₁-x ₂), Relation between α and the h is as shown in table 1.

Table 1

The relation of α and h

Therefore, can be with h * (2 ⁿ+ 1) calculate in advance, (Lookup table is LUT) or among the ROM, then according to Sign (x to be stored in look-up table ₁-x ₂) and

Retrieve corresponding value.In addition, because

Be the integer of n bit, therefore,

Be will

The integral body n bit (the right mend 0) that moves to left.So, Value can obtain by simple concatenation, be about to two n bits

Being connected together gets final product, and does not need extra addition or multiply operation.

So far, tried to achieve and finished M _s={ 2 ⁿ-1,2 ⁿ+ 1}'s is back to conversion.For M ₁And M ₂, need respectively to M ₁₂={ 2 ²ⁿ-1,2 ⁿAnd M ₂₂={ 2 ²ⁿ-1,2 ²ⁿ+ 1} carries out the back can obtain final result to conversion.Introduce M below respectively ₁₂And M ₂₂Back to conversion.

For M ₁₂={ 2 ²ⁿ-1,2 ⁿ, get by formula (3)

$X=T+(2^{2n}-1){<{<{(2^{2n}-1)}^{-1}>}_{2^n}(x_3-T)>}_{2^n}$

$=T+(2^{2n}-1){<T-x_3>}_{2^n}$

$=T+2^{2n}{<T-x_3>}_{2^n}-{<T-x_3>}_{2^n}---\left(11\right)$

Make Y=T-x ₃, T is the number of a 2n bit, so the scale-of-two of Y can be expressed as (y _2n-1Y _N-1Y ₁y ₀) ₂Therefore, formula (11) can be rewritten as

Therefore, M ₁Final back to transformation result X can with Splice in (y _N-1Y ₁y ₀) ₂Obtain afterwards.That is to say that the high n bit of X is (y _N-1Y ₁y ₀) ₂, low 2n bit is

Contrast M ₂₂={ 2 ²ⁿ-1,2 ²ⁿ+ 1} and M _s={ 2 ⁿ-1,2 ⁿ+ 1} can find, M ₂₂Afterwards can adopt and be similar to M to conversion _sMethod realize.Therefore

Wherein

And Sign (T-x ₃),

With

Relation as shown in table 2.

Table 2

With

Relation

In like manner, calculate in advance

Be stored among LUT or the ROM.

Also can be by with two 2n bits

Being stitched together obtains, and does not need extra additions and multiply operation.

In the present invention, at first according to M ₁And M ₂Characteristics, promptly they have one group of common sub-remainder base M _S={ 2 ⁿ-1,2 ⁿ+ 1} finishes M earlier _SBack to conversion, finish M then on this basis more respectively ₁₃={ 2 ²ⁿ-1,2 ⁿAnd M ₃₃={ 2 ²ⁿ-1,2 ²ⁿ+ 1}'s is back to conversion, and their result promptly is respectively M ₁And M ₃Transformation result.In addition, M ₂₂Form and M _SSimilar, therefore can adopt similar structure to realize, thereby make modular design be reused well, avoided the trouble of design iterations.To based on M ₁Or M ₂The application of this residue number system in digital information processing system played positive effect.

Description of drawings

Fig. 1 is that base is M ₁={ 2 ⁿ-1,2 ⁿ+ 1,2 ⁿResidue number system back to the conversion scantling plan;

Fig. 2 is that base is M _S={ 2 ⁿ-1,2 ⁿThe residue number system of+1} back to the Change-over knot composition;

Fig. 3 is that base is M ₁₂={ 2 ²ⁿ-1,2 ⁿResidue number system back to the Change-over knot composition;

Fig. 4 is that base is M ₂={ 2 ⁿ-1,2 ⁿ+ 1,2 ²ⁿThe residue number system of+1} back to the conversion scantling plan;

Fig. 5 is that base is M ₂₂={ 2 ²ⁿ-1,2 ²ⁿThe residue number system of+1} back to the Change-over knot composition.

Embodiment

Below in conjunction with embodiment foregoing invention content of the present invention is described in further detail.

But this should be interpreted as that the scope of the above-mentioned theme of the present invention only limits to following embodiment.Not breaking away under the above-mentioned technological thought situation of the present invention, according to ordinary skill knowledge and customary means, make various replacements and change, all should comprise within the scope of the invention.

A kind of base is respectively M ₁={ 2 ⁿ-1,2 ⁿ+ 1,2 ⁿAnd M ₂={ 2 ⁿ-1,2 ⁿ+ 1,2 ²ⁿThe residue number system of+1} (RNS) back is to conversion (R2B) device, and it comprises:

The data pretreatment unit, it is based on remainder base M ₁And M ₂Have one group of common sub-remainder base M _S={ 2 ⁿ-1,2 ⁿ+ 1} is with M ₁Be decomposed into M _SAnd M ₁₂={ 2 ²ⁿ-1,2 ⁿ, with M ₂Be decomposed into M _SAnd M ₂₂={ 2 ²ⁿ-1,2 ²ⁿ+ 1};

The back is finished M respectively to converting unit _S, M ₁₂And M ₂₂Back to conversion.

Further, M _Safterwards adopt binary addition, shifting function, LUT etc. to conversion.

Further, above-mentioned M _Safterwards specifically comprise as lower member to conversion equipment:

(Ripple Carry Adder RCA), is used for calculating (x to the ripple carry adder 201 of a n+1 bit ₁-x ₂) value, Xi is meant, for N coprime in twos positive integer m ₁, m ₂..., m _N, make M=m ₁* m ₂* ... * m _N, any one positive integer X in interval [0, M-1], it is about m ₁, m ₂..., m _NRemainder be respectively

I=1,2 ..., N;

Data cutout unit 202 is used to intercept the high n bit of the output of this ripple carry adder, thereby obtains

Data shift cells left 203, it is right to be used for realizing

The operation of n bit moves to left;

Data splicing unit 204 is used for the output result of data interception unit 202 and data shift cells left unit 203 is spliced, thereby obtains $T=x_2+\frac{(x_1-x_2)+\mathrm{\&alpha;}(2^n-1)}{2}\&times;(2^n+1)$ $=x_2+[{\mathrm{MSB}}_1^n(x_1-x_2)+h]\&times;(2^n+1)$ $=x_2+\left[{\mathrm{MSB}}_1^n(x_1-x_2)\right]\&times;(2^n+1)+h\&times;(2^n+1)$ In $\left[{\mathrm{MSB}}_1^n(x_1-x_2)\right]\&times;(2^n+1);$

Data storage cell 205 is used to store h * (2 ⁿ+ 1) LUT, the degree of depth of LUT is 4, width is 2n;

(Carry Save Adder CSA), is used for finishing the carry save adder 206 of a 2n bit $T=x_2+\frac{(x_1-x_2)+\mathrm{\&alpha;}(2^n-1)}{2}\&times;(2^n+1)$ $=x_2+[{\mathrm{MSB}}_1^n(x_1-x_2)+h]\&times;(2^n+1)$ $(=x_2+[{\mathrm{MSB}}_1^n(x_1-x_2)]\&times;(2^n+1)+h\&times;(2^n+1)$ In the operation on last equal sign the right;

The output of the carry save adder 206 of 2n bit is { 2 ⁿ-1,2 ⁿ+ 1}'s is back to transformation result T, and it is made up of two parts: part and SUM and carry CARRY.

Further, the present invention according to Chinese remainder theorem to M ₁₂Carry out the back to conversion.

Further, M ₁₂afterwards further comprise following content to conversion equipment:

A CSA processor, it comprises three input items: the x of n bit ₃Correspond to remainder base 2 ⁿ, the SUM of 2n bit and CARRY are that the back is to transformation result T;

N bit ripple carry adder RCA 302 (1) is used for the result of CSA processor is carried out carry addition;

Data cutout unit 303 is used for the low n bit of the output of interception unit n bit ripple carry adder RCA 302 (1), and its output is

In Y (n bit), its as final output result's high n bit, participates on the one hand on the other hand

In the computing of T-Y, the computing of T-Y is as final output result's low 2n bit.

Further, M ₂₂Adopt and M _SSimilarly structure realizes.

Another object of the present invention is to, provide a kind of base to be respectively M ₁={ 2 ⁿ-1,2 ⁿ+ 1,2 ⁿAnd M ₂={ 2 ⁿ-1,2 ⁿ+ 1,2 ²ⁿThe residue number system of+1} (RNS) back is to conversion (R2B) method, and it comprises following steps:

The data pre-treatment step, it is based on remainder base M ₁And M ₂Have one group of common sub-remainder base M _S={ 2 ⁿ-1,2 ⁿ+ 1} is with M ₁Be decomposed into M _SAnd M ₁₃={ 2 ²ⁿ-1,2 ⁿ, with M ₂Be decomposed into M _SAnd M ₂₂={ 2 ²ⁿ-1,2 ²ⁿ+ 1};

The back is finished M respectively to switch process _S, M ₁₂And M ₂₂Back to conversion.

Further, M _Safterwards adopt binary addition, shifting function, LUT etc. to conversion.

Further, M _Safterwards specifically comprise to switch process:

(Ripple Carry Adder RCA) is used for calculating (x with the ripple carry adder 201 of a n+1 bit ₁-x ₂) value, Xi is meant, for N coprime in twos positive integer m ₁, m ₂..., m _N, make M=m ₁* m ₂* ... * m _N, any one positive integer X in interval [0, M-1], it is about m ₁, m ₂..., m _NRemainder be respectively

I=1,2 ..., N

Intercept the high n bit of the output of this ripple carry adder, thereby obtain

Right

The n bit moves to left;

Above-mentioned intercepting and the output result that moves to left are spliced, thereby obtain

$T=x_2+\frac{(x_1-x_2)+\mathrm{\&alpha;}(2^n-1)}{2}\&times;(2^n+1)$

$=x_2+[{\mathrm{MSB}}_1^n(x_1-x_2)+h]\&times;(2^n+1)$

$=x_2+\left[{\mathrm{MSB}}_1^n(x_1-x_2)\right]\&times;(2^n+1)+h\&times;(2^n+1)$ In $\left[{\mathrm{MSB}}_1^n(x_1-x_2)\right]\&times;(2^n+1);$

Storage h * (2 ⁿ+ 1) in LUT, the degree of depth of LUT is 4, and width is 2n;

Realize by CSA $T=x_2+\frac{(x_1-x_2)+\mathrm{\&alpha;}(2^n-1)}{2}\&times;(2^n+1)$ $=x_2+[{\mathrm{MSB}}_1^n(x_1-x_2)+h]\&times;(2^n+1)$ $=x_2+\left[{\mathrm{MSB}}_1^n(x_1-x_2)\right]\&times;(2^n+1)+h\&times;(2^n+1)$ In the operation on last equal sign the right;

The output of CSA is { 2 ⁿ-1,2 ⁿ+ 1}'s is back to transformation result T, and it is made up of two parts: part and SUM and carry CARRY.

Further, according to Chinese remainder theorem to M ₁₂Carry out the back to conversion.

Further, M ₁₂afterwards specifically comprise to switch process:

A CSA processor is imported 3: the x of n bit ₃Correspond to remainder base 2 ⁿ, the part of 2n bit and SUM and carry CARRY are that the back is to transformation result T;

Result to the CSA processor carries out carry addition;

The low n bit of intercepting abovementioned steps output, its output is In Y (n bit), its as final output result's high n bit, participates on the one hand on the other hand

In the computing of T-Y, the computing of T-Y is as final output result's low 2n bit.

Further, M ₂₂afterwards adopt and M to conversion _SSimilarly step realizes.

Fig. 1 shows base for { 2 ⁿ-1,2 ⁿ+ 1,2 ⁿResidue number system back to the conversion general structure.It is made up of 101 and 102 two big unit.According to formula (9), (10) and table 1, unit 101 is used for finishing M _s={ 2 ⁿ-1,2 ⁿ+ 1}'s is back to conversion.Unit 102 is for to finish M according to formula (12) and (13) ₁₂={ 2 ²ⁿ-1,2 ⁿBack to conversion, thereby obtain { 2 ⁿ-1,2 ⁿ+ 1,2 ⁿFinal back to the result of conversion.

Fig. 2 shows base for { 2 ⁿ-1,2 ⁿThe residue number system of+1} back to the Change-over knot composition is corresponding to the unit among Fig. 1 101.Unit 201 is that (Ripple Carry Adder RCA), is used for calculating (x for the ripple carry adder of a n+1 bit ₁-x ₂) value.The high n bit of the output of unit 202 interceptings 201, thus obtain

It is right that unit 203 is used for realizing

The move to left operation of n bit, unit 204 is used for the output result of unit 202 and unit 203 is spliced, thereby in the formula of obtaining (9)

Unit 205 comes with storage h * (2 ⁿ+ 1) LUT, wherein h has only 4 values, sees Table 1.The degree of depth of LUT is 4, and width is 2n.According to (x ₁-x ₂) sign bit Sign (x ₁-x ₂) and lowest order Can find corresponding h * (2 ⁿ+ 1) value.206 is that (Carry Save Adder CSA), is used for the operation on last equal sign the right in the perfect (9) for the carry save adder of a 2n bit.Because the operation on last the equal sign the right in the formula (9) is obtained by three number additions, is therefore realized by CSA.The output of unit 206 is { 2 ⁿ-1,2 ⁿ+ 1}'s is back to transformation result T, and it is made up of two parts: part and SUM and carry CARRY.

Fig. 3 shows base for { 2 ²ⁿ-1,2 ⁿResidue number system back to conversion implementation structure view, corresponding to the unit among Fig. 1 102.X in three inputs ₃(n bit) corresponds to remainder base 2 ⁿ, two other input SUM and CARRY (being the 2n bit) are 101 output T.Unit 302 (1) is n bit ripple carry adder RCA.Unit 301 (1) is output as the number of 2n bit, and why unit 302 (1) selects the RCA of n bit, is because 303 of unit need the low n bit of the output of interception unit 302 (1).The output of unit 303 is the Y (n bit) in the formula (12), and it is on the one hand as the final high n bit of exporting the result, the computing of the T-Y in the participatory (12) on the other hand.The computing of T-Y is as final output result's low 2n bit.

Fig. 4 shows base for { 2 ⁿ-1,2 ⁿ+ 1,2 ²ⁿThe residue number system of+1} back to the Change-over knot composition.Unit 401 is just the same with unit 101 or Fig. 2 among Fig. 1, is used for finishing base for { 2 ⁿ-1,2 ⁿ+ 1}'s is back to conversion.Unit 402 is used for finishing base for { 2 ²ⁿ-1,2 ²ⁿ+ 1}'s is back to conversion.Because { 2 ²ⁿ-1,2 ²ⁿ+ 1} and { 2 ⁿ-1,2 ⁿ+ 1} similarity, { 2 ²ⁿ-1,2 ²ⁿBack can the employing to conversion of+1} is similar to { 2 ⁿ-1,2 ⁿThe structure of+1} realizes.

Fig. 5 is that base is for { 2 ²ⁿ-1,2 ²ⁿThe residue number system of+1} back to conversion implementation structure view, corresponding to unit among Fig. 4 402.It has three input x ₃, SUM and CARRY, wherein x ₃Corresponding to remainder base 2 ²ⁿ+ 1, SUM and CARRY are part and and the carry of the output T of unit 401.Unit 501 (1) is a 2n+1 bit CSA, is used for T-x in the perfect (13) ₃Computing.Unit 502 (1) is exported to do to merge to unit 501 (1) and is handled, and obtains a 2n+1 bit output.Unit 503 is used for finishing

Operation, the i.e. high 2n bit of interception unit 502 (1).It is right that unit 504 is used for realizing

The operation of the 2n bit that moves to right, unit 505 is used for finishing the concatenation to unit 503 and unit 504 output results, obtains in the formula (13) Unit 506 is that a degree of depth is 4, and width is the LUT of 4n bit, is used for storing

Value; It is by T-x ₃Sign bit Sign (T-x ₃) and lowest order Common retrieval, thus obtain corresponding

With Sign (T-x ₃),

Relation see Table 2.Unit 501 (2) and unit 501 (3) are the CSA of 4n bit, and 502 (2) is the RCA of 4n bit.Their threes are used in the common perfect (13)

Operation, thus M obtained ₂Final back to transformation result.

Claims (10)

1. a base is M ₁={ 2 ⁿ-1,2 ⁿ+ 1,2 ⁿAnd M ₂={ 2 ⁿ-1,2 ⁿ+ 1,2 ²ⁿThe residue number system of+1} (RNS) back is characterized in that to conversion (R2B) device, comprises:

The data pretreatment unit, it is based on remainder base M ₁And M ₂Have one group of common sub-remainder base M _S={ 2 ⁿ-1,2 ⁿ+ 1} is with M ₁Be decomposed into M _SAnd M ₁₂={ 2 ²ⁿ-1,2 ⁿ, with M ₂Be decomposed into M _SAnd M ₂₂={ 2 ²ⁿ-1,2 ²ⁿ+ 1};

The back is finished M respectively to converting unit _S, M ₁₂And M ₂₂Back to conversion.

2. device according to claim 1 is characterized in that: M _Safterwards adopt binary addition, shifting function, LUT to conversion.

3. device according to claim 2 is characterized in that M _Safterwards specifically comprise as lower member to conversion equipment:

The ripple carry adder 201 of a n+1 bit is used for calculating (x ₁-x ₂) value, Xi is meant, for N coprime in twos positive integer m ₁, m ₂..., m _N, make M=m ₁* m ₂* ... * m _N, any one positive integer X in interval [0, M-1], it is about m ₁, m ₂..., m _NRemainder be respectively I=1,2 ..., N;

Data cutout unit 202 is used to intercept the high n bit of the output of this ripple carry adder, thereby obtains

Data shift cells left 203, it is right to be used for realizing

The operation of n bit moves to left;

Data splicing unit 204 is used for the output result of data interception unit 202 and data shift cells left unit 203 is spliced, thereby obtains $T=x_2+\frac{(x_1-x_2)+\mathrm{\&alpha;}(2^n-1)}{2}\&times;(2^n+1)$ $=x_2+[{\mathrm{MSB}}_1^n(x_1-x_2)+h]\&times;(2^n+1)$ $=x_2+\left[{\mathrm{MSB}}_1^n(x_1-x_2)\right]\&times;(2^n+1)+h\&times;(2^n+1)$ In $\left[{\mathrm{MSB}}_1^n(x_1-x_2)\right]\&times;(2^n+1);$

Data storage cell 205 is used to store h * (2 ⁿ+ 1) LUT, the degree of depth of LUT is 4, width is 2n;

(Carry Save Adder CSA), is used for finishing the carry save adder 206 of a 2n bit $T=x_2+\frac{(x_1-x_2)+\mathrm{\&alpha;}(2^n-1)}{2}\&times;(2^n+1)$ $=x_2+[{\mathrm{MSB}}_1^n(x_1-x_2)+h]\&times;(2^n+1)$ $(=x_2+[{\mathrm{MSB}}_1^n(x_1-x_2)]\&times;(2^n+1)+h\&times;(2^n+1)$ In the operation on last equal sign the right;

The output of the carry save adder 206 of 2n bit is { 2 ⁿ-1,2 ⁿ+ 1}'s is back to transformation result T, and it is made up of two parts: part and SUM and carry CARRY.

4. according to claim 1 or 2 or 3 described devices, it is characterized in that: according to Chinese remainder theorem to M ₁₂Carry out the back to conversion.

5. device according to claim 4 is characterized in that: M ₁₂afterwards further comprise following content to conversion equipment:

A CSA processor, it comprises three input items: the x of n bit ₃Correspond to remainder base 2 ⁿ, the part of 2n bit and SUM and carry CARRY are that the back is to transformation result T;

N bit ripple carry adder RCA 302 (1) is used for the result of CSA processor is carried out carry addition;

Data cutout unit 303 is used for the low n bit of the output of interception unit n bit ripple carry adder RCA 302 (1), and its output is In the Y of n bit, its as final output result's high n bit, participates on the one hand on the other hand In the computing of T-Y, the computing of T-Y is as final output result's low 2n bit.

6. according to the wherein any described device of claim 1,2,3,5, it is characterized in that: M ₂₂Adopt and M _SSimilarly structure realizes.

7. a base is M ₁={ 2 ⁿ-1,2 ⁿ+ 1,2 ⁿAnd M ₂={ 2 ⁿ-1,2 ⁿ+ 1,2 ²ⁿThe residue number system of+1} (RNS) back is characterized in that to conversion (R2B) method, comprises following steps:

The data pre-treatment step, it is based on remainder base M ₁And M ₂Have one group of common sub-remainder base M _S={ 2 ⁿ-1,2 ⁿ+ 1} is with M ₁Be decomposed into M _SAnd M ₁₂={ 2 ²ⁿ-1,2 ⁿ, with M ₂Be decomposed into M _SAnd M ₂₂={ 2 ²ⁿ-1,2 ²ⁿ+ 1};

The back is finished M respectively to switch process _S, M ₁₂And M ₂₂Back to conversion.

8. method according to claim 7 is characterized in that: described M _Safterwards specifically comprise to switch process:

(Ripple Carry Adder RCA) is used for calculating (x with the ripple carry adder 201 of a n+1 bit ₁-x ₂) value, Xi is meant, for N coprime in twos positive integer m ₁, m ₂..., m _N, make M=m ₁* m ₂* ... * m _N, any one positive integer X in interval [0, M-1], it is about m ₁, m ₂..., m _NRemainder be respectively

I=1,2 ..., N;

Intercept the high n bit of the output of this ripple carry adder, thereby obtain

Right

The n bit moves to left;

Above-mentioned intercepting and the output result that moves to left are spliced, thereby obtain

$T=x_2+\frac{(x_1-x_2)+\mathrm{\&alpha;}(2^n-1)}{2}\&times;(2^n+1)$

$=x_2+[{\mathrm{MSB}}_1^n(x_1-x_2)+h]\&times;(2^n+1)$

$=x_2+\left[{\mathrm{MSB}}_1^n(x_1-x_2)\right]\&times;(2^n+1)+h\&times;(2^n+1)$ In $\left[{\mathrm{MSB}}_1^n(x_1-x_2)\right]\&times;(2^n+1);$

Storage h * (2 ⁿ+ 1) in LUT, the degree of depth of LUT is 4, and width is 2n;

Realize by CSA $T=x_2+\frac{(x_1-x_2)+\mathrm{\&alpha;}(2^n-1)}{2}\&times;(2^n+1)$ $=x_2+[{\mathrm{MSB}}_1^n(x_1-x_2)+h]\&times;(2^n+1)$ $=x_2+\left[{\mathrm{MSB}}_1^n(x_1-x_2)\right]\&times;(2^n+1)+h\&times;(2^n+1)$ In the operation on last equal sign the right;

The output of CSA is { 2 ⁿ-1,2 ⁿ+ 1}'s is back to transformation result T, and it is made up of two parts: part and SUM and carry CARRY.

9. according to claim 7 or 8 described methods, it is characterized in that: according to Chinese remainder theorem to M ₁₂Carry out the back to conversion, described M ₁₂afterwards specifically comprise to switch process:

A CSA processor is imported 3: the x of n bit ₃Correspond to remainder base 2 ⁿ, the part of 2n bit and SUM and carry CARRY are that the back is to transformation result T;

Result to the CSA processor carries out carry addition;

The low n bit of intercepting abovementioned steps output, its output is In the Y of n bit, its as final output result's high n bit, participates on the one hand on the other hand In the computing of T-Y, the computing of T-Y is as final output result's low 2n bit.

10. according to any described method in the claim 7,8,9, it is characterized in that: M ₂₂afterwards adopt and M to conversion _SSimilarly step realizes.

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