On the Design of an On-line Complex Householder Transform

Transcription

1 On the esign of an On-line Coplex Householder Transfor Robert McIlhenny Coputer Science epartent California State University, Northridge Northridge, CA 9330 Milo s Ercegovac Coputer Science epartent University of California, os Angeles os Angeles, CA ilos@csuclaedu Abstract In this paper, we present an ipleentation for the Coplex Householder Transfor, using coplex nuber on-line arithetic, based on adopting a redundant coplex nuber syste (RCNS) to represent coplex operands as a single nuber We present coparisons with (i) a real nuber on-line arithetic approach, and (ii) a real nuber arithetic parallel approach, to deonstrate a signi cant iproveent in cost I INTROUCTION The Householder Transfor [4] is an iportant operation in nuerous signal processing applications, including QR decoption and array processing [7] When the eleents of the atrix are coplex nubers, it is denoted the Coplex Householder Transfor (CHT) [2] The CHT is applied to a colun vector x to zero out all the eleents except the rst one Given a colun vector x =[x,,x k ] T C k, where x = x e jθ with θ R, the basic steps of the CHT algorith are: (i) de ne a colun vector u = x + e jθ x 2 e, where e T =[, 0,,0]; (ii) de ne the k k CHT B as: B = I 2 u H u uuh () (iii) apply the atrix B to the vector x to produce a colun vector y = Bx, in which all eleents are zeroed out except for the rst eleent, ie B B 2 B k B 2 B 22 B 2k B k B k2 B kk x x 2 x k = In these steps, I is the k k identity atrix and u H is the coplex conjugate transpose of u The paraeter e jθ = x, where x xx is the coplex conjugate of x The paraeter x 2 is the nor of x of degree two, in which x 2 = x x + + x kx k The product uh u produces a single real value in which u H u = u u + u 2u u ku k, since u is a colun vector and u H is a row vector The product uu H is a k k atrix V, in which each real eleent v ij = u i u j y 0 0 (2) II COMPEX NUMBER ON-INE FOATING-POINT ARITHMETIC On-line arithetic [3] is a class of arithetic operations in which all operations are perfored digit serially, in a ost signi cant digit rst (MSF) anner Several advantages, copared to conventional parallel arithetic, include: (i) ability to overlap dependent operations, since on-line algoriths produce the output serially, ost-signi cant digit rst, enabling successive operations to begin before previous operations have copleted; (ii) low-bandwidth counication, since interediate results pass to and fro odules digitserially, so connections need only be one digit wide; and (iii) support for variable precision, since once a desired precision is obtained, successive outputs can be ignored One of the key paraeters of on-line arithetic is the on-line delay, de ned as the nuber of digits of the operand(s) necessary in order to generate the rst digit of the result, which is gnerally one cycle after the result is coputed Each successive digit of the result is generated one per cycle This is illustrated in Figure, with on-line delay δ =4 The latency of an on-line arithetic operator, assuing -digit precision is then δ + input copute output Fig δ=4 On-line delay of a function Coplex nuber on-line arithetic [6] uses a class of online arithetic operators on coplex nuber operands For ef cient representation, a Redundant Coplex Nuber Syste (RCNS) [] is adopted A RCNS is a radix rj syste, in which digits are in the set { a,,0,,a}, where r 2 and r 2 /2 a r 2 Such a nuber syste can be denoted RCNS rj,a A Redundant Coplex Nuber Syste with r =2, a =3denoted RCNS 2j,3, allows ease of the de nition of priitive on-line arithetic odules, as well as ease of conversion to and fro other representations /06/$

2 This nuber syste was introduced as Quarter-iaginary Nuber Syste in [5] At the binary level, the digits will be represented using borrow-save encoding, in which each digit x k { 3, 2,, 0,, 2, 3} is represented as 4 bits (x + k,,x k,,x+ k,0,x k,0 ) such that x k =2x + k, 2x k, + x+ k,0 x k,0 This is an extension of radix 2 borrow-save encoding, in which each digit x k {, 0, } is represented as 2 bits (x + k,x k ) such that x k = x + k x k The throughput of a radix 2j on-line arithetic operator is the sae as for the radix 2 ipleentation of a coplex arithetic operator A radix 2j on-line arithetic operator generates real and iaginary digits in alternate cycles, with each radix 2j digit corresponding to two radix 2 digits The equivalent radix 2 ipleentation of a coplex on-line arithetic operator generates real and iaginary digits in the sae cycle This is deonstrated below for the output of two radix 2j borrow-save encoded digits z =(z, +,z,,z+,0,z,0 ) and z 2 =(z 2, +,z 2,,z+ 2,0,z 2,0 ), and four radix 2 borrow-save encoded digits, consisting of real digits z R, =(z + R,,z R, ) and z R,2 =(z + R,2,z R,2 ) and iaginary digits z I, =(z + I,,z I, ) and z I,2 =(z + I,2,z I,2 ) Radix-2: RCNS z + R, z + 2j,3 : R,2 z - R, z - R,2 z + z + I, I,2 z z Ī, Ī,2 Fig 2 z +, z + 2, z -, z - 2, z + z +,0 2,0 z - z -,0 2,0 Throughput of radix 2 and radix 2j digits Since the throughputs are equal (differing only in respective on-line delays), a radix 2j output can be converted to a radix 2 forat for input to radix 2 on-line arithetic operators, and vice versa, at the digit level The algorith below deonstrates the conversion fro a radix 2j borrow-save digit z k =(z + k,,z k,,z+ k,0,z k,0 ) to equivalent individual real and iaginary radix 2 borrow-save digits z R,k =(z + R,k,z R,k ) and z I,k =(z + I,k,z I,k ) Radix 2j to Radix 2 Borrow-save Conversion if k od 4=0 then (z + R,k,z R,k )=(z+ k,0,z k,0 ) (z + I,k,z I,k )=(z+ k+,0, z k+,0 ) else if k od 4= then (z + R,k,z R,k )=(z+ k+,0, z k+,0 ) (z + I,k,z I,k )=(z+ k,0, z k,0 ) else if k od 4=2 then (z + R,k,z R,k )=(z+ k,0, z k,0 ) (z + I,k,z I,k )=(z+ k+,0,z k+,0 ) else if k od 4=3 then (z + R,k,z R,k )=(z+ k+,0,z k+,0 ) (z + I,k,z I,k )=(z+ k,0,z k,0 ) The algorith below deonstrates conversion fro real and iaginary radix 2 borrow-save digits z R,k =(z + R,k,z R,k ) and z I,k = (z + I,k,z I,k ) to a radix 2j borrow-save digit z k = (z + k,,z k,,z+ k,0,z k,0 ) Radix 2 to Radix 2j Borrow-save Conversion if k od 4=0 then (z + k,,z k,,z+ k,0,z k,0 )=(z+ R,k,z R,k,z+ R,k,z R,k ) else if k od 4= then (z + k,,z k,,z+ k,0,z k,0 )=(z+ I,k, z I,k, z+ I,k, z I,k ) else if k od 4=2 then (z + k,,z k,,z+ k,0,z k,0 )=(z+ R,k, z R,k, z+ R,k, z R,k ) else if k od 4=3 then (z + k,,z k,,z+ k,0,z k,0 )=(z+ I,k,z I,k,z+ I,k,z I,k ) Using a radix 2j representation, a oating-point coplex nuber x =(X R + jx I ) (2j) ex can be noralized with regard either to the real coponent X R or the iaginary coponent X I, depending on which has larger absolute value The exponent e x is shared between the real and iaginary coponent A radix 2j fraction x is considered noralized if 2 ax( X R, X I ) < The noralization algorith which takes as input the generated output digit z k, the output exponent e z and the on-line delay for the arithetic operation δ is shown below NORM(z k,e z,δ) done =0 while not(done) loop if k =(δ 2) and z k 0then e z = e z +2 done = if k =(δ ) and z k 0and not(done) then e z = e z + done = else if k δ and z k =0and not(done) then e z = e z else if (k δ and z k 0) then done = end if end loop Although RCNS 2j,3 allows exibility in representation, there are also several drawbacks: Handling digits 3 and 3 requires producing signi cand ultiples 3X and 3X, requiring an extra addition step A signi cand X with fractional real and iaginary coponents X R and X I can have integer digits, such as (322) 2j = j, which can coplicate ensuring coplex signi cands within the range ax( X R, X I ) < Several recoding algoriths to handle these issues are described, including: (i) digit-set recoding; and (ii) ostsigni cant-digit recoding 39

3 igit-set recoding initially recodes a RCNS 2j,3 digit x k { 3,,3} into a pair of digits (t k 2,w k ),inwhicht k 2 {, 0, } and w k { 2,,2} such that x k = 4t k 2 + w k ThenaRCNS 2j,2 digit χ k is coputed as χ k = t k +w k In order to restrict χ k { 2,,2}, two cases of pairs of values ust be prevented: (i) t k =, w k = 2, (ii) t k =, w k = 2 Todoso,x k+2 is exained If x k+2 2 and x k =2, which could allow the rst case, x k is recoded as (, 2), otherwise as (0, 2) In the sae way, if x k+2 2 and x k = 2, which could allow the second case, x k is recoded as (, 2), otherwise as (0, 2) Then it is assured that χ k { 2,,2} The digit-set recoding algorith SREC is shown below SREC(x k,x k+2 ) (, ) if x k = 3 (, 2) if x k = 2 and x k+2 2 (0, 2) if x k = 2 and x k+2 < 2 (0, ) if x k = (t k 2,w k )= (0, 0) if x k =0 (0, ) if x k = (0, 2) if x k =2and x k+2 > 2 (, 2) if x k =2and x k+2 2 (, ) if x k =3 χ k = t k + w k In order to handle carries produced when perforing operations on signi cands consisting of RCNS 2j,3 digits, ostsigni cant-digit recoding recodes ost-signi cant residual digits w,w 0 {, 0, } of respective weights (2j) =2j and (2j) 0 =, and digits w,w 2 { 3,,3}, of respective weights (2j) and (2j) 2, into digits ω,ω 2 { 3,,3} of respective weights (2j) and (2j) 2 The algorith MSREC for recoding general digits w k 2 and w k into digit ω k is shown next MSREC(w k 2,w k ) ω k = 3 if (w k 2 =0and w k = 3) or (w k 2 =and w k =) 2 if (w k 2 =0and w k = 2) or (w k 2 =and w k =2) if (w k 2 =0and w k = ) or (w k 2 =and w k =3) 0 if w k 2 =0and w k =0 if (w k 2 =0and w k =)or (w k 2 = and w k = 3) 2 if (w k 2 =0and w k =2)or (w k 2 = and w k = 2) 3 if (w k 2 =0and w k =3)or (w k 2 = and w k = ) III CHT IMPEMENTATION The CHT produces a coplex colun vector y k in which y is the only non-zero eleent Siplifying the coputation results in x y = e jθ x x 2 = x + + x kx k x x (3) The ipleentation requires k coplex ultipliers (CMUT), k real adders (RA), real divider (RIV), a real square root unit (RSQRT), a unit to negate a digit (NEG), and a coplex-real ultiplier (CRMUT) as shown in Figure 3 Since the product of a coplex-conjugate ultiplier is a real nuber, radix 2j to radix 2 converters will be used to convert the outputs into radix 2 representation for input to the real adders ikewise, radix 2 to radix 2j converters will be used to convert the output of the real square root unit to radix 2j representation for input to the coplexreal ultiplier, which produces the coplex output y The recurrence algoriths and designs of the radix 2j on-line oating-point coplex-conjugate ultiplier and the radix 2j on-line oating-point coplex-real ultiplier are described next x k x k * CMUT RA Fig 3 x 2 x 2 * CMUT RA RIV RA RSQRT x x * CMUT CRMUT y NEG Coputational graph of CHT A Radix 2j on-line oating-point coplex-conjugate ultiplication Radix 2j oating-point coplex-conjugate ultiplication (z = xx )isde ned such that given inputs x = (X R + jx I ) (2j) ex and x = (X R jx I ) (2j) ex, the output z =(Z R + jz I ) (2j) ez is produced such that Z R = XR 2 + X2 I Z I =0 (4) e z =2e x The recurrence forula for radix 2j on-line ultiplication (z = xy) is the following: W [k] = (2j)(W [k ] z [ k ]) +(2j) δ+ (X[k]y k+δ + Y [k ]x k+δ ) z k = W E [k] (5) 320

4 where W E [k] is the low-precision estiate of the even-indexed (real) coponent of W [k] For radix 2j on-line oating-point coplex-conjugate ultiplication (z = xx ), since { x k+δ = xk+δ if k + δ is even { x k+δ if k + δ is odd WE [k] if k + δ is even z = 0 if k + δ is odd the recurrence can be rewritten as (2j)(W [k ]) + (2j) δ+ ((2X R [k ] +x k+δ (2j) k δ+ )x k+δ ) if k + δ is even W [k] = (2j)(W [k ] z k ) +(2j) δ+ ( 2X I [k ] x k+δ (2j) k δ+ )x k+δ ) if k + δ is odd For the ipleentation, two types of odular slices are required An odd-indexed slice M 2k (k = to /2 ) consists of one borrow-save digit ultiplier, a 2: borrowsave digit adder, a digit-wide latch, a ip- op, a bit-wide ip- op, a digit-wide ip- op, a TWICE unit for coputing 2X[k ], a 3-to- for appropriately appending digit x k to vector 2X[k ], and a 2-to- for appropriately shifting the residual An even-indexed slice M 2k (k = to /2 ) consists of a digit-wide latch, a bit-wide ip- op, and a TWICE unit for coputing 2X[k ] A ag bit eo controls switching between odd-indexed and even-indexed slices (eo =for an odd-indexed slice, and eo =0for an even-indexed slice) The CONJ unit generates digits of x The digit x k is recoded into digit set { 2,,2} using one SREC unit The ost-signi cant digit of the recurrence is deterined using one MSREC unit, which perfors output digit selection as well as handles ost signi cant carry-out bits of the adder The MUT2E unit ultiplies the operand exponent by two to produce the output exponent e z The NORM unit noralizes the result by updating the output exponent e z The design of a -digit signi cand and e-bit exponent radix 2j on-line oating-point coplex-conjugate ultiplier unit is shown in Figure 4 The nuber of individual odule types utilized, the cost per odule type, and the total overall cost are suarized in Table I Assuing =24and e =8, the cost is 224 CB slices The on-line delay is δ =9 B Radix 2j on-line oating point coplex-real ultiplication Radix 2j oating-point coplex-real ultiplication (z = xy) isde ned such that given coplex input x = (X R + jx I ) (2j) ex and real input y = Y (2) ey, the output z = (Z R + jz I ) (2j) ez is produced such that (6) (7) Z R = X R Y Z I = X I Y e z = e x + e y (8) TABE I COST OF RAIX 2j ON-INE FOATING-POINT COMPEX-CONJUGATE MUTIPIER x k SREC e x CONJ MUT2E e z eo Fig 4 START NORM e z Module Count CB slices MUT2E e CONJ 4 SREC 2 3-to to- 2 BS ult ( ) 2 2 BS adder (2:) 2 2 MSREC 0 NORM 2e Total cost e +26 M MSREC z k-δ 2: "0" M 2 M 3 2: M - TWICE TWICE TWICE TWICE TWICE Radix 2j on-line oating-point coplex-conjugate ultiplier For radix 2j on-line oating-point coplex-real ultiplication, the recurrence fro Equation 5 can be rewritten as: (2j)(W [k ] z k ) +(2j) δ+ (X[k]y k+δ + Y [k ]x k+δ ) if k + δ is even W [k] = (2j)(W [k ] z k ) +(2j) δ+ (Y [k ]x k+δ ) if k + δ is odd (9) For the ipleentation two types of odular slices are required An odd-indexed slice M 2k (k = to /2 ) consists of one borrow-save digit ultiplier, a 2: borrowsave digit adder, a digit-wide latch, a bit-wide ip- op, and a digit-wide ip- op An even-indexed slice M 2k (k = to /2 ) consists of two borrow-save digit ultipliers, a 3: borrow-save digit adder, two digit-wide latches, a bit-wide ip- op, and a digit-wide ip- op The digits x k and y k are recoded into the digit set { 2,,2} using two SREC units The ost-signi cant digits of the recurrence are deterined using two MSREC units, which perfor output digit selection 2: M "" "0" 32

5 as well as handling the ost signi cant carry-out bits of the adder The AE unit adds the operand exponents to produce the output exponent e z The NORM unit noralizes the result by updating the output exponent e z The design of a -digit signi cand and e-bit exponent radix 2j on-line oating-point coplex-real ultiplier unit is shown in Figure 5 The nuber of individual odule types utilized, the cost per odule type, and the total overall cost are suarized in Table II Assuing =24and e =8, the cost is 356 CB slices The on-line delay is δ =9 TABE II COST OF RAIX 2j ON-INE FOATING-POINT COMPEX-REA MUTIPIER Module Count CB slices AE e SREC BS ult ( ) 2 6 BS adder (3:) 2 4 BS adder (2:) 2 2 MSREC 2 20 NORM 2e Total cost 2 +3e digit (or bit) signi cands and 8-bit exponents, as shown in Table III TABE III COMPARISON OF CB COSTS FOR CHT k Radix 2j Radix 2 Radix 2 on-line on-line parallel The delay of the proposed radix 2j on-line network, and the alternative radix 2 on-line network and the radix 2 parallel network are copared for the ipleentation of CHT unit for various values of k In each case, we assue oatingpoint operands consisting of 24-digit (or bit) signi cands and 8-bit exponents, as shown in Table IV TABE IV COMPARISON OF CYCE ATENCIES FOR CHT k Radix 2j Radix 2 Radix 2 on-line on-line parallel IV CONCUSION We have deonstrated a new approach for the ipleentation of a CHT unit, based on using coplex nuber online arithetic odules which adopt a redundant coplex nuber syste (RCNS) for ef cient representation Signi cant iproveent in cost in coparison to a radix 2 on-line approach and a radix 2 parallel approach, as well as a significant reduction in latency in coparison to a radix 2 parallel approach have been shown This otivates the research into other application to utilize coplex on-line arithetic Fig 5 Radix 2j on-line oating-point coplex-real ultiplier Three approaches for the design of the CHT are copared: (i) a radix 2j approach which uses a cobination of radix 2j on-line arithetic odules for coplex inputs and radix 2 online arithetic odules for real inputs; (ii) a radix 2 approach which strictly uses radix 2 on-line arithetic odules; and (iii) a radix 2 parallel approach which uses the Xilinx library of oating-point parallel arithetic operators [8] The results in ters of cost and latency are shown next The cost of the proposed radix 2j on-line network, and the alternative radix 2 on-line network and the radix 2 parallel network are copared for the ipleentation of CHT unit which operates on a k-digit vector x, for various values of k In each case, we assue oating-point operands consisting of REFERENCES [] T Aoki, Y Ohi, and T Higuchi, Redundant coplex nuber arithetic for high-speed signal processing, 995 IEEE Workshop on VSI Signal Processing, Oct 995, pp [2] K- Chung and W-M Yan, The coplex householder transfor, IEEE transactions on signal processing, Vol 45, no 9, Sept 997, pp [3] M Ercegovac and T ang, igital arithetic, Morgan Kaufann Publishers, 2004 [4] AS Householder, Unitary triangularization of a nonsyetric atrix, Journal of the ACM, 5 (4), 58, pp [5] E Knuth, The art of coputer prograing, Vol 2, 973 [6] R McIlhenny, Coplex nuber on-line arithetic for recon gurable hardware: algoriths, ipleentations, and applications, Ph issertation, University of California, os Angeles, 2002 [7] CFT Tang, KJR iu, SF Hsieh, and K Yao, VSI algoriths and architectures for coplex householder transforation with applications to array processing, University of Maryland, College Park, Technical Report TR 9-84, 99, pp -36 [8] Xilinx Corporation, Xilinx ata Book,