Truncated Squarers with Constant and Variable Correction

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1 Please veify that ) all pages ae pesent, 2) all figues ae acceptable, 3) all fonts and special chaactes ae coect, and ) all text and figues fit within the Tuncated Squaes with Constant and Vaiable Coection E. Geoge Waltes III a, Michael J. Schulte b, and Mak G. Anold a a CSE Dept., Lehigh Univesity, Bethlehem, PA b Dept. of ECE, Univesity of Wisconsin-Madison, Madison, WI ABSTRACT This pape descibes tuncated squaes, which ae specialized squaes with a potion of the squaing matix eliminated. Rounding eo and eos due to matix eduction ae quantified and analyzed. Constant and vaiable coection techniques ae pesented that minimize eithe the mean eo o the maximum absolute eo as equied by the application. Aea and delay estimates ae pesented fo a numbe of designs, as well as eo statistics obtained both analytically and numeically by exhaustive simulation. As an example, one design of a 6-bit tuncated squae using constant coection is. % faste and equies 27.9 % less aea than a compaable standad squae with tue ounding. The ange of eo fo this tuncated squae is.892 to ulps, compaed to ±.5 ulps fo the standad squae. Keywods: tuncated squaes, tuncated multiplies, compute aithmetic, low powe. INTRODUCTION Squaing is an impotant opeation in many digital signal pocessing applications including speech ecognition, wieless communication, Vitebi decoding, patten matching, and image pocessing. Squaing is also used in quadatic intepolatos fo function appoximation. In addition to high thoughput, many of these applications also have demanding equiements fo low powe consumption. Tuncated squaes ae an attactive design option wheeve ounded squaes ae used. Tuncated squaes ae an extension of specialized squaes and tuncated multiplies. 2 By eliminating seveal of the least significant columns of the squaing matix that contibute little to the final esult, the squaes aea, delay, and powe consumption ae significantly educed. Although some additional eo is intoduced into the computation, this eo can be constained to ± unit in the last place ulp), which is compaable to the ±.5 ulp eo intoduced by ounding in a conventional squae. This pape pesents tuncated squaes, beginning with a bief desciption of two s complement specialized squaes in Section 2. Tuncated squaes ae pesented in Section 3, with equations quantifying the aveage eo and the eo bounds. Section pesents a technique to offset ounding and eduction eo by adding a constant. Section 5 pesents a technique to offset these eos by adding a vaiable value in addition to a small coection constant. Aea and delay estimates fo vaious designs ae given in Section 6 and compaed to conventional ound to neaest squaes. Concluding emaks ae given in Section 7. Although the pape assumes the input opeands ae integes, mino modifications can me made to design squaes that suppot opeands that ae factions o mixed numbes. Futhe autho infomation: Send coespondence to E.G.W.) E.G.W.: waltesg@ieee.og, Addess: P.O. Box 275, Owigsbug, PA, 796 M.J.S.: schulte@eng.wisc.edu, Addess: 5 Engineeing Dive, Madison, WI 5376 M.G.A.: manold@eecs.lehigh.edu, Addess: 9 Memoial Dive West, Bethlehem, PA, SPIE USE, V p. of ) / Colo: No / Fomat: Lette/ AF: Lette / Date: :9:5

2 Please veify that ) all pages ae pesent, 2) all figues ae acceptable, 3) all fonts and special chaactes ae coect, and ) all text and figues fit within the a 7a a 6a a 5a a a a 3a a 2a a a a a a 7a a 6a a 5a a a a 3a a 2a a a a a a 7a 2 a 6a 2 a 5a 2 a a 2 a 3a 2 a 2a 2 a a 2 a a 2 a 7a 3 a 6a 3 a 5a 3 a a 3 a 3a 3 a 2a 3 a a 3 a a 3 a 7a a 6a a 5a a a a 3a a 2a a a a a a 7a 5 a 6a 5 a 5a 5 a a 5 a 3a 5 a 2a 5 a a 5 a a 5 a 7a 6 a 6a 6 a 5a 6 a a 6 a 3a 6 a 2a 6 a a 6 a a 6 a 7a 7 a 6a 7 a 5a 7 a a 7 a 3a 7 a 2a 7 a a 7 a a 7 a) Oiginal matix a 7 a 6 a 5 a a 3 a 2 a a a 7a 6 a 7a 5 a 7a a 7a 3 a 7a 2 a 7a a 7a a 6a a 5a a a a 3a a 2a a a a 6a 5 a 6a a 6a 3 a 6a 2 a 6a a 5a a a a 3a a 2a a 5a a 5a 3 a 5a 2 a a 2 a 3a 2 a a 3 b) Reduced matix a 7a 6 a 7a 6 a 6a 5 a 6a 5 a 5a a 5a a a a 3 a 3a 2 a 3a 2 a 2a a 2a a a a a a a 7a 5 a 7a a 7a 3 a 7a 2 a 7a a 7a a 6a a 5a a a a 3a a 2a a 6a a 6a 3 a 6a 2 a 6a a 5a a a a 3a a 5a 3 a 5a 2 a a 2 c) Modified matix Figue. 8-bit signed squaing matices. 2. TWO S COMPLEMENT SPECIALIZED SQUARERS Specialized squaes ae multiplies that ae optimized fo the case whee the multiplie and the multiplicand ae the same. This section descibes two s complement specialized squaes as pesented by Wies et al. If A is an n-bit intege in two s complement fom, then and the squae of A is A =a n a a a )= a n 2 n + a i 2 i, ) A 2 = a n a n j= a i a j 2 i+j a i a n 2 n+i a n a i 2 n+i. 2) Using techniques fo two s complement tee multiplies pesented by Bickestaff et al., 2 2) can be ewitten as A 2 = a n a n j= a i a j 2 i+j + a i a n 2 n+i + a n a i 2 n+i +2 2n +2 n. 3) An 8-bit two s complement squaing matix based on 3) is given in Figue a). The a i a j bits with i = j compise the antidiagonal, which is indicated using boldface type. Since a i a j = a j a i, the potion of the matix above the antidiagonal is equal to the potion below it, 3 so the top and bottom potions can be combined using the identity a i a j + a j a i =2a i a j. Using this obsevation and the SPIE USE, V p.2 of ) / Colo: No / Fomat: Lette/ AF: Lette / Date: :9:5

3 Please veify that ) all pages ae pesent, 2) all figues ae acceptable, 3) all fonts and special chaactes ae coect, and ) all text and figues fit within the n n add to ound keep squae output) discad Figue 2. 2-bit squae with tue ounding to neaest. identity a i a i = a i, the matix can be educed to the one shown in Figue b). This matix can be expessed as n i A 2 = a i 2 2i + a i a j 2 i+j+ + a n a i 2 n+i +2 2n +2 n. ) i= j= The educed matix can be futhe modified to educe its maximum height. The tems a,a 2,...,a can be combined with othe patial poduct bits using the following identity: a i + a i a i =2a i a i + a i a i. 5) If n is even, the identity +a i + a i a i = a i a i +2a i 6) is used to modify the, a n/2,anda n/2 a n/2 tems in the 2 n column of Figue b). Finally, the identities a i + a i a i =2a i a i + a i a i 7) +a i a i =2a i a i + a i a i 8) ae used to modify the two most significant columns. The a i a i tem in the 2 2n column esulting fom 8) is discaded. The final matix is given in Figue c), and is expessed as A 2 = a + n/2 i= ai a i 2 2i + a i a i 2 2i+) + i=n/2+ ai a i 2 2i + a i a i 2 2i+) + a n a 2 2n + a n a a n/2 2 n+ + a n/2 a n/2 2 n i 2 a n a j 2 n+j + a i a j 2 i+j+. + j= i=2 j= 9) If n is odd, an identity diffeent than 6) is used, and the final matix is slightly diffeent. This is descibed in detail by Wies et al. Figue 2 depicts a 2-bit specialized squae with ounding to neaest, whee esults exactly halfway between two numbes ae ounded up. A is added to the squaing matix in the 2 n column fo ounding and the patial poducts ae added to poduce a sum. The n least significant bits of the sum ae discaded, poducing an n-bit esult that is coectly ounded to neaest. While this poduces the coectly ounded esult, it does not poduce the tue esult due to ounding eo. Defining eo as the output value minus the tue value, the ange of this ounding eo, E nd,is 2 n +5<E nd < 2 n n 6), ) which is appoximately ±.5 ulps elative to the output. Note that the lowe bound of the ounding eo is slightly less negative than that fo a ounded multiplie, because some combinations of bits in the squaing matix ae not possible. This iegulaity also causes the aveage ounding eo to be slightly lage fo a squae than a multiplie. SPIE USE, V p.3 of ) / Colo: No / Fomat: Lette/ AF: Lette / Date: :9:5

4 Please veify that ) all pages ae pesent, 2) all figues ae acceptable, 3) all fonts and special chaactes ae coect, and ) all text and figues fit within the k not fomed Figue 3. 2-bit tuncated squaing matix, =. k maximum value Figue. Maximum eduction eo, =. 3. TRUNCATED SQUARERS Tuncated squaes ae an extension of tuncated multiplies. 2 In a tuncated squae, seveal of the least significant columns of the patial poducts ae not fomed. In this pape, denotes the numbe of unfomed columns and k denotes the numbe of columns that ae fomed but discaded in the final esult. In an n-bit squae ounded to n-bits, + k = n. Figue 3 depicts a 2-bit tuncated squae with =. In this example, 32 % of the patial poduct bits ae not fomed. As with tuncated multiplies, eliminating patial poducts fom the squaing matix educes the aea of the squae in thee ways. Fist, the logic gates and associated wiing equied to fom the patial poducts ae eliminated. Second, the adde cells and associated wiing that would othewise be equied to add the patial poducts ae also eliminated. Finally, a shote final cay-popagate adde is equied fo computing the final esult, which also tends to impove the oveall delay of the squae. The eduction in aea is significant fo both tee eduction schemes and designs employing aays of adde cells. In addition to educing aea, the eduction in logic and the consequent eduction in switching activity esults in significant powe savings as well. As shown by Waltes and Schulte, these aea and powe eductions extend to hadwae acceleatos that use tuncated aithmetic units, lagely due to shote wodlengths of intemediate esults. Eliminating patial-poduct bits intoduces a eduction eo, E, into the output. E low and E high ae the lowe and uppe bounds on the eduction eo, espectively. The uppe bound is zeo, which occus when each of the unfomed patial poduct bits would have been s. Thus, the ange of E is E low E. ) The lowe bound occus when most of the unfomed bits in the squaing matix would have been s. Examination of Figue c) shows that it is impossible fo all the bits to be, because a i a i and a i a i cannot both be. Figue shows the case whee the lowe bound of the eduction eo occus fo a 2-bit tuncated squae with =. Caeful analysis shows that if is even, and if is odd, E low = E low = 2 q= /2 q= q +) 2 2q+ +q ) 2 2q), 2) q 2 2q +2 2q ) ) q= 2 2 q q= 2 2q. 3) SPIE USE, V p. of ) / Colo: No / Fomat: Lette/ AF: Lette / Date: :9:5

5 Please veify that ) all pages ae pesent, 2) all figues ae acceptable, 3) all fonts and special chaactes ae coect, and ) all text and figues fit within the k 2 expected value These equations can be expessed in closed fom as Figue 5. Aveage eduction eo, =. { 2 +2 if is even E low = ) if is odd. ) The aveage eduction eo, E avg, is the additive invese of the sum of the aithmetic mean of each unfomed patial poduct bit. Assuming that the squae opeand, A, has a unifom pobability distibution, each bit in A has an equal pobability of being a o a. Figue 5 shows the expected values of each bit fo a 2-bit tuncated squae with =. It can be shown that if is even, then and if is odd, then E avg = E avg = /2.5 q= 2 which can be expessed in closed fom as E avg = q= q +) 2 2q+ + q 2 2q), 5) q +) 2 2q+ + q 2 2q) 2, 6) 2 if is even if is odd. Values fo E low and E high ae tabulated in Table fo vaious values of. The equations in this section can be conveted to ulps by dividing each tem by 2 +k, which is 2 n fo n-bit squaes ounded to n-bits. As an example, conside a 6-bit squae with = 3 ounded to 6-bits. The ange of eduction eo in ulps is.62 E, and the aveage eduction eo is -.77 ulps. Since the least significant half of the squaing matix of both unsigned and two s complement squaes ae identical, the equations in this pape ae also valid fo unsigned squaes povided n. They ae also valid fo signed and twos complement squaes whee n is odd, povided n.. CONSTANT CORRECTION TRUNCATED SQUARERS The constant coection technique fo tuncated squaes is an extension of wok by Schulte and Swatzlande, who pesent a constant coection method fo tuncated multiplies. Schulte and Swatzlande show that the total eo is a combination of the ounding eo, E nd, and the eduction eo, E. Figue 6 depicts a tuncated squae using constant coection. With this method, a coection constant is added to the squaing matix to offset the ounding eo and the eduction eo. Assuming each of the bits in the output have an equal pobability of being a o a, the expected value of the ounding eo is 7) E nd = 2 +k +2. 8) SPIE USE, V p.5 of ) / Colo: No / Fomat: Lette/ AF: Lette / Date: :9:5

6 Please veify that ) all pages ae pesent, 2) all figues ae acceptable, 3) all fonts and special chaactes ae coect, and ) all text and figues fit within the k not fomed coection constant keep squae output) discad Figue 6. 2-bit tuncated squae with constant coection, =. Equations fo the ange of eduction eo and the aveage eduction eo ae given in Section 3. Using these equations, the constant can be selected to minimize eithe the aveage eo o the maximum absolute eo of the tuncated squae output... Minimizing aveage eo The aveage eo of a tuncated squae with no coection is the sum of E avg and E nd. A coection constant appoximating the additive invese of the aveage eo is selected using the equation E avg 2 +k +2 ) C cc avg = ound 2, 9) whee ound ) indicates ound to the neaest intege, and E avg is calculated using Equation 7. The ounding of the coection constant is pefomed because the least significant bits of the coection constant would have no effect on the final esult. Due to this ounding, the aveage eo of the tuncated squae is minimized but is not zeo. As an example, conside a 2-bit squae with = as in Figue 6. Using Equation 7, The aveage eo is found to be -9.5, which is. in binay. Applying Equation 9, the coection constant is chosen to be 372, which is in binay. is added to the squaing matix as shown in Figue Minimizing maximum absolute eo The aveage eduction eo of a tuncated squae is oughly one quate of the maximum absolute eo, so minimizing the aveage eo does not usually minimize the maximum absolute eo. In applications whee this is desiable, the coection constant is chosen using the midpoint of the ange of eduction eo and the expected ounding eo: E low + E high 2 +k +2 C cc abs = ound ) Since E high is zeo in this case, this equation simplifies to C cc abs = ound E low k ) 2 Continuing the example of a 2-bit squae with =, the coection constant to minimize maximum absolute eo would be 96, which is in binay. Thus, would be added to the squaing matix in Figue 6 athe than. SPIE USE, V p.6 of ) / Colo: No / Fomat: Lette/ AF: Lette / Date: :9:5

7 Please veify that ) all pages ae pesent, 2) all figues ae acceptable, 3) all fonts and special chaactes ae coect, and ) all text and figues fit within the k not fomed vaiable coection value coection constant keep squae output) discad Figue 7. 2-bit tuncated squae with vaiable coection, =. 5. VARIABLE CORRECTION TRUNCATED SQUARERS The vaiable coection technique fo tuncated squaes is an extension of wok by King and Swatzlande. 6 Figue 7 depicts a tuncated squae using vaiable coection. In this method, the patial poduct bits in the 2 column ae fomed and added to the 2 column as a vaiable coection value. This can be viewed as ounding the unfomed patial poducts at bit position 2. Fo example, the top ow of unfomed bits in Figue 7 would be ounded to the 2 column by adding a to the 2 column, esulting in the most significant unfomed bit being moved to the 2 column. If the squaing matix is educed using a paallel tee stuctue, the vaiable value can incease the height of the matix such that an additional eduction stage is equied, thus inceasing the delay. When implemented as an aay of full adde cells, the vaiable value bits can added as a caies into the adde cells on the edge of the tuncated aay. King and Swatzlande discuss this fo vaiable coection tuncated multiplies. With this technique, the aveage eduction eo is diffeent than the aveage eduction eo given by Equation 7. With vaiable coection, E avg is the expected value of the value in the 2 column plus the aveage eduction eo due to the unfomed bits, given by 7). If is even, the aveage eduction eo is If is odd, the aveage eduction eo is Simplifying these equations, E avg = ) E avg = ) ) E avg = if is even if is odd. The ange of eduction eo with the vaiable coection method is difficult to detemine analytically, so it was done numeically using exhaustive bit accuate simulations. The ange of eo is bounded by E low and E high, these values ae tabulated fo vaious values of in Table. Fo compaison, the ange of eduction eo and aveage eduction eo ae given fo tuncated squaes with no coection as well. In ode to impove the vaiable coection method, a constant can be added in addition to the vaiable value. As with constant coection, the value can be chosen to eithe minimize the aveage eo o the maximum absolute eo. 2) SPIE USE, V p.7 of ) / Colo: No / Fomat: Lette/ AF: Lette / Date: :9:5

8 Please veify that ) all pages ae pesent, 2) all figues ae acceptable, 3) all fonts and special chaactes ae coect, and ) all text and figues fit within the No Coection With Vaiable Coection Value E low E high E avg E low E high E avg Table. Tabulated values fo eduction eo, E. Range of eduction eo is E low E E high. E avg denotes the aveage aithmetic mean) eduction eo. 5.. Minimizing aveage eo The aveage eo is minimized using the same technique used with constant coection tuncated squaes. In addition to the vaiable value, the following constant is added to the squaing matix: E avg 2 +k +2 ) C vc avg = ound 2. 25) This is the same as Equation 9, except the value fo E avg is given by Equation 2 athe than Equation 7. As an example, conside a 2-bit tuncated squae with =. With vaiable coection, E avg is 7.5. Fom 25) C vc avg is found to be 2, which is in binay. Figue 7 shows this constant being added to the squaing matix Minimizing maximum absolute eo The maximum absolute eo is minimized in the same manne as fo constant coection. Following Equation 2, E low + E high 2 +k +2 C vc abs = ound ) 2 2 C vc abs is calculated using values fo E low and E high fom the vaiable coection columns of Table, and added to the squaing matix in addition to the vaiable value. Continuing with the example of a 2-bit tuncated squae with =, E low and E high ae -7 and 383 espectively. Using Equation 26, C vc abs is 2, which is the same as the value used to minimize the aveage eo. As it tuns out, this was the case fo all of the vaiable coection squae designs developed fo this pape; the value that minimizes the aveage eo also minimizes the maximum absolute eo. This is a significant advantage of the vaiable coection method ove the constant coection method. SPIE USE, V p.8 of ) / Colo: No / Fomat: Lette/ AF: Lette / Date: :9:5

9 Please veify that ) all pages ae pesent, 2) all figues ae acceptable, 3) all fonts and special chaactes ae coect, and ) all text and figues fit within the 6. EXPERIMENTAL RESULTS 6.. Methodology Aea and delay estimates wee obtained using the Leonado synthesis tool and the TSMC25.25 micon CMOS standad cell libay. VHDL models fo 8,, 2,, 6, 2, 2, and 32-bit squaes wee geneated automatically using a softwae tool witten in Java. Fo each wodlength, models fo constant coection and vaiable coection squaes with vaying values of wee geneated, as well as a tue ound-to-neaest squae to be used as a baseline fo compaison. Fo each model, a paallel tee stuctue using Reduced Aea eduction 2 is used to educe the squaing matix to two ows, which ae then added by a cay-lookahead adde to poduce the final output. A epesentative subset of the data collected is pesented in Table 2, which includes 8, 6, 2, and 32-bit squaes. Fo each of the models, the constant was chosen to minimize the aveage eo. Thus, in the table, C is C cc avg fo constant coection and C vc avg fo vaiable coection designs. Aea and delay estimates ae given fo each model. Fo tuncated squaes, the impovement in aea and delay is given using the tue ound-to-neaest tue tn) squae as a baseline. Eo statistics ae given fo each model, including the low and high eo values, the ange of eo E high E low ), the aveage eo, and the standad deviation of eo, σ E. The coection constants and eo statistics ae given in ulps of the final output. Fo the 8, 6, and 2-bit models, eo statistics wee obtained by exhaustive bit-accuate simulation. These simulations veified the equations pesented in this pape fo eduction eos. Fo 32-bit models, simulations wee done using 2 2 andomly selected opeands. Since 32-bit simulations wee not exhaustive, values in the table fo low and high eos may not be the tue values, because such values only occu fo a small numbe of opeands. Howeve, values in the table fo aveage eo and standad deviation of eo should be faily accuate Analysis Tuncated squae designs that maintain output eo within ± ulp ae indicated by boldface type fo 8, 6, and 2-bit squaes. Constant coection designs given in these tables use a constant that was chosen to minimize aveage eo, so they may show a ange of eo geate than ± ulp i.e., 8-bit constant coection squae). Howeve, changing the constant to minimize the maximum absolute eo would bing the eo to within ± ulp, and this change would have minimal impact on the aea and delay estimate. 32-bit designs ae not indicated because they wee not simulated exhaustively. Looking at the indicated designs, significant aea and delay impovements ae achieved. Aea impovement anges fom 9.5 % to 3.5 % and delay eduction anges fom 5. % to 3. %, while limiting the eo to ± ulp. Although not estimated hee, consideable powe savings can be expected as well, due to the eduction in logic cells and consequent elimination of thei switching activity. Compaing constant coection squaes to vaiable coection squaes, the esults show that constant coection has an advantage in delay and possibly an advantage in aea. This is tue fo paallel tee implementations, due to the inceased matix height of the vaiable coection technique. These esults wee obtained using automatically geneated models with few optimizations. Vaiable coection squaes would benefit moe fom hand optimization than constant coection squaes, due to thei moe iegula squaing matix. Theefoe, with optimized designs aea would be simila and the diffeence in delay would be educed. In an adde aay implementation, vaiable coection squaes would be expected to have compaable o bette aea and delay chaacteistics than constant coection squaes. 7. CONCLUSIONS Tuncated squaes, which ae an extension of specialized squaes and tuncated multiplies, have been pesented. Constant coection and vaiable coection techniques ae pesented and analyzed. Constant coection tuncated squaes add a coection constant to offset ounding and eduction eo, while vaiable coection tuncated squaes add a vaiable value deived fom the unfomed patial poduct bits as well as a small constant. With constant coection, the constant is chosen eithe to minimize the aveage eo o the maximum absolute eo. Due to the eo distibution, only one o the othe can be minimized. Fo most designs using vaiable SPIE USE, V p.9 of ) / Colo: No / Fomat: Lette/ AF: Lette / Date: :9:5

10 Please veify that ) all pages ae pesent, 2) all figues ae acceptable, 3) all fonts and special chaactes ae coect, and ) all text and figues fit within the coection, howeve, a constant can be chosen that minimizes both eos simultaneously. Reductions in aea up to 3.5 % and delay up to 3. % can be achieved while maintaining eo within ± ulp. Coesponding deceases in powe consumption can be expected. If additional eo can be toleated in the taget application, futhe aea, delay, and powe impovements can be achieved by inceasing the numbe of unfomed columns. Since othewise unfomed patial poduct bits ae used to geneate the vaiable coection value, the aea fo vaiable coection squaes with unfomed columns is compaable to the aea fo constant coection squaes having unfomed columns. Fo designs having simila aea savings, vaiable coection and constant coection have compaable eo pefomance. Fo squaes using a paallel tee eduction scheme, vaiable coection squaes tend to be somewhat slowe than constant coection due to the inceased matix height. Fo aay implementations, howeve, the speeds would be simila. REFERENCES. K. E. Wies, M. J. Schulte, L. P. Maquette, and P. I. Balzola, Combined Unsigned and Two s Complement Squaes, in Poceedings of the 33d Asiloma Confeence on Signals, Systems, and Computes, 2, pp , Pacific Gove, CA), Octobe E. E. Swatzlande, J., Tuncated Multiplication with Appoximate Rounding, in Poceedings of the 33d Asiloma Confeence on Signals, Systems, and Computes, 2, pp. 8 83, Pacific Gove, CA), Octobe Y. C. Lim, Single-Pecision Multiplie with Reduced Cicuit Complexity fo Signal Pocessing Applications, IEEE Tansactions on Computes,, pp , Octobe M. J. Schulte and E. E. Swatzlande, J., Tuncated Multiplication with Coection Constant, in VLSI Signal Pocessing, VI, pp , IEEE Pess, Eindhoven, Nethelands), Octobe S. S. Kidambi, F. El-Guibaly, and A. Antoniou, Aea-Efficient Multiplies fo Digital Signal Pocessing Applications, IEEE Tansactions on Cicuits and Systems II: Analog and Digital Signal Pocessing, 3, pp. 9 95, Febuay E. J. King and E. E. Swatzlande, J., Data-Dependent Tuncation Scheme fo Paallel Multiplies, in Poceedings of the 3st Asiloma Confeence on Signals, Systems, and Computes, 2, pp , Pacific Gove, CA), Novembe J. M. Jou, S. R. Kuang, and R. D. Chen, Design of Low-Eo Fixed-Width Multiplies fo DSP Applications, IEEE Tansactions on Cicuits and Systems II: Analog and Digital Signal Pocessing, 6, pp , June M. J. Schulte, J. E. Stine, and J. G. Jansen, Reduced Powe Dissipation Though Tuncated Multiplication, in Poceedings of the IEEE Alessando Volta Memoial Wokshop on Low-Powe Design, pp. 6 69, Como, Italy), Mach L.-D. Van, S.-S. Wang, and W.-S. Feng, Design of the Lowe Eo Fixed-Width Multiplie and Its Application, IEEE Tansactions on Cicuits and Systems II: Analog and Digital Signal Pocessing, 7, pp. 2 8, Octobe 2.. K. E. Wies, M. J. Schulte, and D. McCaley, FPGA Resouce Reduction Though Tuncated Multiplication, in th Intenational Confeence on Field Pogammable Logic and Applications, pp , Belfast, Nothen Ieland, UK), August 2.. K.-S. Chong, B.-H. Gwee, and J. S. Chang, Low-voltage Micopowe Asynchonous Multiplie fo Heaing Instuments, in Poceedings of the IEEE Intenational Symposium on Cicuits and Systems,, pp , Phoenix, AZ), May K. Bickestaff, M. J. Schulte, and E. E. Swatzlande, J., Paallel Reduced Aea Multiplies, IEEE Jounal of VLSI Signal Pocessing, 9, pp. 8 9, T. C. Chen, A Binay Multiplication Scheme Based on Squaing, IEEE Tansactions on Computes, C-2, pp , 97.. E. G. Waltes III and M. J. Schulte, Design Tadeoffs Using Tuncated Multiplies in FIR Filte Implementations, in Poceedings of the SPIE: Advanced Signal Pocessing Algoithms, Achitectues, and Implementations XII, 79, pp , Seattle, WA), July 22. SPIE USE, V p. of ) / Colo: No / Fomat: Lette/ AF: Lette / Date: :9:5

11 Please veify that ) all pages ae pesent, 2) all figues ae acceptable, 3) all fonts and special chaactes ae coect, and ) all text and figues fit within the 8-bit squaes coection C Aea Delay Impovement Eo Statistics ulps) n k method ulps) gates) ns) Aea Delay low high ange mean σ E 8 8 tue tn / baseline constant / %.7% constant / %.% constant / %.6% constant % 22.7% vaiable 3/ % -.9% vaiable / %.6% vaiable % 5.% vaiable %.% bit squaes coection C Aea Delay Impovement Eo Statistics ulps) n k method ulps) gates) ns) Aea Delay low high ange mean σ E 6 6 tue tn / n/a n/a constant 9/ % 6.3% constant 5/ %.% constant 3/ % 9.5% constant % 5.2% constant % 6.% vaiable 7/ % 2.3% vaiable 3/ % 6.6% vaiable / % 5.3% vaiable % 7.% vaiable % 6.6% bit squaes coection C Aea Delay Impovement Eo Statistics ulps) n k method ulps) gates) ns) Aea Delay low high ange mean σ E 2 2 tue tn / n/a n/a constant 5/ % 3.% constant 3/ % 3.6% constant % 8.2% constant -/ % 8.8% constant % 2.% vaiable 7/ %.7% vaiable 3/ %.5% vaiable / %.3% vaiable % 2.% vaiable % 3.2% bit squaes coection C Aea Delay Impovement Eo Statistics ulps) n k method ulps) gates) ns) Aea Delay low high ange mean σ E tue tn / n/a n/a constant 9/ % 5.2% constant / % 6.7% constant 7/ % 8.8% constant -/ % 2.9% constant % 2.3% constant % 22.9% vaiable 5/ % 5.% vaiable 7/ % 2.% vaiable 3/ %.% vaiable / % 5.% vaiable 5 2..% 7.% vaiable % 2.5% Table 2. Aea and delay estimates fo vaious squaes, along with eo statistics. SPIE USE, V p. of ) / Colo: No / Fomat: Lette/ AF: Lette / Date: :9:5

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