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Improving accuracy in logarithmic multiplication using operand decomposition

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Material Information

Title:
Improving accuracy in logarithmic multiplication using operand decomposition
Physical Description:
Book
Language:
English
Creator:
Venkataraman, Mahalingam
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla.
Publication Date:

Subjects

Subjects / Keywords:
Logarithmic number systems
Digital signal processing
Mitchell's algorithm
Convolution
Average error percentage
Dissertations, Academic -- Computer Engineering -- Masters -- USF   ( lcsh )
Genre:
government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
ABSTRACT: The arithmetic operations such as multiplication and division in binary number system are computationally complex in terms of area, delay and power. Logarithmic Number Systems (LNS) offer a viable alternative combining the simplicity of fixed point number systems and the precision of floating point number systems. However, the computations in LNS result in some loss of accuracy and thus, are limited to mostly signal processing applications; where in certain amount of error is tolerable. In LNS, the cost of computations can be tradeoff with the level of accuracy needed. The Mitchell algorithm proposed incitemitchell, is a simple approach commonly used for logarithmic multiplication. The method involves a high error margin due to a piecewise straight line approximation of the logarithm curve. Thus, several methods have been proposed in the literature for improving the accuracy of Mitchell's algorithm.
Thesis:
Thesis (M.S.C.P.)--University of South Florida, 2005.
Bibliography:
Includes bibliographical references.
System Details:
System requirements: World Wide Web browser and PDF reader.
System Details:
Mode of access: World Wide Web.
Statement of Responsibility:
by Mahalingam Venkataraman.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 50 pages.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001680981
oclc - 62494261
usfldc doi - E14-SFE0001140
usfldc handle - e14.1140
System ID:
SFS0025461:00001


This item is only available as the following downloads:


Full Text
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Improving accuracy in logarithmic multiplication using operand decomposition
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ABSTRACT: The arithmetic operations such as multiplication and division in binary number system are computationally complex in terms of area, delay and power. Logarithmic Number Systems (LNS) offer a viable alternative combining the simplicity of fixed point number systems and the precision of floating point number systems. However, the computations in LNS result in some loss of accuracy and thus, are limited to mostly signal processing applications; where in certain amount of error is tolerable. In LNS, the cost of computations can be tradeoff with the level of accuracy needed. The Mitchell algorithm proposed incite[mitchell], is a simple approach commonly used for logarithmic multiplication. The method involves a high error margin due to a piecewise straight line approximation of the logarithm curve. Thus, several methods have been proposed in the literature for improving the accuracy of Mitchell's algorithm.
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JI ‡<\j"–'“K/{,A"&&"\nD" £G%0*+$$#%&?\,e+$#$&+r7#iS‹rF& 6 0-'&-r+$!"T$#%'-'-/->#>!%C#>?AiR"2@_›£G%0 +$!"#%'-:?@#%0r&#Ur 6 0-["022052/r&2'#%0JrF#+$!"T$#%’?1#%0-§-[20#>0/r052@_ £G%00#%M2€->0527#>B&r2*#%0+$!"$#% r2#%0rF#+$!"$#%& '&#-o_e(7$#T%05++ 05-05rC#>052Ÿ-h+$00#%M2#>7"&5'"#>0/#%0/+$!"#%r2ŸrC#+!"$#%¦jzwaD-r &$050iQ->0[->#>!%C#u+r0:"jM'"#$!r)?#%0L+$!"T$#%k0_DXU$!0G¦"_¦L-%niJ#%0[?A&r#$!r+ 6 +$M;,2"T9?–#%0I(*#%05++wvb-=+$!"$#%/oI+#$&+"#!r4+$#%_ Input B Input A Logarithm Calculator Logarithm Calculator A x B Zero Detector Calculator Antilogarithm Adder XU$!0'¦"_¦"_nXr#$!r&+ ¤R+$;43o"T?[(*#%05++@+$#%¦jn £G%0d(*$#T%05++pvb-J+$!"$#%r2rF#+$!"$#%&9++"#$!r&-=05E$052-%&$?\#rer&2!rF#r 10"#$!r-!r&+$_N=(€.HKŸ8UKI&+$0505rF#"#$!r‹?)(*#%05++wvb-+$$#% 6 ->052Ž+$!"#%' e+$#$&+"#$!r4-R01#>052rqM¦a ¦w_:šr#%05->0H"!T%05-a#%0H+$!"$#%/-R?–#%05-0HrF 6 0"0'+&+"#>052->0&"T"#>05+$‘r2€#%05r22052@_£G%0rC#+$!"$#%?O#%0/-''"#$!r!$05-d#%0 M2#)?R#%0”rF 6 0-_/£J%0 6 +$M;€+->05EF05-H0720#>05#>#>05r-0'0!#>&# $?)rFœ?)#%0€r&#'r 6 0-/-/0_(*$#T%05++Qr%-'&"10¦jzwaG1!->052#%0*?A!++$niJr ->05E05rF#+@M0520“£D 6 +$0'¦"_¨!=?\QI+#$&+$MrI#WiO”r 6 0-=05ar$#++$I+!2052 iJ$#%l#%0/iQ2#%AiO2l-$0no?=#%04r&#)n0Tr2@aD?\e05%‹-%?\#_,£G%0”B&r+:"+0'?=#%05->0

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£D 6 +$0'¦"_¨q_1(*$#T%05++pvb-=u!"$#%/H(*&+$#$&+"#$!r”+$#% L #>0l¦']SNR++"#>0NMPORQ+$052r!r0dn!-$#$!r4?:rF 6 0Gd L #>07q]SNR++"#>0NMPSTQ+$052&r!r0)1!-$#$!r,?Ur 6 0G¤ U MPOdr2VMWSR0?A0-=#>'#%0d%"T#>0->#H?–#%0I+$!"$#%/-R?U07]SNR++"#>0YX O Q05'rr&"#=?U#%0)10r&2d L #>0*]SNR++"#>0YXZSTQ05'rr&"#=?D#%0H10r2I U XZOIr2[XZSR0?A0-=#>'#%0d'rF#--&"#=?D#%0H+$!"$#%y?:ˆr2¤a&->00dB!0'¦"_¨q L #>07t][J105r2-G!"#r'1!rF#=rF& 6 0-]\O^Q CMWOP_)XZOzOr2`\S]Q CMPSa_)XZST U 2&2#%0d+$!"T$#%y"+05-Gr2,#%05r++"#>0d#%0drF#+$!"$#%& L #>07]\ O STQb\ O0,]SNR++"#>0)#%0HM2#edgfYhiQrC#+$&C\ O ST !rF#>0-["0G-#>052're0!-#>0-jM O r26MWS_UJ&n05r&2I#%0=05'rrJ&"#:?h#%0=r&#UrF 6 0-a XZOr2kXZSGiJ#%VMWOr&2kMPSQ-#%0'rF#--”&"#Q?\o#%0+$!"$#%&'-al\Oer&2k\O_o22*#%05->0 +$!"#%"+05-'r2205M20#%0*T%"#>0->#"+0*?#%005-+##>Ÿr-0#'rœ!r0Œ¦"vm rF#>”"&"#>0dn!-$#$!r?:#%0I05-+#_R£J%0H05'rr&"#J?:#%0e2205205-+$#\ O SG-J#%05r "105r2052/'052"#>05+$"?\#>0R#%-L!r0e¦"vmT_SQr”0M'&+$0J?AR++-#>"#ro#%0o 6 j0M0520 -Q-%jiQr7r*B!04¦"_¨q_XU$!0d-%niJ-#%0I+!"$#%'HI&+$#$&+"#$!r,?L205/+ur 6 0-I¦s r2¦5t_ 18 = 00010010 60 = 00111100 log 18 = 100.0010 log 60 = 101.1110 log 18 + log 60 = 1010.0000 Result = 1000000000 Decimal = 1024 XU$!0'¦"_¨q_1(*#%05++wvb-= !"$#%&(*&+$#$&+"#$!r,P:&+$0 š#r 6 0,->005rŒ?A!#%04B!04#%"#aL#%0,+"#$!rŒ?Q#%04'!->#-$!r$BhrC#e!r0¦"vm 6 $# 20#>0/r05-'#%07rF#>00”h"#”r2#%0*05'rr 6 $#6 05!0#%0*?A#$!r&"#”?\nmo8p ¦5r r2qmo8p "sM_£G%0#WiOŸr 6 0-rmo8p ¦5r‘r2qmo8p "s"0#%05rœ2&2052@_œ !r0l¦"vm-h+052 !r‹#%04B&r+OM2# 6 Œ205M2r7#%0rC#>00"+0,?G#%0,-''"#$!r‹r2Œ#%0,05/rr ?A#$!r”&"#L-S"105r2052''052"#>05+"?A#>0O#%-[!r0e¦"vmT_L£G%0J 6 j0o2---$!r'-%jiJ-L#%"#

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|e*{,H‰e„o–E 4„‚O{)‰e„… /4} (7rC2!$#+M-$!r+M05--ror2'!rF#>!+->M->#>05'-[rC!+$0G%055e-0=?n"#%0#O10> "#$!r-L+$;0=22$#!r@a6 #>#!r@a!I&+$#$&+"#$!rIr&22$M-$!r@_–£G%0=->0=?hB052'n!rC#UrF& 6 0 ->M->#>05’?AL#%05->0G10"#$!r-L-:?“->#Lr2/05->0a 6 #[r#[1niS0S0/$05rF#L?ALI&+$#$&+$erF#>05r-$0 "&+"#$!r-Lq"sw_DXUM052H1!rF#ur 6 0-u+->Q%&50O#%0L 6 +$05?n0jiŽr2I-+r_–£G%)->0!-U!r0r'r 6 +$0Rr2I&+$#052J"&+"#$!r-=zw_:XU+$!"#ro1!rF#:rF 6 0:->-#>05''rF#rŒ*05+"#$0/&05-$!rŸr2Ÿj0!0#%0”n0jir&2Ÿ-+r*& 6 +$05'-5_*gJniS00ja !"#r71!rF#Ir 6 0->M->#>05'-e05E$04!&+$0Œ%"2iR"0,r2‹!r-0,041jiO0#%r BM0521!rF#Rr 6 0G->-#>05'-_ u!"$#%'Or 6 0D->M->#>05“–^oK& "10-:#%0R2zrF#"05-[? 6 #%B052e1!rF#Dr2&!"#r 1!rF#JrF& 6 0Q-->#>05/-_KiR"#>5+r20r27cHr!6 ‘qMa–¦5q"wahiO00e#%0dBT->#J#>,2005+$#%0 ->07?Hu^)Kœr2$!#+=B&+$#>0rlr2XDXU£"&+"#$!r-05->105#$05+$_™–^oK"0*h"#+"T+$ -#>052€#>*->05-a–iJ%0005-$!r05E$05'05rC#-H"0'+$05--Hr2‘e+$#&+"#$!r-H'0?\05EF05rF# #%r”22$#$!r _–u^)K 6 ->052/e+$#$h+"#$!r0#%M2-[r 6 0 6 !2+e+--$B052-S+$F;/# 6 +$0 6 ->052r2(7$#%05++pvb-'+$T$#% 6 ->052I&+$#$&+"#$!r@_‹£G%0*+"#>#>040#%M2œ%-0‹-'++ j0T%052 6 #e%-€%$!%Ÿ01005rC#"0_‹K00+Rj0505rF#-%50 6 005rŒ1!->052Ÿ#> (7$#%05++pvb-=+$$#%& 6 ->052I&+$#$&+"#$!r@_Uk#zM!r!d2"9-%niJra->05+$05#>05205->05"T% 2!r0Hr,#%-G!rF#>0#-R!$05r7r,B!0Hq_¦"_ XU$!0pq_¦jD+--$B05-u+!"$#%'Le+$#$&+"#$!ro#>o# 6 +$0 6 ->052e+$!"$#%&'-–r&2e(7$#%05++pvb6 ->052+$!"T$#%'-_U£G%0R+$niŒn0%052?1(*#%05++wvb-D+$#%ˆ'";05-:$#DM 6 +$0O?\:2!$#+F-$!r+ M05--r_LKMr0#%0H'r4?A&-R?u#%-O#%05--=-O!r(*#%05++wvb-=+$$#%a#%0o+F;4,# 6 +$0 6 ->052™"!T%05-*-,!r&+$M-$#>052-10B&++œr›#%-T%"#>05_ KM05!r2+$a#%0l05#$!r 0#%M2-”#>œ(*$#T%05++pvb-,+$T$#%r05+$aJ2&$2052"&5'"#$!r—r2# 6 +$0?I05#$!r

PAGE 18

Look up Table based Brubaker and Becker, 1975 [7] Kmetz, 1986, [13] Maenner, 1987, [16] Lai and Wu, 1991, [14] Lewis, 1994, [15] Arnold et al., 2003, [4] Mitchell Algorithm (MA)based Mitchell, 1962 [17] Improving MA Accuracy Divided Approximation Combet et al., 1965 [8] Hall et al., 1970 [10] Sangregory et al., 1999 [22] Abed and Siferd, 2000 [1] Abed and Siferd, 2003 [2] Correction Term based Mitchell, 1962, [17] Mclaren, 2003, [9] This Work, 2005 Operand Decomposition Multiplication Using Logarithms XU$!0dq_¦"_&£DzM!r!d?U !"T$#%¤R->052(*+#$&+"#!r "+00#%M2-H"00M&+r052€r€20#+p_£G%0T%"#>0-#-H?O#%05->0'0#%2-oiO00/-#2$052 "0?“++$4r2! 6 r052,0”$05rC#+”iJ#%,!G10r2205!1!-$#$!r->#>T"#>0_ E1p ‚S‰  e&'GF—‚S&"hz\nD"lA‡vuD\nHD“A!z\n re05"+$d"&!%#>)+$!"#%'SI+#$&+"#!rH#+052d!&+0#>0R+!"$#%z+0S# 6 +$05->#>052re'05_:£G%-–iR-:%jiO005aFr0/$05rF#DrI#>0T'-U?&1jiO0[r2%"2iO"0=!h+$0M#W_ ;*€£D 6 +$0“uV£Q 6 ->052€rC#>0n!+"#$!r#>05T%rE05-n0527#> 6 0Ir0/05rC#)->#>"#>07#> 052&0'uV£-$0"#H#%0'!->#d?OT_'£G%0&$!r00Tr4iS;€r€#%-"05?O–VQ£ 6 ->052 rF#>01!+"#$!r@a@iR-H2056 052 6 €PG_bPJ_DKiR"#>5+r20ja rœqzw_'£G%0'#%-Hrœqz![05->05rF#>052 +$#%'-S?\R10?A'r)?A! 6 -J"$#%&0#Qn0T"#$!r-o“22$#$!r aF6 #>#$!r@aMI&+$#$&+$ "#$!rer2d2&$-$!rh&iJ$#%H#%0Lu^)Kn_K00+C–VQ£ 6 ->052IrC#>0n!+"#$!r)#>05%&rE05-Sba¦5MaC¦5Ma"q"s" ?A!++$jiO052#%-a 6 ,205rF#$?ArI&#'$052–VQ£k-$05-=?AJ'->105$B052_ (*0*0505rC#+$œŸr0i?A?H+$!"$#%/"$#%'0##>0052—-'#%07e+$#205r-$!r&+ +$!"#%'rF& 6 0->M->#>05 p(73–^oK&€¦5MaH¦5rz 6 050‘1&+"5_’£J%0(*3ou^)K—jM205¦s

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Adder C A Logarithm Logarithm Logarithm Logarithm Input Y Input X Operand Decomposition Adder X x Y Adder D B Zero Detector Zero Detector C x D A x B Antilogarithm Antilogarithm lg D lg C lg B lg A XU$!0d_¨q_nXr#!r+@¤R+$;3"?[.o3Q(*#%05++wvb-JQ%#>05#0 0M&+r052ŒrŒ20#+ 6 l#%0#%-r@aGqMa:zw_‘£G%0.10r2Œ305!1!->052w.o3HW(*#%05++wvb+$!"#%'Ge+$#&+"#$!r/+$$#%¢-S-%jiQr”r4£D 6 +$0_¨q_SgJ00#%010r2-L™r2Š"0 205!1!->052#>Ia@¤QauNr2€3¢-r”#%0M0520I!$05r‘r->05#$!r_¦"_d£J%0e205!1!->052 10r2-="0H#%0He+$#$&+$052”-0&""#>05+$r22&2052,#>0#%0#>' 6 #r#%0HB&r+12&#_ XU$!0o_¨e-%niJ-Rr,0M'&+$0o++-#>"#$!r? #%010r&2”205!'n!-0524+!"$#%'Je+$# &+"#$!r@_–-[-%jiJr'r#%0=B!0aC#%0GI&+$#$&+r2-–›2 v¦5ror2rŠ™›2@v¨"sH"0G205!1!->052 rF#>/?A!Qr 6 0-J)’2 v¨qan¤Go2@v¦5auNG)2@v”r273Ho2@vbs!q_J£G%0I205!1!->052710r2-Q"0 #%05rŒI&+$#$&+052‘->0&""#>05+$Ÿ-r(*$#T%05++-e+$!"#% e+$#$h+"#$!r@_£G%0/05-&+$#-d?=#%05->0 205!1!->052,&2&#-S-¦jo2@v¨qr&24-/q!o2@v¨rre"0o#%05r2&20524#>0#=#%0oB&r+nM2# ‹|ŒGŽˆ2 v¦s!r"sM_[£G%0R-0=+$!"$#%&'Le+$#&+"#$!rd0M'&+$0R-U-%jiQrerdB!0R_)iQ$#%!# 10r2205!'n!-$#$!r@_D£G%0RB&r+M2#:re#%-:->0=05EF&+#>H2@v¦s!qzHiJ#%er'0:?@t_¨q! qM¦

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£D 6 +$0d_¦"_n.)3J(7$#T%05++pvb-Gu!"$#%/H(*&+$#$&+"#$!r,Q+$T$#% K#>0l¦'][3Q05!'n!-0d#%0Hr 6 0T-Rˆr&2[ŠˆrC#>'Ia&¤anN›r2*3I U MPOdr2VMWS="0H#%0H%"#>0T->#)?U#%0H+!"$#%'-G?Dˆr2¤ K#>07q]SNR++"#>0NMPOrQ+$052r!r0dn!-$#$!r4?:rF 6 0Gd K#>07]SNR++"#>0NMPSTQ+$052&r!r0)1!-$#$!r,?Ur 6 0G¤ U XZONlXZC1[C†Y†C?Y#D“DC†“D7C„D:†1 „† K#>0*]SNR++"#>0YXZOrQ05'rr&"#=?U#%0)10r&2d K#>07t]SNR++"#>0YXZSTQ05'rr&"#=?D#%0H10r2I K#>07][J105r2-G!"#r'1!rF#=rF& 6 0-]\O^Q CMWOP_)XZOzOr2`\S]Q CMPSa_)XZST U „kC†x“D7C0P]VC†n18™?~rCY†C?x“D7C„ K#>0,]\ O STQb\O c \ST K#>07r]SNR++"#>0 O SQrC#+$&C\ O ST K#>07][Z–0?\ ->#>0&-Jq'r2?AJ205!1!->052e+$#&+r2&-RN›r&23ˆr2->#>0 #%0H"+05-=rZMWGr2VMr" K#>0l¦sM]RNR++"#>0YXZ=r&2lXnO-Jr->#>0&-=/r&2*t K#>0l¦¦"]SJ105r2-J!"#r1!rF#=rF& 6 0-]\TQ CMP XZRr2`\NQ CMr XnT K#>0l¦5q]@\ NQb\ Ac \" K#>0l¦5t]RNR++"#>0 ^QrC#+&C\ jT U N„J& C“™7jDC†^&DN.D1“¦[™7C l?1~CxD„ K#>0l¦5]SZ[M2#].‹fYŽQ O S c Product (X, Y) = X Y = Antilog (Sum1) + Antilog (Sum2) = 1080 lg C = 0101.011000 lg D = 0001.000000 Sum2 = lg C + lg D = 0110.011000 Antilog(Sum2) = 0000000001011000 = d'88 Antilog(Sum1) = 0000001111100000 = d'992 D = b'{00000010} = d'02 C = b'{00101100} = d'44 Sum1 = lg A + lg B = 1001.111100 lg B 0100.000000 lg A = 0101.111100 B = b'{00010000} = d'16 A = b'{00111110} = d'62 Y = b'{00111100} = d'60 X = b'{00010010} = d'18 XU$!0H_¨_@.o3Q(*$#T%05++pvb-JP:&+$0 "5'"#>05+$_)gQ05r0I#%0e10r&27205!'n!-$#$!r*"&!%‘+05"+$&j05-#%0 ?)+$!"$#%&'e+$#$&+"#$!r@_Ž£G%0j0"00/1005rF#'?H(7$#%05++-'+$!"#%',e+$# &+"#$!r@a:"?\#>010r&2Ÿ205!'n!-$#$!r052052Œ#>q_¦j rŒn0505rC#?J"!r2ŸCt!4_ X#%05ahr#%0er0#Q->05#$!r@aniS0I1!->0d+$#%'-J#>”0”$05rC#+,! 6 r0H10r2*205! qq

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X = b'{00010010} = d'18 Y = b'{00111100} = b'60 lg X = 0100.001000 lg Y = 0101.111000 Sum = lg x + lg Y = 1010.000000 X Y = Antilog (Sum) = 0000001000000000 X Y = d'1024 XU$!0d__n(*$#T%05++pvb-JP:&+$0 1!-$#$!riQ$#%#%0S0S05#$!r/"&!%05-L05->05rC#>052”r#%0J0M$!-[T%"#>0L#>I?A#%0 20505-0H#%0H0G1005rC#"0I?U(7$#T%05++pvb-G+$#% 6 -052+$!"$#%&'oI&+$#$&+"#$!r@_ I@JI |Ih"kƒUAD“u"/…–”\n…w F–'GF{nHHDnGyu“"nAh rŒ#%-e->05#$!r@a[iO041!->0,r0iy+$#% #>‘! 6 r0”10r&2Ÿ205!'n!-$#$!rŸiJ$#% 6 052 vb-qz@2$M2052”"jM'"#$!r4"!T%,?\=+$!"#%'QI+#$&+"#!r@_u£G%0o2$M2052”" jM'"#$!rrji%-o00 6 +$#W,-#%0#WiO->0&""#>0'e+$#$h+"#$!r-Jr 6 0j052 iJ$#%,2$n005rF#R05#$!r05E"#$!r&-S?A=#%05$='rF#--Ih"#_[gJ00)#%0on0Tr2-Or2¦Š"0 205!1!->052#>Ia@¤QauNr2€3¢-r”#%0M0520I!$05r‘r->05#$!r_¦"_d£J%0e205!1!->052 10r2-Ia:¤¢r&2Na:3 "0#%05rŽI+#$&+$052l->0&""#>05+$‹-r*#%0*(7$#T%05++pvb-+$!"#%' e+$#$&+"#$!reM0520_U£G%0=&2&#-U?1#%05->0Je+$#$h+"#$!r-: O S[r2 J"0G#%05r”22052 #>0#H#%0'B&r+U2&#‹ f^Ž”_'£G%0'+$#%iR-dr&$#++$--052€#>*&->0#WiO7->0h""#>0 05E"#$!r-U#>H"&5'"#>0=#%0='rF#--o&"#:r+$!"$#%&<++"#$!r?AS.o3QW3GI_C£G%-U6 0 ->0an#%0e2$M2052"5'"#$!r*M0520d-!0#05-Qj005-#'"#>0#%0I/rC#--"+0d? #%0H+$!"T$#%'-Rr2,#%0o0Gr05->05-=r”#%0)1!-$#>0o-20'“%$!%,r0!"#0)z+0nT_LgJniS00ja -e+"#!r-Sr2"#>0o#%"#=#%0H-0)05EF&"#$!r-R?A 6 #%,#%0H205!1!->052e+$#$&+"#$!r-L" M20 6 0#>#>0J0Jr0_RgQ05r0r#%-=05->05"T%@aniS0e->0H#%0d?\!++jiJr/qnw0!$!r7?Ar&#$!r-=?A #%0d205!1!->052e+$#$&+"#$!r-_:£G%0HF0/05rC#-J?\G#%05->0eqnw0!!r?“r#$!r-="0d205rF#$B052 ?A!ƒ0M#>05r-$0#>+:r2‘0I-e+"#$!r-5_e£G%05->0-e+"#$!r-#";05-I#%0"+05-)?OM2# O Ser2ŒM2#N ,rF#>!rF#iJ$#%Œ50"0*n0T05rC#e0r2Ÿ0r0,-e#%0 q

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