Suggested Resolution: Counter As Per 06-1924R0

December 2006 doc.: IEEE 802.11-06/1924r0

IEEE P802.11 Wireless LANs

LB84-submission-simplifying-formulas

Date: 2006-12-19

Author(s): Name Company Address Phone email Matam Industrial Park Assaf Kasher Intel Corporation +972-4-8651547 [email protected] Haifa 31015, Israel om

Abstract This document attempts to resolve CID 12164 by simplifying formulas in the document (D1.07), continuing the effort of 06-0935r3

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lb84 simlifying formulas page 1 Assaf Kasher, Intel Corporation December 2006 doc.: IEEE 802.11-06/1924r0

This document attemts to resolve CID 12164 by simplifying some formulas in the document: The waveform equations (1) Define a function uppercaseUpsilon_k(BW) as a generalization of the existing such as (20-10), inside uppercaseUpsilon. For BW=20MHz, this is always 1, For 40MHz, this is 1 then j. table n66, etc etc, are In this way, the two summations over disjoint k may be simplified to a single unnecessarily ugly. They summation over all of k. (2) Replace the 47 by a named constant (3) Use a contain much duplication named constant for the 90deg rotation in the HT-SIG, to avoid confusion with and therefore are much other j factors floating around. (4) The summation are sometimes from some more difficult to verify negative N to some positive N, and sometimes 0 to some positive M. Pick a than they should be. convention and stick with it. For instance (20-53) is particularly ugly (5) The separate summations over data tones and pilot tones are ugly and would be better if merged into a single summation. Define a new function, basically the X of p210 line 3, which is the multiplexing of data and pilot subcarriers - basically the spatial mapping input. (6) Order the sqrt(1/N) term and w(t) term consistently

Suggested Resolution: Counter as per 06-1924r0

Discussion:

1) k was defined in 06-935r3, it is not usefull in equations in which data is duplicated over the two adjacent channels.

2) This submission creates a new variable Dk,n which exists for k=-NSR:NSR and takes the values from dk,n which exists for k=0:NSD-1. This enables using a single summation for the data and the pilots and therefore simplifies equations. 3) It is now possible simplify the duplication in frequency so as little as possible is duplicated. The 90° rotation in table n59 is now clearer and simpler.

4) The constant NSR is used when its value in table n52 is the value used in the equation. This does not happen in all of the “legacy” part of a mixed format packet or in HT and non-HT duplicate packet. 5) The sqrt(1/N) and windowing term were ordered consitently were possible 6) An error was fixed in equation 21-58 (summation over symbol index was missing)

In general, I think that together with 06-935r3, the equations are much simpler and clearer.

TGn Editor: Remove table n56 on page 229 (draft D1.07)

Modulation method Conversion to time domain signal on Tx chain iTX

20 MHz transmission on several (i ) 1 transmit chains rTX ( t) = w( t) L- SIG TSYM Tone NTX N L- SIG 47 iTX dkexp( j 2p M( k)D F( t - T GI - T CS )) + k =0 26 iTX p0 Pkexp( j 2p kD F( t - T GI - T CS )) k=-26

40 MHz transmission on several (i ) 1 transmit chains rTX ( t) = w( t) L- SIG TSYM Tone NTX N L- SIG 47 iTX dkexp( j 2p ( M( k) - 32) D F( t - T GI - T CS )) + k=0 47 iTX +j dkexp( j 2p ( M( k) + 32) D F( t - T GI - T CS )) + k =0 26 p Pexp j 2p ( k- 32) D t - T - T iTX + 0 k( ( F( GI CS )) k=-26

jexp j 2p ( k+ 32) D t - T - T iTX ( F( GI CS ))))

M( k ), Pk are defined in 17.3.5.9.

p0 is the first pilot value in the sequence defined in 17.3.5.9.

TGn Editor Replace the following part of lines 29:31 in page 228 of section 21.3.9.3

denoted by dk , k = 0 47 . The conversion of these complex numbers into a time domain signal is decribed in the Tabel n56 (Generation of the Non-hT SIGNAL Field) With

denoted by dk , k = 0 47 . The time domain waveform of the Non-HT SIGNAL field in 20MHz transmission is: 26 (i ) 1 i rTX ( t) = w( t)� ( D D p P)exp j 2p k - t - T T TX L- SIGTone TSYM k 0 k( F( GI CS )) k 26 NTX N L- SIG =- In a 40MHz transmission the time domain waveform is: 1 26 r(iTX ) ( t) = w( t)� ( D - p P) exp j 2p ( k D 32) - t - T + T i TX L- SIGTone TSYM k 0 k( ( F( GI CS )) k 26 NTX N L- SIG =-

jexp j 2p ( k+ 32) D t - T - T iTX ( F( GI CS ))) where: lb84 simlifying formulas page 3 Assaf Kasher, Intel Corporation December 2006 doc.: IEEE 802.11-06/1924r0

0k = 0,北 7, 21 D = k d otherwise Mr ( k ) k+26 - 26＃ k - 22 k+25 - 20＃ k - 8 k+24 - 6＃ k - 1 Mr ( k ) = k+ 23 1＃ k 6 k+ 22 8＃ k 20 k+ 21 22＃ k 26

Pk are defined in 17.3.5.9.

p0 is the first pilot value in the sequence defined in 17.3.5.9.

TGn Editor: remove table n59 on page 232 (draft D1.07) Modulation method Conversion to time domain signal on Tx chain iTX

20 MHz transmission 1 1 riTX ( t) = w( t - nT ) HT- SIGTone TSYM SYM n 0 NTX N HT- SIG = 47 iTX j dk, nexp( j 2p M( k)D F( t - nT SYM - T GI - T CS )) k=0 26 iTX +pn+1 P kexp( j 2p k D F( t - nT SYM - T GI - T CS )) k =-26

1 40 MHz transmission i 1 rTX ( t) = w( t - nT ) HT_ SIGTone TSYM SYM n 0 NTX N HT- SIG = 47 d jexp j 2p M k- 32 D t - nT - T - T iTX k, n( ( ( ( ) ) F( SYM GI CS )) k =0

-expj 2p M k + 32 D t - nT - T - T iTX ( ( ( ) ) F( SYM GI CS ))) 26 +p Pexp j 2p k - 32 D t - nT - T - T iTX n+1 k( ( ( ) F( SYM GI CS )) k =-26

+jexp j 2p k + 32 D t - nT - T - T iTX ( ( ) F( SYM GI CS ))))

TGn Editor: Replace the following part of lines 49-52 of page 231 of section 21.3.9.4.5

two groups of 48 complex numbers: dk, n , k= 0 47, n = 0,1 . The conversion of these into the time domain signal is according to Table n59 (Modulcation of the HT SIGNAL field in an HT mixed format packet)

With two groups of 48 complex numbers: dk, n , k= 0 47, n = 0,1 . The time domain waveform for the HT SIGNAL field in an HT mixed format packet in a 20MHz transmission is: 1 1 riTX ( t) = w( t - nT ) HT- SIGTone TSYM SYM n 0 NTX N HT- SIG = 26 iTX ( jDk, n+ p n+ 1 P k)exp( j 2p k D F( t - nT SYM - T GI - T CS )) k=-26 For a 40MHz transmission the time domain waveform is: 1 1 riTX ( t) = w( t - nT ) HT_ SIGTone TSYM SYM n 0 NTX N HT- SIG = 26 jD+ p Pexp j 2p k - 32 D t - nT - T - T iTX ( k, n n+ 1 k)( ( ( ) F( SYM GI CS )) k=-26

+jexp j 2p k + 32 D t - nT - T - T iTX ( ( ) F( SYM GI CS ))) where: 0k = 0,北 7, 21 D = k, n d otherwise Mr ( k), n r M( k ) is defined in 21.3.9.3

TGn Editor: replace equation 21-24 on page 241 (D1.07) 1 1 riTX ( t) = w( t - nT ) HT- SIGTone TSYM SYM n 0 NSTS N HT- SIG =

47 NSTS j轾 Q P dexp j 2p M kD t - nT - T - T iSTS 邋 M( k )[ HTLTF]i ,1 k , n( ( ) F( SYM GI CS )) 臌 iTX, i STS STS k=0 iSTS = 1

NSR NSTS +p Q P Pexp j 2p k D t - nT - T - T iSTS n[ k]i, i[ HTLTF] i ,1 k( F( SYM GI CS )) TX STS STS k=-NSR iSTS =1 With

1 1 riTX ( t) = w( t - nT ) HT- SIGTone TSYM SYM n 0 NSTS N HT- SIG =

26 NSTS Q P jD+ p Pexp j 2p k D t - nT - T - T iSTS 邋 [ k]i, i[ HTLTF] i ,1 ( k, n n)k ( F( SYM GI CS )) TX STS STS k=-26 iSTS = 1

TGn Editor: replace line 46 from page 241 of (D1.07):

M(k), Pk, and pn are defined in 17.3.5.9, With

Pk and pn are defined in 17.3.5.9,

Dk,n is defined in 21.3.9.4.3 lb84 simlifying formulas page 5 Assaf Kasher, Intel Corporation December 2006 doc.: IEEE 802.11-06/1924r0

TGn Editor: replace equation 21-25 on page 242 (D1.07) 1 1 riTX ( t) = w( t - nT ) HT- SIGTone TSYM SYM n 0 NSTS N HT- SIG =

47 NSTS P j轾 Q dexp j 2p M k- 32 D t - nT - T - T iSTS 邋 [ HTLTF]i ,1 M( k )- 32 k , n( ( ( ) ) F( SYM GI CS )) STS ( 臌 iTX, i STS k=0 iSTS = 1

iSTS -轾QM( k )+ 32 d k , nexp( j 2p ( M( k) + 32) D F( t - nT SYM - T GI - T CS )) 臌 iTX, i STS )

26 NSTS + p P P Qexp j 2p k- 32 D t - nT - T - T iSTS n邋 [ HTLTF]i,1 k[ k-32 ] i , i ( ( ) F( SYM GI CS )) STS( TX STS k=-26 iSTS = 1

+j Qexp j 2p k + 32 D t - nT - T - T iSTS [ k+32 ]i, i ( ( ) F( SYM GI CS )) TX STS )) With 1 1 riTX ( t) = w( t - nT ) HT- SIGTone TSYM SYM n 0 NSTS N HT- SIG =

26 NSTS P jD+ p P Qexp j 2p k - 32 D t - nT - T - T iSTS 邋 [ HTLTF]i,1( k, n n k) [ k- 32 ] i , i ( ( ) F( SYM GI CS )) STS( TX STS k=-26 iSTS = 1

+j Qexp j 2p k + 32 D t - nT - T - T iSTS [ k+32 ]i, i ( ( ) F( SYM GI CS )) TX STS )

Dk,n is defined in 21.3.9.4.3

TGn Editor: replace equation 21-53 on the top of page 260 (D1.07)

1 NSYM -1 riTX ( t )= w ( t - nT ) HT- DATATones TSYM SYM n 0 NSTS N HT- DATA =

NSD-1 N STS 轾Q d exp j 2p M ( k )D t - T - nT - T iSTS + 邋 M( k ) k , iSTS , n( F( GI SYM CS )) 臌 iTX, i STS k=0 iSTS = 1

NSR N STS k iSTS pn+ z邋 [ Q k] P( i , n ) exp( j 2p kD F( t - T GI - nT SYM - T CS ) iTX, i STS STS k=- NSR iSTS =1 With

1 NSYM -1 riTX ( t )= w ( t - nT ) HT- DATATones TSYM SYM NSTS N HT- DATA n=0

NSR N STS Q D + p Pk exp j 2p k D t - T - nT - T iSTS 邋 [ k]i, i ( k, n n+ z(iSTS , n) ) ( F( GI SYM CS )) TX STS k=- NSR i STS =1 TGn Editor: replace the following lines (19-32) from page 260 of (D1.07): M(k) in the 20 MHz case is

k- 28 0＃ k 6 k- 27 7＃ k 19 k- 26 20＃ k 25 M( k ) = , and (20-1) k- 25 26＃ k 31 k- 24 32＃ k 44 k- 23 45＃ k 51

0k = 0,北 7, 21 D = k, n d otherwise Mr ( k), n k+28 - 28＃ k - 22 k+27 - 20＃ k - 8 k+26 - 6＃ k - 1 Mr ( k ) = , and k+ 25 1＃ k 6 k+ 24 6＃ k 20 k+ 23 22＃ k 28 TGn Editor: replace equation 21-55 on page 261 (D1.07) 1 NSYM -1 riTX ( t )= w ( t - nT ) HT- DATATones TSYM SYM n 0 NSTS N HT- DATA = N SD -1 2 NSTS 轾Q d exp j 2p M ( k )D t - T - nT - T iSTS + 邋 M( k ) k , iSTS , n( F( GI SYM CS )) 臌 iTX, i STS k=0 iSTS = 1

0 NSTS p Q Pk exp j 2p kD t - T - nT - T iSTS + n+ z邋 [ k]i, i ( iSTS , n ) ( F( GI SYM CS )) TX STS k=- NSR iSTS =1

NSD-1 N STS j轾 Q d exp j 2p M ( k )D t - T - nT - T iSTS + 邋臌 M( k )i, i k , iSTS , n( F( GI SYM CS )) N TX STS k= SD iSTS =1 2

NSR N STS jp Q Pk exp j 2p kD t - T - nT - T iSTS n+ z邋 [ k]i, i ( iSTS , n ) ( F( GI SYM CS )) TX STS k=1 iSTS = 1 With lb84 simlifying formulas page 7 Assaf Kasher, Intel Corporation December 2006 doc.: IEEE 802.11-06/1924r0

1 NSYM -1 riTX ( t )= w ( t - nT ) HT- DATATones TSYM SYM NSTS N HT- DATA n=0

NSR N STS Q D + p Pk � exp j 2p k - t - T - nT T iSTS 邋 [ k]i, i ( k n+ z(iSTS , n) ) k( F( GI SYM CS )) TX STS k=- NSR i STS =1 TGn Editor: replace the following lines (44-58) from page 261 of (D1.07): M(k) in the 40 MHz case is

禳k- 58 0＃ k 4 镲k- 57 5＃ k 31 镲镲k- 56 32＃ k 44 镲镲k- 55 45＃ k 53 M( k ) = 睚 , and (20-2) 镲k- 52 54＃ k 62 镲k- 51 63＃ k 75 镲镲k- 50 76＃ k 102 镲铪k- 49 103＃ k 107

With In the 40MHz case: 0k = 0,北 1, 11, 北 25, 53 D = k d otherwise Mr ( k ) k+58 - 58＃ k - 54 k+57 - 52＃ k - 26 k+56 - 24＃ k - 12 k+55 - 10＃ k - 2 Mr ( k ) = , and k+ 52 2＃ k 10 k+ 51 12＃ k 24 k+ 50 26＃ k 52 k+ 49 54＃ k 58

TGn Editor: replace equation 21-57 on page 262 (D1.07)

1 NSYM -1 riTX ( t) = w( t - nT ) HT- DATATones TSYM SYM NDuplicate n=0

47 轾Q dexp j 2p M k- 32 D t - nT - T + 臌 M( k )- 32i ,1 k , n( ( ( ) ) F( SYM GI )) k =0 TX 47 j轾 Q dexp j 2p M k+ 32 D t - nT - T + 臌 M( k )+ 32i ,1 k , n( ( ( ) ) F( SYM GI )) k =0 TX

pn+ z[ Q k -32 ] P kexp( j 2p ( k- 32) D F( t - nT SYM - T GI )) + iTX ,1 k=- NSR

jpn+ z[ Q k +32 ] P kexp( j 2p ( k+ 32) D F( t - nT SYM - T GI )) iTX ,1 k=- NSR With 1 NSYM -1 riTX ( t) = w( t - nT ) HT- DATATones TSYM SYM g NDuplicate n=0

(Dk, n+ p n+ z P k) [ Q k - 32 ] exp( j 2p ( k - 32) D F( t - nT SYM - T GI )) ( iTX ,1 k=- NSR

+j[ Qk+32 ] exp( j 2p ( k + 32) D F( t - nT SYM - T GI )) iTX ,1 ) TGn Editor: replace line 36 from page 262 of (D1.07):

Dk,n is defined in 21.3.9.4.3 TGn Editor: replace equation 21-58 on page 263 (D1.07)

1 riTX ( t- nT) = w( t - nT ) LEG- DUP SYMTones TSYM SYM NDuplicate

47 iTX dk, nexp( j 2p ( M( k) - 32) ( D F( t - nT SYM - T GI - T CS )) + k =0 47 iTX j dkexp( j 2p ( M( k) + 32) ( D F( t - nT SYM - T GI - T CS )) + k =0 26 iTX pn P kexp( j 2p ( k- 32) D F( t - nT SYM - T GI - T CS )) + k =-26 26 iTX jpn P kexp( j 2p ( k+ 32) D F ( t -nTSYM - T GI - T CS )) k =-26 With

1 NSYM -1 riTX ( t- nT) = w( t - nT ) LEG- DUP SYMTones TSYM SYM n=0 NDuplicate 26 jD+ p Pexp j 2p k - 32 D t - nT - T - T iTX ( k, n n k)( ( ( ) F( SYM GI CS )) k =-26

+jexp j 2p k + 32 D t - nT - T - T iTX ( ( ) F( SYM GI CS ))) lb84 simlifying formulas page 9 Assaf Kasher, Intel Corporation December 2006 doc.: IEEE 802.11-06/1924r0

M( k ) , Pk , and pn and are defined in 17.3.5.9,

Pk and pn and are defined in 17.3.5.9,

Dk,n is defined in 21.3.9.4.3

lb84 simlifying formulas page 10 Assaf Kasher, Intel Corporation