OFDM channel estimation by singular value decomposition

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(1)/&$- CHANNEL ESTIMATION BY SINGULAR VALUE DECOMPOSITIONc /VE %DFORSY -AGNUS 3ANDELLY *AN *AAP VAN DE "EEKY 3ARAH +ATE 7ILSONZ 0ER /LA "¶RJESSONY Y. $IVISION OF 3IGNAL 0ROCESSING ,ULE¥ 5NIVERSITY OF 4ECHNOLOGY 3 ,ULE¥ 37%$%.. Z. 3CHOOL OF %LECTRICAL AND #OMPUTER %NGINEERING 0URDUE 5NIVERSITY 7EST ,AFAYETTE

(2) ). 53!. c 4HIS WORK HAS BEEN PRESENTED IN PART AT THE 6EHICULAR 4ECHNOLOGY #ONFERENCE 64#Ú IN !TLANTA

(3) 'EORGIA

(4) !PR Ô-AY

(5)

(6) PP .

(7)

(8) !BSTRACT )N THIS PAPER WE PRESENT AND ANALYSE LOW RANK CHANNEL ESTIMATORS FOR ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING /&$- USING THE FREQUENCY CORRELATION OF THE CHANNEL ,OW RANK AP PROXIMATIONS BASED ON THE DISCRETE &OURIER TRANSFORM $&4 HAVE BEEN PROPOSED BUT THEY SUdER FROM POOR PERFORMANCE WHEN THE CHANNEL IS NOT SAMPLE SPACED 7E APPLY THE THEORY OF OPTIMAL RANK REDUCTION TO LINEAR MINIMUM MEAN SQUARED ERROR ,--3% ESTIMATORS AND SHOW THAT THESE ESTIMATORS

(9) WHEN USING A çXED DESIGN

(10) ARE ROBUST TO CHANGES IN CHANNEL COR RELATION AND SIGNAL TO NOISE RATIO 3.2 4HE PERFORMANCE IS PRESENTED IN TERMS OF UNCODED SYMBOL ERROR RATE 3%2 FOR A SYSTEM USING 1!-.

(11)

(12) #ONTENTS )NTRODUCTION. . 3YSTEM DESCRIPTION 3YSTEM MODEL #HANNEL MODEL 3CENARIO . . ,INEAR CHANNEL ESTIMATION ACROSS TONES ,--3% ESTIMATION /PTIMAL LOW RANK APPROXIMATIONS %STIMATOR COMPLEXITY . . %STIMATOR PERFORMANCE AND DESIGN 2ANK REDUCTION 3%2 PERFORMANCE UNDER MISMATCH )NCORRECT CHANNEL CORRELATION )NCORRECT 3.2 . . . 'ENERIC LOW RANK ESTIMATOR 0ERFORMANCE GAIN #OMPARISON TO &)2 çLTERS 4HE USE OF TIME CORRELATION . . #ONCLUSIONS. . ! /PTIMAL RANK REDUCTION. . " #HANNEL CORRELATION MATRICES. . # %STIMATOR MEAN SQUARED ERROR. . . . . . . . . . . . . . . . . . . . . . . . .

(13)

(14) #HAPTER )NTRODUCTION 7IRELESS DIGITAL COMMUNICATION SYSTEMS USING MULTI AMPLITUDE MODULATION SCHEMES

(15) SUCH AS QUADRATURE AMPLITUDE MODULATION 1!-

(16) REQUIRE ESTIMATION AND TRACKING OF THE FADING CHAN NEL )N GENERAL

(17) THIS MEANS A MORE COMPLEX RECEIVER THAN FOR DIdERENTIAL MODULATION SCHEMES

(18) SUCH AS DIdERENTIAL PHASE SHIFT KEYING $03+

(19) WHERE THE RECEIVERS OPERATE WITHOUT A CHANNEL ESTIMATE ;= )N ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING /&$- SYSTEMS

(20) $03+ IS APPROPRIATE FOR RELATIVELY LOW DATA RATES

(21) SUCH AS IN THE %UROPEAN DIGITAL AUDIO BROADCAST $!" SYSTEM ;= (OWEVER

(22) FOR MORE SPECTRALLY EbCIENT /&$- SYSTEMS

(23) COHERENT MODULATION IS MORE APPROPRI ATE 4HE STRUCTURE OF /&$- SIGNALLING ALLOWS A CHANNEL ESTIMATOR TO USE BOTH TIME AND FRE QUENCY CORRELATION 3UCH A TWO DIMENSIONAL ESTIMATOR STRUCTURE IS GENERALLY TOO COMPLEX FOR A PRACTICAL IMPLEMENTATION 4O REDUCE THE COMPLEXITY

(24) SEPARATING THE USE OF TIME AND FREQUENCY CORRELATION HAS BEEN PROPOSED ;= 4HIS COMBINED SCHEME USES TWO SEPARATE &)2 7IENER çLTERS

(25) ONE IN THE FREQUENCY DIRECTION AND THE OTHER IN THE TIME DIRECTION )N THIS PAPER WE PRESENT AND ANALYSE A CLASS OF BLOCK ORIENTED CHANNEL ESTIMATORS FOR /&$-

(26) WHERE ONLY THE FREQUENCY CORRELATION OF THE CHANNEL IS USED IN THE ESTIMATION 7HATEVER THEIR LEVEL OF PERFORMANCE

(27) IT MAY BE IMPROVED WITH THE ADDITION OF A SECOND çLTER USING THE TIME CORRELATION ;

(28) = 4HOUGH A LINEAR MINIMUM MEAN SQUARED ERROR ,--3% ESTIMATOR USING ONLY FREQUENCY CORRELATION HAS LOWER COMPLEXITY THAN ONE USING BOTH TIME AND FREQUENCY CORRELATION

(29) IT STILL REQUIRES A LARGE NUMBER OF OPERATIONS 7E INTRODUCE A LOW COMPLEXITY APPROXIMATION TO A FREQUENCY BASED ,--3% ESTIMATOR THAT USES THE THEORY OF OPTIMAL RANK REDUCTION /THER TYPES OF LOW RANK APPROXIMATIONS

(30) BASED ON THE DISCRETE TIME &OURIER TRANSFORM $&4

(31) HAVE BEEN PROPOSED FOR /&$- SYSTEMS BEFORE ;

(32)

(33) = 4HE WORK PRESENTED IN THIS PAPER WAS INSPIRED BY THE OBSERVATIONS IN ;=

(34) WHERE IT IS SHOWN THAT $&4 BASED LOW RANK CHANNEL ESTIMATORS HAVE LIMITED PERFORMANCE FOR NON SAMPLE SPACED CHANNELS AND HIGH 3.2S !FTER PRESENTING THE /&$- SYSTEM MODEL AND OUR SCENARIO IN 3ECTION

(35) WE INTRODUCE THE ESTIMATORS AND DERIVE THEIR COMPLEXITIES IN 3ECTION 7E ANALYSE THE SYMBOL ERROR RATE 3%2 PERFORMANCE IN 3ECTION WHERE WE ALSO DISCUSS DESIGN CONSIDERATIONS 4HE PROPOSED LOW RANK ESTIMATOR IS COMPARED TO OTHER ESTIMATORS IN 3ECTION AND A SUMMARY AND CONCLUDING REMARKS APPEAR IN 3ECTION . .

(36) .

(37) #HAPTER 3YSTEM DESCRIPTION . 3YSTEM MODEL. &IGURE DISPLAYS THE /&$- BASE BAND MODEL USED IN THIS PAPER 7E ASSUME THAT THE USE OF A CYCLIC PREçX #0 ;= BOTH PRESERVES THE ORTHOGONALITY OF THE TONES AND ELIMINATES INTER SYMBOL INTERFERENCE )3) BETWEEN CONSECUTIVE /&$- SYMBOLS &URTHER

(38) THE CHANNEL FS ~ IS ASSUMED TO BE SLOWLY FADING

(39) SO IT IS CONSIDERED TO BE CONSTANT DURING ONE /&$- SYMBOL 4HE NUMBER OF TONES IN THE SYSTEM IS -

(40) AND THE LENGTH OF THE CYCLIC PREçX IS + SAMPLES. &IGURE "ASE BAND MODEL OF AN /&$- SYSTEM Ú#0Ú DENOTES THE CYCLIC PREçX 5NDER THESE ASSUMPTIONS WE CAN DESCRIBE THE SYSTEM AS A SET OF PARALLEL 'AUSSIAN CHANNELS

(41) SHOWN IN &IGURE

(42) WITH CORRELATED ATTENUATIONS GJ 4HE ATTENUATIONS ON EACH TONE ARE GIVEN BY J GJ &

(43) J - `

(44) - 3R WHERE & a IS THE FREQUENCY RESPONSE OF THE CHANNEL F S ~ DURING THE /&$- SYMBOL

(45) AND 3R IS THE SAMPLING PERIOD OF THE SYSTEM )N MATRIX NOTATION WE DESCRIBE THE /&$- SYSTEM AS X 7G M. . WHERE X IS THE RECEIVED VECTOR

(46) 7 IS A MATRIX CONTAINING THE TRANSMITTED SIGNALLING POINTS ON ITS DIAGONAL

(47) G IS A CHANNEL ATTENUATION VECTOR

(48) AND M IS A VECTOR OF IID COMPLEX

(49) ZERO MEAN

(50) 'AUSSIAN NOISE WITH VARIANCE }M .

(51) &IGURE 4HE /&$- SYSTEM

(52) DESCRIBED AS A SET OF PARALLEL 'AUSSIAN CHANNELS WITH CORRELATED ATTENUATIONS. . #HANNEL MODEL. 7E ARE USING A FADING MULTI PATH CHANNEL MODEL ;=

(53) CONSISTING OF , IMPULSES F ~ . ` 8. ,. mJ p ~ ` ~J 3R

(54). . J . WHERE mJ ARE ZERO MEAN

(55) COMPLEX 'AUSSIAN

(56) RANDOM VARIABLES

(57) WITH A POWER DELAY PROçLE t ~J )N THIS PAPER WE HAVE USED , IMPULSES AND TWO VERSIONS OF THIS CHANNEL MODEL q 3YNCHRONIZED CHANNEL 4HIS IS A MODEL OF A PERFECTLY TIME SYNCHRONIZED /&$- SYS TEM

(58) WHERE THE çRST FADING IMPULSE ALWAYS HAS A ZERO DELAY

(59) ~

(60) AND OTHER FADING IMPULSES HAVE DELAYS THAT ARE UNIFORMLY AND INDEPENDENTLY DISTRIBUTED OVER THE LENGTH OF THE CYCLIC PREçX 4HE IMPULSE POWER DELAY PROçLE

(61) t ~J "D`~J ~QLR

(62) DECAYS EXPO NENTIALLY ;= q 5NIFORM CHANNEL !LL IMPULSES HAVE THE SAME AVERAGE POWER AND THEIR DELAYS ARE UNIFORMLY AND INDEPENDENTLY DISTRIBUTED OVER THE LENGTH OF THE CYCLIC PREçX. . 3CENARIO. /UR SCENARIO CONSISTS OF A WIRELESS 1!- /&$- SYSTEM

(63) DESIGNED FOR AN OUTDOOR ENVIRON MENT

(64) THAT IS CAPABLE OF CARRYING DIGITAL VIDEO 4HE SYSTEM OPERATES AT K(Z BANDWIDTH AND IS DIVIDED INTO TONES WITH A TOTAL SYMBOL PERIOD OF xS

(65) OF WHICH xS IS THE CYCLIC PREçX /NE /&$- SYMBOL THUS CONSISTS OF SAMPLES - +

(66) FOUR OF WHICH ARE CONTAINED IN THE CYCLIC PREçX + 4HE UNCODED DATA RATE OF THE SYSTEM IS -"ITSEC 7E ASSUME THAT ~QLR SAMPLE FOR THE SYNCHRONIZED CHANNEL. .

(67) #HAPTER ,INEAR CHANNEL ESTIMATION ACROSS TONES )N THE FOLLOWING WE PRESENT THE ,--3% ESTIMATE OF THE CHANNEL ATTENUATIONS G FROM THE RECEIVED VECTOR X AND THE TRANSMITTED DATA 7 7E ASSUME THAT THE RECEIVED /&$- SYMBOL CONTAINS DATA KNOWN TO THE ESTIMATOR Ô EITHER TRAINING DATA OR RECEIVER DECISIONS 4HE COMPLEXITY REDUCTION OF THE ,--3% ESTIMATOR CONSISTS OF TWO SEPARATE STEPS )N THE çRST STEP WE MODIFY THE ,--3% BY AVERAGING OVER THE TRANSMITTED DATA

(68) OBTAINING A SIMPLIçED ESTIMATOR )N THE SECOND STEP WE REDUCE THE NUMBER OF MULTIPLICATIONS REQUIRED BY APPLYING THE THEORY OF OPTIMAL RANK REDUCTION ;=. . ,--3% ESTIMATION. 4HE ,--3% ESTIMATE OF THE CHANNEL ATTENUATIONS G

(69) IN

(70) GIVEN THE RECEIVED DATA X AND THE TRANSMITTED SYMBOLS 7 IS ;= t. r. B G 1GG }M 77' KLLRD 1GG. WHERE. s` u`. . B G. X X X ` 7`X aaa W W W `. . 3. -. KR. B G KR. . -. IS THE LEAST SQUARES N O ,3 ESTIMATE OF G

(71) }M IS THE VARIANCE OF THE ADDITIVE CHANNEL NOISE AND 1GG $ GG' IS THE CHANNEL AUTOCORRELATION 4HE SUPERSCRIPT a' DENOTES (ERMITIAN TRANSPOSE )N THE FOLLOWING WE ASSUME

(72) WITHOUT LOSS OF GENERALITY

(73) THAT THE VARIANCES OF THE N O CHANNEL ATTENUATIONS IN G ARE NORMALIZED TO UNITY

(74) IE $ JGJ J 4HE ,--3% ESTIMATOR IS OF CONSIDERABLE COMPLEXITY

(75) SINCE A MATRIX INVERSION IS NEEDED EVERY TIME THE TRAINING DATA IN 7 CHANGES 7E REDUCE THE COMPLEXITY OF THIS ESTIMATOR BY AVERAGING OVERNTHE TRANSMITTED DATA ;=

(76) IE WE REPLACE THE TERM 77' ` IN WITH O ITS EXPECTATION $ 77' ` !SSUMING THE SAME SIGNAL CONSTELLATION ON ALL TONES AND EQUAL. N. O. PROBABILITY ON ALL CONSTELLATION POINTS

(77) WE HAVE $ 77' ` $ FJWJ JG (

(78) WHERE ( IS THE IDENTITY MATRIX $EçNING THE AVERAGE SIGNAL TO NOISE RATIO AS 2-1 $ FJWJ JG }M

(79) WE OBTAIN A SIMPLIçED ESTIMATOR ` n B B

(80) G 1GG 1GG. ( G KR 2-1 .

(81) WHERE. N. O. N. n $ JWJ J $ JWJ J. O. IS A CONSTANT DEPENDING ON THE SIGNAL CONSTELLATION )N THE CASE OF 1!- TRANSMISSION

(82) n n "ECAUSE 7 IS NO LONGER A FACTOR IN THE MATRIX CALCULATION

(83) THE INVERSION OF 1GG 2-1 ( DOES NOT NEED TO BE CALCULATED EACH TIME THE TRANSMITTED DATA IN 7 CHANGES &URTHERMORE

(84) IF 1GG AND 2-1 ARE KNOWN BEFOREHAND OR ARE SET TO çXED NOMINAL VALUES

(85) THE MATRIX 1GG 1GG. n (` NEEDS TO BE CALCULATED ONLY ONCE 5NDER THESE CONDITIONS THE ESTIMATION REQUIRES 2-1 MULTIPLICATIONS PER TONE 4O FURTHER REDUCE THE COMPLEXITY OF THE ESTIMATOR

(86) WE PROCEED WITH LOW RANK APPROXIMATIONS BELOW. . /PTIMAL LOW RANK APPROXIMATIONS. /PTIMAL RANK REDUCTION IS ACHIEVED BY USING THE SINGULAR VALUE DECOMPOSITION 36$ ;= 4HE 36$ OF THE CHANNEL AUTOCOVARIANCE MATRIX IS 1GG 4c4' . . WHERE 4 IS A UNITARY MATRIX CONTAINING THE EIGENVECTORS AND c IS A DIAGONAL MATRIX

(87) CONTAINING THE SINGULAR VALUES w w w w w w- ` ON ITS DIAGONAL )N !PPENDIX ! IT IS SHOWN THAT THE OPTIMAL RANK O ESTIMATOR IS B 4a 4' G G O O B KR WHERE aO IS A DIAGONAL MATRIX WITH THE VALUES pJ . . wJ n J 2-1. w. . J O ` J O - ` . . 6IEWING THE ORTHONORMAL MATRIX 4' AS A TRANSFORM

(88) THE SINGULAR VALUE wJ OF 1GG IS THE CHANNEL POWER VARIANCE CONTAINED IN THE JSG TRANSFORM COEbCIENT AFTER TRANSFORMING THE ,3 B 3INCE 4 IS UNITARY

(89) THIS TRANSFORMATION CAN BE VIEWED AS ROTATING THE VECTOR G B ESTIMATE G KR KR SO THAT ALL ITS COMPONENTS ARE UNCORRELATED ;= 4HE DIMENSION OF THE SPACE OF ESSENTIALLY TIME AND BAND LIMITED SIGNALS LEADS US TO THE RANK NEEDED IN THE LOW RANK ESTIMATOR )N ;= IT IS SHOWN THAT THIS DIMENSION IS ABOUT !3 WHERE ! IS THE ONE SIDED BANDWIDTH AND 3 IS THE TIME INTERVAL OF THE SIGNAL !CCORDINGLY

(90) THE MAGNITUDE OF THE SINGULAR VALUES OF 1GG SHOULD DROP RAPIDLY AFTER ABOUT + LARGE VALUES

(91) WHERE + IS THE LENGTH OF THE CYCLIC PREçX ! 3R

(92) 3 +3R AND !3 + 7E PRESENT THE CHANNEL POWER CONTAINED IN THE çRST COEbCIENTS IN &IGURE 4HE CALCULATIONS ARE BASED ON OUR SCENARIO AND THE TWO CHANNEL MODELS THE SYNCHRONIZED AND THE UNIFORM 4HE MAGNITUDE OF THE CHANNEL POWER DROPS RAPIDLY AFTER ABOUT J

(93) IE COEbCIENTS

(94) WHICH IS CONSISTENT WITH THE OBSERVATION THAT THE DIMENSION OF THE SPACE SPANNED BY 1GG IS APPROXIMATELY +

(95) THAT IS

(96) IN THIS CASE ! BLOCK DIAGRAM OF THE RANKÒ ESTIMATOR IN IS SHOWN IN &IGURE

(97) WHERE THE ,3 ESTIMATE IS CALCULATED FROM X BY MULTIPLYING BY 7` . 3INCE WE ARE DEALING WITH (ERMITIAN MATRICES THE wK S ARE ALSO EIGENVALUES (OWEVER

(98) WE USE THE TERMINOLOGY OF THE 36$ SINCE IT IS MORE GENERAL AND CAN BE USED IN OPTIMAL RANK REDUCTION OF NON (ERMITIAN MATRICES 4HE TRANSFORM IN THIS SPECIAL CASE OF LOW RANK APPROXIMATION IS THE +ARHUNEN ,OEVE AKA (OTELLING TRANSFORM OF H. .

(99) &IGURE 2ELATIVE CHANNEL POWER

(100) wJ %JGJ J

(101) IN THE TRANSFORM COEbCIENTS FOR THE TWO EXAMPLE CHANNELS. &IGURE "LOCK DIAGRAM OF THE RANKÒ CHANNEL ESTIMATOR. . %STIMATOR COMPLEXITY. 4HE LIMITING FACTOR OF THE RANKÒ ESTIMATORS IS AN ERROR âOOR

(102) SEE 3ECTION 4O ELIMINATE THIS ERROR âOOR UP TO A GIVEN 3.2 WE NEED TO MAKE SURE OUR ESTIMATOR RANK IS LARGE ENOUGH 4HIS PROMPTS AN ANALYSIS OF THE COMPUTATIONAL COMPLEXITY OF THE RANKÒ ESTIMATOR 4HE IMPLEMENTATION WE HAVE CHOSEN IS BASED ON WRITING AS A SUM OF RANK MATRICES

(103) WHICH GIVES US THE EXPRESSION B G O. ` 8. O. B !G pJ TJ T' KR J. J. $. ` 8. O. $. B PJ TJ G KR. %. . J. %. B B IS THE %UCLIDIAN INNER PRODUCT 4HE LINEAR COMBINATION WHERE PJ pJ TJ AND TJ G T' G KR KR J OF O VECTORS OF LENGTH - ALSO REQUIRES O- MULTIPLICATIONS 4HE ESTIMATION THUS REQUIRES OMULTIPLICATIONS AND THE TOTAL NUMBER OF MULTIPLICATIONS PER TONE BECOMES O )N COMPARISON WITH THE FULL ESTIMATOR

(104) WE HAVE MANAGED TO REDUCE THE NUMBER OF MULTIPLICATIONS FROM - TO O PER TONE 4HE SMALLER O IS

(105) THE LOWER THE COMPUTATIONAL COMPLEXITY

(106) BUT THE LARGER. .

(107) THE APPROXIMATION ERROR BECOMES &OLLOWING THE ANALYSIS IN 3ECTION

(108) WE CAN EXPECT A GOOD APPROXIMATION WHEN O IS IN THE RANGE OF SAMPLES IN THE CYCLIC PREçX

(109) WHICH IS USUALLY MUCH SMALLER THAN THE NUMBER OF TONES

(110) - ! LEGITIMATE QUESTION AT THIS POINT IS WHAT HAPPENS FOR A SYSTEM WITH MANY TONES AND MANY SAMPLES IN THE CYCLIC PREçX 4HE NUMBER OF CALCULATIONS PER TONE CAN BE CONSIDERABLE IF A RANKÒ ESTIMATOR IS USED DIRECTLY ON ALL TONES IN THE SYSTEM /NE SOLUTION TO THIS PROBLEM IS A PARTITIONING OF THE TONES INTO REASONABLE SIZED BLOCKS AND

(111) AT A CERTAIN PERFORMANCE LOSS

(112) PERFORM THE ESTIMATION INDEPENDENTLY IN THESE BLOCKS "Y DIVIDING THE CHANNEL ATTENUATIONS INTO * EQUALLY SIZED BLOCKS

(113) THE BANDWIDTH IN EACH BLOCK IS REDUCED BY A FACTOR * 2EFERRING AGAIN TO THE DIMENSION OF THE SPACE OF ESSENTIALLY TIME AND BANDLIMITED SIGNALS ;=

(114) THE EXPECTED NUMBER OF ESSENTIAL BASE VECTORS IS REDUCED FROM + TO +* (ENCE THE COMPLEXITY OF THE ESTIMATOR DECREASES ACCORDINGLY 4O ILLUSTRATE THE IDEA

(115) LET US ASSUME WE HAVE A SYSTEM WITH - TONES AND A + SAMPLE CYCLIC PREçX 4HE UNIFORM CHANNEL CORRELATION BETWEEN THE ATTENUATIONS GL AND GM IN THIS SYSTEM IS

(116) SEE !PPENDIX "

(117) QLM . . Ì {+ L`M `D L`MI {+ -. IF L M IF L M. . 4HIS ONLY DEPENDS ON THE DISTANCE BETWEEN THE TONES

(118) L ` M

(119) AND THE RATIO BETWEEN THE LENGTH OF THE CYCLIC PREçX AND THE NUMBER OF TONES

(120) +- 4HE TONE SYSTEM CAN BE DESCRIBED BY . . . . . . . X 7 G M

(121) X G M 7 THAT IS

(122) AS PARALLEL TONE SYSTEMS

(123) XJ 7J GJ MJ J 7E HAVE THE SAME CHANNEL CORRELATION IN EACH SUBSYSTEM AS WE HAVE IN THE TONE SCENARIO IN THIS PAPER +- "Y ESTIMATING THE CHANNEL ATTENUATIONS GJ IN EACH SUB SYSTEM INDEPENDENTLY

(124) WE NEGLECT THE CORRELATION BETWEEN TONES IN DIdERENT SUB SYSTEMS

(125) BUT OBTAIN THE SAME -3% PERFORMANCE AS IN OUR TONE SCENARIO. .

(126) #HAPTER %STIMATOR PERFORMANCE AND DESIGN 7E PROPOSE A GENERIC LOW RANK FREQUENCY BASED CHANNEL ESTIMATOR

(127) IE THE ESTIMATOR IS DESIGNED FOR çXED

(128) NOMINAL VALUES OF 3.2 AND CHANNEL CORRELATION (ENCE

(129) WE NEED TO ANALYSE HOW THE RANK

(130) CHANNEL CORRELATION AND 3.2 SHOULD BE CHOSEN FOR THIS ESTIMATOR SO THAT IT IS ROBUST TO VARIATIONS IN THE CHANNEL STATISTICS

(131) IE MISMATCH !S A PERFORMANCE MEASURE

(132) WE USE UNCODED SYMBOL ERROR RATE 3%2 FOR 1!- SIGNALLING 4HE 3%2 IN THIS CASE CAN BE CALCULATED FROM THE MEAN SQUARED ERROR -3% WITH THE FORMULAE IN ;=. . 2ANK REDUCTION. N. O. 4HE MEAN SQUARED ERROR

(133) RELATIVE TO THE CHANNEL POWER $ JGJ J

(134) OF THE RANKÒ ESTIMATOR IS MAINLY DETERMINED BY THE CHANNEL POWER CONTAINED IN THE TRANSFORM COEbCIENTS AND CAN BE EXPRESSED

(135) SEE !PPENDIX #

(136) ` ` O8 n -8 LRD O wJ ` pJ . wJ p. - J 2-1 J - JO. . WHERE wJ AND pJ ARE GIVEN BY AND RESPECTIVELY 4HE -3% IS A MONOTONICALLY DECREASING FUNCTION OF 2-1 AND CAN BE BOUNDED FROM BELOW BY THE LAST TERM

(137). ` -8 LRDO wJ v LRD O

(138) - JO. . WHICH IS THE SUM OF THE CHANNEL POWER IN THE TRANSFORM COEbCIENTS NOT USED IN THE ESTIMATE 4HIS -3% âOOR

(139) LRDO WILL GIVE RISE TO A ERROR âOOR IN THE SYMBOL ERROR RATES 4HE ERROR âOOR IS THE MAIN LIMITATION ON THE COMPLEXITY REDUCTION ACHIEVED BY OPTIMAL RANK REDUCTION !S AN ILLUSTRATION

(140) &IGURE DISPLAYS THE 3%2 RELATIVE TO THE CHANNEL VARIANCE

(141) FOR THREE DIdERENT RANKS

(142) AS A FUNCTION OF THE 3.2 4HE RANKS CHOSEN ARE O

(143) AND

(144) AND THE CHANNEL USED IN THE EXAMPLE IS THE SYNCHRONIZED CHANNEL 4HE CORRESPONDING 3%2 âOORS ARE SHOWN AS HORIZONTAL LINES &OR O

(145) THE 3%2 âOOR IS RELATIVELY SMALL

(146) AND THE 3%2 OF THE RANK` ESTIMATOR IS COMPARABLE TO THE ORIGINAL

(147) FULL RANK ESTIMATOR IN THE RANGE TO D" IN 3.2 "Y CHOOSING THE APPROPRIATE RANK ON THE ESTIMATOR

(148) WE CAN ESSENTIALLY AVOID THE IMPACT FROM THE 3%2 âOOR UP TO A GIVEN 3.2 7HEN WE HAVE FULL RANK

(149) O -

(150) NO 3%2 âOOR EXISTS .

(151) &IGURE ,OW RANK ESTIMATOR SYMBOL ERROR RATE AS A FUNCTION OF 3.2

(152) WITH RANKS O

(153) AND #ORRESPONDING 3%2 âOORS SHOWN AS HORIZONTAL LINES 3YNCHRONIZED CHANNEL "ASED ON THE CHANNEL POWERS PRESENTED IN &IGURE

(154) WE SHOW THE CORRESPONDING 3%2 âOORS

(155) RELATIVE TO THE CHANNEL VARIANCE

(156) IN &IGURE !FTER ABOUT RANK` THE 3%2 âOOR DECREASES RAPIDLY 7E ARE THEREFORE ABLE TO OBTAIN A GOOD ESTIMATOR APPROXIMATION WITH A RELATIVELY LOW RANK. &IGURE %STIMATOR 3%2 âOOR AS A FUNCTION OF ESTIMATOR RANK #IRCLES SHOW THE 3%2 âOORS APPEARING IN &IGURE . .

(157) . 3%2 PERFORMANCE UNDER MISMATCH. )N PRACTICE

(158) THE TRUE CHANNEL CORRELATION AND 3.2 ARE NOT KNOWN 4O GET A GENERAL EXPRESSION FOR THE ESTIMATOR 3%2

(159) WE DERIVE IT UNDER THE ASSUMPTION THAT THE ESTIMATOR IS DESIGNED G FOR CORRELATION 1GG AND SIGNAL TO NOISE RATIO 2-1

(160) BUT THE TRUE VALUES ARE 1EGEG AND 2-1

(161) E DENOTES A CHANNEL WITH DIdERENT STATISTICS THAN G 4HIS ALLOWS US TO RESPECTIVELY

(162) WHERE G ANALYSE THIS ESTIMATORÚS SENSITIVITY TO DESIGN ERRORS 5NDER THESE ASSUMPTIONS

(163) THE RELATIVE -3% OF THE RANKÒ ESTIMATE BECOMES

(164) SEE !PPENDIX #

(165) ` ` O8 n -8 LRD O xJ ` pJ G pJ. xJ - J - JO 2-1. . WHERE xJ IS THE J SG DIAGONAL ELEMENT OF 4' 1EGEG 4

(166) CF )T CAN BE INTERPRETED AS THE VARIANCE E UNDER CORRELATION MISMATCH SINCE OF THE TRANSFORMED CHANNEL

(167) 4' G $. |r. sr s' } ' E E 4 G 4 G 4' 1EGEG 4 '. E ARE NO LONGER UNCORRELATED (OWEVER DUE TO )T SHOULD BE NOTED THAT THE ELEMENTS OF 4' G THE FACT THAT THE POWER DELAY PROçLE IS SHORT COMPARED TO THE /&$- SYMBOL

(168) THE çRST O ELEMENTS CAN BE EXPECTED TO CONTAIN MOST OF THE POWER 4HIS PROPERTY WILL ENSURE ONLY A SMALL PERFORMANCE LOSS WHEN THE ESTIMATOR IS DESIGN FOR WRONG CHANNEL STATISTICS )F RANKÒ ESTIMATORS ARE USED IN A REAL SYSTEM

(169) THE SENSITIVITY TO MISMATCH IN BOTH CHANNEL CORRELATION AND 3.2 ARE IMPORTANT 7E WILL SHOW THAT A RANKÒ ESTIMATOR BASED ON THE UNIFORM CHANNEL MODEL AND A NOMINAL 3.2 CAN BE USED AS çXED GENERIC ESTIMATOR WITH ONLY A SMALL LOSS IN AVERAGE PERFORMANCE 7E DIVIDE THE MISMATCH ANALYSIS INTO TWO PARTS çRST WE ANALYSE THE 3%2 WHEN WE HAVE A MISMATCH IN CHANNEL CORRELATION AND LATER WE ANALYSE THE 3%2 WHEN WE HAVE A MISMATCH IN 3.2. . )NCORRECT CHANNEL CORRELATION. G &ROM

(170) WITH NO 3.2 MISMATCH 2-1 2-1

(171) BUT INCORRECT CHANNEL CORRELATION

(172) 1GG 1EGEG

(173) WE OBTAIN THE PERFORMANCE FOR THE CORRELATION MISMATCH CASES 7E COMPARED THE PER FORMANCE OF OUR CHANNEL ESTIMATOR IN TWO MISMATCH SITUATIONS I USING THE A UNIFORM CHANNEL WHEN THE TRUE CHANNEL MODEL WAS THE SYNCHRONOUS CHANNEL AND II USING THE SYNCHRONOUS CHAN NEL WHEN THE TRUE CHANNEL MODEL WAS THE UNIFORM CHANNEL 4HE RESULTING CHANNEL ESTIMATES THAT WERE USED IN THE DETECTION OF THE DATA PRODUCED NO NOTICABLE DIdERENCE IN SYMBOL ERROR RATES Ô LESS THAN D" CHANGE IN EdECTIVE 3.2 FOR AN AVERAGE 3.2 UP TO D" (OWEVER

(174) WHEN BOTH THE CHANNEL 3.2 AND THE CHANNEL CORRELATION MATRIX ARE MISMATCHED

(175) THE NOMINAL DESIGN 3.2 BECOMES MORE IMPORTANT 4HIS CAN BE SEEN IN &IGURE

(176) WHERE WE PRESENT THE RESULTING SYMBOL ERROR RATE FOR RANK ESTIMATORS &OR THE MISMATCHED CASES

(177) MARKED WITH ÚOÚ

(178) THE UNIFORM DESIGN IS MORE ROBUST

(179) IE THE ERROR IN CASE OF MISMATCH IS LOWER 7ITH THE RESTRIC TION THAT THE TRUE CHANNEL HAS A POWER DELAY PROçLE SHORTER THAN THE CYCLIC PREçX

(180) DESIGNING FOR A UNIFORM POWER DELAY PROçLE CAN BE SEEN AS A MINIMAX DESIGN. .

(181) &IGURE -3% FOR CORRECT AND MISMATCHED DESIGN 4HE LATTER IS MARKED WITH CIRCLES o . . )NCORRECT 3.2. &INALLY WE EVALUATE THE SENSITIVITY TO MISMATCH IN DESIGN 3.2 FOR A RANK ESTIMATOR 7HEN THERE IS NO MISMATCH IN CHANNEL CORRELATION

(182) AND NOMINAL 3.2S OF

(183) AND D" ARE USED IN THE DESIGN

(184) THE SENSITIVITY TO 3.2 MISMATCH IS NOT THAT LARGE (OWEVER

(185) IN &IGURE

(186) WE PRESENT THE 3%2 FOR THE SAME RANK ESTIMATORS

(187) BUT WITH THE DIdERENCE THAT THE TRUE CHANNEL CORRELATION IS MISMATCHED WITH THE DESIGN CORRELATION )N THIS SECOND CASE

(188) THERE IS A CLEAR DIdERENCE BETWEEN THE TWO DESIGNS THE HIGHER THE NOMINAL DESIGN 3.2

(189) THE BETTER THE OVERALL PERFORMANCE OF THE ESTIMATOR IN THE RANGE TO D" IN 3.2 )T SHOULD BE NOTED THAT A ,--3% ESTIMATOR DESIGNED FOR A LARGE 3.2 APPROACHES THE ,3 ESTIMATOR. &IGURE 2ANK ESTIMATOR 3%2 WHEN 3.2S OF

(190) AND D" ARE USED IN THE DESIGN 4HE ESTIMATORS ARE DESIGNED FOR INCORRECT CHANNEL CORRELATION. .

(191) #HAPTER 'ENERIC LOW RANK ESTIMATOR )F WE WANT A ROBUST GENERIC CHANNEL ESTIMATOR DESIGN FOR /&$- SYSTEMS

(192) OF THE LOW RANK TYPE

(193) THE ANALYSIS IN THE PREVIOUS SECTION SUGGESTS THE USE OF THE UNIFORM CHANNEL CORRELATION AND A RELATIVELY HIGH 3.2 AS NOMINAL DESIGN PARAMETERS 4HE DESIGN OF SUCH AN ESTIMATOR ONLY REQUIRES KNOWLEDGE ABOUT THE LENGTH OF THE CYCLIC PREçX

(194) THE NUMBER OF TONES IN THE SYSTEM AND THE TARGET RANGE OF 3.2S FOR THE APPLICATION )F THE RECEIVER CANNOT AdORD AN ESTIMATOR THAT INCLUDES TRACKING OF CHANNEL CORRELATION AND 3.2

(195) THIS CHANNEL ESTIMATOR WORKS REASONABLY WELL FOR çXED 3.2 AND CHANNEL CORRELATION. . 0ERFORMANCE GAIN. &OR THE SCENARIO USED IN THIS PAPER

(196) 3EC

(197) WE CHOOSE A RANK ESTIMATOR WITH UNIFORM DESIGN AND 2-1 D" 4HE PERFORMANCE OF THIS ESTIMATOR IS PRESENTED IN &IG

(198) WHERE THE 3%2 FOR THE ,3 ESTIMATE AND KNOWN CHANNEL ARE ALSO SHOWN !S CAN BE SEEN

(199) THE LOW RANK ESTIMATOR IS D" BETTER THAN THE ,3 ESTIMATOR AND LESS THAN D" FROM THE KNOWN CHANNEL. . #OMPARISON TO &)2 çLTERS. !N ALTERNATIVE TO USING LOW RANK ESTIMATORS TO SMOOTH THE CHANNEL ESTIMATES IS TO USE A &)2 çLTER INSTEAD (ENCE WE WILL COMPARE OUR PROPOSED LOW RANK ESTIMATORS TO &)2 çLTERS OF THE SAME COMPLEXITY 4HE &)2 çLTERS ARE O TAPS 7IENER çLTERS ;=

(200) IE O MULTIPLICATIONS PER TONE THAT ARE DESIGNED FOR THE SAME CHANNEL CORRELATION AND 3.2 AS THE LOW RANK ESTIMATORS &IGURE SHOWS THE 3%2 FOR RANKÒ ESTIMATORS IN COMPARISON WITH &)2 çLTERS OF THE SAME COMPUTATIONAL COMPLEXITY 7HEN THE COMPLEXITY IS MULTIPLICATIONS PER TONE ! THE RANKÒ ESTIMATOR HAS ABOUT D" ADVANTAGE IN 3.2 OVER THE &)2 çLTER IN THE RANGE OF 3.2S SHOWN 7HEN THE NUMBER OF CALCULATIONS GOES DOWN TO MULTIPLICATIONS PER TONE " THE 3%2 âOOR OF THE RANKÒ ESTIMATOR BECOMES VISIBLE AND THE &)2 çLTER PERFORMS BETTER AT 3.2S ABOVE D" (OWEVER

(201) IT SHOULD BE NOTED THAT THE PERFORMANCE OF THE LOW RANK ESTIMATORS DEPEND HEAVILY OF THE SIZE OF THE CYCLIC PREçX )F THE CYCLIC PREçX WERE TO BE DECREASED RELATIVE TO THE /&$SYMBOL

(202) THE LOW RANK ESTIMATOR WOULD INCREASE ITS PERFORMANCE 4HIS IS DUE TO THE FACT THAT THE ÞDIMENSIONÞ OF THE CHANNEL WHOSE DURATION IS ASSUMED TO BE SHORTER THAN THE CYCLIC PREçX DECREASES AND CAN THUS BE REPRESENTED WITH FEWER COEbCIENTS /N THE OTHER HAND

(203) IF THE CYCLIC .

(204) &IGURE 3%2 FOR 1!- TRAINING DATA AND A SYNCHRONIZED CHANNEL 4HE GENERIC RANK` ESTIMATOR

(205) DESIGNED FOR A UNIFORM CHANNEL AND D" IN 3.2

(206) IS COMPARED TO THE ,3 ESTIMATOR AND KNOWN CHANNEL AT THE RECEIVER. &IGURE 3%2 COMPARISON BETWEEN THE RANKÒ ESTIMATORS AND &)2 7IENER çLTERS OF THE SAME COMPLEXITY "OTH ESTIMATORS ARE DESIGNED FOR THE UNIFORM CHANNEL AND D" 3.2 ! MULTIPLICATIONS PER TONE AND " MULTIPLICATIONS PER TONE. PREçX INCREASES IN SIZE

(207) MORE COEbCIENTS ARE NEEDED TO AVOID LARGE APPROXIMATION ERRORS (ENCE

(208) WHETHER OR NOT THE LOW RANK ESTIMATOR IS BETTER THAN THE &)2 çLTER DEPENDS ON THE RELATIVE SIZE OF THE CYCLIC PREçX AND THE ALLOWED COMPLEXITY .

(209) . 4HE USE OF TIME CORRELATION. 4HE LOW RANK ESTIMATOR PRESENTED IN THIS PAPER IS BASED ON FREQUENCY CORRELATION ONLY BUT THE TIME CORRELATION OF THE CHANNEL CAN ALSO BE USED 4HE TWO DIMENSIONAL ,--3% ESTIMATOR CAN BE SIMPLIçED USING THE SAME TECHNIQUE WITH RANK REDUCTION AS DESCRIBED HERE (OWEVER

(210) IN ;= IT IS SHOWN THAT SUCH AN ESTIMATOR GIVES AN INFERIOR PERFORMANCE FOR A çXED COMPLEXITY (ENCE

(211) IT SEEMS THAT SEPARATING THE USE OF FREQUENCY AND TIME CORRELATION IS THE MOST EbCIENT WAY OF ESTIMATING THE CHANNEL /THER APPROACHES TO USE THE TIME CORRELATION IS EG TO USE A DECISION DIRECTED SCHEME ;= OR &)2 çLTERS ;

(212) = 4HE FORMER CAN BE USED IN A SLOW FADING ENVIRONMENT

(213) WHERE IT OdERS GOOD PERFORMANCE FOR A MINIMAL COMPLEXITY AND THE LATTER IS PREFERRED IN CASE OF FAST FADING )T IS POSSIBLE TO USE A BANK OF &)2 çLTERS AND CHOOSE THE MOST APPROPRIATE ACCORDING THE ESTIMATED $OPPLER FREQUENCY ;=. .

(214) .

(215) #HAPTER #ONCLUSIONS 7E HAVE INVESTIGATED LOW COMPLEXITY LOW RANK APPROXIMATIONS OF THE ,--3% CHANNEL ESTI MATOR FOR NON SAMPLE SPACED CHANNELS 4HE INVESTIGATION SHOWS THAT AN ESTIMATOR ERROR âOOR

(216) INHERENT IN THE LOW RANK APPROXIMATION

(217) IS THE SIGNIçCANT LIMITATION TO THE ACHIEVED COMPLEXITY REDUCTION 7E SHOWED THAT A GENERIC LOW RANK ESTIMATOR DESIGN

(218) BASED ON THE UNIFORM CHANNEL CORRELATION AND A NOMINAL 3.2

(219) CAN BE USED IN OUR TONE SCENARIO #OMPARED WITH THE FULL ,--3%

(220) THERE IS ONLY A SMALL LOSS IN PERFORMANCE UP TO A 3.2 OF D" BUT A REDUCTION IN COMPLEXITY WITH A FACTOR -O &OR SYSTEMS WITH MORE SUBCHANNELS THIS GAIN IS EVEN LARGER 4HE GENERIC ESTIMATOR DESIGN ONLY REQUIRES KNOWLEDGE ABOUT THE LENGTH OF THE CYCLIC PREçX

(221) THE NUMBER OF TONES IN THE SYSTEM AND THE TARGET RANGE OF 3.2S FOR THE APPLICATION 7E ALSO COMPARED LOW RANK ESTIMATORS TO &)2 çLTERS ACROSS THE TONES 4HE COMPARISON SHOWED THAT AT LOW COMPLEXITIES AND HIGH 3.2S THE &)2 çLTERS IS THE PREFERABLE CHOICE

(222) DUE TO THE ERROR âOOR IN THE LOW RANK APPROXIMATION (OWEVER

(223) IF WE CAN ALLOW UP TO MULTIPLICATIONS PER TONE IN OUR SCENARIO

(224) THE LOW RANK ESTIMATOR IS MORE ADVANTAGEOUS !LSO

(225) THE LOW RANK ESTIMATORS IMPROVE THEIR PERFORMANCE AS THE CYCLIC PREçX DECREASES IN SIZE. .

(226) .

(227) !PPENDIX ! /PTIMAL RANK REDUCTION 4HE OPTIMAL RANK REDUCTION IS FOUND FROM THE CORRELATION MATRICES N. O. B' 1 1GBGKR $ GG GG KR N. O. B G B' 1. 1BGKRBGKR $ G KR GG KR. AND THE 36$. n ( 2-1. `. . 1GBGKR 1BG BG 0#0' KR KR. !. WHERE 0 AND 0 ARE UNITARY MATRICES AND # IS A DIAGONAL MATRIX WITH THE SINGULAR VALUES C w C w a a a w C- ` ON ITS DIAGONAL 4HE BEST LOW RANK ESTIMATOR ;= IS THEN . B 0 G O . #O . . ` GB . 0' 1G. !. . L S GL S. KR. WHERE #O IS THE O b O UPPER LEFT CORNER OF # IE WE EXCLUDE ALL BUT THE O LARGEST SINGULAR n VECTORS )N THIS PAPER WE HAVE 1GBGKR 1GG AND 1BGKRBGKR 1GG 2-1 ( AND WE NOTE THAT THEY ' SHARE THE SAME SINGULAR VECTORS

(228) IE THE ONES OF 1GG 4c4 4HUS

(229) WE MAY EXPRESS ! AS . 4c4. . . n 4 c. ( 4' 2-1. '. . n 4c c. ( 2-1. `. `. . 4' 0#0' `. . n 0 0 4 AND # c c. ( 2-1 4HE RANK O ESTIMATOR ! NOW BECOMES B G. O. 4 . 4. #O #O . . . 4. '. . . . n 4 c. ( 4' 2-1 `. n c. ( 2-1. . `. B 4 4 G KR '. . B G KR. aO . B 4' G KR.

(230) WHERE aO IS THE O b O UPPER LEFT CORNER OF . `. n ac c. ( 2-1. . . w w- ` CH@F aaa n n w 2-1 w- ` 2-1. .OTE THAT 0 0 SINCE WE ARE ESTIMATING THE SAME TONES AS WE ARE OBSERVING IE SMOOTHING AND AN EIGENVALUE DECOMPOSITION COULD BE USED TO ACHIEVE OPTIMAL RANK REDUCTION )N THE GENERAL CASE WHEN EG PILOT SYMBOL ASSISTED MODULATION ;= IS USED AND THERE ARE KNOWN SYMBOLS PILOTS ON ONLY A PART OF THE SUBCHANNELS

(231) WE HAVE 0 0 SINCE 1GBGKR AND 1BGKRBGKR DONÚT SHARE THE SAME SINGULAR VECTORS THE MATRICES ARE NOT EVEN OF THE SAME SIZE (ENCE

(232) THE MORE GENERAL 36$ MUST BE USED WHICH MOTIVATES THE NOMENCLATURE IN THIS ARTICLE. .

(233) !PPENDIX " #HANNEL CORRELATION MATRICES 5SING THE CHANNEL MODEL IN

(234) THE ATTENUATION ON TONE J BECOMES GJ . ` 8. ,. J. mH DÌ { - ~H

(235). H. AND THE CORRELATION MATRIX FOR THE ATTENUATION VECTOR

(236) G

(237) N. O. 1GG $ GG' :QLM < CAN BE EXPRESSED AS ~J ÚS INDEPENDENT QLM . :. aaa. : ,9 `. ` : 8. ,. H. J . E~J ~J . , ` 8. t ~ D` . L ` M H C~ C~, `. I {~. H. H. L`M E~H ~H t ~H DÌ {~H - C~H

(238). ". WHERE t~ IS THE MULTI PATH INTENSITY PROçLE AND E~J ~J IS THE PROBABILITY DENSITY FUNCTION OF ~J 4HE CORRELATION MATRICES OF THE THREE CHANNELS USED IN THIS PAPER ARE CALCULATED BELOW q 3YNCHRONIZED CHANNEL 4HE PROBABILITY DISTRIBUTIONS FOR THE DELAYS ARE E~ ~ p ~

(239) + IF ~H : +< E~H ~H H ,

(240) OTHERWISE AND THE POWER DELAY PROçLE IS t ~ " D`~ ~QLR 3UBSTITUTING IN "

(241) AND NORMALIZING QJJ TO UNITY

(242) GIVES US QLM . +. ,. t. `. I { L`M ~QLR - . ` D` ~QLR +. r. {I. + s. + , ` ~QLR ` D` ~QLR . L`M -. u. .

(243) q 5NIFORM CHANNEL 4HE PROBABILITY DISTRIBUTIONS FOR THE DELAYS ARE . E~H ~H . + IF ~H : +<

(244) H ,

(245) OTHERWISE. AND THE POWER DELAY PROçLE IS CONSTANT t ~ " 3UBSTITUTED IN "

(246) AND NORMALIZING QJJ TO UNITY

(247) GIVES US IF L M `M ` I {+ LQLM `D L`M IF L M I {+ -. .

(248) !PPENDIX # %STIMATOR MEAN SQUARED ERROR )N THIS APPENDIX WE DERIVE THE -3% OF THE RANKÒ ESTIMATOR IN 7E ALSO PRESENT THE -3% âOOR

(249) WHICH BOUNDS THE ACHIEVABLE -3% FROM BELOW IN LOW RANK APPROXIMATIONS OF THE ,--3% ESTIMATOR 4O GET A GENERAL EXPRESSION FOR THE MEAN SQUARED ERROR FOR THE RANKÒ APPROXIMATION OF THE ,--3% ESTIMATOR

(250) WE ASSUME THAT THE ESTIMATOR HAS BEEN DESIGNED E HAS THE FOR CHANNEL CORRELATION 1GG AND SIGNAL TO NOISE RATIO 2-1

(251) BUT THE REAL CHANNEL G G CORRELATION F 1GG AND THE REAL SIGNAL TO NOISE RATIO IS 2-1 &ROM AND

(252) WE HAVE n ` B E E WHERE THE NOISE TERM M E 7 M HAS THE AUTOCOVARIANCE MATRIX 1E GKR G M

(253) 2-1 ( 4HE ME M E B ESTIMATION ERROR DO G ` GO OF THE RANKÒ ESTIMATOR IS . . DO 4 ( `. aO . . E `4 4' G. . aO . . E

(254) 4' M. #. AND THE MEAN SQUARED ERROR IS LRD O . N O 3Q@BD $ DO D'

(255) O -. #. 4O SIMPLIFY THE EXPRESSION WE USE THAT E AND M E ARE UNCORRELATED

(256) HENCE THE CROSS TERMS ARE CANCELLED IN THE EXPECTATION q G r. q 3Q@BD 4 4' 3Q@BD ! ;=. s. 3Q@BD. IF 4 IS A UNITARY MATRIX

(257) AND 3Q@BD . ! 3Q@BD. 0. q 3Q@BD # # J @JJ CJ WHEN # IS A DIAGONAL MATRIX WITH THE ELEMENTS CJ ON ITS DIAGONAL AND NOT NECESSARILY A DIAGONAL MATRIX HAS DIAGONAL ELEMENTS @JJ 5SING # IN #

(258) THE MEAN SQUARED ERROR BECOMES . . 3Q@BD 4 ( ` LRD O . 4. aO . . . aO . 4' 1EMEM4. . . 4' 1EGEG4. aO . '. . . 4' . (`. . aO . '. 4'.

(259) -. . ` 8. O. J . xJ ` pJ . ` 8 O. . . -. J. ` 8. -. . xJ !. JO. ` n O8 pJ G - J 2-1. ` n -8 xJ ` pJ G pJ. xJ - JO 2-1 . #. WHERE xJ IS THE CHANNEL POWER IN THE JTH TRANSFORM COEbCIENT

(260) IE

(261) THE JTH DIAGONAL ELEMENT F 4 4HE -3% CAN BE LOWER BOUNDED

(262) LRD O w LRD O BY WHAT WE CALL OF THE MATRIX 4' 1 GG THE -3% âOOR ` -8 LRD O xJ - JO r. s. )F THERE IS NO MISMATCH IN 2-1 OR CHANNEL CORRELATION

(263) WE HAVE xJ CH@F 4' 1GG 4 wJ G 2-1 AND THE -3% BECOMES AND 2-1 ` ` O8 n -8 LRD O wJ ` pJ . p. wJ - J 2-1 J - JO. .

(264) "IBLIOGRAPHY ;= *' 0ROAKIS $IGITAL COMMUNICATIONS 0RENTICE (ALL

(265) RD EDITION

(266) ;= 2ADIO BROADCASTING SYSTEMS $IGITAL !UDIO "ROADCASTING $!" TO MOBILE

(267) PORTABLE AND çXED RECEIVERS %43

(268) %43) Ô %UROPEAN 4ELECOMMUNICATIONS 3TANDARDS )NSTITUTE

(269) 6ALBONNE

(270) &RANCE

(271) &EBRUARY ;= 0ETER (¶HER 4#- ON FREQUENCY SELECTIVE LAND MOBILE FADING CHANNELS )N 0ROC 4IRRENIA )NT 7ORKSHOP $IGITAL #OMMUN

(272) 4IRRENIA

(273) )TALY

(274) 3EPTEMBER ;= 3ARAH +ATE 7ILSON

(275) 2 %LLEN +HAYATA

(276) AND *OHN - #IOb 1!- MODULATION WITH ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING IN A 2AYLEIGH FADING ENVIRONMENT )N 0ROC )%%% 6EHIC 4ECHNOL #ONF

(277) VOLUME

(278) PAGES Ô

(279) 3TOCKHOLM

(280) 3WEDEN

(281) *UNE ;= !HMAD #HINI -ULTICARRIER MODULATION IN FREQUENCY SELECTIVE FADING CHANNELS 0H$ THESIS

(282) #ARLETON 5NIVERSITY

(283) /TTAWA

(284) #ANADA

(285) ;= *OHN - #IOb 0ERSONAL COMMUNICATION

(286) ;= *AN *AAP VAN DE "EEK

(287) /VE %DFORS

(288) -AGNUS 3ANDELL

(289) 3ARAH +ATE 7ILSON

(290) AND 0ER /LA "¶RJESSON /N CHANNEL ESTIMATION IN /&$- SYSTEMS )N 0ROC )%%% 6EHIC 4ECHNOL #ONF

(291) VOLUME

(292) PAGES Ô

(293) #HICAGO

(294) ),

(295) *ULY ;= ! 0ELED AND ! 2UIZ &REQUENCY DOMAIN DATA TRANSMISSION USING REDUCED COMPUTATIONAL COMPLEXITY ALGORITHMS )N 0ROC )%%% )NT #ONF !COUST

(296) 3PEECH

(297) 3IGNAL 0ROCESSING

(298) PAGES Ô

(299) $ENVER

(300) #/

(301) ;= 0ETER (¶HER ! STATISTICAL DISCRETE TIME MODEL FOR THE 73353 MULTIPATH CHANNEL )%%% 4RANS #OMMUN

(302) Ô

(303) .OVEMBER ;= ,OUIS , 3CHARF 3TATISTICAL SIGNAL PROCESSING $ETECTION

(304) ESTIMATION

(305) AND TIME SERIES ANALY SIS !DDISON 7ESLEY

(306) ;= /VE %DFORS

(307) -AGNUS 3ANDELL

(308) *AN *AAP VAN DE "EEK

(309) 3ARAH +ATE 7ILSON

(310) AND 0ER /LA "¶R JESSON !NALYSIS OF $&4 BASED CHANNEL ESTIMATORS FOR /&$- 2ESEARCH 2EPORT 45,%!

(311) $IV OF 3IGNAL 0ROCESSING

(312) ,ULE¥ 5NIVERSITY OF 4ECHNOLOGY

(313) 3EPTEMBER ;= ( * ,ANDAU AND ( / 0OLLAK 0ROLATE SPHERIODAL WAVE FUNCTIONS

(314) &OURIER ANALYSIS AND UNCERTAINTY Ô ))) 4HE DIMENSION OF THE SPACE OF ESSENTIALLY TIME AND BAND LIMITED SIGNALS "ELL 3YSTEM 4ECH *

(315)

(316) .

(317) ;= 3ARAH +ATE 7ILSON $IGITAL AUDIO BROADCASTING IN A FADING AND DISPERSIVE CHANNEL 0H$ THESIS

(318) 3TANFORD 5NIVERSITY

(319) #!

(320) !UGUST ;= -AGNUS 3ANDELL AND /VE %DFORS ! COMPARATIVE STUDY OF PILOT BASED CHANNEL ESTIMATORS FOR WIRELESS /&$- 2ESEARCH 2EPORT 45,%!

(321) $IV OF 3IGNAL 0ROCESSING

(322) ,ULE¥ 5NIVERSITY OF 4ECHNOLOGY

(323) 3EPTEMBER ;= *AMES + #AVERS !N ANALYSIS OF PILOT SYMBOL ASSISTED MODULATION FOR 2AYLEIGH FADING CHANNELS )%%% 4RANS 6EHIC 4ECHNOL

(324) Ô

(325) .OVEMBER ;= (ENRY ,I AND *AMES + #AVERS !N ADAPTIVE çLTERING TECHNIQUE FOR PILOT AIDED TRANSMISSION SYSTEMS )%%% 4RANS 6EHIC 4ECHNOL

(326) Ô

(327) !UGUST ;= 'ILBERT 3TRANG ,INEAR !LGEBRA AND )TS !PPLICATIONS !CADEMIC 0RESS

(328) ND EDITION

(329) . .

(330)

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