Infinite Length MMSE Decision Feedback Equalization
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1 Infnte Lengt SE ecson Feedbac Equalzaton FE N * * Y F Z ' Z SS ˆ Y Q N b...
2 Infnte-Lengt ecson Feedbac Equalzer as reoval ^ Y Z Feedforward Flter Feedbac Flter - Input to Slcer - Z Y Assung prevous decsons are correct Error Sequence Z Y E Z Y ^
3 SE-FE revous ecsons are assued correct FFF sapes ISI nto a causal part post-cursor ISI tat can be cancelled by te strctly causal FF tat can be cancelled by te strctly causal FF Error Sequence Analyss Y Z E Y Y Y
4 Optu Flter Coeffcents p Ortogonalty rncple ] [ Y E E YY Y R R R LE SE YY Y R R Q F
5 ole-zero Interpretaton of FE Under deal decsons assupton, transfer functon of FE feedbac loop s / Hence, overall transfer functon of FE s /. erefore, n FE, te flters collaborate to syntesze a pole-zero approxaton of te cannel nverse wc s ore accurate as ore degrees of freedo tan a sngle-flter as n LE 5
6 Error Auto-Correlaton ] [ E E E R ee ] [ ee were Y E E Y LE SE LE SE LE SE *, R n FE SE ee, Q F Intutvely : we sould coose te feedbac flter to wten te error sequence so tat te nput to te slcer s te desred nforaton sybol + wte nose
7 7
8 FE Optzaton Spectral Factorzaton Q F s postve real nuber s canoncal causal, R ee, FESE varance of error onc, r ee, nnu pase n n n Snce / s also a onc polynoal
9 FE Optzaton p wt equalty ff n FE SE Q causal non F Q Note te Feedforward flter coeffcents are not te sae as te LE coeffcents
10 FE Optzaton An Alternatve forula for ln Q e j F ln d e ln j e j proof setc : wrte as rato of products second ter st of order pole - zero sectons etals soon! d
11 FE Optzaton p j ln ln relaton use and d e Q F j j d e Q ln F Q e F FE SE FE SE E SEFE
12 Anoter forula for Zero order ter * Q F g F F g F g
13 Specal Case: Q ISI F Q For no ISI, bas!! F F FE SE F free ISIs snce cannel, F FE U SE S free ISI s
14 FE as Analyss y Y E Z Uncorrelated wt N Q nose F nose nose F nose
15 FE as Analyss e bas ter factor tat ultples s [ ] * * F SEFE SEFE SEFE, U SEFE F SEFE Specal Case ZF FE, set Q ; ZF-FE transfer functon
16 ZF-FE * Q Output of Feedforward Flter : * Q FFF converts cannel Q nto a canoncal flter ISI ll d b t f db flt! q j wose ISI s cancelled by te feedbac flter! ln e d e Q n n FE ZF j not based s FE ZF F FE ZF
17 ZF-FE Feedforward Flter Acton Cannel Ipulse Response Feedforward Flter Ipulse Response - Equvalent Cannel at Feedforward Flter Output 7
18 SE-FE Exaple Note : we get te sae results for =+.9.9 snce we assue analog atced flter so Q F Q γ noncausal realzed wt delay would be te sae! Feedforward Flter Ipulse Response
19 FE Exaple Cont d. 633 Canoncal feedbac tap gven by SEFE n SEFE SE FE, U Loss fro F 8.4.6d d c. f. 4.3d for SE - LE F
20 ZF-FE Exaple Q * * / Loss of.8 ZF FE d.6d fro F One feedbac tap of -.9 etaled roof for Salz Forula ans to forer EE6353 student Saab Sanaye not a recoended bed-te readng!: log Q e Q j F F g g d log * log g e j g e j d
21 Salz Forula R.H.S log j j de γ logge g e πj e j j Contour Integrals! log γ log gg πj d s te unt dsc d logγ logg πj πj logg d Z - d dz Z and - - ZZ
22 Salz Forula Hence : RHS R.H.S log γ j d log g log g πj z dz z z Snce g and g z are bot n pase ence tey are bot analytc R.H.S log log g j log g j but g g ; ence log log Q e j F d
23 nu-ase Cannels Consder -ray ultpat cannel c c were s dfferental pat delay n sybol perods. e zeros of satsfy te relaton / nu pase c c sorter delay pat as larger agntude Effects of fadng, sadowng, and reflectons cause cannel to alternate between nu and non-nu pase nu-pase cannels are easer to equalze! c c c 3
24 Zero-Forcng FE Zero Forcng FE Specal case of SE-FE by lettng Specal case of SE FE by lettng Spectral factorzaton * * * * Q Feedbac flter s and feedforward flter s / * * / Infnte-lengt ZF-FE s unbased For nu-pase cannels : Feedbac flter s 3 Cobned atced flter and feedforward flter s a scalar gan! * * * * / * * * * *
25 SE-FE vs. ZF-FE 5 ecson ont =+.9^ F Infnte Lengt SE FE Infnte Lengt ZF FE SE-FE s superor to ZF-FE at low ot structures becoe dentcal at g At very low were nose donates ISI, SE-FE converges to a atced flter Input d
26 FE Error ropagaton ost analyses assue correct past decsons for tractablty accurate at g A sngle decson error n FE results n ncorrect estate of post-cursor ISI possbly causng future decson errors Long feedbac flter exacerbates error propagaton On cannels w/ spectral nulls, te perforance advantage of FE over LE far outwegs effects of error propagaton recodng tecnque s used to elnate error propagaton wen cannel s nown at transtter
27 recodng Idea : ove FE feedbac secton to transtter were no decson errors occur Cauton: sple ovng of feedbac loop / to transtter ncreases transtted power degrades FE Soluton : Use odulo artetc to lt te ncrease n transtted power proposed by olnson & Harasa
28 recodng Used to elnate error propagaton n FE by ovng feedbac flter to transtter Requres cannel nowledge at transtter Consder te ZF-FE were and were Q recoded Input Sequence Y Q N 8
29 recodng Analyss g y Feedforward Flter Output Z Y N Q N 9
30 recodng Analyss g y N were N R were Q N R Q n wte! n n 3
31 olnson Harasa recodng roble: ranst ower s boosted Soluton: odulo Operator t d t t d d t s a real nuber wle and d are ntegers unforly-dstrbuted between and Ex :d, 4 4 t t 4 t 8 8 denotes largest nteger less tan or equal to Sawtoot functon t
32 ropertes of Γ x p Y Y recodng at Input Y Y were Input to odulo operator b Output of odulo operator 3 b were
33 recoded ZF-FE Input recoded Output wt Cannel N Q Y d d ZF FE F df d Flt O t t N Y Z * * recoded ZF-FE Feedforward Flter Output N N 33 n b Z
34 After odulo operaton at FFF output n b Z n b b a n b b a n 34 n
35 were a follows fro n ' n sgnal plus nose ' wat s te pdf of n a b a b ranst ower : a a b b Orgnal A sgnal :? - d d d unfor n, d ower ncrease s as 35
36 recodng Suary y ranstter x odulo Operator x odulo Operator x odulo - SE FE SE FE, U x ' odulo o ec. Recever ^ x d x x x d d For -ary constellaton wt dstance d. x denotes largest nteger less tan or equal to x For d nput, precoder output s d and unforly dstrbuted over d, d resultng n slgt power ncrease -
37 recodng Suary Elnates FE error propagaton p Allows us to cobne FE wt codng scees unle conventonal FE wc requres nstantaneous relable decsons wc are only avalable after decoder delay! Requres perfect nowledge of feedbac flter pulse response possble wt te-nvarant nvarant or slowly te varyng cannels. Estated at recever and sent bac va reverse cannel at expense of rate loss recodng slgtly ncreases transtted power by factor of /- for QA
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