LINEAR EQUALIZERS & NONLINEAR EQUALIZERS. Prepared by Deepa.T, Asst.Prof. /TCE

LINEAR EQUALIZERS & NONLINEAR EQUALIZERS Prepared by Deepa.T, Asst.Prof. /TCE

Eqalzers The goal of eqalzers s to elmate tersymbol terferece (ISI) ad the addtve ose as mch as possble. Itersymbol terferece(isi) arses becase of the spreadg of a trasmtted plse de to the dspersve atre of the chael, whch reslts overlap of adjacet plses. I Fg there s a for level plse ampltde modlated sgal (PAM), x(t). Ths sgal s trasmtted throgh the chael wth mplse respose h(t). The ose (t) s added. The receved sgal r(t) s a dstorted sgal.

Eqalzato sg Classfcato of Eqalzato MLSE (Maxmm lkelhood seqece estmato) Flterg Trasversal flterg Zero forcg eqalzer Mmm mea sqare error (MSE) eqalzer Decso feedback Usg the past decsos to remove the ISI cotrbted by them Adaptve eqalzer Lectre 6 3

Categores of Eqalzato Eqalzers are sed to overcome the egatve effects of the chael. I geeral, eqalzato s parttoed to two broad categores;. Maxmm lkelhood seqece estmato (MLSE) whch etals makg measremet of chael mplse respose ad the provdg a meas for adjstg the recever to the trasmsso evromet. (Example: Vterb eqalzato). Eqalzato wth flters, ses flters to compasate the dstorted plses. The geeral chael ad eqalzer par s show Fgre..

Depedg o the tme atre These type of eqalzers ca be groped as preset or adaptve eqalzers. Preset eqalzers assme that the chael s tme varat ad try to fd H(f) ad desg eqalzer depedg o H(f). The examples of these ADAPTIVE EQUALIZERS are zero forcg eqalzer, mmm mea sqare error eqalzer, ad desco feedback eqalzer. Adaptve eqalzers assme chael s tme varyg chael ad try to desg eqalzer flter whose flter coeffcets are varyg tme accordg to the chage of chael, ad try to elmate ISI ad addtve ose at each tme. The mplct assmpto of adaptve eqalzers s that the chael s varyg slowly.

Eqalzato Step waveform to sample trasformato Demodlate & Sample Step decso makg Detect r(t) Freqecy dow coverso Recevg flter Eqalzg flter z(t ) Threshold comparso mˆ For badpass sgals Compesato for chael dced ISI Receved waveform Basebad plse (possbly dstored) Basebad plse Sample (test statstc) Lectre 6 6

Eqalzato ISI de to flterg effect of the commcatos chael (e.g. wreless chaels) Chaels behave lke bad lmted flters H c ( f ) H c ( f ) e jθ ( c f ) No costat ampltde Ampltde dstorto No lear phase Phase dstorto Lectre 6 7

Eqalzato Techqes Fg.3 Classfcato of eqalzers

Eqalzer Techqes Lear trasversal eqalzer (LTE, made p of tapped delay les as show Fg.4) Fg.4 Basc lear trasversal eqalzer strctre Fte mplse respose (FIR) flter (see Fg.5) Ifte mplse respose (IIR) flter (see Fg.5)

Eqalzato by trasversal flterg Trasversal flter: x(t) A weghted tap delayed le that redces the effect of ISI by proper adjstmet of the flter taps. N z( t) c x( t τ ) N,..., N k N,..., N N τ τ τ τ c N cn + cn cn z(t) Coeff. adjstmet Lectre 6 0

Trasversal eqalzg flter Zero forcg eqalzer: The flter taps are adjsted sch that the eqalzer otpt s forced to be zero at N sample pots o each sde: Adjst { c } N N z( k) 0 k k 0 ±,..., ± N Mea Sqare Error (MSE) eqalzer: The flter taps are adjsted sch that the MSE of ISI ad ose power at the eqalzer otpt s mmzed. Adjst { c } N N m E [ ] ( z( kt ) a ) k Lectre 6

Lear vs. Nolear Eqalzato Techqes Two geeral categores lear ad olear eqalzato Lear No feedback path to adapt the eqalzer, the eqalzato s lear No lear fed back to chage the sbseqet otpts of the eqalzer, the eqalzato s olear

LINEAR EQUALIZERS

Zero forcg Lear Eqalzers Desg E(z) so that ISI s totally removed. Mmm mea sqare error (MMSE) Desg E(z) to mmze the mea sqare error (MSE) MSE k ε k ( c ) k cˆ k k 4

Strctre of a Lear Trasversal Eqalzer [5] dˆ E k N N C * y k T N π ω π j t T F( e ) + N [ ] π e() T o dω F(e j ωt ) :freqecy respose of the chael o N o :ose spectral desty

Strctre of a Lattce Eqalzer [6 7] Fg.7 The strctre of a Lattce Eqalzer

Characterstcs of Lattce Flter Advatages Nmercal stablty Faster covergece Uqe strctre allows the dyamc assgmet of the most effectve legth Dsadvatages The strctre s more complcated

Lear Eqalzers System modelg Zero forcg solto Example K ad {0, 0., 0.9, 0.3, 0.}. Calclate {e }. K K e ĉ + K K K K c c e e K K K K ˆ ˆ ) (... 0 0... 0 0 0 0... 0 ) ( ) (... 0 0 ) ( M M M M 8 0 0, 0, ˆ k k c k

Lear Eqalzers Zero forcg eqalzer Nose ehacemet 9

MMSE Lear Eqalzers 0

[ ] ( ) [ ] [ ] [ ] [ ] [ ] [ ] [ ] R p c 0 e e ee e e e + + T T T H T T T T E E c E E c E c E c c E E c c c * * * * * * *, 0 : Let, ˆ ε ε ε Lear Eqalzers Mmm Mea Sqare Error (MMSE) Solto e [ H ] [ H c] Example K 3, {0.00, 0.07, 0.749,.0000, 0.67, 0.0558, 0.008}. Calclate {e } ad the calclate c ±0, ±, ±,, ±6. Col [0.00, 0.07, 0.749,.0000, 0.67, 0.0558, 0.008 zeros(, 6)]; Row [0.00, zeros(, 6)]; U toepltz(c, Row); R U'*U; P U'*c; %c [0 0 0 0 0 0] ; e [ 0.06 0.008 0.659 0.9495 0.38 0.0670 0.069];

Lear Eqalzers Algorthms To redce the comptato complexty of Least mea sqare algorthm (LMS) μ determes covergece ad resdal errors Recrsve least sqares algorthm (RLS) λ determes the trackg ablty of the RLS eqalzers. ( ) ) (,, * * T T T c e k e e R k R R R R k + + λ λ + + ˆ ˆ ˆ ˆ μ e e e R p p R e (~ (K+) 3 operatos!)

NON LINEAR EQUALIZERS

Nolear Eqalzato Used applcatos where the chael dstroto s too severe Three effectve methods [6] Decso Feedback Eqalzato (DFE) Maxmm Lkelhood Symbol Detecto Maxmm Lkelhood Seqece Estmator (MLSE)

Nolear Eqalzato DFE Basc dea : oce a formato symbol has bee detected ad decded po, the ISI that t dces o ftre symbols ca be estmated ad sbstracted ot before detecto of sbseqet symbols Ca be realzed ether the drect trasversal form (see Fg.8) or N N 3 as a dˆ C y + F d lattce flter k N * k T π m π π k [ ] e() exp{ T o l[ ] dω} E jωt T F( e N ) + N o

Nolear Eqalzer DFE Fg.8 Decso feedback eqalzer (DFE)

Nolear Eqalzato DFE Predctve DFE (proposed by Belfore ad Park, [8]) Cossts of a FFF ad a FBF, the latter s called a ose predctor ( see Fg.9 ) Predctve DFE performs as well as covetoal DFE as the lmt the mber of taps FFF ad the FBF approach fty The FBF predctve DFE ca also be realzed as a lattce strctre [9]. The RLS algorthm ca be sed to yeld fast covergece

Nolear Eqalzer DFE Fg.9 Predctve decso feedback eqalzer

Nolear Eqalzers Decso feedback eqalzer (DFE) 9

DFE Eqalzer Fg.5 Tapped delay le flter wth both feedforward ad feedback taps

Nolear Eqalzato MLSE MLSE tests all possble data seqeces (rather tha decodg each receved symbol by tself ), ad chooses the data seqece wth the maxmm probablty as the otpt Usally has a large comptatoal reqremet Frst proposed by Forey [0] sg a basc MLSE estmator strctre ad mplemetg t wth the Vterb algorthm The block dagram of MLSE recever (see Fg.0 )

Zero forcg DFE Nolear Eqalzers MMSE DFE 3

Nolear Eqalzers Maxmm lkelhood seqece estmato ( ) { } + N L N L N L c f c MLSE c f pdf c f 0 0 / 0 arg m ˆ exp ) ; ( σ πσ f c 33

Nolear Eqalzers Maxmm lkelhood seqece estmato Example 34

Nolear Eqalzers Maxmm lkelhood seqece estmato 35

Nolear Eqalzer MLSE Fg.0 The strctre of a maxmm lkelhood seqece eqalzer(mlse) wth a adaptve matched flter MLSE reqres kowledge of the chael characterstcs order to compte the matrcs for makg decsos MLSE also reqres kowledge of the statstcal dstrbto of the ose corrptg the sgal