ELEC E7210: Communication Theory. Lecture 4: Equalization

Transcription

1 ELEC E7210: Communication Theory Lecture 4: Equalization

2 Equalization Delay sprea ISI irreucible error floor if the symbol time is on the same orer as the rms elay sprea. DF: Equalization a receiver signal processing metho aiming to alleviate the ISI problem cause by elay sprea.

3 Equalization issues (1) igh ata rate applications are more sensitive to elay sprea equalization is one of the most challenging issues in such applications Both signal an noise pass through the equalizer esign must balance ISI mitigation an noise enhancement To main stages: training (learn the frequency (impulse) response of the raio channel) an tracing (upating the estimate of the channel frequency (impulse) response ) challenging for rapily varie channels can be implemente at baseban, carrier frequency or intermeiate frequency often are implemente igitally

4 Equalization (1) Aim: to mitigate effects of the channel Can be performe in TD or FD Types Equalizer Nonlinear Linear DFE ML Symbol Detector MLSE Structures Transversal Lattice Transversal Lattice Transversal Channel Estimator

5 Analog equalizer Noise enhancement ( f ) n(t) eq ( f ) Channel Equalizer ' s( t) n ( t) Demo. f eq 1/ f We observe in the FD f S Y r ( f ) N f

6 Equalizer structures Typically transversal or lattice structure Commonly are implemente igitally Pulse shape g(t) ( t) g t T RF front en s c(t) ISI channel (t) RF front en Matche filter g ( t) y(t) Equalizer ˆ Decision evice ˆ - + Tap upating Equivalent representation

7 ISI free transmission(1) Equivalent impulse response Discrete version t g t c t g t h ) ( ) ( ) ( g S g t T t h t t h t y ) ( ) ( n n n g nt t n n h h n n h t y y s ISI ata esire 0 ) (

8 ISI free transmission(2) ISI free if h( n ) 0 n i.e. Let 0 h( ) h ( f ) FT h( t) The concept of the fole spectrum is introuce ( f ) 1 T s n f n T s ISI free, iff the fole spectrum is flat, i.e. f h0

9 Linear equalizers N tap transversal filter, the tap number balance beteen the accuracy on the one han an complexity an elay on the other han ( z) ZF an MMSE equalizers: ZF: Y ( z) D z eq N 1 i0 i z i z N z 1 z ZF z g

10 MMSE Equalizer(1) i The eights are chosen to minimize E ˆ 2 ˆ N 1 i0 y i i For hite noise a stanar Wiener filtering problem, but noise is colore ith the poer spectrum N G 1 z 0 m / 2 yn Noise hitener 1 / G m 1 / z vn ˆ eq z ˆ z eq

11 MMSE Equalizer(2) The equalizer output The graient 1 0 ˆ N i T i i v v Re 2 Re ˆ T T E E E E J v vv v vv 0 J T N E E J J J v vv

12 MMSE Equalizer(3) Setting J 0 yiels opt 1 Evv Ev J min 2 1 E vv E v E v

13 For an equalizer of infinite length MMSE Equalizer(4) i i hj i N j i g j 0 Taing z transform ˆ eq z z N G 1 z 0 / We obtain ˆ eq z G 1/ z z N0 eq z 1 z N0

14 Nonlinear Equalizers: DFE (1) y(t) y Feeforar filter W(z) + - ˆ Decision evice ˆ Feebac filter V(z) ˆ 0 N i y i N i 2 i v ˆ i 1 1 i

15 DFE (2) Typical criteria for selecting the coefficients: ZF (removes all ISI) MMSE (minimizes expecte MSE beteen the original symbol an DFE output Drabac: is characterize by error propagation that cannot be improve by channel coing since the feebac path operates on coe channel symbols before ecoing seriously egraes error rate performance on channels ith lo SNR

16 Other equalization techniques MLSE is optimal but implementation complexity is very high sub optimal algorithms (reuce the number of surviving sequences in the Viterbi algorithm or reuce the number of symbols spanne by ISI via pre processing or ecision feebac in the Viterbi etector) Can be use turbo ecoing principle turbo equalizers. It iterates beteen the MAP equalizer (computes APP of the transmitte symbol given past channel outputs) an ecoer (computes the LLR associate ith the transmitte symbol given past channel outputs) to etermine the transmitte symbol. The APP an LLR represent the soft information exchange beteen the equalizer an ecoer in turbo iterations

17 Aaptive equalizers: training an tracing (1) Equalizer esign requires nolege about the channel impulse (frequency) response Generally the channel response is time variant the system must perioically estimate the channel (base on a training (pilot) sequence that is non at the both Tx an Rx) an upate the equalizer coefficients perioically.this process equalizer training or aaptive equalization. The equalizer can also use the etecte ata to ajust its coefficients. This process equalizer tracing. Blin equalizers o not use training; they ajust their coefficients by using the etecte ata an possibly channel statistics.

18 Aaptive equalizers: training an tracing (1) Equalizer esign requires nolege about the channel impulse (frequency) response Generally the channel response is time variant the system must perioically estimate the channel (base on a training (pilot) sequence that is non at the both Tx an Rx) an upate the equalizer coefficients perioically.this process equalizer training or aaptive equalization. The equalizer can also use the etecte ata to ajust its coefficients. This process equalizer tracing. Blin equalizers o not use training; they ajust their coefficients by using the etecte ata an possibly channel statistics.

19 Aaptive equalizers: training an tracing (3) Matrix inversion is require To reuce complexity, LMS algorithm is applie N N ˆ y y T 1 N 0 The choice of ictates the convergence spee an stability of the algorithm. For goo performance of the LMS algorithm, is typically small an convergence is typically slo. 1...

20 Aaptive equalizers: training an tracing (4) The number of taps of DF is proportional to the elay sprea Aaptation algorithms: length an perioicity of the training sequence affects the spectral efficiency Aaption must be possible at the highest Doppler frequency The higher complexity is, the higher spee of convergence of the aaptive algorithm is

21 Aaptive equalizers: training an tracing Comparison of algorithms (5) Number of Algorithms Multiply Convergence Avantages Disavantages (for DFE) Operations Least Mean 2N + 1 ~10-100N Lo computational Slo convergence, Square (LMS) complexity epens on channel Kalman 2.5N N ~N Fast convergence, igh Recursive Least goo tracing ability computational Squares (RLS) complexity Fast Kalman 20N + 5 ~N Fast convergence an goo tracing Coul be unstable

22 Frequency omain equalization Can be applie to both single an multi carrier signals Assumes transformation of the receive signal into the frequency omain (DFT) In the time omain r( t) h( t) s( t) Cyclic prefix : linear convolution circular convolution R S N ~ R R 2 ˆ ˆ