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Equalization

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# Equalization

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## Equalization

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1. Equalization • Compensates for ISI created by multipath within the time dispersive channels • Condition for ISI • Modulation BW > coherence bandwidth • Compensates the average range of channel amplitude and delay characteristics • Must be adaptive due to random and time varying channel EC 1451 Mobile Communication

2. Operating modes • Training • To determine the filter coefficients for Minimum BER • PN sequence or fixed prescribed pattern is sent • Tracking • Adaptive recursive algorithm tracks the changing channel when user data is received • Equalizer is said to converged when it is properly trained EC 1451 Mobile Communication

3. (1) (2) (3) EC 1451 Mobile Communication

4. (4) • The desired output from the equalizer is x(t) • Assume nb(t)=0 and in order to force • In the frequency domain: • Equalizer is an inverse filter of the channel • When the channel is frequency selective: • Equalizer enhances low amplitude frequency components and vice versa • When the channel is time varying: • Equalizer tracks the channel variations (5) EC 1451 Mobile Communication

5. Training a generic adaptive equalizer (Transversal filter) EC 1451 Mobile Communication

6. (6) (7) (8) (9) (10) (11) Defining the input signal to the equalizer as a vector yk, Output of the adaptive equalizer is a scalar given by A weight vector can be written as Using Eq. (6) and (8), Eq. (7) can be written as When the desired equalizer output is known (i.e. dk=xk), the error signal ek is given by From Eq.(9), EC 1451 Mobile Communication

7. (12) (13) (14) (15) (16) mean square error: cross correlation vector Input correlation matrix: (N+1) x (N+1) From (14) and (15), equation (13 ) can be written as EC 1451 Mobile Communication

8. Equalizers in a communications receiver • Noise nb(t) makes equation (4) hard to realize • Instantaneous combined frequency response will not be always flat, resulting some prediction error • In digital form, Eq.(2) can be expressed as • Prediction error is • Mean squared error is an important cost function (17) (18) EC 1451 Mobile Communication

9. Survey of Equalization Techniques EC 1451 Mobile Communication

10. Linear Traversal Equalizer (LTE) structure • Consists of tapped delay lines, with tappings spaced a symbol period (Ts) apart • Transfer function will be a function of • Filter can be FIR (feed forward taps only) or IIR (both feed forward and feedback taps) EC 1451 Mobile Communication

11. Tapped delay lines with both feed forward and feed backward taps EC 1451 Mobile Communication

12. Linear Equalizers(1) Transversal filter (FIR filter) EC 1451 Mobile Communication

13. Transversal filter (FIR) output before a decision is made is Minimum MSE is given by EC 1451 Mobile Communication

14. (2) Lattice Filter Equalizer EC 1451 Mobile Communication

15. Input signal yk is transformed into a set of N intermediate forward and backward error signals, fn(k) and bn(k) respectively • fn(k) and bn(k) are used as input to the tap multipliers and are used to calculate the updated coefficients • Recursive equations: EC 1451 Mobile Communication

16. Advantages and disadvantages • Numerical ability • Faster convergence • Dynamic assignment of effective length of the equalizer • Not time dispersive channel- fraction of stages used • Time dispersive channel – length can be increased • More complicated structure than transversal filter EC 1451 Mobile Communication

17. Nonlinear Equalization • Used when the channel distortion is too severe • Linear equalizers do not perform well on channels which have deep spectral nulls in the passband • Three methods: • Decision Feedback Equalization (DFE) • Maximum Likelihood Symbol Detection • Maximum Likelihood Sequence estimation (MLSE) EC 1451 Mobile Communication

18. Decision Feedback Equalization (DFE) Direct Transversal form EC 1451 Mobile Communication

19. Consists of feed forward filter (FFF) and feed back filter (FBF) • Equalizer has N1+N2+1 taps in the feed forward filter and N3 taps in the feedback filter and its output can be expressed as EC 1451 Mobile Communication

20. Minimum mean squared error a DFE can achieve is • Minimum MSE for a DFE is smaller than that of LTE unless |F(ejωT)| is a constant • MSE is suitable for well distorted channel or exhibits null in the spectrum • LTE is well behaved when the channel spectrum is flat EC 1451 Mobile Communication

21. Maximum Likelihood Sequence Estimation (MLSE) Equalizer EC 1451 Mobile Communication