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## Equalization

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**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**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**(1)**(2) (3) EC 1451 Mobile Communication**(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**Training a generic adaptive equalizer (Transversal filter)**EC 1451 Mobile Communication**(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**(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**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**Survey of Equalization Techniques**EC 1451 Mobile Communication**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**Tapped delay lines with both feed forward and feed backward**taps EC 1451 Mobile Communication**Linear Equalizers(1) Transversal filter (FIR filter)**EC 1451 Mobile Communication**Transversal filter (FIR) output before a decision is made is**Minimum MSE is given by EC 1451 Mobile Communication**(2) Lattice Filter Equalizer**EC 1451 Mobile Communication**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**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**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**Decision Feedback Equalization (DFE)**Direct Transversal form EC 1451 Mobile Communication**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**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**Maximum Likelihood Sequence Estimation (MLSE) Equalizer**EC 1451 Mobile Communication