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Minimum Mean Square Error (MMSE) Equalizer Linear equalizer. Aims at minimizing the variance of the difference between the transmitted data and the signal at the equalizer output. This effectively equalizes the freq. selective channel. First, consider the infinite length filter case: The output of the equalizer iswhere the equalized channel IR isSpring'09ELE 739 - Channel Equalization1MMSE Equalizer Infinite Length The difference between the Tx.ed data and the equalizer output is: and the MMSE cost function is: This is a quadratic function with a unique minimum Take derivative w.r.t. wj and equate to 0 to find this minimum. UsingSpring'09ELE 739 - Channel Equalization2MMSE Equalizer Principle of orthogonality:The necessary and sufficient condition for the cost function J to attain its minimum value is, for the corresponding value of the estimation error [n] to be orthogonal to each input sample t[n] that enters into the estimation of the desired response at time n. Error at the minimum is uncorrelated with the filter input! In other words, nothing else can be done for the error by just observing the filter inputs. A good basis for testing whether the linear filter is operating in its optimum condition.ELE 739 - Channel Equalization 3 Spring'09MMSE Equalizer Corollary:If the filter is operating in optimum conditions (in the MSE sense)When the filter operates in its optimum condition, the filter output z[n] and the corresponding estimation error [n] are orthogonal to each other.x[n][n]z[n]Spring'09 ELE 739 - Channel Equalization 4MMSE Equalizer We can calculate the MMSE equalizer by either minimizing J over w:or using the principle of orthogonality:which gives us the Wiener-Hopf EquationsACF of the WMF outputSpring'09Cross-CF of the Tx.ed data and the WMF output5ELE 739 - Channel EqualizationOptimum Equalizer It can easily be shown thatAndTaking the z-transform of the eqn. at the top, we getAlternatively, incorporating the WMF into the MMSE equalizer, we getSpring'09ELE 739 - Channel Equalization6MMSE vs. ZF MMSE: ZF:MMSE suppresses noise, besides equalizing the channel. MMSE will not let infinite noise as ZF does when the channel has a spectral null.As noise becomes negligible N00 MMSE and ZF becomes identical. When N0=0, MMSE cancels ISI completely (ZF cancels for all SNR values) When N0 0, residual ISI and noise will be observed at the output of the MMSE equalizer.Spring'09ELE 739 - Channel Equalization7MMSE - Performance What is the value of Jmin?Due to the principle of orthogonality,, then=b0The summation is a convolution evaluated at shift zero.Spring'09ELE 739 - Channel Equalization8MMSE - Performance Then No ISI X(ejT)=1 Note that, Furthermore, output SNR is No ISI Same as ZF.Spring'09ELE 739 - Channel Equalization9MMSE Performance Example 1: The effective channel has two taps, Spectrum is(has a null at =/T when )When we evaluate the integral of b0, Jmin becomesWhen, Jmin and output SNR areNo ISI ELE 739 - Channel Equalization 10Spring'09MMSE - Performance Example 2: Let the equiv. channel have exponentially decaying taps, a0 Spring'09ELE 739 - Channel Equalization25Canonical Form of the Error-Performance Surface Transformations may significantly simplify the analysis, Use Eigendecomposition for R Then Let Substituting back into Ja vectorCanonical formThe transformed vector v is called as the principal axes of the surface.ELE 739 - Channel Equalization 26Spring'09Canonical Form of the Error-Performance Surfacew2 wo J(wo)=Jmin J(w)=c curve Jmin Q Transformation w1 v1 (1) v2 (2) J(v)=c curveSpring'09ELE 739 - Channel Equalization27


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