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Introduction to equalization

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This presentation is about Equalizer and its design principles, with application point of view.

Introduction to equalization

  1. 1. Introduction Equalizer Design Presented By : Harshit Srivastava
  2. 2. • Communication system model • Need for equalization (ISI) • equalizer design • Zero Forcing Equalizer • Minimum Mean Square • Conclusion Outline
  3. 3. filter channel Receiver filter gT(t) gR(t)c(t) )(*)(*)()( tgtctgth RT Y(t) S[i] Y(tm) A digital communication system requires transmit and receive filter A digital communication system requires transmit and receive filter. Transmit Filter shapes transmitted signal to meet spectral requirements. Receive filter accomplishes two roles: Recover symbol sent and limit noise effect )(*)()( tctgth T )()( tThtg TR Where constitute for matched filter )(tgR
  4. 4. Inter Symbol Interference • When a system is designed according to matched filter criterion, there is no inter-symbol interference(ISI). • When sampling at t with period T we get back same signal. • At these h(t)0 Problem when h(t) = ∞, bandwidth and c(t) is limited. This will always lead to ISI with, Tail of other symbols response overlaps Which can lead to lower performance with synchronization error. Tb 2Tb 3Tb 4Tb 5Tb t 6Tb
  5. 5. • Therefore in channels ISI cannot be avoided so: • and are designed as matched pairs. • These effects of 𝑐(𝑡) is countered by means of an equalizer. )(tgT )(tgR Tx filter Channel )(tn Rx. filter DetectorEqualizer Regardless of ISI, the role of transmit filter is important to fit the transmitted spectrum into the appropriate spectral mask. This requires the usage of longer duration waveforms and spectral occupancy reduction
  6. 6. Equalization • Equalization is all about compensating other effects that distorts the signal • Equalizers are basically used to remove the ill effect generated in signal through channel by using inverse filter • For example if we have a properly shaped transmit pulse as a sinc function which has no ISI but when transmitted it would be combined with additive noise which resembles ISI.
  7. 7. Equalization Process • The natural way to compensate the ISI effect is by using inverse filter if we know the channel impulse response. 𝐻(𝑒 𝑗𝑤) = 𝐻(𝑒−𝑗𝑤) • That means convolution of the channel response and the equalizer response must be equal to 1
  8. 8. Methods to meet requirement • Two main techniques are employed to meet filter coefficient, • Automatic Synthesis and adaptation. • In automatic Synthesis typically equalizer just compares the received signal to original stored signal to determine error and coefficient of an inverse filter.(ZFE and LMS) • In adaptation the equalizer attempts to minimize an error signal based on the difference between the output of the equalizer
  9. 9. Zero Forcing Equalizer • In we have a transfer function of equalizer h 𝑓 , the simplest way to remove ISI is to setting a transfer function inverse of it, h 𝑓 = 1/𝑐(𝑓). • This is known as zero forcing equalizer. • Basically it applies the inverse frequency response of the channel to the received signal. • Let’s say a ZF equalizer has tap coefficients W which are chosen to minimize the peak distortion of the equalized channel, defined as, 𝐷 𝑝 = 1 |𝑞 𝑑1| 𝑛=0,𝑛≠𝑑 𝑁+𝐿−1 |𝑞 𝑛 − 𝑞 𝑛| • Where 𝑞 𝑛=( 𝑞0 … . . 𝑞 𝑋) X= 𝑁 + 𝐿 − 1, is the desired channel and the delay d is a positive integer 𝑑 = 𝑑1 + 𝑑2.
  10. 10. • Condition if , 𝐷 𝑝 < 1, then 𝐷 𝑝 is minimized by N taps values which cause 𝑞 𝑗 = 𝑞 𝑗 for 𝑑1−𝑑2 ≤ 𝑑1 + 𝑑2. • Therefore if the initial distortion is greater than unity the ZF is not guaranteed to minimize the peak distortion. • For the case when 𝑞0 = 𝑒 𝑑 𝑇 the equalized channel is given by, 𝑞 = 𝑞0, . , 𝑞 𝑑1−1, . 0. . , 0,1,0. . . 0 𝑞 𝑑1+𝑁, … , . 𝑞 𝑁+𝐿−1 𝑇 In this case the equalizer forces zeroes into the equalized channel and, hence, the name “zero forcing equalizer”
  11. 11. Equalizer Tap Solution • For a known channel impulse response, the tap gains of the ZF equalizer can be found by the direct solution of the set of linear equations. Therefore for that we for matrix • 𝑃 = 𝑝 𝑑1 , … … 𝑝 𝑑 , … . , 𝑝 𝑁 + 𝑑1 − 1 and the vector, 𝑞= 𝑞 𝑑1 … . . 𝑞 𝑀 T where M=N+𝑑1-1 Then the vector of the optimal tap gains, 𝑤 𝑜𝑝, satisfies 𝑤𝑜𝑝 𝑇 = 𝑞 𝑇 → 𝑤𝑜𝑝 = 𝑃−1 𝑇 𝑞
  12. 12. Drawbacks in ZFE • ZFE strategy suffers from the noise enhancing issue at high frequencies. • In which it relies on perfect estimation of ℎ(𝑛). • The noise has not been taken into account at all. • Since ZFE is generally an inverse filter it applies high gain at upper frequency which increases noise. • The training signal is basically an impulse which is inherently a low signal, which corresponds to low SNR. • MMSE tries to reach a trade-off between noise and ISI effects.
  13. 13. Minimum Mean Square Error • A FIR filter can equalize the worst case ISI only when the peak distortion is small1. • The MMSE gives the filter coefficients to keep a MSE between the output of the equalizer and the desired signal. • The MMSE equalizer requires a training sequences (d(t)). • 𝑦(𝑡) and 𝑣(𝑡) are signals affected by noise 1. Peak distortion: magnitude of the difference between the output of the channel and the desired signal The aim is to minimize: + Noise n(t) 𝜀 = 𝐸 𝑣 𝑡 − 𝑑 𝑡 2 )(td )( fGT )( fC )( fGR )( fHeq )(td MMSE Block Diagram
  14. 14. MSE vs. equalizer coefficients 1c 2c quadratic multi-dimensional function of equalizer coefficient values MMSE aim: find minimum value directly (Wiener solution), or use an algorithm that recursively changes the equalizer coefficients in the correct direction (towards the minimum value of 𝜀 )! Illustration of case for two real-valued equalizer coefficients (or one complex-valued coefficient) 𝜀 = 𝐸 𝑣 𝑡 − 𝑑 𝑡 2 𝜀
  15. 15. • MMSE criterion can be read as 𝜀 = 𝐸 𝑣 𝑡 − 𝑑 𝑡 2 Where 𝑣(𝑡) = 𝑦(𝑡) ∗ ℎ 𝑒𝑞(𝑡) = 𝑛=0 𝑁 𝑏 𝑛 𝑦(𝑡 − 𝑛𝑇) • The minimization parameters are accordingly, 𝜕𝜀 𝜕𝑏 𝑚 = 0 = 2E[(v t − d t ) 𝜕𝑣(𝑡) 𝜕𝑏 𝑚 ] m=0,……,N,N even • Where it has been taken into account the linearity of the E[.] and 𝜕 𝜕𝑥 , and the fact that expected value is taken over noise distribution, independently of their filter coefficient. • This leads to the next orthogonality condition.
  16. 16. • Therefore the error sequence between the output of the equalizer and the desired signal and the received data sequence should be statistically orthogonal. 𝐸 𝑣 𝑡 − 𝑑 𝑡 𝑦(𝑡 − 𝑚𝑇) = 0 = 𝑅 𝑦𝑣 𝑚𝑇 − 𝑅 𝑦𝑑 𝑚𝑇 = 0 • 𝑚 = 0, … . . 𝑁 • 𝑅 𝑦𝑣 𝑚𝑇 = 𝐸 𝑦 𝑡 𝑣 𝑡 + 𝑚𝑇 • 𝑅 𝑦𝑑 𝑚𝑇 = 𝐸 𝑦 𝑡 𝑑 𝑡 + 𝑚𝑇 • All random process involved are considered widesense stationary and jointly wss • A process is wss when its mean and covariance do not vary with time. • Two random Process A and B are jointly wss when they are wss and their cross correlation only depends on the time difference. • 𝑅 𝐴𝐵 𝑡, 𝑡′ = 𝑅 𝐴𝐵(𝑡 − 𝑡′)
  17. 17. • 𝑅 𝑦𝑣 𝑚𝑇 can be written as • 𝑅 𝑦𝑣 𝑚𝑇 = 𝐸 𝑦 𝑡 𝑣 𝑡 + 𝑚𝑇 • = 𝐸 𝑦 𝑡 𝑛=0 𝑁 𝑏 𝑛 𝑦 𝑡 + 𝑚 − 𝑛 𝑇 = 𝑛=0 𝑁 𝑏 𝑛 𝑅 𝑦( 𝑚 − 𝑛 𝑇) M=0,….,N • Where 𝑅 𝑦(𝑡′) is the autocorrelation of 𝑦 𝑡 . • This set of equations can be written in vector matrix form as, • 𝑅 𝑦 𝑏 𝑀𝑀𝑆𝐸 = 𝑅 𝑦𝑑 • They are known as Wiener-Hopf equations. • The filter coefficients can be finally calculated as • 𝑏 𝑀𝑀𝑆𝐸 = 𝑅 𝑦 −1 𝑅 𝑦𝑑
  18. 18. • Note the similarities with the process with ZFE strategy. • We calculate a matrix depending on the channel response, and a vector depending on the expected response. • The matrix is inverted and we get a solution. • Note that the matrices and vectors have statistical meaning. • The minimum mean square error we arrive at is given by: 𝜀 𝑚𝑖𝑛 = 𝐸 𝑑 𝑡 2 − 𝑅 𝑦𝑑 𝑇 𝑏 𝑀𝑀𝑆𝐸 = 𝑅 𝑦𝑑 𝑇 𝑅 𝑦 −1 𝑅 𝑦
  19. 19. Conclusion • Equalization is a process implemented at receiver that is mandatory for almost any modern digital communication. • There is a variety of equalization strategies and possible implementation • Adaptive, blind, ML and so on. • It is not a closed a field and it is subject to ongoing research and improvements. • There is not an all-powerful equalization technique, it all depends on the kind of channel, hardware availability, target performance, tolerable delay, and so on. • Equalizers are not left alone: FEC systems can also contribute to compensation of residual ISI effects.
  20. 20. References • Ziemer, R.E., and Peterson, R.L., Introduction to Digital Communication, Macmillan, 1992.Simon Haykin - Adaptive Filter Theory. • David Smalley, Equalizer Design, Atlanta Regional technology Center, Texas instrument • Lovrich, A. and Simar, R., “Implementation of FIR/IIR Filters with the TMS32010/TMS32020”, Digital Signal Processing Applications with the TMS320 Family, Volume 1, Texas Instruments, 1989. • Proakis, J.G., “Adaptive Equalization for TDMA Digital Mobile Radio”, IEEE Transactions on Vehicular Technology, Volume 40, No. 2, May 1991. • S.Kay – Statistical Signal Processing – Estimation Theory • John G.Proakis – Digital Communications.
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