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Digital Communications Fredrik Rusek

Digital Communications Fredrik Rusek. Chapter 10, adaptive equalization and more Proakis-Salehi. Brief review of equalizers. Channel model is Where f n is a causal white ISI sequence , for example the C- channel , and the noise is white. Brief review of equalizers.

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Digital Communications Fredrik Rusek

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  1. Digital CommunicationsFredrik Rusek Chapter10, adaptive equalization and more Proakis-Salehi

  2. Briefreviewofequalizers Channel model is Wherefnis a causalwhiteISI sequence, for example the C-channel, and the noise is white

  3. Briefreviewofequalizers Letustakea look on howtocreatefnagain Addnoise Where is fn ?

  4. Briefreviewofequalizers Letustakea look on howtocreatefnagain Addnoise Optimal receiver front-end is a matched filter

  5. Briefreviewofequalizers Letustakea look on howtocreatefnagain Addnoise Optimal receiver front-end is a matched filter What is the statistics ofxk and vk ?

  6. Briefreviewofequalizers Letustakea look on howtocreatefnagain Addnoise Optimal receiver front-end is a matched filter What is the statistics ofxk and vk ? Xk has Hermitiansymmetry Cov[vkv*k+l]=xl xk is not causal, noise is not white!

  7. Briefreviewofequalizers Letustakea look on howtocreatefnagain Addnoise Noisewhiteningstrategy 1 Noisewhitener The noisewhitener is using the factthat the noise has xk as covariance fk is nowcausal and the noise is white

  8. Briefreviewofequalizers Noisewhiteningwithmoredetail Define Then Choosing the whitener as willyield a channelaccordingto The noisecovariancewill be flat (independent ofF(z)) becauseof the followingidentity

  9. Briefreviewofequalizers Noisewhiteningstrategy 2. Important. • In practice, oneseldomlysees the matched filter followed by the whitener. • Hardware implementation of MF is fixed, and cannotdepend on the channel • Howtobuild the front-end? • Desires: • Should be optimal • Should generate whitenoise at output

  10. Briefreviewofequalizers From Eq (4.2-11), weknowthatif the front end creates is an orthonormal basis, then the noise is white

  11. Briefreviewofequalizers From Eq (4.2-11), weknowthatif the front end creates is an orthonormal basis, then the noise is white We must thereforechoose the front-end, call it z(t), suchthat Eachpulse z(t-kT) nowconstitutesone dimension φk(t) The root-RC pulses from the last lectureworkswell

  12. Briefreviewofequalizers Noisewhiteningstrategy 2. Important. • In practice, oneseldomlysees the matched filter followed by the whitener. • Hardware implementation of MF is fixed, and cannotdepend on the channel • Howtobuild the front-end? • Desires: • Should be optimal • Should generate whitenoise at output OK! Buthowtoguaranteeoptimality?

  13. Briefreviewofequalizers Fourier transform ofreceivedpulse H(f) This is bandlimited since the transmit pulseg(t) is bandlimited

  14. Briefreviewofequalizers ChooseZ(f) as H(f) In thiswayz(t) creates a complete basis for h(t) and generates whitenoise at the same time • LTE and other practical systems arechoosing a front-end suchthat • Noise is white • Signal ofinterestcan be fullydescribed

  15. Briefreviewofequalizers Addnoise Optimal receiver front-end is a matched filter

  16. Briefreviewofequalizers Addnoise Receiver front-end is a constant and not dependent on the channel at all. Z(f)

  17. Briefreviewofequalizers Linearequalizers. Problem formulation: Given apply a linear filter to get back the data In With We get

  18. Briefreviewofequalizers Linearequalizers. Problem formulation: Given apply a linear filter to get back the data In With We get Zero-forcing MMSE min

  19. Briefreviewofequalizers Non-linear DFE. Problem formulation: Given apply a linear filter to get back the data Ik Previouslydetected symbols DFE - MMSE min

  20. Briefreviewofequalizers Comparisons Output SNR of ZF Error (J) of MMSE Error (J) of DFE-MMSE

  21. Tomlinson-Harashimaprecoding (relatedtodirty-paper-coding) Assume MPAM (-(M-1),…(M-1)) transmission, and the simple channelmodely=x+n Assumethatthere is a disturbance at the channel y=x+n+pM, p an integer The recivercanremove the disturbance by mod(y,M)=mod(x+n+pM,M)=x+w, Where w has a complicated distribution. However, w=n,ifn is small.

  22. Tomlinson-Harashimaprecoding (relatedtodirty-paper-coding) Assume MPAM (-(M-1),…(M-1)) transmission, and the simple channelmodely=x+n Assumethatthere is a disturbance at the channel y=x+n+pM, p an integer The recivercanremove the disturbance by mod(y,M)=mod(x+n+pM,M)=x+w, Where w has a complicated distribution. However, w=n,ifn is small. -3 3 M (=4) Let the be x+n (i.e., received signal withoutanydisturbance

  23. Tomlinson-Harashimaprecoding (relatedtodirty-paper-coding) Assume MPAM (-(M-1),…(M-1)) transmission, and the simple channelmodely=x+n Assumethatthere is a disturbance at the channel y=x+n+pM, p an integer The recivercanremove the disturbance by mod(y,M)=mod(x+n+pM,M)=x+w, Where w has a complicated distribution. However, w=n,ifn is small. Add the disturbance -3 3 -3+4p 3+4p M (=4) Let the be x+n (i.e., received signal withoutanydisturbance

  24. Tomlinson-Harashimaprecoding (relatedtodirty-paper-coding) Assume MPAM (-(M-1),…(M-1)) transmission, and the simple channelmodely=x+n Assumethatthere is a disturbance at the channel y=x+n+pM, p an integer The recivercanremove the disturbance by mod(y,M)=mod(x+n+pM,M)=x+w, Where w has a complicated distribution. However, w=n,ifn is small. -3 3 -3+4p 3+4p M (=4) Nothingchanged, i.e., w=n Nowcompute mod( ,4)

  25. Tomlinson-Harashimaprecoding (relatedtodirty-paper-coding) Assume MPAM (-(M-1),…(M-1)) transmission, and the simple channelmodely=x+n Assumethatthere is a disturbance at the channel y=x+n+pM, p an integer The recivercanremove the disturbance by mod(y,M)=mod(x+n+pM,M)=x+w, Where w has a complicated distribution. However, w=n,ifn is small. -3 3 M (=4) But, in thiscasewehave a difference

  26. Tomlinson-Harashimaprecoding (relatedtodirty-paper-coding) Assume MPAM (-(M-1),…(M-1)) transmission, and the simple channelmodely=x+n Assumethatthere is a disturbance at the channel y=x+n+pM, p an integer The recivercanremove the disturbance by mod(y,M)=mod(x+n+pM,M)=x+w, Where w has a complicated distribution. However, w=n,ifn is small. Add the disturbance -3 3 -3+4p 3+4p M (=4)

  27. Tomlinson-Harashimaprecoding (relatedtodirty-paper-coding) Assume MPAM (-(M-1),…(M-1)) transmission, and the simple channelmodely=x+n Assumethatthere is a disturbance at the channel y=x+n+pM, p an integer The recivercanremove the disturbance by mod(y,M)=mod(x+n+pM,M)=x+w, Where w has a complicated distribution. However, w=n,ifn is small. -3 3 -3+4p 3+4p M (=4) Will be wronglydecoded, seldomlyhappens at high SNR though Nowcompute mod( ,4)

  28. Tomlinson-Harashimaprecoding (relatedtodirty-paper-coding) Howdoesthis fit in withISI equalization? Supposewewantto transmit Ikbutthatweapplyprecoding and transmits ak Or in terms of z-transforms Meaningofthis is that ISI is pre-cancelled at the transmitter Sincechannelresponse is F(z), all ISI is gone

  29. Tomlinson-Harashimaprecoding (relatedtodirty-paper-coding) Howdoesthis fit in withISI equalization? Supposewewantto transmit Ikbutthatweapplyprecoding and transmits ak Or in terms of z-transforms Meaningofthis is that ISI is pre-cancelled at the transmitter Problem is thatif F(z) is small at some z, the transmitted energy is big (this is the same problem as with ZF-equalizers)

  30. Tomlinson-Harashimaprecoding (relatedtodirty-paper-coding) Howdoesthis fit in withISI equalization? Supposewewantto transmit Ikbutthatweapplyprecoding and transmits ak Or in terms of z-transforms Meaningofthis is that ISI is pre-cancelled at the transmitter If A(z) is big, it meansthat the akarealsoverybig Problem is thatif F(z) is small at some z, the transmitted energy is big (this is the same problem as with ZF-equalizers)

  31. Tomlinson-Harashimaprecoding (relatedtodirty-paper-coding) Howdoesthis fit in withISI equalization? Supposewewantto transmit Ikbutthatweapplyprecoding and transmits ak Or in terms of z-transforms Add a disturbancethatreduces the amplitudeof ak. bk is chosen as an integerthatminimizes the amplitudeof ak

  32. Tomlinson-Harashimaprecoding (relatedtodirty-paper-coding) Howdoesthis fit in withISI equalization? Supposewewantto transmit Ikbutthatweapplyprecoding and transmits ak Or in terms of z-transforms Add a disturbancethatreduces the amplitudeof ak. bk is chosen as an integerthatminimizes the amplitudeof ak

  33. Tomlinson-Harashimaprecoding (relatedtodirty-paper-coding) Howdoesthis fit in with ISI equalization? Supposewewantto transmit Ikbutthatweapplyprecoding and transmits ak Or in terms of z-transforms Add a disturbancethatreduces the amplitudeof ak. bk is chosen as an integerthatminimizes the amplitudeof ak Channel ”removes” F(z), modulus operation ”removes” 2MB(z)

  34. Chapter 10

  35. Objectives • So far, weonlyconsidered the casewhere the channelfnwasknown in advance • Nowweconsider the casewhen the channel is unknown, but a training block ofknown data symbols are present • Weaim at establishinglow-complexity adaptive methodsfor finding the optimal equalizer filters • Thischapter has manyapplicationsoutsideof digital communications

  36. 10.1-1: Zero-forcing Weconsider a ZF-equalizerwith 2K+1 taps Withfinitelength, wecannotcreate sincewe do not haveenoughDoFs Instead (seebook), we try toachieve Howtoachievethis?

  37. 10.1-1: Zero-forcing Weconsider a ZF-equalizerwith 2K+1 taps Withfinitelength, wecannotcreate sincewe do not haveenoughDoFs Instead (seebook), we try toachieve Howtoachievethis? Consider

  38. 10.1-1: Zero-forcing Weconsider a ZF-equalizerwith 2K+1 taps Withfinitelength, wecannotcreate sincewe do not haveenoughDoFs Instead (seebook), we try toachieve Howtoachievethis?

  39. 10.1-1: Zero-forcing Weconsider a ZF-equalizerwith 2K+1 taps Withfinitelength, wecannotcreate sincewe do not haveenoughDoFs Instead (seebook), we try toachieve Howtoachievethis? For We get

  40. 10.1-1: Zero-forcing Let be the j-thtapof the equalizer at time t=kT. A simple recursivealgorithm for adjustingthese is is a small stepsize is an estimateof For We get

  41. 10.1-1: Zero-forcing Let be the j-thtapof the equalizer at time t=kT. A simple recursivealgorithm for adjustingthese is Initial phase. Training present is a small stepsize is an estimateof The above is doneduring the trainingphase. Once the trainingphase is complete, the equlizer has convergedtosomesufficientlygood solution, so that the detected symbols can be used. This is the trackingphase(no known data symbols areinserted). Trackingphase. No training present. Thiscancatch variations in the channel

  42. 10.1-2: MMSE. The LMS algorithm Again, wehave a 2K+1 tapequalizertoadaptivelysolve for Expanding J(K) gives Where c is a columnvectorofequalizertaps (tosolve for) and v is the vectorofobserved signals. It turnsoutthat E(v*v)= E(Ik*v)= (2K+1)x(2K+1) matrix T (2K+1) vector

  43. 10.1-2: MMSE. The LMS algorithm Usingthis, we get J(K)=1 – 2Re(ξ*c)+c*Γc Where c is a columnvectorofequalizertaps (tosolve for) and v is the vectorofobserved signals. It turnsoutthat E(v*v)= E(Ik*v)= Set gradient to 0 ξ* ξ (2K+1)x(2K+1) matrix T (2K+1) vector

  44. 10.1-2: MMSE. The LMS algorithm Usingthis, we get J(K)=1 – 2Re(ξ*c)+c*Γc Now, wewould like toreachthis solution without the matrix inversion. In general, wewould like tohave a recursivewaytocompute it Set gradient to 0 ξ* ξ

  45. 10.1-2: MMSE. The LMS algorithm Wecanformulate the followingrecursivealgorithm

  46. 10.1-2: MMSE. The LMS algorithm Wecanformulate the followingrecursivealgorithm Small stepsize (moreaboutthis later) Equalizer at time t=kT Vectorofreceived symbols Gradient vector

  47. 10.1-2: MMSE. The LMS algorithm Wecanformulate the followingrecursivealgorithm WheneverGk = 0, the gradient is 0 and the optimal point is reached (since J(K) is quadratic and thereforeanystationarypoint is a global optimum)

  48. 10.1-2: MMSE. The LMS algorithm Wecanformulate the followingrecursivealgorithm Basic problem: The gradient depends on Γ and ξ, whichareunknown (depends on channel) As a remedy, weuseestimates

  49. 10.1-2: MMSE. The LMS algorithm Wecanformulate the followingrecursivealgorithm Basic problem: The gradient depends on Γ and ξ, whichareunknown (depends on channel) As a remedy, weuseestimates The estimatorof the gradient is unbiased

  50. 10.1-2: MMSE. The LMS algorithm Wecanformulate the followingrecursivealgorithm Basic problem: The gradient depends on Γ and ξ, whichareunknown (depends on channel) As a remedy, weuseestimates LMS algorithm (veryfamous) The estimatorof the gradient is unbiased

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