Advanced Signal Processing in Communications

Transcription

1 Avance Signal Processing in Communications Lecture 2 Morten Jeppesen & Joachim Dahl {mje,ja}@cpk.auc.k DICOM, Aalborg University DICOM 2 p./26

2 Minimum Variance Estimation DICOM 2 p.2/26

3 Problem formulation Recall the linear moel: where is complex white Gaussian. It is critical that has full column rank. We wish to obtain the estimate of as the solution to without any restrictions or prior knowlege of This is a maximumlikelihoo problem. For the linear moel, this coincies with a leastsquares solution. We will follow the leastsquares approach.. DICOM 2 p.3/26

4 $ ' % ( Complex scalar an vector ifferentiation For complex scalar ifferentiation:, we efine complex Vectorifferentiation is efine as: #"!!! With these efinitions it is straightforwar to show that, ' +) &*) & % & % DICOM 2 p.4/26

5 +, + Leastsquares solution : We wish to fin the (global) minimum of, Applying the three rules from before, we get.,! +., + to / / is obtaine by equating The global minimizer zero: 3,2 + DICOM 2 p.5/26

6 ) ) % % ) ; : < < ) :; < EF D C 8, B + = : ) 6 G < +) ) The pseuoinverse Let be an arbitrary matrix with rank seen that only has a solution if If 8. We have., +), then the pseuoinverse (of imensions ) is efine as where the pseuoinverse, the LS solution is!!! 3 >? B!!! 3 >A@. In terms of < % If, +) then the pseuoinverse is given as! 3 ),2 ) DICOM 2 p.6/26

7 Applications to channel estimation DICOM 2 p.7/26

8 L Q PW QXW X B Z 6 Signal moel Recall the FIRfilter moel: I "N IJ"M IJ" IJ" The output of the filter is Q P 6 2Y P 6V R S T U Q P 6O where B Q P 6V Q P 6V for is the known transmitte signal (we assume ) an we wish to estimate. Q P 6 DICOM 2 p.8/26

9 e e a a iii M e e L M N L N a M iii N M iii & Reworke signal moel Put in matrixvector form, we have Ia Ia Ia I [ \^\^\^\^\\]\\^\^\^\\]\\`_ h ff]ff]fff]ff^f^f^ff`g Ia I Ib ff]ff^f^f`g I [ \\]\\^\^\`_ k ff]ff^f^f`g Ia I Ib [ \\]\\^\^\`_ L f^f^f^f^ff]ff^f^f^ff]ff`g [ \\]\\]\\\]\\^\^\^\\`_ Ia Ib Ib Ic Ic b Ic j Ic Ic. The optimum channel estimate is m l or & l, l m l + DICOM 2 p.9/26

10 t s q Example: Measure UMTS channel We have a measure impulse response: real part of channel imaginary part of channel We have npo an we choose rpo. The SNR is 2B. DICOM 2 p./26

11 u u u Importance of probing signal Below we see the powerspectrum of ( ) a binary pseuoranom sequence an ( ) a binary alternating sequence with istinct zeros: powerspectrum [B] ra powerspectrum [B] ra DICOM 2 p./26

12 The estimate impulse response real part of channel real part of channel imaginary part of channel imaginary part of channel (a) Pseuoranom probing signal (b) Alternating probing signal We see a clear avantage of using a pseuoranom probing signal. DICOM 2 p.2/26

13 Applications to econvolution DICOM 2 p.3/26

14 v L c " a e L a a iii iii iii L a e e e N M L iii N M a h a L N M iii L a iii a a iii iii iii Signal moel Again we consier the output of a FIR filter IJ"N k I{ L { IJ" h IJ"M wyx z is I ". Assuming that I " only now is assume known an we wish to estimate only nonzero for, we get IJ" Ia L Ia Ib L Ia I Ib [ \\^\^\^\\]\\\]\\]\\^\^\^\\]\\^\^\^\\]\\\]\`_ Ia Ia L f^f^f^ff]f`g I Ia I Ic [ \^\^\^\\]\`_ ff^f^f^ff]fff]ff]ff^f^f^ff]ff^f^f^ff]fff]f`g Ia I f^f^f^ff]f`g I [ \^\^\^\\]\`_ k f^f^f^ff]f`g [ \^\^\^\\]\`_ Ib L Ib L bk Ic bk Ic Ib L Ib L DICOM 2 p.4/26

15 } j } } } o r s Ÿ Š ˆ ž Zeroforcing equalizer As before the estimate of is &ƒ & ~ }o This is the optimal solution to ŠŒ Ž ˆ ~ }o i.e. without restrictions on owever, if c s s A suboptimal solution to this problem is. the problem is much harer to solve. &š & u o ~œ o This solutions is calle the Zeroforcing equalizer, as it forces the ISI to zero. The rawback is a egraation of the SNR. DICOM 2 p.5/26

16 s Ex: equalization of UMTS channel We use the measure UMTS channel for a burst of bits. We assume that the channel is perfectly known. r o 2 SNR=B 2 SNR=2B imaginary part imaginary part real part real part 2 SNR=3B 2 SNR=4B imaginary part imaginary part real part real part DICOM 2 p.6/26

17 Array application DICOM 2 p.7/26

18 Ê «ª Á Á ¹ Array application Recall the array moel from lecture Á À ÁAË AÀ Omniirectional antenna Linear array Equiistance: Plane waves wih incient angle: Æ Ç È É Å Ä ¹ J ¹ À ºÃ AÀ º ¾ Á AÀ¾ º»½¼ A ²³ ²³µ ± J «ª A Á AÀ Âà Á AÀ  ¾ Á AÀ¾ »½¼ DICOM 2 p.8/26

19 Array application We have 4 equal powere users with 4 istinct electrical angles We now want to use the ZF spatial equalizer to estimate the signals from the 4 users. DICOM 2 p.9/26

20 Array application The array manifol of the ZF exhibits nulls at interferers irrespective of the noise level > Nulling beamformer. 2 Array manifol for user 2 Array manifol for user 2 Power [B] 2 Power [B] Array manifol for user 3 2 Array manifol for user 4 Power [B] 2 Power [B] DICOM 2 p.2/26

21 Array application Scatterplots at SNR=5b (blue) an SNR=25B (re). 2 2 Quarature Quarature Inphase Inphase 2 2 Quarature Quarature Inphase Inphase DICOM 2 p.2/26

22 Extensions of moel to coloure noise DICOM 2 p.22/26

23 Ì = Î Ì Ñ Ñ PÎ Ò Ò Ñ Coloure noise In the linear moel ÍÌ we mae the important assumption that Gaussian noise, i.e. that is white, D Ï 3!!! +ÐÏ Q P Ì Ì If instea Q is coloure Gaussian with covariance for an arbitrary ermitian, we factor it as e.g. with a Cholesky factorization. DICOM 2 p.23/26

24 3 Ò Ó 3 Õ Î Ò PÎ Coloure noise We next transform the signal moel as Ô The covariance of the filtere noise is Q Ò P Ì Ì Q Thus the noise in the moifie signal moel is white, an the results from before now apply. It can be shown, that the LS solution for the moifie moel using a whitening transformation is still optimal. DICOM 2 p.24/26

25 Summary of lecture DICOM 2 p.25/26

26 Summary In this lecture we have seen: ow to obtain the LS solution to complex form. That this solution is optimal when in is analogous. ow it is relate to the pseuoinverse solution. ow to apply this solution to channel estimation. zeroforcing equalization. nulling beamforming. DICOM 2 p.26/26