Decision-Point Signal to Noise Ratio (SNR)

Transcription

1 Decision-Point Signal to Noise Ratio (SNR) Receiver Decision ^ SNR E E e y z Matched Filter Bound error signal at input to decision device Performance upper-bound on ISI channels Achieved on memoryless channels and with one-shot transmission on ISI channels (matched filter reception) SNR MFB e S h N 0 z See Cioffi Notes Chapter 3, Section 3. 1

2 Matched Filter n(t) (t) Q : what is optimum h(t) that maimizes output SNR at t=t for AWGN channel & one-shot transmission? y(t) h(t) t = T Signal energy = ( t ( ) ht ( ) t T ) = Noise energy = ( nt ( ) ht ( ) ) t T = SNR ( ( t), h( T t) ) h n h * with equality iff h ( t ) ( T t ) n (t) is transmit pulse shape inner product maimized when angle=0 Cauchy-Schwartz Inequality Matched filter does not increase signal energy but combines freq. components coherently * H( f ) X ( f ) e j ft j ft H( f ) X( f ) e X( f ) e j 1 f t Output due to frequency component is j e f1 ( t T ) X( f ) 1 which is maimized by sampling output at t=t

3 Biased vs. Unbiased Receivers z ' Biased Receiver : e ' E[ z ' ] where 0 1 Biased : Eample : ' e 0.5 Unbiased : ' 3. 1 ' z ˆ z ˆ 3 ˆ arg min 1, 3 arg min 1, 3 z 1 ' ' z : scale decision regions : scale input to slicer Although consists of filtered noise and residual ISI, we will use the Euclidean distance metric for simplicity 3

4 Biased Receivers 4

5 Bias is Due to MMSE design rule The output of an unbiased receiver has the form z e where E[ z ] SNR U def S e z z ( e ) Scaling unbiased receiver output leads to a biased rcvr that has higher h SNR but is suboptimum from Prob. error point of view Bias common to all MMSE receivers and must be removed SNR ( ) What is optimum? E[ z ] 1 e 1 that maimizes SNR of a biased receiver SNR( ) 0 opt S S e z e SNRU 1/( 1 ) S 1 1 SNR U is input to decision device in a biased rcvr S : SNR U e Bias effects are more pronounced at low SNR and diminish at high SNR 5 Substituting this optimum value of opt in the epression for biased SNR, we get

6 Biased Vs. Unbiased Receivers Substituting optimum values for SNR SNR U 1 General Rule Biased receiver has higher h SNR but suboptimum from a probability of error point of view (since decision regions are based on the signal constellation not its scaled version) Biased receiver can have SNR higher than MFB!! The optimum biased and unbiased receivers have same settings; differ by scaling of input to decision device 6

7 ISI Mitigation Schemes Filtering i : One (linear equalizer) or two filters (decision feedbac equalizer) mitigate ISI Maimum Lielihood Sequences Estimation (MLSE) : treats ISI channel as finite-state machine (uses memory to advantage) Multicarrier Modulation (MCM) : Channel frequency response divided into large number of parallel, independent, and memoryless subchannels each decoded separately (divide and conquer!) 7

8 Filtering Techniques Linear Equalizer y,..., y N Decision Feedbac y y Equalizer N Filter,...,,..., 1 N b Filters N N b received bloclength no. processed decisions Design Issues : How many filter taps are needed d? How to compute optimum filter tap settings? How to determine optimum decision delay? 8

9 Maimum Lielihood Sequence Estimation Treats ISI channel as finite-state machine with M states Eample : BPSK on (1+D) channel (M=, 1 ) /1 0/-1 -/-1 +1 Output/input /+1-1 State Diagram y 1 z /1 0/-1 -/-1 0/+1 Trellis Diagram Describes time evolution of states In presence of noise, received sequence not equal to +, -, 0. Minimize Euclidean distance (cost metric) output y current input 1 z state (previous input) 9

10 Maimum lielihood Sequence Estimation (MLSE) Idea : Given the received sequence, find the most liely path through the trellis Model : Y ( D) H ( D) X ( D) Z( D) H ( D) h h D... h D 0 1 v FIR E y( ) z( ) h Channel SNR E z( ) Assumptions : 1. All inputs are equally liely. FIR channel and LTI 3. Additive white Gaussian noise (otherwise, apply noise - whitening filter) z 10

11 Maimum lielihood Sequence Estimation (MLSE) { y( )}is the output of a finite - state machine over the comple field C X ( 1) Let ( ) X ( ) s be the "state"of the system (previous inputs), X ( v) Maimum Lielihood Sequence Eti Estimation (Detection) ti MLSE/MLSD Xˆ ( D) arg ma P( Y ( D) X ( D)) X ( D) Note : this is similar to problem of decoding convolutional codes use Viterbi algorithm (dynamic programming) 11

12 Maimum Lielihood Sequence Estimation (MLSE) Since z(d) is IID logp( Y ( D) X ( D)) Maimize Let X ( ) X ( v) u( D) H ( D) X Gaussian log or equivalently on s() P ( y( ) z ( h( )* ( ))) [log( ) y( ) h( )* ( ) Minimize ( D) u( ) depends on and s( 1) ] 1

13 Maimum lielihood Sequence Estimation (MLSE) Denote the log lielihood function by Γ[ S ( D )] log{ P ( y ( ) ( ( )* h ( ))} 1 z Dynamic Programming Principle 1 Suppose we new that the state s( ) at time was Sj. Then, for any allowable state sequence S( D) that starts with S(0) 0 and passes through state Sj at time, the log lielihood is given by : [ S( D)] L 0 [ S( D)] 0 Additive Metric [ S( D)] Summary : on a trellis diagram, any state s metric at time is equal to the minimum sum of previous state metrics and transition costs over all M possible transitions to this state t L 13

14 Survivor Sequences For each of the states, there are M possible inputs Keep the one with the smallest cost (called survivor) Let sˆ j ( D) be any allowable state sequence from time to time that t has maimum log lielihoodlih metric [ s( D)] starting ti call sˆ state [ s ˆ j ( D )] 0 j from 0 among all allowable state sequences s ( ) and ending at s ( ) s We ( D) the survivor at time corresponding to s j. Consequently, we store one survivor sequence for each state and its corresponding log lielihood metric for j {1,..., M v }. j. 14

15 Mathematical Description of Viterbi Algorithm Notation ti : 1) State inde " i": i 0,1,..., M v ) State metric for state " i" at time " ": Ci, 3) Set of previous states to state " i": J i 4) Noiseless output in going from state " 5) Branch metric in going g from state " ~ j, i, y y ( j i) 1 With Cardinality M j" to state " i": ~ y ( j j" to state " i" at time " " i) 15

16 Mathematical Description of Viterbi Algorithm Recursions : Initial condition : C i, 1 Recursions : For End i 0,1,..., M For End C i v 0, State inde time inde min [ C j, 1 j, i,, i jj ] : Over all M previous states to state i One survivor path per state Compleity eponential in 16

17 Viterbi Detection Eample (1+D) Channel y 1 n ; 1 Received Sequence : 0.08,.1,-1.1, Estimated Input Sequence : +1, +1, -1, -1 Estimated Noise Sequence : 0.08, , -1.1,

18 Remars In principle, the Viterbi algorithm terminates at K. However, we can mae a decision on a particular state if all the survivors agree on that state. In practice, decision can be typically made with 4v-5v symbols MLSE also wors if (D) itself is the output of a finitestate machine (e.g. convolutional code). In this case, the two state machines can be combined and we can perform joint equalization and decoding MLSE minimizes sequence error probability and involves a forward recursion. Another (more complicated) algorithm nown as BCJR minimizes symbol error probability and involves a forward + bacward recursion (not discussed here, project?) 18

19 Application : MLSE Detection for Magnetic Recording Channels For large magnetic recording densities, ISI etends over several bit intervals (0 for D=4) Use linear equalizer to equalize h(t) to a short target response (channel shortening) and detect using Viterbi algorithm (PRML) Most common choice for target response is the partial response family ( 1 D )( 1 D ) n n 01,,..., 4