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# Correlative level coding

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### Correlative level coding

1. 1. Baseband Pulse Transmission Correlative-Level Coding Baseband M-ary PAM Transmission Tapped-Delay-Line Equalization Eye Pattern Hyeong-Seok Yu Vada Lab. gargoyle@vada1.skku.ac.kr 1
2. 2. Correlative-Level Coding Correlative-level coding (partial response signaling)  adding ISI to the transmitted signal in a controlled manner Since ISI introduced into the transmitted signal is known, its effect can be interpreted at the receiver A practical method of achieving the theoretical maximum signaling rate of 2W symbol per second in a bandwidth of W Hertz Using realizable and perturbation-tolerant filters 2
3. 3. Correlative-Level CodingDuobinary Signaling Dou : doubling of the transmission capacity of a straight binary system Binary input sequence {bk} : uncorrelated binary symbol 1, 0 +1 ak =  if symbol bk is 1 ck = ak + ak −1 −1 if symbol bk is 0 3
4. 4. Correlative-Level CodingDuobinary Signaling  Ideal Nyquist channel of bandwidth W=1/2Tb H I ( f ) = H Nyquist ( f )[1 + exp(− j 2πfTb )] = H Nyquist ( f )[exp( jπfTb ) + exp(− jπfTb )] exp(− jπfTb ) = 2 H Nyquist ( f ) cos(πfTb ) exp(− jπfTb ) 1, | f |≤ 1 / 2Tb H Nyquist ( f ) =  0, otherwise 2 cos(π fTb ) exp(− jπ fTb ), | f |≤ 1/ 2Tb HI ( f ) =   0, otherwise sin(πt / Tb ) sin[π (t − Tb ) / Tb ] hI (t ) = + πt / Tb π (t − Tb ) / Tb Tb2 sin(πt / Tb ) = πt (Tb − t ) 4
5. 5. Correlative-Level CodingDuobinary Signaling The tails of hI(t) decay as 1/|t|2, which is a faster rate of decay than 1/|t| encountered in the ideal Nyquist channel. ^ Let a represent the estimate of the original pulse ak as k conceived by the receiver at time t=kTb ^ ^ ak = ck − ak −1 Decision feedback : technique of using a stored estimate of the previous symbol Propagate : drawback, once error are made, they tend to propagate through the output Precoding : practical means of avoiding the error propagation phenomenon before the duobinary coding 5
6. 6. Correlative-Level CodingDuobinary Signalingd k = bk ⊕ d k −1  symbol 1 if either symbol bk or d k −1 is 1dk =   symbol 0 otherwise {dk} is applied to a pulse-amplitude modulator, producing a corresponding two-level sequence of short pulse {ak}, where +1 or –1 as before ck = ak + ak −1  0 if data symbol bk is 1 ck =  ±2 if data symbol bk is 0 6
7. 7. Correlative-Level CodingDuobinary Signaling |ck|=1 : random guess in favor of symbol 1 or 0 If | ck |< 1, say symbol bk is 1 If | ck |> 1, say symbol bk is 0 7
8. 8. Correlative-Level CodingModified Duobinary Signaling Nonzero at the origin : undesirable Subtracting amplitude-modulated pulses spaced 2Tb second ck = ak + ak −1 H IV ( f ) = H Nyquist ( f )[1 − exp(− j 4π fTb )] = 2 jH Nyquist ( f ) sin(2π fTb ) exp(− j 2π fTb )  2 j sin(2π fTb ) exp(− j 2π fTb ), | f |≤ 1/ 2Tb H IV ( f ) =   0, elsewhere sin(π t / Tb ) sin[π (t − 2Tb ) / Tb ] hIV (t ) = − π t / Tb π (t − 2Tb ) / Tb 2Tb2 sin(π t / Tb ) = π t (2Tb − t ) 8
9. 9. Correlative-Level CodingModified Duobinary Signaling precoding d k = bk ⊕ d k − 2  symbol 1 if either symbol bk or d k − 2 is 1 =  symbol 0 otherwise 9
10. 10. Correlative-Level CodingModified Duobinary Signaling |ck|=1 : random guess in favor of symbol 1 or 0 If | ck |> 1, say symbol bk is 1 If | ck |< 1, say symbol bk is 0 10
11. 11. Correlative-Level CodingGeneralized form of correlative-level coding |ck|=1 : random guess in favor of symbol 1 or 0 Type of N w0 w1 w2 w3 w4 comments class I 2 1 1 Duobinary II 3 1 2 1 III 3 2 1 –1 IV 3 1 0 –1 Modified V 5 -1 0 2 0 -1 N −1  t  h(t ) = ∑ wn sin c  − n  n  Tb  11
12. 12. Baseband M-ary PAM Trans.  Produce one of M possible amplitude level  T : symbol duration  1/T: signaling rate, symbol per second, bauds  Equal to log2M bit per second  Tb : bit duration of equivalent binary PAM : T = Tb log 2 M  To realize the same average probability of symbol error, transmitted power must be increased by a factor of M2/log2M compared to binary PAM 12
13. 13. Tapped-delay-line equalization Approach to high speed transmission  Combination of two basic signal-processing operation  Discrete PAM  Linear modulation scheme The number of detectable amplitude levels is often limited by ISI Residual distortion for ISI : limiting factor on data rate of the system 13
14. 14. Tapped-delay-line equalization Equalization : to compensate for the residual distortion Equalizer : filter  A device well-suited for the design of a linear equalizer is the tapped- delay-line filter  Total number of taps is chosen to be (2N+1) N h(t ) = ∑ w δ (t − kT ) k =− N k 14
15. 15. Tapped-delay-line equalization P(t) is equal to the convolution of c(t) and h(t) N p(t ) = c(t ) ∗ h(t ) = c(t ) ∗ ∑ w δ (t − kT ) k =− N k N N = ∑ w c(t ) ∗ δ (t − kT ) = ∑ w c(t − kT ) k =− N k k =− N k nT=t sampling time, discrete convolution sum N p (nT ) = ∑ w c((n − k )T ) k =− N k 15
16. 16. Tapped-delay-line equalization Nyquist criterion for distortionless transmission, with T used in place of Tb, normalized condition p(0)=1 1, n = 0 1, n=0 p (nT ) =  = 0, n ≠ 0 0, n = ±1, ± 2, .....,± N Zero-forcing equalizer  Optimum in the sense that it minimizes the peak distortion(ISI) – worst case  Simple implementation  The longer equalizer, the more the ideal condition for distortionless transmission 16
17. 17. Adaptive Equalizer The channel is usually time varying  Difference in the transmission characteristics of the individual links that may be switched together  Differences in the number of links in a connection Adaptive equalization  Adjust itself by operating on the the input signal Training sequence  Precall equalization  Channel changes little during an average data call Prechannel equalization  Require the feedback channel Postchannel equalization synchronous  Tap spacing is the same as the symbol duration of transmitted signal 17
18. 18. Adaptive Equalizer Least-Mean-Square Algorithm Adaptation may be achieved  By observing the error b/w desired pulse shape and actual pulse shape  Using this error to estimate the direction in which the tap-weight should be changed Mean-square error criterion  More general in application  Less sensitive to timing perturbations an : desired response, en : error signal, yn : actual response Mean-square error is defined by cost fuction ε = E en  2   18
19. 19. Adaptive Equalizer Least-Mean-Square Algorithm Ensemble-averaged cross-correlation ∂ε  ∂e   ∂y  = 2 E en n  = −2 E en n  = −2 E [ en xn − k ] = −2 Rex (k ) ∂wk  ∂wk   ∂wk  Rex (k ) = E [ en xn − k ] Optimality condition for minimum mean-square error ∂ε =0 for k = 0, ± 1,...., ± N ∂wk 19
20. 20. Adaptive Equalizer Least-Mean-Square Algorithm Mean-square error is a second-order and a parabolic function of tap weights as a multidimentional bowl-shaped surface Adaptive process is a successive adjustments of tap-weight seeking the bottom of the bowl(minimum value ε min ) Steepest descent algorithm  The successive adjustments to the tap-weight in direction opposite to the vector of gradient ∂ε / ∂wk )  Recursive formular (µ : step size parameter) 1 ∂ε wk (n + 1) = wk (n) − µ , k = 0, ± 1,...., ± N 2 ∂wk = wk (n) − µ Rex (k ), k = 0, ± 1,...., ± N 20
21. 21. Adaptive Equalizer Least-Mean-Square Algorithm Least-Mean-Square Algorithm  Steepest-descent algorithm is not available in an unknown environment  Approximation to the steepest descent algorithm using instantaneous estimate ) Rex (k ) = en xn − k ) ) wk (n + 1) = wk (n) + µ en xn − k  LMS is a feedback system  In the case of small µ, roughly similar to steepest descent algorithm 21
22. 22. Adaptive Equalizer Operation of the equalizer Training mode  Known sequence is transmitted and synchorunized version is generated in the receiver  Use the training sequence, so called pseudo-noise(PN) sequence Decision-directed mode  After training sequence is completed  Track relatively slow variation in channel characteristic Large µ : fast tracking, excess mean square error 22
23. 23. Adaptive Equalizer Implementation Approaches Analog  CCD, Tap-weight is stored in digital memory, analog sample and multiplication  Symbol rate is too high Digital  Sample is quantized and stored in shift register  Tap weight is stored in shift register, digital multiplication Programmable digital  Microprocessor  Flexibility  Same H/W may be time shared 23
24. 24. Adaptive Equalizer Decision-Feed back equalization Baseband channel impulse response : {hn}, input : {xn} yn = ∑hk xn −k k = h0 xn + ∑hk xn −k + ∑hk xn −k k <0 k >0 Using data decisions made on the basis of precursor to take care of the postcursors  The decision would obviously have to be correct 24
25. 25. Adaptive EqualizerDecision-Feed back equalization  Feedforward section : tapped- delay-line equalizer  Feedback section : the decision is made on previously detected symbols of the input sequence  Nonlinear feedback loop by decision device ) (1) ) (1) ) (1)  wn   xn  wn +1 = wn +1 − µ1en xn cn =  ) (2)  vn =  )  en = an − cn vn T ) (2) ) (2) )  wn   an  wn +1 = wn +1 − µ1en an 25
26. 26. Eye Pattern Experimental tool for such an evaluation in an insightful manner  Synchronized superposition of all the signal of interest viewed within a particular signaling interval Eye opening : interior region of the eye pattern In the case of an M-ary system, the eye pattern contains (M-1) eye opening, where M is the number of discreteamplitude levels 26
27. 27. Eye Pattern 27