Accurate Statistical Analysis of High-Speed Links with PAM-4 Modulation

Transcription

1 EPEPS IBIS Summit San Jose, California October 8, 05 Accurate Statistical Analsis of High-Speed Links with PAM-4 Modulation Vladimir Dmitriev-Zdorov Mentor Graphics Corp.,

2 Wh statistical analsis of SERDES links? Unlike bit-b-bit analsis, statistical simulation can evaluate ver low BER (e- to e-5 and beond) in a reasonable time First statistical/worst case techniques appeared decades ago in Communication Electronics, however, the do not meet the requirements of the modern technolog Among other modern approaches, [,] described the framework that allows accurate statistical analsis of LTI channels with transmit and receive jitter and edge asmmetr. In [3], it was shown that the method can also consider data-dependent jitter, a set of datadependent edge transitions, and correlated input patterns. All that was applied to PAM- (NRZ) modulation. In this work, we generalize these results on PAM-4 modulation [] M. Tsuk, D. Dvorscsak, C.S. Ong, and J. White, An electrical-level superimposed-edge approach to statistical serial link simulation, IEEE/ACM International Conf. on Computer-aided Design Digest of Technical Papers, pp , 009. [] V. Stojanovic, M. Horiwitz, Modeling and analsis of high-speed links, Proc. IEEE Custom Integrated Circuit Conf., pp , Sept 003 [3] V. Dmitriev-Zdorov, Accurate statistical analsis of SERDES links considering correlated input patterns, data-dependent edge transitions, and transmit jitter, DesignCon 05. e-7 e- Bit-b-bit simulation Confidence worsens with fewer samples available at lower probabilit Statistical analsis

3 Major elements of statistical analsis (general formulation), The signal at the output of a linear channel can be represented as n ( t RX ) ( bk bk ) S( t kt TX, k k ) Where T - smbol interval, b k, b k- are smbol values which are {-,} for NRZ and {-, - /3, /3, } for PAM4; S(t) is the channels step response, and TX, RX are transmit and receive jitter. If statistical properties of the input pattern (i.e. distribution of smbol coefficients) and jitter doesn t change in time, (t) is a random variable whose distribution is T-periodic in time. Hence, it can be characterized b a collection of M> PDF or CDF functions. Each one should be computed for t=nt m, m [0,T]. For ever m, the problem reduces to finding na single distribution of the variable ) ( b b ) S ( ), n( m k k n k TXk, k with S S[( n kt ) x] n k m 3

4 In absence of transmit jitter, the samples of the step response are discrete values. The set of samples is defined b the timing offset m within unit interval. S(t) Major elements of statistical analsis (samples of the step response) With random transmit jitter, the samples become random values themselves, with their individual distributions (PDFs shown in green). P(t)=S(t)-S(t-T bit ) 0 T bit In presence of deterministic Tx jitter, the samples or PDFs should be taken at unequal distances: Sine jitter: Dut-ccle distortion: S(t) S(t) 0 0 4

5 Basic computation flow for PAM- For the selected set of samples or vertical PDFs, the cross-section of an ee can be built b a series of shifts (or convolutions) and summations. The samples, k= Q, of the edge/step response are taken in reverse S order. Q-k (-S ) A A No random Tx jitter A Q A 0.5 k A0 k 0.5 A k A0 k (S ) A0 A0 A0 Q A k A k Ee densit plot With random Tx jitter A0 k A0 k k= k=3 k=5 k=0 k=30 5

6 Basic computation flow for PAM-4 The same samples of the step response or same vertical PDFs are used in a 4-level structure (-S ) A3 (/ a )V Q-k (/a) A3 Without random Tx jitter, the shifts of PDFs between the logical states happen in both directions, with increments equal (/3)S Q-k. (-S /3) (S /3) (S ) A A A0 A A A0 With random Tx jitter, PDFs are modified b wa of convolution with scaled/normalized vertical PDFs V Q-k (). Such scaling/normalization is made b appling the factor a, so that V Q-k () becomes (/ a )V Q-k (/a). Negative a means verticall flipped PDF, larger a means expanding along -axis. As in the discrete case, depending on the direction and the magnitude of the transition, a can be /3, 4/3, and. 6

7 What effects can we simulate statisticall? Random transmit (directl) and receive jitter (added b convolution), with different PDFs Deterministic transmit jitter (b uneven sampling of the step response) and receive jitter (b convolution of resulting ee/ber) Different shape of the edges (rising/falling asmmetr for PAM-, or 6 PAM-4 transition edges with different shapes) Can we consider the effects that are data-dependent, such as correlation of the input bit values, data-dependent jitter and edge transitions)? 7

8 Simulation flow as a multistep Markov chain: consider dependenc on more preceding bits, and data-dependent edge transitions. Example of PAM- modulation, N=3. (/)v Q-k (/) (/)v Q-k (-/) A,k z 0 A,k z 7 A 0,k z A 0,k z 6 A 0,k z 5 z A 0,k A 00,k z 4 z 3 A 00,k A 0,k z 3 z 4 A 0,k A 00,k z z 5 A 00,k A 00,k z z 6 A 00,k A 000,k z 0 z 7 A 000,k 8

9 Same with PAM-4 modulation, N= A 33,k A 3,k A 3,k A 30,k A 3,k A,k A,k A 0,k A 3,k A,k A,k A 0,k A 03,k A 0,k A 0,k A 00,k z 4 z 5 z 8 z 6 z 9 z z 7 z 0 z 6 z3 z z 4 5 z z 5 z 4 z 3 z z9 0 z 8 z 7 z 6 z 5 z 4 z 3 z z z 0 (/ a )V Q-k (/a) z 0 z z z 3 A 33,k A 3,k A 3,k A 30,k A 3,k A,k A,k A 0,k A 3,k A,k A,k A 0,k A 03,k A 0,k A 0,k A 00,k In this structure, we can consider identical or individual transition edges. In the last case, all transitions should be evaluated, assuming dependence on the current bit b n and a number of previous bits b n-, b n-. Convolution should be performed with the vertical PDF v a Q k b n ( / ) b a n, bn There are 64 transition probabilit factors z i total. With statistical smmetr, their number is halved. Among those 3, onl 4 are independent (considering 8 normalization conditions). With 4 independent correlation factors measured, we build a sstem of equations and solve it for unknown transition probabilities. 9

10 Statistical Ee Diagrams, PAM-4 modulation, N= PRBS5->Gra->PAM-4, Identical transitions N= Vertical cross-sections of all three PRBS5->Gra->PAM-4, Different transition edges, N= Uncorrelated binar input ->Gra->PAM-4, Different edges, N= 0

11 Sampling thresholds for PAM-4: we can find them from peak distortion Assume:. Identical transition edges. Uncorrelated input pattern (or correlated with statistical smmetr) 3. No jitter, noise, crosstalk and no non-linear equalization. Then the vertical PDFs at different levels can be described in terms of the pulse response samples: 3 pk bk pk k K pk bk pk 3 k K pk bk pk 3 k K 0 pk bk pk k K Here, b k ={, /3} are discrete random variables x sampling position within UI p k (x) cursors of the pulse response for a given offset

12 Sampling thresholds for PAM-4: PAM4_UpperThreshold First, we find the threshold between levels 3 and (PAM4_UpperThreshold). Let s find the worst margins for the distributions at these levels. These are: 3 _ wrst( x) pk pk k K pk _ wrst () x pk () x () x 3 kk Median 3 [ 3_ wrst( x) _ wrst( x)] pk 3 VertEeSiz e 3 [ 3_ wrst( x) _ wrst( x)] pk pk 3 k K As we see, the optimal sampling threshold is a function of the horizontal offset. Its dependence is proportional to the shape of the pulse response (near its peak)

13 The mutuall-worst margins for levels and are: Sampling thresholds for PAM-4: PAM4_CenterThreshold pk _ wrst () x pk () x () x 3 kk pk _ wrst () x pk () x () x 3 kk Median [ _ wrst () x _ wrst ()] x 0 (As expected) VertEeSiz exxx px p kx ()[ _ wrst () _ wrstk ()] () ( ) 3 kk (Same as before) 3

14 The mutuall-worst margins for levels and 0 are: Sampling thresholds for PAM-4: PAM4_LowerThreshold pk _ wrst () x pk () x () x 3 kk pk ( _0 wrst () x pk () x x ) kk Median 0 [ _ wrst () x _0 wrst ()] x pk () x 3 (Opposite to upper threshold) VertEeSiz exxx px p kx 0 ()[ _ wrst () _0 wrstk ()] () ( ) 3 kk (Same as before) 4

15 Sampling thresholds for PAM-4: Can be found from the pulse response of the channel. Ee median (and the optimal sampling threshold) is a function of the sampling position x 0. It can be found as 3 p K ( x 0 ). Under assumed conditions, the vertical openings for all 3 ees are identical. 3. For LTI smmetrical channels and no impairments, the optimal threshold values can be determined from the single bit response (can be found from equalized impulse response of the channel). 5

16 Sampling thresholds for PAM-4: Can be found from the pulse response of the channel, but Uncorrelated crosstalk and noise will not change the relationships found for the ee median and margins (medians still repeat the shape of the pulse response, vertical ee openings are same in the vertical cross-section But, Tx and Rx jitter ma invalidate them (the top and bottom ees are most affected) Even for the balanced ee opening, we ma decide to modif upper or lower sampling threshold, if the timing margin is tighter than the voltage allowance. For example, in the case below we trade some of the vertical space for improved horizontal margin (green versus red sampling) 6

17 Sampling thresholds for PAM-4: For non-lti channels, we can build vertical histogram For non-lti channels, we use bit-b-bit analsis and call GetWave function. If each call processes sufficient number of bits (e.g. -4 thousand), Rx AMI DLL (or EDA tool) can build a vertical histogram from the waveform sampled at the selected location - e.g. dictated b clock times - and find the best threshold values b analzing this histogram. Or, it can find thresholds and offsets from other considerations, based on specific knowledge about device/slicers When simulation advances, the thresholds are updated from the modified waveform etc. 7

18 Finding SER and BER from statistical analsis In statistical analsis, accurate SER and BER evaluation is possible if we store/keep the branches of the ee densit plot separatel. Then, for a given sample position we can find the source and effect of the error and convert SER into BER SER = P(3)[P( 3)P( 3)P(0 3)] P()[P(3 )P( )P(0 )] P()[P(3 )P( )P(0 )] P(0)[P(3 0)P( 0)P( 0)]. We can convert SER into BER if we know the mapping (e.g. Gra code): 0 <=> 00 <=> 0 <=> 3 <=> 0 BER = P(0)[P( 0)*P(0 0)*P(00 0)*] P()[P(0 )*P(0 )*P(00 )*] P(0)[P(0 0)*P( 0)*P(00 0)*] P(00)[P(0 00)*P( 00)*P(0 00)*] Each of the four summands in SER and BER can be found from considering one branch of the ee densit

19 The effects we can consider in statistical simulation with PAM-4 modulation Accurate consideration of ISI and crosstalk Transmit jitter, random and deterministic, and receive jitter Edge asmmetr, or dependence of the edge shapes on a number of preceding bits Correlated input pattern The effect of data-dependent jitter and noise Simulation is fast: tpicall takes a minute or less The branches of ee densit are formed separatel, this allows accurate BER evaluation PAM-4 thresholds are new for IBIS-AMI in IBIS Version 6. 9