Method for analytically calculating BER (bit error rate) in presence of non-linearity. Gaurav Malhotra Xilinx

Transcription

1 Method for analytically calculating BER (bit error rate) in presence of non-linearity Gaurav Malhotra Xilinx

2 Outline Review existing methodology for calculating BER based on linear system analysis. Link model with ISI, Crosstalk, Jitter, Noise. Model of nonlinearity based on power series. Modification of PDF in presence of nonlinearity. BER results for a typical high speed link. Link model with multiple linear & NL blocks.

3 Linear system : Link Model Crosstalk AFE (CTLE + DFE) d d [+1-1] Tx Channel Rx - RJ - DCD - PSIJ - SJ - Jitter enhancement - ISI / reflections AWGN - RJ - DCD - PSIJ - SJ [+1-1] Equivalent linear model: Signal and impairments can be referred to the slicer input. d d [+1-1] h(n) Tx + Channel + RxAFE Signal(t) Joint pdf (per sampling phase): pdf(signal) pdf(awgn ) pdf(xtalk ) pdf(isi) Crosstalk Crosstalk(t) AWGN n(t) x(t) - RJ - DCD - PSIJ - SJ [+1-1] Objective is to determine joint PDF F X x of signal + impairments [ x(t) ] at the decision point. BER k = P error = k P error d k P(d k ), where P error 1 = F X x 1 SS S Taking timing jitter into account : BER = BER k F k k

4 LTI Systems: Link BER methodology Joint pdf for Each phase Voltage noise Bath Tub curve Timing noise Joint pdf (per sampling phase): pdf(awgn) pdf(xtalk) pdf(isi) Joint pdf : pdf(rj) pdf(sj).. Conditional pdf BER = BER k F k k

5 Recap goal If we can accurately determine the probability distribution at the decision point, we can calculate BER. BER k = P error = k P error d k P(d k ), where SS P error 1 = F X x 1 S Taking timing jitter into account: BER = BER k F k k GOAL: to determine PDF (overall/joint including all impairments AND nonlinearity) at the decision point.

6 Modeling of nonlinearity Common model of NL: Y = n X n n Observed to be very close to real circuits. I N P U T Actual Circuit / System LTI system Model LTI system Model Volterra series Model X Power series polynomial Y = O U T P U T n n X n

7 Modeling of nonlinearity Input Circuit model (Known ) Actual Circuit / System Y M = [X 1 X 2 X n ] Linear Model H(f) Design specification (Known ) X NL (?) Y = n n X n 1 n No NL modeling = M 1 Y Error = Y Y y 2 / error 2 (db) Up to 3 rd order Up to 5 th order Up to 7 th order Design specification (say pole-zero model) is known. Input, X, Y, Y are time domain signals. Only NL terms { n } are unknown. Matrix inversion (zero forcing) though not optimum, but gives a good estimate of NL terms.

8 Modeling of nonlinearity No NL modeling Up to 3 rd order Up to 5 th order No NL modeling Up to 3 rd order Up to 5 th order Up to 7 th order Adding higher order terms in estimation reduces error in modeling due to NL. Up to 7 th order

9 Modification of PDF in presence of NL Let y = g(x) represent the output of a non-linear function whose input is x. The PDF of Y, F Y (y) can be determined in terms of PDF of X as: [Probability, Random variables and Stochastic Processes: Athanasios Papoulis, Section 5-2] x g(x) y y g(x 1 ) g(x 2 ) g(x 3 ) pdf of x = F X x x g(x) x 1 x 2 x 3 pdf of y = F Y y = F X x 1 g x 1 + F X x 2 g x 2 + F X x n g x n Simplification for monotonic functions: F Y y = dx dy F X x

10 Modification of PDF : AWGN Example Y= X + X 3 F Y y = dx dy F X x d d [+k -k] NL [+1-1] AWGN X= Signal +AWGN F X x = pdf(signal) pdf(awgn) Note the warping of PDF in accordance with dx dy

11 AWGN Example : Simulation VS Analysis Y = X + X 3 F Y y = dx F dy X x 1 = 1+3 X 2 F X x d [ ] NL d [ ] AWGN X= Signal +AWGN F X x = pdf(signal) pdf(awgn) PDF can be obtained analytically or by running a bit-by-bit simulation. Both methods give the same result. Analytically computing BER is much faster. This is the method we will adopt for this presentation.

12 AWGN Example : PAM2 VS PAM4 In general we expect higher order modulations to suffer more from NL. Outer points in constellation dominate BER. *** Detection rule may be modified to take advantage of (known) nonlinearity. This paper assumes that same detection rule (minimum distance) as is used for linear system analysis is used for calculating BER in presence of non-linearity.

13 Link Model: typical high speed link Equivalent model with NL: Linear components convolve. d [+1-1] TX + package + connector Tx + package + card + connector + CTLE Crosstalk AWGN Crosstalk CTLE Y= X + X 3 F Y y NL = dx dy F X x d d [+1-1] Channel NL [+1-1] AWGN - RJ [UI/64] (rms) - DCD [UI/32] X= Signal +AWGN F X x = pdf(signal) pdf(awgn) pdf(xtalk) pdf(isi) CTLE (Analog front end) is a significant source of NL. = -0.3; Output = Input 0.3 * Input 3 CTLE output referred Xtalk : Xtalk_Out(f) = Xtalk_In(f) * CTLE(f)

14 Link Model: typical high speed link d [+1-1] TX + package + connector Crosstalk AWGN CTLE NL Y = X + X 3 BASELINE: PAM2 VS PAM4 Start with the same BER, compare the effect of NL PAM-4 PAM-2 Bandwidth of insertion loss, crosstalk, AWGN and CTLE for PAM2 are half that of PAM4. Jitter is specified as a fraction of UI, so that automatically adjusts for signaling rate. Since the crosstalk channel is not flat, we had to make small adjustment on gain of crosstalk channel to make the baseline BER (without NL) the same for both PAM2 & PAM4.

15 BER results: typical high speed link Bandwidth (Nyquist) UI BER without NL BER with NL PAM2 F N 1/(2* F N ) 1e-25 1e-23 PAM4 F N / 2 2/(2* F N ) 1e-25 1e-20

16 Link Model: Multiple NL blocks d Tx + package + card + connector + CTLE Crosstalk F Y y = dx dy F X x F A a = F Dfe dfe 1 F Y y 1 Summer F B b (LTI method) F Z z = db dz F B b d [+1-1] LTI1 NL1 LTI3 NL2 [+1-1] AWGN X= Signal +AWGN F X x = pdf(signal) pdf(awgn) pdf(xtalk) pdf(isi) LTI2 d [+1-1] F Dfe dfe DFE F Dfe dfe 1 = {Tap1 x pdf(d) } {Tap2 x pdf(d) } PDF transformation Linear block : Convolution Nonlinear block : F Y y = F X x 1 g x 1 + F X x 2 g x 2 + F X x n g x n

17 Summary Presented methodology for calculating BER of a link in presence of nonlinearity. Modification of PDF. Static nonlinearity model using power series polynomial considered. Work ongoing to model nonlinearity using Volterra series. Higher order modulations are more susceptible to NL. Quantified the loss for a typical NL, typical high speed link.