Method for analytically calculating BER (bit error rate) in presence - PowerPoint PPT Presentation

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

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.

Linear system : Link Model Crosstalk AFE (CTLE + DFE) d’ d Rx Channel Tx [ + 1 -1] [ + 1 -1] - Jitter ‘enhancement’ - RJ - ISI / reflections AWGN - RJ - DCD - DCD - PSIJ - PSIJ - SJ - SJ Crosstalk’ Crosstalk(t) Equivalent linear model: d’ d • h(n) Signal and impairments [ + 1 -1] [ + 1 -1] can be referred to the Signal(t) x(t) slicer input. Tx + Channel + RxAFE  - RJ’ AWGN’ - DCD’ n(t) - PSIJ’ Joint pdf (per sampling phase): - SJ’ pdf(Signal)  pdf(AWGN’)  pdf(xtalk’)  pdf(ISI) • Objective is to determine joint PDF 𝐺 𝑌 𝑦 of signal + impairments [ x(t) ] at the decision point. 𝑇𝑇 𝐶𝐹𝑆  𝑙 = 𝑄 𝑓𝑠𝑠𝑝𝑠 = 𝑄 𝑓𝑠𝑠𝑝𝑠 𝑒 𝑙 𝑄(𝑒 𝑙 ) , where 𝑄 𝑓𝑠𝑠𝑝𝑠 1 = 𝐺 𝑌 𝑦|1 S • 𝑙 −∞ Taking timing jitter into account : 𝐶𝐹𝑆 = 𝐶𝐹𝑆  𝑙 𝐺  𝑙 • 𝑙

LTI Systems: Link BER methodology Joint pdf for Each phase Bath Tub curve Voltage noise Timing noise Joint pdf (per sampling phase): Joint pdf : pdf(AWGN)  pdf(xtalk)  pdf(ISI) pdf(RJ)  pdf(SJ)  .. Conditional 𝐶𝐹𝑆 = 𝐶𝐹𝑆  𝑙 𝐺  𝑙 pdf 𝑙

Recap goal • If we can accurately determine the probability distribution at the decision point, we can calculate BER. – 𝐶𝐹𝑆  𝑙 = 𝑄 𝑓𝑠𝑠𝑝𝑠 = 𝑄 𝑓𝑠𝑠𝑝𝑠 𝑒 𝑙 𝑄(𝑒 𝑙 ) , where 𝑙 𝑇𝑇 𝑄 𝑓𝑠𝑠𝑝𝑠 1 = 𝐺 𝑌 𝑦|1 S −∞ – Taking timing jitter into account: 𝐶𝐹𝑆 = 𝐶𝐹𝑆  𝑙 𝐺  𝑙 𝑙 • GOAL: to determine PDF (overall/joint including all impairments AND nonlinearity) at the decision point.

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

Modeling of nonlinearity Circuit model (Known  ) Actual Circuit 𝑍 Input 𝑁 = [𝑌 1 𝑌 2 … 𝑌 𝑜 ]   / System  1 ⋮ = 𝑁 −1 𝑍  𝑜 𝑍′ =  𝑜 𝑌 𝑜 𝑌 Linear Model NL 𝑜  H(f) (?) Error = 𝒁 – 𝒁′  y 2 /  error 2 (dB) Design specification (Known  ) Up to 3 rd Up to 5 th Up to 7 th No NL modeling order order order 11 23 46 51 • Design specification (say pole-zero model) is known. Input, X, Y , Y’ are time domain signals. Only NL terms {  𝑜 } are unknown. • • Matrix inversion (zero forcing) though not optimum, but gives a good estimate of NL terms.

Modeling of nonlinearity No NL modeling Up to 3 rd order Up to 5 th order Up to 3 rd Up to 5 th Up to 7 th No NL modeling order order order 11 23 46 51 • Adding higher order terms in estimation reduces error in modeling due to NL. Up to 7 th order

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 ] g(x) y x y g(x 2 ) g(x 1 ) g(x 3 ) x x 3 x 1 x 2 pdf of y = pdf of x = g(x) 𝑍 𝑧 = 𝐺 |𝑕′ 𝑦 1 | + 𝐺 𝑌 𝑦 1 |𝑕′ 𝑦 2 | + ⋯ 𝐺 𝑌 𝑦 2 𝑌 𝑦 𝑜 𝐺 𝑌 𝑦 𝐺 |𝑕′ 𝑦 𝑜 | 𝑇𝑗𝑛𝑞𝑚𝑗𝑔𝑗𝑑𝑏𝑢𝑗𝑝𝑜 𝑔𝑝𝑠 𝑛𝑝𝑜𝑝𝑢𝑝𝑜𝑗𝑑 𝑔𝑣𝑜𝑑𝑢𝑗𝑝𝑜𝑡: 𝑒𝑦 𝐺 𝑍 𝑧 = | 𝑒𝑧 | 𝐺 𝑌 𝑦

Modification of PDF : AWGN Example Y= X +  X 3 𝑒𝑦 𝐺 𝑍 𝑧 = | 𝑒𝑧 | 𝐺 𝑌 𝑦 d’ d NL [ + k -k] [ + 1 -1] AWGN X= Signal +AWGN 𝑌 𝑦 = pdf(Signal)  pdf(AWGN) 𝐺 Note the ‘warping’ of PDF in accordance 𝑒𝑦 with | 𝑒𝑧 |

AWGN Example : Simulation VS Analysis Y = X +  X 3 𝑒𝑦 𝐺 𝑍 𝑧 = | 𝑒𝑧 | 𝐺 𝑌 𝑦 1 = | 1+ 3  X 2 | 𝐺 𝑌 𝑦 d’ d NL [+0.5 -0.5] [+0.5 -0.5] AWGN X= Signal +AWGN 𝐺 𝑌 𝑦 = 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.

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) non- linearity. 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.

Link Model: typical high speed link Crosstalk’ TX + d package + CTLE NL [ + 1 -1] connector AWGN’ Y= X +  X 3 𝑒𝑦 𝐺 𝑍 𝑧 = | 𝑒𝑧 | 𝐺 𝑌 𝑦 Crosstalk Tx + package + card + Equivalent model with NL: connector + CTLE • Linear components d d’ NL Channel convolve. [ + 1 -1] [ + 1 -1] AWGN - RJ [UI/64] (rms) - DCD [UI/32] X= Signal +AWGN 𝑌 𝑦 = pdf(Signal)  pdf(AWGN) 𝐺  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)

Link Model: typical high speed link Crosstalk’ TX + d package + CTLE NL [ + 1 -1] connector Y = X +  X 3 AWGN’ 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.

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

Link Model: Multiple NL blocks 𝐺 𝑍 𝑧 = 𝐺 𝑎 𝑨 = 𝐺 𝐶 𝑐 𝑒𝑦 | 𝑒𝑧 | 𝐺 𝑌 𝑦 𝐺 𝐵 𝑏 = 𝑒𝑐 | 𝑒𝑨 | 𝐺 𝐶 𝑐 (LTI method) 𝐺 𝐸𝑔𝑓 𝑒𝑔𝑓| − 1  𝐺 𝑍 𝑧|1 Crosstalk Tx + package + card + Summer connector + CTLE d d’ NL1 LTI3 NL2 LTI1 [ + 1 -1] [ + 1 -1] 𝐺 𝐸𝑔𝑓 𝑒𝑔𝑓 AWGN LTI2 DFE 𝐺 𝐸𝑔𝑓 𝑒𝑔𝑓| − 1 = X= Signal +AWGN 𝑌 𝑦 = pdf(Signal)  pdf(AWGN) 𝐺 {Tap1 x pdf(d) }   pdf(xtalk)  pdf(ISI) {Tap2 x pdf(d) } … d’ [ + 1 -1] PDF transformation • Linear block : Convolution • Nonlinear block 𝐺 𝑌 𝑦 1 𝐺 𝑌 𝑦 2 𝐺 𝑌 𝑦 𝑜 : 𝐺 𝑍 𝑧 = |𝑕′ 𝑦 1 | + |𝑕′ 𝑦 2 | + ⋯ |𝑕′ 𝑦 𝑜 |

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.