advanced topics in circuit design high-speed electrical...

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EE290C - Spring 2004Advanced Topics in Circuit DesignHigh-Speed Electrical Interfaces

Lecture #5Adaptive Equalization in High-Speed LinksVladimir StojanovicStanford University and Rambus Inc.2/3/2004

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System overview

Transmit and Receive Equalization Need to set the equalizer coefficients adaptivelyAlgorithms limited by hardware issues

Very limited precision in the RxTx output swing constraint

The key is minimum hardware cost

Linear transmit equalizer

Decision-feedback equalizer

SampledData

Deadband Feedback taps

Tap SelLogic

TxData

Causaltaps

Anticausal taps

Channel

2

3

Agenda

System overview/requirementsStandard (unconstrained) adaptive equalizationHardware constraint driven adaptationAlternative algorithms/cost functions

4

Adaptive filteringMany algorithms to choose from

Steepest-descentLMS, RLSInterior point method based (most recent, fastest, handle

non-linear cost functions)Typically ranked by convergence speed

Our channel has VERY slow changes so o.k. to use simple, slow algorithms

The key is hardware simplicityLeast Mean Squares (LMS) is one of the simplest algorithms (B. Widrow)

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5

Goal is to minimize the Mean-Square Error - E(e(n)2 )

Follow the negative gradient of the mean-square error (linear in w)Actually, approximate the mean with instantaneous value

-x (n-∆)

-e (n)x (n)

equalizer wchannel Pu (n)

u (n), e (n)

)(ˆ nx

Adaptive linear Rx EQ

)(2

21 nw

wnn eEww ∇−=+

µ

)(2

)(22

)(ˆ)()()(),()(ˆ

1

21

2

nueww

eww

nueweee

nxnxenxPnunxwnx

nwnn

nww

nn

nn

nnw

n

T

µ

µ

+=

∇−=

−=∂∂=∇

−∆−===

+

+

P

6

Equalizer adapts until received signal u(n) and error e(n) are orthogonal, and any further update is useless.Problem

How to apply the algorithm to our architecture? (transmit equalizer)

-x (n-∆)

-e (n)x (n)

equalizerchannelu (n)

u (n), e (n)

)(ˆ nx

Adaptive linear Rx EQ

nnwnn ueww µ+=+1

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7

Adaptive Tx equalizer

How do we generate the error signal e(n)?Where do we get u(n)?

-x (n-∆)

-e (n)x (n)

channelequalizery (n)

u (n), e (n)back channel

)(ˆ nx

8

Agenda

System overview/requirementsStandard (unconstrained) adaptive equalizationHardware constraint driven adaptation

Obtaining the error signalAlternative algorithms/cost functions

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Error signal

Hard to generate high-resolution error signalUse the sign-sign variant of the algorithm

Ref. Level (dLev)

Initial eye

errorinitp-p

nn xdLeve −=

nx

slicerthreshold

nx

)( nesign

dLev

slicerthreshold

)( nxsign

dLev

)()(1 nnwnn usignesignstepww +=+

10

Fully parallel error generation (2PAM)

-dlevel

+dlevel

0

0

1

2

)ˆ(xsign

)( Lesign

)( Hesign

x̂

)(xsign

1 0

1

0

)(errorsign

)(datasign

on “live” data

Intput data is 2PAMThresholds of samplers 0 and 2 (+/- dLevel) at reference levels of the received 2PAM signal Samplers 0 and 2 evaluate the signal error with respect to the reference level (dLevel) settingSampler 1 determines the sign of the received data (which is also the received data itself)

2 extra samplers (0,2)

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11

Fully parallel error generation (4PAM)

4PAM data, need 4 extra samplersIn both cases, huge sampler overhead

-2dlevel/3

+2dlevel/3

0

0

1

2

Ther

mo-co

ded s

ymbo

l valu

e

x̂

-dlevel/3

+dlevel

+dlevel/3

x̂

-dlevel

Erro

r sign

s

Extra 4 samplers (4PAM example)

12

Serialize to minimize the overhead

0 0.5 1 1.5

x 10-10

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

ERR

MSB

x̂

)ˆ(xSign

)( neSign

Just use one reference samplerFilter the updates based on the received data

Data filtering effectively creates an indicator function added to the LMS adaptive algorithm:

ILMS covers the positive halfplane

dlevel_00

0

)ˆ()()0(

1 nnwLMSnn

nLMS

xsignesignstepIwwxI

⋅+=≥=

+

)

00

10

0

7

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Agenda


Obtaining the error signalMissing signal in Tx Equalization u(n)

Alternative algorithms/cost functions

14

Solution for u(n) – (1) Alternate equalized and unequalizedtransmission

Channel response is pre-computed by the channel itself Simple update:

-x (n-∆)

-e (n)channelx (n)

u (n)back channel

)(ˆ nx

-x (n-∆)

-e (n)x (n)


u (n), e (n)back channel

)(ˆ nx

)()(1 nnwnn usignesignstepww +=+

Requires synchronization (of data pattern & sample point) when switching between Phase 1 and Phase 2.

Significant overhead for synchronization.

Phase 1Send the data sequence through unequalized channel to get u(n)

Phase 2Repeat the data sequence through equalized channel to get e(n), and update the equalizer.

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Solution for u(n) – (2) Give up MMSE and do ZF only

Simple update:Use the instead of for ZF adaptation from the start

sign( ) can have a high BER in the early part of adaptationNeed to use block averaging before the update

)ˆ()(1 nnwnn xsignesignstepww +=+

nx̂ nu

-x (n-∆)

-e (n)x (n)


, e (n)back channel

)(ˆ nx

Decision directed equalization

nx̂

)(ˆ nx

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Agenda


Obtaining the error signalMissing signal in Tx Equalization u(n)Transmit output swing constraint


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Transmitter output swing constraint

Output swing constraintSymbol is received attenuated

Need the second loop toAmplify the received symbol to restore the original value Adjust the expected symbol level

Need to normalize equalizer taps after each update

-e (n)x (n) channelequalizery (n)

u (n), e (n) back channel

output swing

)(ˆ nxg ⋅

)( ∆−− nx

)(ˆ nx g

0 0.5 1 1.5 2 2.5-25

-20

-15

-10

-5

0

frequency [GHz]

Atte

nuat

ion

[dB

]

equalized

unequalized

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Getting around limited gain



output swing

)(ˆ nx

)( ∆−⋅ nxdataLevel

Hard to get ~30-50GHz Gain-Bandwidth product in CMOS Use adaptive data levels

For error generationFor slicer thresholds

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Reference loop

Data level reference loop

The dataLevel loop is similar to that of gain update)ˆ()(1 nngainnn xsignesignstepgg +=+



output swing

)(ˆ nx

)( ∆−⋅ nxdataLevn

e (n) < 0

0

0)(ˆ >nx

dataLevn

)ˆ()(1 nndataLevnn xsignesignstepdataLevdataLev −=+

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Reference data level

… …

dLevinitdLevmid

dLevend

Initial eye Mid-way equalized Equalized

Tracks the mean value of received signal around chosen constellation point

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Dual-loop algorithmReference loop

Equalizer loop

One more thing left to considerMake the algorithm obey the output Tx constraint

)ˆ()(1 nndataLevnn xsignesignstepdataLevdataLev −=+

)ˆ()(1 nnwnn xsignesignstepww +=+

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Peak transmitter output constraint translates to

Straightforward implementation

Perturbs the convergence a little, but not the optimal pointOptimal point for TxEq equalizer is at the maximum output swing ( w l1 norm surface)

Normalizing Tx EQ

1

max1 )(

nnnnn

updatewWupdateww

+⋅+=+

)ˆ()( nnwnxsignesignstepupdate =

max1Www nn <==∑

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Taylor Series Approximation Method

Hardware efficient scaling approximation

)max

()()(

)max

1()(1

,max

max1

1)(

max

max)(

1

max)(1

,max1

WresidualW

nupdatenwnupdatenw

WresidualW

nupdatenwnw

WresidualWif

WresidualWnupdatenw

residualWW

Wnupdatenw

nupdatenw

Wnupdatenwnw

then

WnupdatenwresidualW

let

⋅+−+=

−⋅+≈+

<<

+⋅+=

+⋅+=

+⋅+=+

−+=

(Taylor Series Approximation)

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Hardware implementationThe derived formula can be implemented very efficiently in hardware:

uses binary addition

is right shift by log2(Wmax) bits, as long as Wmax is power of 2. (i.e. 128)

Wresidual/Wmax - right shift by log2(Wresidual/Wmax) bits, if result is a power of 2 Wresidual can be from -5 to +5

Round Wresidual to ±0, 1, 2 or 4. When Wresidual is ±3, we alternate the rounding between ±2 and ±4.

)max

()()(1 WresidualW

nupdatenwnupdatenwnw ⋅+−+≈+

)( nupdatenw +

max

1W

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Peak Power Main Tap Adjustment MethodAll taps, except the main will be updated normally:

wn(i) is the ith tap at time n

Main tap (or 2nd tap) adjustment will only depend on peak output swing.

Alternative hardware efficient scaling

5,4,3,1),()()(1 =+=+ iiupdateiwiw nnn

∆−=

≥∆+=

≤

+

+

)2()2(

)2()2(

1

1

1

1

nn

uppern

nn

lowern

wwWwelseifww

Wwif

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Choose Wupper and Wlower to minimize dithering of the main tap (Wupper - Wlower) expected dithering of ||wn||1.

(e.g. Wupper is set to128, and Wlower is set to 120, for Wmax=128).∆ sets the adjustment step on the main tap.

∆ can be a function of the updates (i.e. ∆n = ||wn||1 – Wmax).

No multiplication required.Transmitter might not utilize peak swing

Maximum gap is Wupper - Wlower

In cases where ∆ is hardcoded, ||wn||1 can go above Wmax during equalizer adaptation

Alternative scaling contd.

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Agenda


Obtaining the error signalMissing signal in Tx Equalization u(n)Transmit output swing constraintConvergence results


28

Adaptive ZF EQ– Results

Convergence example, 5taps Tx EqLeft plot shows the convergence of TX EQ Taps after about 200 updates. Right figure shows the convergence of Data Level.

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29

Dual loop convergence – 3 tap example

Hard to estimate analyticallySims and experimental results show

Both loops are stable within wide range 0.1 – 100x of relative speeds

0 20 40 60 80 1000

20

40

60

80

100

# updates

code

[lsb

]

(a)

(b)

(c)

(a) dLev speed 1x, eq speed 1x (b) dLev speed 10x, eq speed 1x (c) dLev speed 1x, eq speed 10x

dLev learning curve

0 20 40 60 80 100-50

050

100

code

[lsb

]

0 20 40 60 80 100-50

050

100

code

[lsb

]

0 20 40 60 80 100-50

050

100

# updates

code

[lsb

]

(a)

(b)

(c)

equalizer tap learning curves

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Tx Eq. tap + data level adaptation

0 500 1000-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1rawequalized-learning

0 500 1000-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5referenceequalized-last

0 200 400 600 800 1000 1200-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Eq tap learning curve

0 200 400 600 800 1000 1200-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Eq tap learning curve

0 500 1000-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1rawequalized-learning

0 500 1000-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5referenceequalized-last

Dispersion only Dispersion + reflections

Receivedsignal

Eq taps

Reflections are treated as proportional noise –do not affect the eq. tap nor data level adaptation

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Agenda


Min. BER driven adaptation

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Min-BER driven 2PAM AdaptationOptimize equalizer directly for min BERAdaptive version – AMBER – (Yeh & Barry)Main idea – update only when in error

)()(IminBER1 nwnn xsignnstepww +=+

-1

+1

0

)(IminBER n

Received constellation

Very slow for low BER targetsNeed to narrow the indicator funciton

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Implementing Min-BER 2PAM Adaptation

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

trap

MSB

x̂

)ˆ(xSign

For faster updates, add a trap zoneSimilar to data-filtered LMS, the MIN-BER driven 2PAM adaptation uses an indicator function (IMIN-BER).

(With adaptive sampler sitting at trap) This indicator function can be created using data filter of:(MSB = 1) && (ERR = 0)

+trap

0

)()0(

1 nwBERMINnn

nBERMIN

xsignstepIwwtrapxI

)

)

⋅+=≤≤=

−+

−

10

00

0

trap)ˆ( trapxSign −

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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-40

-35

-30

-25

-20

-15

-10

-5

0trap 0.1trap 0.05m mseEqhighNoisem mseEqlowNoise

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-2.5

-2

-1.5

-1

-0.5

0trap 0.1trap 0.05m mseEqhighNoise

Margin [V]

Log10 BER Log10 BER

Margin [V]

sigmaA=50mVsigmaR=5mV sigmaA=50mV

sigmaR=50mV

sigmaA – noise used during adaptationsigmaR – actual noise of the system at which the BER curves are used

Slow convergence for small BER targets, have to use trap zone or add extra noise

Simulations with two trap settings (100mV and 50mV), signal levels around 200mV

Min-BER vs LMS(MMSE)

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35

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02trap 0.1trap 0.05 m mseEqlowNoise

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1-10

-9

-8

-7

-6

-5

-4

-3

-2

-1trap 0.1trap 0.05mm seEq

[V] [V]

Log10 pISI pISI

Residual ISI probability distributionminBER eq algorithm tries to shape the ISI distributions such as to minimize the BER

36

Agenda


Min. BER driven adaptationBalancing data and edge errors

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37

Voltage only equalization – (ZFE)

Clean voltage marginsBad timing marginsZFE causes multi-modal edge ISI jitterMultiple edge histogram peaks => bad for CDRWith jitter on tx and rx clocks voltage margin is not all that matters

0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6

x 1 0- 1 0

-0 . 5

-0 . 4

-0 . 3

-0 . 2

-0 . 1

0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

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ZFE pre-emphasis causes multi-modal edges

While sample points are “clean” of ISI, the ISI shifts almost to quadrature (right at the edges)This impacts the timing of the crossing between the next symbol and the one after next symbol

250 300 350 400 450

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Equalized (g) and Equalized with refl canc. (r) differential pulse response 5..5..5-17..1.6e-010 simdir/simdir52/v0.mdat

ISI peaks that hit the symbol edges

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39

Control both data and edge samples

ZFE “looks” only at data samples and minimizes the errorNeed equalization algorithm that “looks” at both the data and edge samples and minimizes the errorForm a total cost function as:MSEtot=alpha*MSEdata+(1-alpha) *MSEedgeFor equalizer tap calculation need pulse response samples at both data and edge samples

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Error function formulationEasy for data samples:ed(n)= data(n-delay)-data_received(n)For edges, first formulate the target:edge_target(n)=0.5*(data(n-delay)+data(n-delay+1)Then form the error:ee(n)=edge_target(n)-data_received(n-1/2)

ed

ee

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Cost function

MSEtot=alpha*mean(ed2)+(1-alpha)*mean(ee2)

Need to balance weight alpha to tradeoff voltage and timing marginsUltimate answer is to tie alpha to the BER through Txand Rx jitter numbers and other voltage noise

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For ZF (one phase algorithm)Instead of udn, use received data xnInstead of uen, use a filter for edge transitions, i.e. when xn+xn-1=0

)()()()(1

enenwe

dndnwdnn

usignesignstepusignesignstepww

+++=+

Adaptive EQ data and edge

)()(1 dndndnn usignesignstepdataLevdataLev −=+

Simple update for MMSE (two phase algorithm) ud, ue, from phase 1ed, ee from phase 2

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0 0.5 1 1.5 2 2.5 3 3.5

-15

-10

-5

0

GHz

|H| [

dB]

VS5 - H(f) [dB]: channel, goal, eq, eq+channel

o.k. but why does this work?

0 0.5 1 1.5 2 2.5 3 3.5-15

-10

-5

0

GHz|H

| [dB

]

VS5 - H(f) [dB]: channel, goal, eq, eq+channel

The amount of pre-emphasis at sample times that “cleans” the data samples is traded for the error in edge samplesEqualizer is “told” not to attenuate low frequencies as much

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Oversampled frequency response

0 0.5 1 1.5 2 2.5 3

x 109

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Raw (b), Equalized (g) and Equalized with refl canc. (r) frequency response)

0 0.5 1 1.5 2 2.5 3

x 109

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Raw (b), Equalized (g) and Equalized with refl canc. (r) frequency response)

Data ZFE Data-Edge ZFE

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Equalized pulse responses

250 300 350 400 450

0

0.05

0.1

0.15

0.2

0.25


250 300 350 400 450

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18


Data ZFE Data-Edge ZFE

460 0.5 1 1.5

x 10-10

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x 10-10

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Eye diagrams

Eye/schmooW=131/128ps

Eye/schmooH=161/165mV

Eye/schmooH=157/157mV

Eye/schmooW=140/136ps

Bimodaledges Unimodal

edges

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47

Raw pulse response

5.2 5.4 5.6 5.8 6 6.2 6.4 6.6

x 10-9

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

sec

sbR

Raw pulse response 5..5..5-17..1.6e-010 simdir/simdir52/v0.mdat

48

We’ve covered todaySystem overview/requirementsStandard (unconstrained) adaptive equalizationHardware constraint driven adaptation

Obtaining the error signalMissing signal in Tx Equalization u(n)Transmit output swing constraintConvergence results

Alternative algorithms/cost functionsMin. BER driven adaptationBalancing data and edge errors

After we learn about clock and data recovery and other system issues, we will get back to this in the context of link system performance (BER) analysis

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To probe furtherB. Widrow et al,, “Stationary and nonstationary learning characteristics of the LMS adaptive filter,” Proc. IEEE, vol. 64, no. 8, pp. 1151-1162, 1976.V. Stojanović, G. Ginis and M. A. Horowitz, "Transmit Pre-emphasis for High-Speed Time-Division-Multiplexed Serial Link Transceiver," IEEE International Conference on Communications, pp. 1934 -1939, May 2002.J.T. Stonick et al, "An adaptive pam-4 5-Gb/s backplane transceiver in 0.25-µµµµm CMOS," IEEE J. Solid-State Circuits, vol. 38, no. 3, March 2003, pp. 436-443. C-C. Yeh and J. R. Barry, “Adaptive Minimum Bit-Error Rate Equalization for Binary Signaling,” IEEE Transactions on Communications, vol. 48, no. 7, July 2000V. Stojanovic, A. Ho, B. Garlepp, F. Chen, J. Wei, E. Alon, C. Werner, J. Zerbe, and M.A. Horowitz, “Adaptive Equalization and Data Recovery in a Dual-Mode (PAM2/4) Serial Link Transceiver,” submitted to IEEE Symposium on VLSI Circuits, June 2004.

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