System Identification Approachapplied to

Drift Estimation.

Vrije Universiteit BrusselFaculteit IngenieurswetenschappenVakgroep ELECPleinlaan 2, B-1050 Brussels, Belgium

Frans VerbeystRik PintelonYves RolainJohan SchoukensTracy Clement

2

Motivation = phase calibration

LSNA phase calibration

requires

requires

a calibrated oscilloscope

a calibrated HPR

3

Sampling oscilloscopes “reality”

drift

jitter

mismatches, connector saver

time base errors

distortiontk k.t

vertical errors

offset

nonlinearity

dynamics

measure,estimate,compensate

vertical cal plug-in

avoid: usesmall signal

measure and compensate

nose2noseEOS

4

Enhanced drift estimation

old:

new:

5

200 400 600 800freqGHz0.1

0.2

0.3

0.4

0.5varimx106 V2

freq (GHz)

var(im) (x10-6 V2)

200 400 600 800freqGHz0.1

0.2

0.3

0.4

0.5varrex106 V2

freq (GHz)

var(re) (x10-6 V2)

10 20 30 40 50 60 70 80freqGHz

0.4

0.2

0.2

0.4

re,im(re,im)

freq (GHz)

Circular complex noise ?

5000 realizations(jitter only, stdev 1 ps)

10 11 12 13 14remV

10.750.5

0.25

0.250.5

0.751

immVim (mV)

re (mV)

9.5 10 10.5 11remV

1

2

3

immV

re (mV)

im (mV)

5.5 6 6.5 7 7.5remV

4.5

3.5

3

2.5

2

1.5

1

immVim (mV)

re (mV)

1.5 1 0.5 0.5 1 1.5remV

1.5

1

0.5

0.5

1

1.5

immV

re (mV)

im (mV)

6

Covariance info: time - frequency

5000 realizations(jitter only, stdev 1 ps)Cov[y(t)] Cov[Y()]

Fourier transform, separate real and imag. part

10 20 30 40 50 60 70 80freqGHz

0.5

0.4

0.3

0.2

0.1

0.1

0.2

re,im

10 20 30 40 50 60 70 80freqGHz

0.5

0.4

0.3

0.2

0.1

0.1

0.2

re,im

freq (GHz)

uncertainty on LS estimate

construct WLS estimator

relevant value of cost

(re,im)

7

Comparison

0 1000 2000 3000 4000

2

1

0

1

2

K. Coakley and P. Hale, “Alignment of Noisy Signals,” IEEE Transactions on Instrumentation and Measurement,

Vol. 50, No. 1, February 2001

EstimatorRMS prediction error

95% conf. interval

naive cross-correlation ~ 0.25

complete cross-correlation ~ 0.15

naive LS 0.16 0.14 .. 0.19

enhanced LS 0.16 0.14 .. 0.19

enhanced WLS 0.08 0.07 .. 0.09

Case of moderate jitter and small additive noise

(arbitrary units)

8

Measurements

impulselaser

calibratedO/E

samplingoscilloscope

trigger2nd O/E

Ch1/3impulselaser

impulselaser

calibratedO/E

calibratedO/E

samplingoscilloscope

trigger2nd O/E

Ch1/3calibratedO/E

impulselaser

2nd O/E

samplingoscilloscope

trigger

Ch1/3

0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6tns

0.02

0.02

0.04

0.06

0.08

ytV

t (ns)

y(t) (V)

0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6tns

0.02

0.02

0.04

0.06

0.08

ytV

t (ns)

y(t) (V)

9

100 200 300 400 500

realizationindex

0.5

1

1.5

2

estimateddriftps

uncertainty reduced by factor of 2 !

Measurements

100 200 300 400 500

realizationindex

0.5

1

1.5

2

estimateddriftps

100 200 300 400 500

realizationindex

0.5

1

1.5

2

estimateddriftps

naive LS

enhanced LSenhanced WLS

estimateddrift (ps)

realizationindex

expected value WLS cost:2.M.K - p

p = 2.M + K

M = K = 500

expected value = 498500 ± 1997actual value = 500346

10

Conclusions

• System identification framework

• Outperforms any other published technique

• Proper weighting: relevant value of cost decreased uncertainty on estimated drift: factor 2

• Estimation of drift in presence of jitter