Perfect data reconstruction Perfect data reconstruction algorithm of interleaved ADCalgorithm of interleaved ADC

Dr. Fang XuTeradyne, Inc. Boston, MA, U.S.A.

Presentation OutlinePresentation Outline

Purpose

Time Interleaved ADC

The reconstruction algorithm

Experiment

Conclusions

PurposePurpose

Test instruments are built with available parts Instrument development time is long Instruments are designed for testing future products Performance gap needs be solved by design

Year

Performance(frequency, bits)

Instrument architecture reduces Performance gap

State of art device performance

Future

ProductInstrument

Design-in

Time Interleaved ADC’sTime Interleaved ADC’s

Capture of a continuous time domain waveform

ADC7ADC6

ADC5ADC4

ADC3ADC2

ADC1ADC0

Clock generation

Interleaved samples

Time Interleaved Real ADC’sTime Interleaved Real ADC’s

ADC’s and analog sections have differentoffset, gain and phase

Gain and phase vary with frequencyUp-to 20 dB measured for gain !

Samples are not uniformly distributed Need advanced algorithm to reconstruct signal

Relative gain/phase (timing) error vs. 1st ADC @199.99 MHz

5 dB/div 50 ps/div

Time Interleaved Real ADC’sTime Interleaved Real ADC’s

ADC’s and analog sections have differentoffset, gain and phase

Gain and phase vary with frequencyUp-to 20 dB measured for gain !

Samples are not uniformly distributed Need advanced algorithm to reconstruct signal

ADC7ADC6

ADC5ADC4

ADC3ADC2

ADC1ADC0

Input

Clock generation

Datacorrection

reconstruction

FFT of Capture Before CorrectionFFT of Capture Before Correction

H2

offset

gain/phase

-120

-80

-40

Fi = 199.990200 MHz,Fs = 1.494220800 Gsamples/sSNR=20 dBc,Non harmonic spur=-25 dBc

100 200 300 400 500 600 700

Offset Discrepancy ArtifactsOffset Discrepancy Artifacts

100 200 300 400 500 600 700

Repetitive noise pattern Spurs at integer FsEasy to remove

H2

offset

-120

-80

-40

gain/phase

Gain Discrepancy ArtifactsGain Discrepancy Artifacts

0

Repetitive amplitude modulation Spur at ± input tone to integer FsNeed advanced algorithm

100 200 300 400 500 600 700

H2

offset

-120

-80

-40

gain/phase

Phase/Timing Discrepancy ArtifactsPhase/Timing Discrepancy Artifacts

0

Repetitive phase modulation Spur at ± input tone from integer FsNeed advanced algorithm

H2

offset

-120

-80

-40

gain/phase

100 200 300 400 500 600 700

Sampling and Aliasing at FsSampling and Aliasing at Fs

Aliased in frequency domain without Hermitian symmetry

Redundant information with Hermitian symmetry

Alias

Alias

Family of Mutually Aliased FrequenciesFamily of Mutually Aliased Frequencies

Repetitive amplitude/phase modulation Spur at ± input tone from integer Fs

That is a subset of whole spectrum

-40

gain/phase

100 200 300 400 500 600 700

We call this subset of frequencies including that of signal

A family of mutually aliased frequencies (FMAF)

Frequencies number equals the number of ADCs

Vector notation: iNMiMNiNkNikNiNi XXXXXX )12/(*

)2/(*

)(* ,,,,,

-20

Frequency Domain ReconstructionFrequency Domain Reconstruction

FsInput signal spectrumto be reconstructed

ADC7ADC6

ADC5ADC4

ADC3ADC2

ADC1ADC0

Clock generation

Fs/2

Spectrum at output of each ADC

Matrix of linear system FMAF

Orthogonal components outside FMAF Porous matrix (lot of 0)

Sampling with Hermitian symmetry Small matrix for each FMAF

iNMMikNMiMiNMiMNM

iNMmikNmimiNmiMNm

iNMikNiiNiMN

R

HHHHH

HHHHH

HHHHH

)12/,(1,1,1*

,1*

2/,1

)12/,(,,*

,*

2/,

)12/,(0,0,0*,0

*2/,0

.....

.....

H

Matrix RepresentationMatrix Representation

ADC7ADC6

ADC5ADC4

ADC3ADC2

ADC1ADC0

Clock generation

Fs/2

iNM

ikN

i

iN

iNkN

iMN

R

X

X

X

X

X

X

)12/(

*

*)(

*)2/(

ˆ

X

iM

im

i

R

X

X

X

,1

,

,0

~.

~.

~

~X

RRR XXH~ˆ

Fs

To be reconstructed Input signal spectrum

Within a family of mutually aliased frequencies

Hm,j

Unknowns and Knowns in EquationUnknowns and Knowns in Equation

Fs

Component at frequency i

iNM

ikN

i

iN

iNkN

iMN

R

XX

X

X

X

X

)12/(

*

*)(

*)2/(

ˆ

X

iM

im

i

R

X

X

X

,1

,

,0

~.

~.

~

~X

Unknown:All frequency

components within a FMAF

Captured data of all converters

Fs/2

Captured data

of converter m

at frequency i

iX

imX ,

~

iNMMikNMiMiNMiMNM

iNMmikNmimiNmiMNm

iNMikNiiNiMN

R

HHHHH

HHHHH

HHHHH

)12/,(1,1,1*

,1*

2/,1

)12/,(,,*

,*

2/,

)12/,(0,0,0*,0

*2/,0

.....

.....

H

Interpretation of Matrix CoefficientsInterpretation of Matrix Coefficients

Each coefficient is complex gain relative to system clock of a converter at a specific frequencyIt includes information on amplitude (flatness)and phase (group delay, clock delay)

To solve equation, each coefficient needs to be measured

Hm,j

Complex gain of converter

m for input frequency i

ADC7 FFT

Fs

ADC6 FFTADC5 FFT

ADC4 FFTADC3 FFT

ADC2 FFTADC1 FFT

ADC0 FFT

Input

N/2 times MxM linear equations

Order of data

Frequency Domain ReconstructionFrequency Domain Reconstruction

Solving linear equations for each FMAFReorder data according to Hermitian symmetry

RRR XHX~ˆ 1

-120

-100

-80

-60

-40

-20

Mag

nitu

de (

dBF

S)

Correction Result of Captured SignalCorrection Result of Captured Signal

Fi = 199.9902 MHz, Fs = 1.4942208 Gmples/sBefore correctionSNR= 20dBc, Non harmonic spur= -25dBcAfter correctionSNR= 54dBc, Non harmonic spur= -78dBc

100 200 300 400 500 600 700

DiscussionsDiscussions

Performance54dBc SNR @750MHZ BW = 142dBc/Hz

limited by signal generator-78dBc Spur –20dB dispersion better than

SFDR of ADC Hardware stability limitation

Ex: A 0.1% converter gain change will limit performance level to about -60dB

This does not cover non-linear distortion Application limitation

DFT can only start when entire segment of waveform has been captured

This method is a better fit for applications which do not need real time capture

ConclusionsConclusions

Solution based on general model of ADC

Gain and phase are functions of frequency

Complete mathematical resolution

Validation by data captured on hardware

Results exceed expectation

Base of high-performance instruments

Perfect data reconstruction Perfect data reconstruction algorithm of interleaved ADCalgorithm of interleaved ADC

Questions and Answers

? And !