Equalization of Time Errors in Time Interleaved ADC System - Part II: Analysis and Examples

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ADC System – Part II: Analysis and Examples

Jonas Elbornsson

,

Fredrik Gustafsson

Jan-Erik Eklund

Control & Communication

Department of Electrical Engineering

Link¨

opings universitet

, SE-581 83 Link¨

oping, Sweden

WWW:

http://www.control.isy.liu.se

E-mail:

jonas@isy.liu.se

,

fredrik@isy.liu.se

3rd March 2003

AUTOMATIC CONTROL

COM

MUNICATION SYSTEMS LINKÖPING

Report no.:

LiTH-ISY-R-2495

Submitted to

Technical reports from the Control & Communication group in Link¨oping are available athttp://www.control.isy.liu.se/publications.

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Abstract

In the accompanying paper a method for blind (i.e., no calibration needed) estimation and compensation of the time errors in a time inter-leaved ADC system was presented. In this paper we evaluate this method. The Cramer-Rao bound is calculated, both for additive noise and random clock jitter. Monte-Carlo simulations have also been done to compare to the CRB. Finally, the estimation method is validated on measurements from a real time interleaved ADC system with 16 ADCs.

Keywords: A/D conversion, nonuniform sampling, equalization, estimation

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Equalization of Time Errors in Time Interleaved

ADC System –Part II: Analysis and Examples

Jonas Elbornsson, Fredrik Gustafsson, Jan-Erik Eklund

Abstract— In [1] a method for blind (i.e., no calibration

needed) estimation and compensation of the time errors in a time interleaved ADC system was presented. In this paper we evaluate this method. The Cramer-Rao bound is calculated, both for additive noise and random clock jitter. Monte-Carlo simulations have also been done to compare to the CRB. Finally, the estimation method is validated on measurements from a real time interleaved ADC system with 16 ADCs.

Index Terms— A/D conversion, nonuniform sampling,

equal-ization, estimation

I. INTRODUCTION

T

HIS is the second part in a series of two articles de-scribing a method to estimate time errors in a time interleaved A/D converter (ADC) system. In this part we will calculate the Cramer-Rao bound (CRB) for the time error estimates and compare it to simulations. We will also present some measurement results. But first we will describe the time interleaved ADC system and briefly review the time error estimation method presented in [1].

Time interleaved ADCs [2], [3] can be used to increase the sample rate, see Figure 1. The time interleaved ADC system works as follows:

• The input signal is connected to all the ADCs.

• Each ADC works with a sampling interval of M Ts, where

M is the number of ADCs in the array and Ts is the

desired sampling interval.

delay, Ts sampling ADC0 ADC1 ADC2 ADC_M−1 u clock y0 y1 y2 y_M−1 y M U X

Fig. 1. A time interleaved ADC system. M parallel ADCs are used with the same master clock. The clock is delayed by the nominal sampling interval to each ADC. The outputs are then multiplexed together to form a signal sampled M times faster than the output from each ADC.

• The clock signal to the ith ADC is delayed with iTs. This

gives an overall sampling interval of Ts.

Due to the manufacturing process, all the ADCs in the time interleaved array are not identical. This means that mismatch errors will occur in the system. Three kinds of mismatch errors will occur:

• Time errors (static jitter)

The delay times of the clock between the different ADCs are not equal. This means that the signal will be periodically but non-uniformly sampled.

• Amplitude offset errors

The ground level differs between the different ADCs. This means that there is a constant amplitude offset in each ADC.

• Gain errors

The gain, from analog input to digital output, differs between the different ADCs.

The errors listed above are static or slowly time varying. This means here that the errors can be assumed to be constant for the same ADC from one cycle to the next over an interval of some million samples.

With a sinusoidal input, the mismatch errors can be seen in the output spectrum as non harmonic distortion. With input signal frequency ω0, the gain and time errors cause distortion at the frequencies

i

Mωs± ω0, i = 1, . . . , M− 1

where ωs is the sampling frequency. The offset errors cause

distortion at the frequencies

i

Mωs, i = 1, . . . , M− 1

An example of an output spectrum from an interleaved ADC system with four ADCs with sinusoidal input signal is shown in Figure 2. This distortion causes problems for instance in a radio receiver where a weak carrier cannot be distinguished from the mismatch distortion from a strong carrier. It is there-fore important to remove the mismatch errors. A method for blind (i.e., no calibration signal is needed) adaptive estimation of the time errors is presented in [1].

Apart from the static errors listed before, there are also ran-dom errors due to for instance thermal noise and quantization, which are different from one sample to the next. These errors do not have anything to do with the parallel structure of the ADC and are impossible to estimate because of their random behavior. However, the random errors are important to study for the robustness of the estimation algorithm, and to calculate lower bounds on the estimation accuracy. The random errors in an A/D converter that are discussed in this paper are

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2 0 1 2 3 4 5 6 −50 −40 −30 −20 −10 0 10 20 30 40 50

Normalized angular frequency

Signal power [dB]

ADC output spectrum

signal component offset error distortion time and gain error distortion

Fig. 2. Simulated output spectrum from interleaved ADC system with four ADCs. The input signal is a single sinusoid. The distortion is caused by mismatch errors.

• Quantization noise

This is a deterministic error, if the input signal is known. However, for most signals it can be treated as additive white noise uncorrelated with the input signal and with uniform distribution [4].

• Random jitter

Due to noise in the clock signal there is a random error on the sampling instances [5]. These errors can be treated as Gaussian white noise on the sampling instances.

II. NOTATION ANDDEFINITIONS

We will in this section introduce the notation that will be used in this paper. The nominal sampling interval, that we would have without time errors, is denoted Ts. M denotes the

number of ADCs in the time interleaved array, which means that the sampling interval for each ADC is M Ts. The time

error parameters are denoted ∆ti, i = 0, . . . , M − 1. The estimates of these errors are denoted ˆ∆ti, and the true errors are denoted ∆0_t_i. The vector notation ∆t= [∆t0· · · ∆tM−1] is used for all the time error parameters. We use the following notation for the signals involved:

• u(t) is the analog input signal.

• u[k] denotes an artificial signal, sampled without

mis-match errors.

• ui[k], i = 0, . . . , M− 1 denotes the M subsequences of

u[k],

ui[k] = u[kM + i]. (1) • yi[k] i = 0, . . . , M− 1 are the output subsequences from

the M A/D converters, sampled with time errors.

yi[k] = u (kM + i)Ts+ ∆0ti+ e

jitter i [k]

+ ei[k] (2)

Here ejitter_i [k] is the random jitter and ei[k] is

quantiza-tion noise.

• y[k] is the multiplexed output signal from all the ADCs,

y[k] = y(kmodM ) k M ,

where b·c denotes integer part.

• z(∆t)_{[k] denote the output signal, y[k], reconstructed with} the error parameters, ∆t.

• z(∆t)

i [k] are the subsequences of z(∆t)[k]

How the reconstructed signal z(∆t)_{[k] is calculated is} de-scribed in [1]. We assume throughout this paper that u(t) is band limited to the Nyquist frequency of the complete ADC system.

We will next define two concepts for measuring the perfor-mance of an ADC. Assume that the output y[k] of an ADC consists of a signal part s[k], a distortion part d[k], and a noise part e[k]

y[k] = s[k] + d[k] + e[k]

Then the SNDR (Signal to Noise and Distortion Ratio) is defined as SN DR = 10 log₁₀ E{s2_[k]_} E{d2_[k]} + E{e2_[k]} (3)

The SFDR (Spurious Free Dynamic Range) is defined for a sinusoidal input signal as the distance between the signal com-ponent in the spectrum and the strongest distortion comcom-ponent, measured in dB, see Figure 3.

0 1 2 3 4 5 6 −50 −40 −30 −20 −10 0 10 20 30 40 50 SFDR Normalized frequency dB SFDR definition

Fig. 3. Definition of the SFDR.

III. TIMEERRORESTIMATION

We will in this section briefly review the time error estima-tion method presented in [1].

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The time errors are estimated by minimization of the loss function V_tN,(L)(∆t) = L X l=0 MX−1 i=1 i−1 X j=0 ¯ R(N ),(∆t) zi,zi−1 [l]− R (N ),(∆t) zj,zj−1 [l] 2 + MX−1 i=1 i−1 X j=0 1 N N X k=1 z(∆ti) i [k] 2 − z(∆tj) j [k] 22 (4) where ¯ RN,(∆t) zi,zj [l] = 1 N N X k=1 z(∆t) (imodM ) k + i M + l − z(∆t) (jmodM ) k + j M 2 (5) and z(∆t)_{[k] is the output signal reconstructed with the time} error parameters ∆t. The estimation is done according to the

following algorithm:

Algorithm 1 (Interleaved ADC equalization) Initialization:

• Choose a batch size, N , for each iteration.

• Initialize the step size of the stochastic gradient algo-rithm, µt. If the order of magnitude of the mismatch

errors are known, this information can be used for the initialization.

• Initialize the parameter estimates for i = 0, . . . , M− 1

ˆ ∆(0)_t_i = 0

Adaptation:

1) Collect a batch of N data from each ADC, yi[k], i = 0, . . . , M− 1.

2) Calculate the reconstructed signals:

z( ˆ∆

(j) t )

i [k], i = 0, . . . , M− 1

3) Calculate the gradient of the loss function,

∇V(N )

t ( ˆ∆

(j)

t ). The gradients can be calculated

numerically by a finite difference approximation from the loss functions, or by analytically differentiating the loss function. The loss function is defined in (4). 4) Update the parameter estimates

ˆ ∆(j+1)_t = ˆ∆(j)_t − µt ∇V (N ) t ( ˆ∆ (j) t ) max|∇V_t(N )( ˆ∆(j)_t )|

5) If the loss function has increased since the last iteration

V_t(N )( ˆ∆(j+1)_t ) > V_t(N )( ˆ∆(j)_t )

backtrack the step size µt := µt/2 and change the

parameter estimates in step 4) until the loss function decreases. Otherwise double the step size for the next iteration: µt:= 2µt.

6) Return to step 1).

IV. CRAMER-RAOBOUND

The Cramer-Rao bound (CRB) is a lower bound on the variance of the estimated parameters for a given amount of data [6], independent of the estimation algorithm used. Comparing simulations with the CRB gives a measure of how good the estimation algorithm is. The CRB is here calculated assuming known input, which means that a blind estimation algorithm never can reach the CRB. But it still gives a good hint of the estimation performance. We will first calculate the CRB for a general input signal assuming only additive noise on the signal. With stochastic jitter we cannot calculate a general expression for the CRB, but we will calculate the CRB for some special input signals.

A. CRB for additive noise

Assuming only additive noise, the output signal subse-quences are

y0[k] = u(M kTs) + e0[k]

yi[k] = u((M k + i)Ts+ ∆0ti) + ei[k]

i = 1, . . . , M− 1

The noise is here assumed to be white Gaussian

ei[k]∈ N(0, σe)

despite that uniformly distributed noise is a better model of the quantization noise. However, Gaussian noise simplifies the calculations of the CRB and simulations show that, with the same noise variance, the estimation accuracy is approximately the same for Gaussian and uniform noise. The parameterized signal model is ˆ y0[k] = u(M kTs) ˆ yi[k] = u((M k + i)Ts+ ∆ti) i = 1, . . . , M− 1

The negative log-likelihood function [6] is then calculated by taking the logarithm of the probability density function of the noise. − lim N→∞log fe(∆t, y N₎ = lim N→∞ 1 N 1 2σ2 eM M−1_X i=0 N X k=1 yi[k]− ˆyi[k] 2 = 1 2σ2 eM M−1_X i=0 (2σ_u2+ σ2_e)− 2Ru(∆0ti− ∆ti)

Differentiating the log-likelihood function twice with respect to the error parameters gives

d2 d∆2(− lim_N→∞log fe(∆t, y N_{)) =} 1 2σ2 eM ·      −2d2Ru(∆0_t1−∆t1) (d∆_t1)2 · · · 0 .. . . .. ... 0 · · · −2d 2 Ru(∆0_tM−1−∆_tM−1) (d∆_tM−1)2      (6)

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Evaluating (6) at ∆t = ∆0t gives the Fisher information

matrix,

F =− 1

M σ2

e

R00u(0)I(M−1)×(M−1) (7)

The Fisher information matrix gives a lower bound on the covariance of the parameter estimates. If the parameters are estimated from N samples per ADC, the Cramer-Rao bound is

Cov( ˆ∆t)≥ 1

M NF

−1 ₍₈₎

Putting (7) into (8) we get

Var( ˆ∆ti)≥

σ2

e

N Ru00(0)

(9) We can see from (9) that the CRB for the time error depends on the input signal. We will next evaluate the CRB for a few signal examples.

• Sinusoidal input: In radio applications a single

modu-lated sinusoidal carrier is often used. Here we discard the modulation and calculate the CRB for a sinusoidal signal.

u(t) =√2 sin(ωt)

The covariance function is here

Ru(τ ) = cos(ωτ )

which gives the Cramer-Rao bound

Var( ˆ∆ti)≥

σ_e2

N ω2

• Multisine input: In DSL modems and OFDM radio

communications, a sum of several sinusoidal carriers are used. Here we calculate the CRB for a multisine input signal. u(t) = L X i=1 αisin(ωit) 1 2 L X i=1 α2i = 1

The covariance function is

Ru(τ ) = L X i=1 α2 i 2 cos(ωiτ )

Var( ˆ∆ti)≥

2σ2

e

NPL_i=1α2

iω2i • Band limited white noise input:

Here the input is a stochastic process with spectrum

Φu(ω) =

π

ωmax(1−α) αωmax≤ |ω| ≤ ωmax

0 otherwise (10)

This gives the covariance function

Ru(τ ) =

sin(ωmaxτ )− sin(αωmaxτ )

(1− α)ωmaxτ

Var( ˆ∆ti)≥

3σ2

e

N (α2_{+ α + 1)ω}2

max

We can make a few remarks about the examples above.

• When α → 1 in (10) we get the same variance for the band limited white noise input as for the sinusoidal input.

• For low pass filtered white noise, i.e., α = 0 we get

Var( ˆ∆t)≥ 3σ2

e

N ω2

max

That is, the parameter variance is 3 times larger than with a sinusoidal input signal.

• With equal spacing, over an interval, between the fre-quencies in the multisine case,

ωi=

i

Lωmax, i = L

0_{, . . . , L}

we get the Cramer-Rao bound

Var( ˆ∆t)≥ σ2 e(L− L0+ 1)L2 N ω2 maxf (L, L0) f (L, L0) = 1 3((L + 1) 3_{− L}03₎ −1 2((L + 1) 2_{− L}02_{) +}1 6(L + 1− L 0₎ If we have L0=bαLc 0≤ α ≤ 1

and let L → ∞ we get the same result as for the band limited noise case

Var( ˆ∆t)≥

3σ2

e

N (α2_{+ α + 1)ω}2

max

The convergence is quite fast, for instance with 16 tones,

L = 16, L0= 1, we get Var( ˆ∆t)≥ 2.75σ2 e N ω2 max

This means that the CRB for band limited white noise input is a good approximation for most multitone signals. 1) Optimal input signal: Usually we cannot choose the input signal since the estimation should work without a special calibration signal. But it is still interesting to investigate what input signal gives the lowest CRB. For a general input signal we have the CRB for the time error parameter from (9) as

Var( ˆ∆t)≥ −

σ_e2

N R_u00(0)

To find the optimal input signal from a time error estimation point, we should maximize−R00_u(0) over all signals u(t), band

limited to _Tπ

s. We have, for a band limited signal u(t), that

Ru(τ ) = 1 2π Z π/Ts −π/Ts Φu(ω)eiωτdω

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This gives R00_u(τ ) =− 1 2π Z π/Ts −π/Ts ω2Φu(ω)eiωτdω

and to minimize the variance bound we should maximize

max Φu(ω) 1 2π Z π/Ts −π/Ts ω2Φu(ω)dω subject to 1 2π Z π/Ts −π/Ts Φu(ω)dω = 1 and Φu(ω) = 0, ω≥ π Ts Since Φ_u(ω) = 0, ω≥ _Tπ

s we cannot allow the solution

Φu(ω) = π(δ(ω− π Ts ) + δ(ω + π Ts ))

But as ω0→ _Tπ_s, the spectrum

Φu(ω) = π(δ(ω− ω0) + δ(ω + ω0))

tends to the optimal solution. This means that the best input signal is a sinusoid close to the Nyquist frequency.

B. CRB for noise and jitter

Here we will evaluate the CRB with both noise and stochas-tic jitter present. The output signal subsequences are now

y0[k] = u(kM Ts+ ejitter0 [k]) + e0[k] yi[k] = u((kM + i)Ts+ ∆0ti+ e

jitter

i [k]) + ei[k] (11)

i = 1, . . . , M − 1

We assume that both the noise and the random jitter are Gaussian distributed

ei[k]∈ N(0, σe)

ejitter_i [k]∈ N(0, σjitter)

Here we cannot, in general, assume that the output signal at a certain time instance is Gaussian distributed. But if we take a sum over many samples we have, according to the central limit theorem [7], that

¯ y = 1 N M N−1_X k=0 M_X−1 i=0 yi[k] (12)

is Gaussian distributed. If we assume that y[k] is modulo M quasistationary with respect to g(u_i) = ui [1] the mean value

of ε(∆t, yN) is zero, independent of the input signal shape.

However, the variance depends on what input signal we have. We will therefore in the following consider a few special cases. 1) Sinusoidal input: Again we assume that the input signal is normalized to have power equal to one.

u(t) =√2 sin(ω0t)

Assuming that the random jitter and the quantization noise are independent, we have the variance of the output signal as

Var(¯y) = 2 N M N_X−1 k=0 M_X−1 i=0

Varsin(ω0((M k + i)Ts+ ∆0ti+ e

jitter i [k])) + Var(ei[k]) (13)

The last term in (13) is known Var(ei[k]) = σe2, but we have to

calculate the first term, i.e., we need to calculate the variance

Varsin(x + Z)

where x is a constant and

Z ∈ N(0, σ)

First, we calculate the expectation

Esin(x + Z) = Z _∞ −∞sin(x + z) 1 σ√2πe −z2 2σ2dz = sin(x)e−σ22

From this we can calculate the variance

Varsin(x + Z)= Esin2(x + Z)− sin2(x)e−σ2 = 1

2− 1 2E

cos(2(x + Z))− sin2(x)e−σ2 = 1 2− 1 2cos(2x)e −2σ2 − sin2_(x)e−σ2

which gives that the mean output signal variance is

Var(¯y) = 1

2(1− e

−ω0σ2jitter₎ and the error signal

ε(∆t, yN) = N_X−1 k=0 M_X−1 i=0 (√2 sin(ω0((M k + i)Ts+ ∆0ti+ e jitter i [k])) + ei[k]− √

2 sin(ω0((M k + i)Ts+ ∆ti))e−σ 2 jitter/2 is then Gaussian distributed under ∆t= ∆0t

ε(∆t, yN)∈ N(0,

q

M N (1− e−ω02σjitter2 _{+ σ}2

e))

From this we can calculate the negative log-likelihood function

− log fε(∆t, yN) = 1 2M N (1− e−ω02σjitter2 + σ2 e) · N_X−1 k=0 M_X−1 i=0 (√2 sin(ω0((M k + i)Ts+ ∆0ti+ e jitter i [k])) + ei[k]− √

2 sin(ω0((M k + i)Ts+ ∆ti))e−σ 2 jitter/2 = 1 2(1− e−ω20σ2jitter_{+ σ}2 e) ((1 + e−ω20σ 2 jitter/2_{) + σ}2 e − 1 M M_X−1 i=0 2 cos(ω0(∆0i − ∆i))e−ω 2 0σ 2 jitter₎

Differentiating the negative log-likelihood function twice with respect to the time error parameters gives

d2₍_{− log f} ε(∆t, yN)) d∆2 t = 1 1− e−ω20σ2jitter_{+ σ}2 e (−ω02cos(ω0(∆0i − ∆i))e−ω 2 0σ 2 jitter₎ (14)

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Evaluating (14) at ∆t = ∆0t gives the Fisher information

matrix Ft= e−ω20σ 2 jitter_ω2 0 M (1− e−ω20σ2jitter_{+ σ}2 e) I(M−1)×(M−1) (15) From this we can calculate a lower bound on the variance of the parameter estimates

Var( ˆ∆ti)≥ 1− e−ω20σ 2 jitter _{+ σ}2 e N e−ω20σ2jitter_ω2 0 (16) With a first order Taylor expansion of (16) we get

Var( ˆ∆ti)& σ_jitter2 N + σ2 e N ω2 0 (17) i.e., the jitter gives an additional term to the CRB depending only on the jitter noise variance and the number of estimation data.

2) Multisine input: With a multisine input signal

u(t) = L X i=1 αisin(ωit) 1 2 L X i=1 α2_i = 1

we get, with similar calculations as for the sinusoidal case that the error signal

ε(∆t, yN) = N_X−1 k=0 M_X−1 i=0 ( L X l=1 αlsin(ωl((M k + i)Ts+ ∆0ti+ e jitter i [k])) + ei[k]− L X l=1

sin(ωl((M k + i)Ts+ ∆ti))e−σ 2 jitter/2

is Gaussian distributed under ∆t= ∆0t

ε(∆t, yN)∈ N 0, v u u tMN(1 2 L X i=1 α2 i(1− e−ω 2 iσjitter2 _{) + σ}2 e)

From this we can calculate the negative log-likelihood function

− log fε(∆t, yN) = 1 2(1 2 PL i=1α2i(1− e−ω 2 iσ2jitter) + σ2 e) · = (1 2 L X i=1 α2_i(1 + e−ω2iσ 2 jitter) + σ2 e − 1 M M−1_X i=0 L X l=1 α2_lcos(ωl(∆0i − ∆i))e−ω 2 lσ2jitter₎

Differentiating this function twice and evaluating at ∆t= ∆0t

gives the Fisher information matrix

Ft= PL l=1α2le−ω 2 lσ 2 jitter_ω2 l 2M (1 2 PL l=1α2l(1− e−ω 2 lσ 2 jitter_{) + σ}2 e) I(M−1)×(M−1) (18)

From (18) the CRB for a multisine input signal can be calculated Var( ˆ∆ti)≥ 1 2 PL l=1α2l(1− e−ω 2 lσ 2 jitter_{) + σ}2 e N1₂PL_l=1α2 lω2le−ω 2 lσ2jitter ≈ σjitter2 N + 2σ2 e NPL_l=1α2 lω2l (19) Again the jitter gives an additional term to the CRB. The contribution from the jitter to the CRB is independent of the number of tones and hence the same as for the single sinusoidal case (17).

V. SIMULATIONS

To evaluate the performance of the time error estimation method, a time interleaved ADC system has been simulated. In Figure 4 the spectrum of the output signal is shown before and after correction with estimated time errors. Here the input signal is a single sinusoid. We can see here that, after correction, the time errors can not be seen above the noise floor. The convergence rate is different for different

0 1 2 3 4 5 6 −100 −80 −60 −40 −20 0

Signal energy [dB] Before equalization 0 1 2 3 4 5 6 −100 −80 −60 −40 −20 0

Signal energy [dB]

After equalization

Fig. 4. Upper plot: The output spectrum of an interleaved ADC system with time errors. Lower plot: The same spectrum after compensation with estimated time error parameters. The parameters were estimated from 214samples per ADC.

input signals and different number of ADCs, but usually the parameters converge in about 10− 50 iterations. In Figure 5 an example of the convergence of the time error estimates is shown. The simulation is here done with four ADCs and sinusoidal input. The amount of data is here 214 _{samples per} batch. One iteration was done on each batch. In this example the parameters converge in about 20 iterations.

To compare the estimation accuracy with the CRB the minimization has been done on one batch of data instead of updating with new data for each iteration. The estimation algorithm has been tested with different input signals and different signal parameters have been varied. One parameter at a time is changed according to the following list. The default value, used when other parameters are changed, is given inside parentheses.

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0 20 40 60 80 100 10−6 10−5 10−4 10−3 10−2 10−1 Iteration number Estimation error

Time error estimation convergence

Fig. 5. Convergence of time error parameter estimates for ADC system with four ADC (three parameters). The estimation error is here shown in fractions of Ts.

• Sinusoidal input signal

– Angular frequency: ω0∈ [0.01, 3.1] (ω0= 1). – Number of data per ADC: N∈ [23, 216_{] (N = 2}14_). – Number of ADCs: M∈ [2, 16] (M = 4).

– Quantization noise, given as number of bits: n =

[2, 16] (n = 10).

– Jitter variance: σ_jitter2 ∈ [0, 1] (σ_jitter2 = 0). • Multisine input signal

– Maximum angular frequency: ω0∈ [0.01, 3.1] (ω0=

1).

– Number of tones: L∈ [2, 256] (L = 64).

• Low pass filtered white noise

– Cut off frequency: ω0∈ [0.01, 3.1].

• Band pass filtered white noise, band width 10% of cut off frequency

– Cut off frequency: ω0∈ [0.01, 3.1].

The true time error parameters have been generated randomly from a uniform distribution

For i = 1, . . . , M− 1

∆0_t_i ∈ U[−0.1Ts, 0.1Ts]

The standard deviation of the parameter estimation errors have been calculated from 25 Monte-Carlo simulations for each case in the list above. Some of the results from these simulations are shown in the plots described below.

In Figure 6 the root mean square of the estimation error of the time error parameters is shown, as a function of the number of data, N . The input signal is here sinusoidal with input frequency ω0 = 1. For large values of N the simulated parameter standard deviation is about a factor of

10 above the CRB. In Figure 7 the estimation error is shown

with varying input signal frequency instead. We can see here that the estimation works well even close to the Nyquist frequency. For very low frequencies the the input signal is very slowly varying. The output signal will therefore be constant for several samples due to the quantization. This means that

much fewer samples contribute to the loss function and the performance is therefore worse. Figure 8 shows the estimation error as a function of the random jitter variance. We can see here that we get quite good estimates even when the jitter is in the same order of magnitude as the static time errors. In Figure 9 the estimation error is shown for a multisine input signal as a function of the maximum frequency. Figure 10 shows the estimation error with band limited white noise input. The pass band is here between 0.9ωc and ωc and the result is

shown for varying ωc.

100 101 102 103 104 105 10−6 10−5 10−4 10−3 10−2 10−1 Number of data, N

Time estimation error standard deviation

Time errors estimated with sinusoidal input

MC time error CRB time error w/o estimation

Fig. 6. Time estimation error as a function of the number of estimation data compared to the CRB. The input signal is here a single sinusoid with frequency ω0= 1. The simulated values are calculated from 25 Monte Carlo

simulations. 10−2 10−1 100 101 10−6 10−5 10−4 10−3 10−2 10−1 Nyquist

Input signal angular frequency, ω₀

Fig. 7. Time estimation error as a function of input signal frequency compared to the CRB. The input signal is here a single sinusoid. The simulated values are calculated from 25 Monte Carlo simulations.

VI. MEASUREMENTS

To validate the estimation method, the algorithm has also been tested on measured data from a time interleaved A/D

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8 10−5 10−4 10−3 10−2 10−1 100 10−7 10−6 10−5 10−4 10−3 10−2 10−1

Random jitter variance

Fig. 8. Time estimation error as a function of the random jitter variance compared to the CRB. The input signal is here a single sinusoid with frequency ω0= 1. The simulated values are calculated from 25 Monte Carlo

simulations. 10−2 10−1 100 101 10−6 10−5 10−4 10−3 10−2 10−1 Nyquist

Input signal max angular frequency, ω_max

Time errors estimated with multisine input

Fig. 9. Time estimation error as a function of input signal maximum frequency compared to the CRB. The input signal is here a multisine signal with 64 tones. The simulated values are calculated from 25 Monte Carlo simulations.

converter system. The following parameters were used in the measurements

• 16 parallel ADCs with 12-bit precision. • Sampling frequency, fs= 5M Hz.

• Sinusoidal input signal with frequencies between

0.31M Hz and 2.2M Hz.

• 8192 samples per ADC in each batch of data.

Here we have estimated the gain and offset errors also, as described in [8]. The signal generator is not perfect, which means that there is some harmonic distortion in the output spectrum. There are also other errors, besides the mismatch errors, that give distortion in the output signal. An example of an output spectrum is shown in Figure 11. Here we see that the mismatch distortion is small compared to the harmonic distortion. Therefore SFDR or SNDR is not useful

10−2 10−1 100 101 10−6 10−5 10−4 10−3 10−2 10−1 Nyquist

Input signal max angular frequency, ω_c

Time errors estimated with band limited white noise input

Fig. 10. Time estimation error as a function of input signal maximum frequency compared to the CRB. The input signal is here band limited white noise with pass band between 0.9ωc and ωc. The simulated values are calculated from 25 Monte Carlo simulations.

0 0.5 1 1.5 2 2.5 0 20 40 60 80 100 120 Frequency [MHz] Signal Power [dB] Signal component Offset error distortion Time and gain error distortion Harmonic distortion

Fig. 11. Output spectrum from ADC measurement. The signal component is marked by ’o’, the offset error distortion is marked by ’x’ and the gain error distortion is marked by ’*’.

to measure the improvement after compensation for mismatch errors. Instead we study the improvement of the frequency components caused by the mismatch errors. In Figure 12 the same spectrum is shown after compensation with estimated mismatch parameters. The mismatch distortion is here no longer visible above the noise floor. To validate the mismatch error estimation algorithm a parameter estimate was calculated for each input signal frequency and all signals were then compensated with each estimate. In Figure 13 the mean improvement of the gain and time error distortion components is shown. Since the sampling frequency is quite low, the time errors relative to the sampling interval are very small. This means that the time error distortion is very small, especially for low frequency signals, and therefore cannot be improved much. But we still see some improvement after the time error compensation.

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0 0.5 1 1.5 2 2.5 0 20 40 60 80 100 120 Frequency [MHz] Signal Power [dB] Signal component Offset error distortion Time and gain error distortion Harmonic distortion

Fig. 12. Output spectrum from ADC measurement after compensation with estimated mismatch errors. Here the mismatch distortion is no longer visible above the noise floor.

0 0.5 1 1.5 2 2.5 0 5 10 15 20 25 30 35 40 45 50

Input signal frequency [MHz]

Improvement [dB]

Estimated at 0.31MHz Estimated at 0.63MHz Estimated at 2.2MHz

Fig. 13. Gain and time error distortion improvement. The improvement is shown for three sets of estimated parameters, estimated from sinusoidal signals with frequencies 0.31M Hz, 0.63M Hz and 2.2M Hz. The curves marked with ’x’ show the improvement after compensation with only the gain error parameters and the curves marked with ’o’ show the improvement after compensation with both gain and time error parameters.

VII. CONCLUSION

A time interleaved ADC system is a good option to significantly increase the sampling rate of A/D conversion. However, due to errors in the manufacturing process, the ADCs in the time interleaved system are not exactly identical. This means that mismatch errors in time, gain and offset are introduced. The mismatch errors cause distortion in the sampled signal. Calibration of ADCs is time consuming and costly. Further, the mismatch errors may change slowly with for instance temperature and aging. Therefore it is preferable to continuously estimate the mismatch errors while the ADC is used.

In [1] a method for estimation and compensation of the time errors in a time interleaved ADC system was presented. In this paper we have evaluated this method. We have calculated the

Cramer-Rao bound for the estimated parameters, and shown simulation and measurement results. The estimation method is blind, and the CRB is calculated assuming known input, so the CRB cannot be reached. However, the simulation results show that we can come rather close to the CRB even with this blind estimation method.

In a real ADC there are other distortions, besides the mismatch error distortion. The measurement results show that the estimation method works well even if the ADCs are not ideal.

REFERENCES

[1] J. Elbornsson, F. Gustafsson, and J.-E. Eklund, “Equalization of time errors in time interleaved ADC system –Part I: Theory,” 2003, to be submitted to IEEE Transactions on Signal Processing.

[2] W. Black and D. Hodges, “Time interleaved converter arrays,” IEEE

Journal of Solid-State Circuits, vol. SC-15, no. 6, pp. 1022–1029,

December 1980.

[3] Y.-C. Jenq, “Digital spectra of nonuniformly sampled signals: A robust sampling time offset estimation algorithm for ultra high-speed waveform digitizers using interleaving,” IEEE Transactions on Instrumentation and

Measurement, vol. 39, no. 1, pp. 71–75, February 1990.

[4] B. Widrow, I. Kollar, and M.-C. Liu, “Statistical theory of quantization,”

IEEE Transactions on Instrumentation and Measurement, vol. 45, no. 2,

pp. 353–361, April 1996.

[5] R. van de Plassche, Integrated Analog-to-Digital and Digital-to-Analog

Converters. Kluwer Academic Publishers, 1994.

[6] L. Ljung, System Identification, Theory for the user, 2nd ed. Prentice-Hall, 1999.

[7] A. Gut, An Intermediate Course in Probability. Springer-Verlag, 1995. [8] J. Elbornsson, F. Gustafsson, and J.-E. Eklund, “Blind adaptive equaliza-tion of mismatch errors in time interleaved a/d converter system,” 2003, submitted to IEEE Transactions on Circuits and Systems.