Blind estimation of timing and amplitude offsets
in parallel A/D converters
Jonas Elbornsson, Jan-Erik Eklund
Department of Electrical Engineering
Link¨
oping University, SE-581 83 Link¨
oping, Sweden
WWW: http://www.comsys.isy.liu.se
Email: jonas@isy.liu.se
November 8, 2000
REGLERTEKNIK
AUTOMATIC CONTROL
LINKÖPING
Report no.: LiTH-ISY-R-2314
Technical reports from the Automatic Control group in Link¨oping are available by anonymous ftp at the address ftp.control.isy.liu.se. This report is contained in the pdf file 2314.pdf.
Blind estimation of timing and amplitude offsets
in parallel A/D converters
Jonas Elbornsson
November 8, 2000
Abstract
Parallel AD converter structures is one way to increase the sampling rate. Instead of increasing the sample rate in one AD converter, several AD converters with lower sampling rate can be used instead. A problem in these structures is that the time between samples is usually not equal because there are errors in the delays between the AD converters. There are also differences in the ground level in the different AD converters. We will here present a method to estimate the timing and amplitude offsets. The estimation algorithm works without any calibration signal, instead the usual input signal is used. The only assumption that is made about the signal is that most of the energy lies in a low frequency band. The AD converter structure will also be simulated to show that the method works.
1
Introduction
Many digital signal processing applications, such as mobile telephones or ADSL modems, require AD converters with very high sample rate. To achieve high enough sample rates, an array of M AD converters in parallel can be used. Each ADC should work at 1/Mth of the desired sample rate. To avoid time drift between the different ADC’s, the same master clock is used for all ADCs, see figure 1. To get the same sample interval for all samples the nth ADC should be delayed nTs
M s, where Tsis the sample interval for the master sampling clock [8].
If there are errors in the delay times we will get a non-uniformly sampled signal. If these errors are not compensated for, the resolution of the ADC will decrease with several bits [2, 7, 6]. However, if the delay errors are known, the true signal can be recovered [9, 5]. Due to errors in the ground levels in the different converters there can also be differences in the amplitude between different AD-converters. Both these errors are assumed static, so that the error is the same in the same AD-converter from one cycle to the next. There are also random errors in time and amplitude due to thermal noise, that are different from one sample to the next. These errors do not have anything to do with the parallel structure of the AD-converter and are not discussed further here. Figure 2 shows a sine signal that is converted through a parallell AD-converter, where each AD-converter has a fix error in both amplitude and sampling time that is randomly generated. The timing errors are between 0 and 50 percent of the nominal sampling time and the amplitude errors are between 0 and 10 percent of
sampling clock delay ADC1 ADC2 ADC3 ADCM u y
Figure 1: M parallel ADC’s with the same master clock.
the maximum amplitude. There are some presented methods to estimate these errors, see [4, 3]. But these methods assume a known input signal. Here we will present a method for estimation of these errors with unknown input signal.
2
Notation
The analog input signal is denoted u(t). Tsdenotes the nominal sampling time,
that we would have without any errors. M is the number of AD converters in the parallell structure. The time and amplitude offset for the i:th AD converter are denoted ti and Ai respectively. The output from the i:th AD converter is
denoted yi[k] where k is the k:th sample from that AD converter. Each AD
converter form a subsequence, yi[k] = u((kM + i)Ts+ ti) + Ai. The sample
time for each such subsequence is exactly M Ts. These subsequences are merged
to the output signal y[m] = y(m mod M )[bMmc], where b·c denotes integer part.
The differens between samples from AD converter i− 1 and AD converter i is denoted ∆yi[k] = yi[k]− yi−1[k]. We denote by N the number of data points
from each AD converter. We assume that the same number of samples is taken from all the AD converters so that N M is the total number of data. We use the notation ¯E(s(t)) = limn→∞n1
Pn
t=1E(s(t)) for quasistationary signals [10],
where the expectation is taken over possible stochastic parts of s(t).
3
Estimation method
Throughout the report we will assume that the AD-converters are infinite pre-cision sampling units, i.e. we have no amplitude quantization. The estimation of the amplitude and timing offset errors works with unknown input signal. We need some assumptions on the frequency content in the input signal however, see sections 3.1 and 3.2 respectively.
0 5 10 15 20 25 30 −1 −0.5 0 0.5 1 1.5
Effects of errors in sampling time and amplitude offset
ADC with errors ADC without errors
Figure 2: Sampled sinusoid, distorted due to errors in sampling time and am-plitude offset.
3.1
Amplitude offset estimation
There are some identifiability properties that have to be fulfilled to achieve consistent estimates of the amplitude offsets. The first property is about the input signal: the input signal must not be periodic with a period that is an exact multiple of the sample period of one AD-converter (fs/M ). This property
is essential for the algorithm. The second property is needed to avoid bias in the estimates: We must assume either that the mean of the input signal is known or that the amplitude offset of one of the AD-converters is known. If this property is not fulfilled there will be a constant bias in the estimates, but the estimation method will still work except that the output signal will get a different mean value. The mean value of the output from the ith AD-converter will converge to the ith amplitude offset plus the mean of the input signal:
1 N N X k=1 yi[k] = 1 N N X k=1
u((kM + i)Ts+ ti) + Ai→ ¯E(u) + Ai, N → ∞ (1)
If the mean of the input signal, ¯E(u) = m, is known the ith offset can be
estimated as: ˆ Ai= 1 N N X k=1 yi[k]− m (2)
If, instead, one of the amplitude offsets, Aj, is known the mean value of the
input signal can be estimated by:
ˆ m = 1 N N X k=1 yj[k]− Aj (3)
3.2
Timing offset estimation
For the time offset estimation algorithm we need a bandwidth assumption on the input signal, it must at least be bandlimited to the Nyquist frequency. The algorithm is based on the assumption that the signal changes more on average if it is a long time between the samples than if it is a short time between them. Therefore we have to assume that the signal varies slow enough, i.e. has low enough bandwidth. We look again at the difference, ∆yi[k], between to adjacent
samples and make a Taylor exansion around the nominal sampling time of the first AD converter. Here we assume that the amplitude offset errors are zero. This is no restriction since the amplitude offset errors can be estimated and compensated for without any knowledge about the time offset errors.
∆yi[k] = yi[k]− yi−1[k]
= u(kM Ts+ iTs+ ti)− u(kMTs+ (i− 1)Ts+ ti−1)
≈ u(kMTs+ (i− 1)Ts) + (Ts+ ti)u0(kM Ts+ (i− 1)Ts)
− u(kMTs+ (i− 1)Ts) + ti−1u0(kM Ts+ (i− 1)Ts)
= (Ts+ ti− ti−1)u0(kM Ts+ (i− 1)Ts) (4)
Here we cannot look at the mean difference over time since that would converge to zero. Instead we look at the mean squared difference:
ˆ RNi,i−1[0] = 1 N N X k=1 {∆yi[k]}2 = (Ts+ ti− ti−1)2 1 N N X k=1 {u0(kM T s+ (i− 1)Ts)}2 → (Ts+ ti− ti−1)2E{(u0(t))2}, N → ∞ (5)
If the mean value of the input signal derivative is known we can calculate the time offsets from equation (5) using the first time offset as reference, i.e. as-suming t1= 0: ti = i X j=2 ( s ˆ RN j,j−1[0] E{(u0(t))2} − Ts), i = 2, . . . , M (6)
Since the mean value of (u0(t))2 is usually unknown we have to estimate it
from the sampled data. E(u0(t))2 can be estimated as an average over all the
AD-converters: 1 M M X i=1 ˆ RNi,i−1[0] = 1 N M N X k=1 M X i=1 (yi[k]− yi−1[k])2 ≈ 1 N M N X k=1 M X i=1 (Ts+ ti− ti−1)2(u0(kM Ts+ (i− 1)Ts))2 → 1 M M X i=1 (Ts+ ti− ti−1)2E{(u0(t))2} = (Ts2+ 2 M M X i=1 t2i − 2 M M X i=1 titi−1)E{(u0(t))2} (7)
From equation (7) it seems like we have to know the time offset to calculate the estimate of E{(u0(t))2}. We can however get a crude estimate by assuming that
2 M PM i=1t 2 i −M2 PM
i=1titi−1 is small. This is a quite reasonable assumption if
the timing offset errors are small on avarage compared to the nominal sampling time. using this estimate we can calculate a crude estimate of the time offsets using equation (6): t(0)i = i X j=2 ( v u u t RˆNj,j−1[0] 1 M T2 s PM i=1Rˆ N i,i−1[0] − Ts), i = 2, . . . , M (8)
With this estimate of the time offsets we can improve the estimate of E{(u0(t))2}
using equation (7). Then the time offset estimates can be improved by fixed-point iteration [1]: t(l)i = i X j=2 ( v u u t RˆNj,j−1[0] 1 a(l−1) PM i=1RˆNi,i−1[0] − Ts), i = 2, . . . , M where a(l)= M (Ts2+ 2 M M X i=1 (t(l)i ) 2− 2 M M X i=1 t(l)i t (l) i−1) (9)
3.3
Noise sensitivity
The algorithm in the previous section is based on the assumption that adjacent samples are correlated, i.e. that the signal is bandlimited. We will in this section discuss how the estimation is affected by an additive noise term at the output of the AD converter:
yi[k] = u((kM + i)Ts+ ti) + ei[k], ei[k]∈ N(0, λ) (10)
We have here omitted the amplitude offset since the amplitude offset estimation algorithm does not require that the input signal is bandlimited, and therefore works even if the input is only white noise. From equation (4) we now have
∆yi[k]≈ (Ts+ ti− ti−1)u0(kM Ts+ (i− 1)Ts) + ei[k]− ei−1[k] (11)
Since we assume that the noise is independent of u(t) and the different noise sources are independent of each other the cross terms disappear and we have from equation (11): ˆ RNi,i−1= 1 N N X k=1 {∆yi[k]}2→ (Ts+ ti− ti−1)2E{(u0(t))2} + 2λ, N → ∞ (12)
With (12) in (10) we can see that the noise will cause an error in the time error estimates: t(l)i = i X j=2 ( v u u t ( ˆRNj,j−1)y0+ 2λ 1 a(l−1) PM i=1( ˆR N i,i−1)y0+ M a(l−1)2λ − Ts), i = 2, . . . , M where a(l)= M (Ts2+ 2 M M X i=1 (t(l)i )2− 2 M M X i=1 t(l)i t(l)i−1) (13)
1 2 3 4 5 6 7 8 9 10 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 ADC number
Actual and estimated amplitude offsets, sinusoidal input
actual offsets estimated offsets
Figure 3: Amplitude offset estimation, sinusoidal input with frequency ω = 0.02rad/s. Here ( ˆRNi,i−1)y0 = 1 N PN k=1{∆y 0 i[k]} 2
, where yi0[k] denotes the noise-free part of
yi[k]. From the equation above we can see that the estimates are more accurate
when the signal is varying quickly (the derivatives get larger). The accuracy of the first order taylor expansion approximaiton decreases however when the signal is fast varying. This means that if we can choose the frequency content in the input signal there is a trade-off between accuracy of the taylor approximation and accuracy of the time estimation algorithm in the noise corrupted case. If the noise variance, λ, is known, we can compensate for the noise in the algorithm by subtracting 2λ in equation (12).
4
Simulations
Throughout this section we have simulated 10 AD-converters in parallell if noth-ing else is mentioned. We have here assumed that the AD-converters are perfect samplers so there are no other errors than time and amplitude offsets. The pre-cision is assumed to be infinite. The sample interval is set to one since any other sample interval is just a scaling under the conditions assumed above.
4.1
The amplitude offset estimation
To test the performance of the amplitude offset algorithm we have simulated the AD-converter array with amplitude offsets randomly generated between −10% and 10% of the maximum amplitude. In the following simulations we have used 100000 samples of the signals. Figure 3 shows an example of the amplitude estimation. Here a sinusoidal input with frequency ω = 0.02rad/s has been used. If M ω is exactly a multiple of the sampling frequency the estimation algorithm does not work as discussed in section 4.1. We can also see this in Figure 4 where the input signal is a sinusoid with ω = 0.2πrad/s. If we move just slightly away from the exact multiple, however, the algorithm converges again. In Figure 5 we have used an input signal with ω = 0.2001πrad/s and
1 2 3 4 5 6 7 8 9 10 −2 −1.5 −1 −0.5 0 0.5 ADC number
Input signal exactly on multiple of sampling frequency
actual offsets estimated offsets
Figure 4: Amplitude offset estimation, sinusoidal input with frequency ω = 0.2πrad/s.
we can see that the estimation errors are again very small. These simulations show that a “bad” input signal should not be a problem for a real AD-converter since the input signal frequency has to be exactly a multiple of the sampling frequency. To avoid bad offset estimates we can adjust the algorithm so that it only allows the estimate to change slowly, e.g. by using a forgetting factor. Figure 6 shows the estimation errors for sinusoidal input signals of different frequencies up to half the sampling frequency. If we look at the DFT of the output signal before and after estimation and correction we can see that the overtones are completely eliminated, see Figure 7.
4.2
The timing offset estimation
In the simulations of AD-converters with timing offsets we have used random timing offsets between−10% and 10% of the nominal sampling time. We have used 100000 samples in the simulations if nothing else is mentioned. The sam-pling frequency is fs = 1Hz in all the simulations. Figure 8 shows the
re-sult of the timing offset estimation when a sinusoidal input with frequency
ω = 0.02rad/s has been used. The estimation algorithm works best for low
frequencies, but the errors stay below 1% for frequencies up to about 1/3 of the Nyquist frequency. In Figure 9 the root mean square error is plotted after estimation with sinusoidal input signals with frequencies up to the Nyquist fre-quency. If we look at the DFT of the output signal before and after estimation we can also in this case see that all the harmonics are eliminated, see Figure 10. If there is additive white noise on the signal the estimation accuracy decreases, especially for low frequency signals. That is because the derivative of a low frequency signal is small compared to the noise, see Figure 11. The estimation algorithm has also been tested with other signals, for example sums of sinusoids and bandlimited white noise. In Figure 12 the estimation error is shown for estimation from bandlimited white noise with different bandwidths. From all these simulations we can see that the estimation algorithm works well for most
1 2 3 4 5 6 7 8 9 10 −0.04 −0.02 0 0.02 0.04 0.06 0.08 ADC number Input signal near multiple of sampling frequency
actual offsets estimated offsets
Figure 5: Amplitude offset estimation, sinusoidal input with frequency ω = 0.2001πrad/s. 10−3 10−2 10−1 100 10−7 10−6 10−5 10−4 10−3 10−2 10−1
Input signal frequency
Root mean square error
Amplitude offset errors after estimation with sinusoidal input
After estimation Before estimation
Figure 6: Amplitude offset estimation error, sinusoidal input with different fre-quencies. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10−1 100 101 102 103 104 105 frequency DFT of output signal before correction
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10−1 100 101 102 103 104 105 frequency DFT of output signal after estimation and correction
Figure 7: DFT of output signal before and after estimation and correction of amplitude offset errors.
1 2 3 4 5 6 7 8 9 10 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 ADC number
Actual and estimated timing offsets, sinusoidal input
actual time offsets estimated time offsets
Figure 8: Timing offset estimation, sinusoidal input with frequency ω = 0.02rad/s. 10−3 10−2 10−1 100 10−6 10−5 10−4 10−3 10−2 10−1
Input signal frequency
Root mean square error
timing offset errors after estimation with sinusoidal input
After estimation Before estimation
Figure 9: Timing offset estimation, sinusoidal input with frequency up to the nyquist frequency. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10−2 10−1 100 101 102 103 104 105 frequency DFT of output signal before correction
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10−1 100 101 102 103 104 105 frequency DFT of output signal after estimation and correction
Figure 10: DFT of output signal before and after estimation and correction of timing offset errors.
10−3 10−2 10−1 100 10−4
10−3 10−2 10−1
Input signal frequency
Root mean square error
timing offset errors after estimation with sinusoidal noisy input, SNR=60dB
After estimation Before estimation
Figure 11: Timing offset estimation error, sinusoidal with noise, SN R = 60dB.
10−2 10−1 100 101 10−4 10−3 10−2 10−1
Input signal bandwidth
Root mean square error
timing offset errors after estimation with bandlimited noise
After estimation Before estimation
Figure 12: Timing offset estimation error, bandlimited white noise with different bandwidths as input signals.
10−3 10−2 10−1 100 10−3 10−2 10−1 100
Input signal frequency
Root mean square error
Figure 13: timing offset estimation errors for different number of data, between 10 and 1000000. 1 2 3 4 5 6 7 8 9 10 −0.15 −0.1 −0.05 0 0.05 0.1 ADC number Actual and estimated timing offsets, before iteration
actual time offsets estimated time offsets
1 2 3 4 5 6 7 8 9 10 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 ADC number Actual and estimated timing offsets, after one iteration
actual time offsets estimated time offsets
Figure 14: To the left: estimation error before iteration. To the right: estimation error after one iteration.
kinds of signals. If the estimates are updated slowly, e.g. with a forgetting factor close to one, “bad” parts of the signal will not destroy the estimation accuracy. Simulations have also shown that accuracy of the estimates is signif-icantly improved with more data up to about 106 samples, especially for noisy
signals. With more than 106 samples the improvement is quite small however,
see Figure 13. We can also see that the iteration in equation 9 is quite impotant. One iteration is, however, usually enough, see Figure 14.
5
Conclusions
Parallel AD-converters are used for imcreasing the sample rate by letting each AD-converter work at a lower sampling rate. A problem in these AD-converter structures is the synchronization and calibration between the AD-converters. We have presented a method to estimate amplitude and timing offsets in a parallel AD-conveter structure. The estimation works without knowledge of the input signal. The only assumtion that is needed about the signal is that the
energy spectrum should be concentrated to the lower frequencies for the timing offset estimation to work. Simulations show that the amplitude offset estimation works well under general conditions. The timing offset estimation works well for signals bandlimited to about 1/3 of the nyquist frequency. The accuracy is however still improved even if the bandwidth is at the nyquist frequency. If there is too much white noise in the signal the estimation accuracy decreases.
References
[1] G. Dahlquist ˚A. Bj¨ork. Numerical mathematics. July 1997.
[2] S.K. Mitra A. Petraglia. Analysis of mismatch effects among A/D con-verters in a time-interleaved waveform digitizer. IEEE Transactions on
Instrumentation and Measurement, 40(5):831–835, October 1991.
[3] J.J. Corcoran. Timing and amplitude error estimation for time-interleaved analog-to-digital converters. US Patent nr. 5,294,926, October 1992. [4] F.H. Irons D.M. Hummels, J.J. McDonald II. Distortion compensation
for time-interleaved analog to digital converters. In IEEE Instrumentation
an Measurement Technology Conference, 1996, pages 728–731. IEEE, June
1996.
[5] M. Ganshadzahi F. Marvasti, M. Analoui. Recovery of signals from nonuni-form samples using iterative methods. IEEE Transactions on Signal
Pro-cessing, 39(4):872–878, April 1991.
[6] Y-C Jenq. Digital spectra of nonuniformly sampled signals: Digital look-up tunable sinusoidal oscillators. IEEE Transactions on Instrumentation and
Measurement, 37(3):358–362, September 1988.
[7] Y-C Jenq. Digital spectra of nonuniformly sampled signals: Fundamentals and high-speed waveform digitizers. IEEE Transactions on Instrumentation
and Measurement, 37(2):245–251, June 1988.
[8] 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, 39(1):71–75, February 1990.
[9] L. Cuvelier L. Vandendorpe, B. Maison. RLS design of polyphase compo-nents for the interpolation of periodically noonuniformly sampled signals. In International Conference on Acoustics, Speech, and Signal Processing,
1995, volume 2, pages 1229–1232. IEEE, 1995.
[10] L. Ljung. System Identification, Theory for the user. Prentice-Hall, 2
