CoSimulation for Systematic and Statistical Correction of MultiDigitaltoAnalogConvertor Systems
 Author: Park Youngcheol, Yoon Hoijin
 Publish: Journal of electromagnetic engineering and science Volume 17, Issue1, p39~43, 31 Jan 2017

ABSTRACT
In this paper, a systematic and statistical calibration technique was implemented to calibrate a highspeed signal converting system containing multiple digitaltoanalog converters (DACs). The systematic error (especially the imbalance between DACs) in the current combining network of the multiDAC system was modeled and corrected by calculating the path coefficients for individual DACs with wideband reference signals. Furthermore, by applying a Kalman filter to suppress noise from quantization and clock jitter, accurate coefficients with minimum noise were identified. For correcting an arbitrary waveform generator with two DACs, a cosimulation platform was implemented to estimate the system degradation and its corrected performance. Simulation results showed that after correction with 4.8 Gbps QAM signal, the signaltonoiseratio improved by approximately 4.5 dB and the errorvectormagnitude improved from 4.1% to 1.12% over 0.96 GHz bandwidth.

KEYWORD
DigitaltoAnalogConverter , MillimeterWave , Quadrature Modulation , Signal Generator , Wireless System

I. INTRODUCTION
To realize the deployment of 5G systems, methods to calibrate signal generators with wide bandwidths are being studied [1,2]. The highspeed signals required for 5G systems are generally synthesized by incorporating mediumspeed digitaltoanalog converters (DACs) as if the overall speed reaches multiple times the speed of an individual DAC [3]. Many methods are available for increasing the sampling speed. One such method is called time interleaving (TI), in which the output of each DAC in the time domain is combined. It is used in many converting devices such as analogtodigital converters (ADCs) and DACs [4,5]. These systems require a highspeed clock to drive the output switch in such a way that the output current of the lowspeed DAC is directed alternatingly toward the output port; as a result, the final combined current is equivalently composed in accordance with the highspeed sampling clock.
However, in practice, TI devices cause additional errors owing to the nonidealities of the additional process. For example, in TI DACs and ADCs, clock jitter causes random phase errors in the equivalent sampling pulse train, resulting in irregular spurs in the frequency domain [6]. In addition, because high sampling speed in digital converters is often achieved at the expense of the bit resolution, the quantization error cannot be ignored as a random noise source, especially when digital converters with speeds exceeding hundreds of mega samples per second are concerned [7]. On the other hand, systematic errors such as the nonuniform analog characteristic between the lowspeed DACs also degrade the signal quality. Finally, because the calibration process is based on discrete Fourier transform (DFT), it is critical to maintain synchronization between the measurement equipment and to carefully design the reference waveform; otherwise, unintended artifacts are introduced from the windowing and picketfence effects [8].
The author has proposed correction methods for a TI DAC system [2, 9]. However, it showed somewhat limited performance, presumably owing to the residual random noise and lack of accurate deembedding. In this paper, we addressed these issues by using a novel noise model and Kalman filtering for correcting the statistical error. Furthermore, a cosimulation platform was implemented to enable systematic nonideality correction across the time, frequency, and constellation domains.
II. COMBINING OUTPUT WAVEFORMS
Various methods can be used to combine the output of each converter. Among these, the TI technique is widely used because it affords high combining ratio [4–7]. Whereas TI ADCs incorporate tens of ADCs, commercial TI DACs mostly use a combination of only two DACs [4].
In the TI architecture, the bandlimited output waveform
x (t ) is expressed by its discrete form with sampling periodT_{s} :This is the result of TI combination, from which the
m ^{th} individual DAC output is expressed as follows (Fig. 1):where
M is the number of TI DACs.As a result, its Fourier transform is
where
X (ω) is the Fourier transform ofx (t ).Therefore, the TI output
X (ω)_{SUM} is the sum of each DAC output, and each DAC path has a nonuniform transfer coefficientc_{m} as follows:In addition,
X (ω)_{SUM} is expressed as the sum of Nyquist zone imagesX_{k} as follows:where
H_{k} (ω) represents the response at thek ^{th} Nyquist zone.Therefore, the nonuniform path coefficients
c_{m} are found through a comparison withX_{k} Nyquist zone responses.Furthermore,
H_{k} (ω) can be found by dividingX_{k} byX (ωω_{k}) in (5).III. CORRECTION BY KALMAN FILTERING
The finite resolution of DACs causes quantization noise at the output. The
rms noise voltage at the combined output is approximated aswhere
N is the number of bits in a DAC, andV_{pp} is the fullscale voltage.Furthermore, jitter in the sampling clock causes noise power, which is estimated as follows [5]:
where is the variance of the timing jitter.
Squaresumming these noise voltages gives the total random noise voltage as
Because this noise is random and white over the entire sampling frequency, a Kalman filter is applied to reduce the noise power.
A Kalman filter is effective for estimating the state of a discretetime controlled process
x under a linear stochastic difference equation [10]:and the measurement equation
where the process noise covariance
Q and measurement noise covarianceR affect the states ofx and the measurement statez .The process noise
w and measurement noisev are assumed to be independent, white, and normally distributed:When the a priori state estimation error at step
k ,e _{k}^{}, is determined by the difference betweenxk and the a priori estimateThe a posteriori error
e_{k} is defined similarly.The error covariance matrix is defined from
e_{k} as follows:It is then used to update the Kalman gain of state
k ,K_{k} , that minimizes the error.Fig. 2 shows the complete Kalman filter update operation. For applying the Kalman filter to the TI DAC system, the state vector
x was defined as a DFT coefficient of lengthL , and the measurement vectorz was defined as the time domain voltage from the measurement equipment. The measurement noise covariance matrix is defined from the rootsumsquared values of (6) and (7).IV. MEASUREMENT AND COSIMULATION
The path coefficients in Section II are identified by carefully designed waveforms that occupy the frequency band of interest. In this regard, a highspeed 64 quadratureamplitudemodulated (QAM) waveform was synthesized at 2 GHz with speed of 4.8 Gbps and signaltonoiseratio exceeding 43 dB. For identifying the path coefficients, a commercial arbitrary signal generator (AWG) with two 8bit TI DACs was measured using the QAM waveform and a digitizer.
Furthermore, to estimate the signal quality degradation from the random noise and systematic nonideality, a TI DAC system with the same configuration was modeled and implemented on a simulation platform. The simulator calculates the errorvectormagnitude (EVM) as a metric of the signal quality. Fig. 3 shows the implemented system with the QAM reference source and the RF upconverter to 2 GHz. By applying the Kalman filter to the measured waveforms, an accurate set of path coefficients was obtained, and it was loaded on the simulation platform to represent the system nonideality. The Kalman gain was tuned to converge within five measurements. The filter coefficient was used to correct the system nonidealities throughout TI paths. Fig. 4 shows the spectrum of the reference QAM signal and the uncorrected DAC output. The identified path response of the AWG has inband ripple of ±2 dB and rolloff from the sampling aperture. The path response was stored in a file and loaded into the simulation platform for the cosimulation to represent the system artifact in the TI DAC system. As a result, the received signal shows degraded quality, as in Fig. 5(a), in which the detected QAM constellation is overlaid on the reference symbol locations. Because the overall system is assumed to be quasilinear, the systematic error is corrected by the inverse coefficient, which consists of a wideband correction filter. As such, the correction filter is calculated and applied in the cosimulation platform. Figs. 5(b) and 6 show the result of the correction. As shown in these figures, the suggested correction clearly improved the EVM from 4.1% to 1.13%; this is consistent with the SNR improvement of 4.5 dB as obtained by a commercial vector signal analyzing software. The overall mean error in the EVM over 100 symbols was approximately 1.124% ± 0.22%. For the sake of comparison, correction was performed without the Kalman filter, and it showed EVM of about 1.43%.
V. CONCLUSION
This paper presented systematic and stochastic methods for the correction of multigigabitpersecond signals with AWGs that include TI DACs. BY using Kalman filtering, accurate system coefficients between each DAC are identified, and then, correction filter coefficients to overcome nonuniform path responses are calculated. Through a cosimulation of the AWG system, we improved the EVM from 4.1% to 1.13% for a 4.8 Gsps, 64QAM modulated signal with 2GHz carrier frequency. As an extension of this study, this approach could be useful to calibrate multiple TI converting systems to compensate for both systematic and stochastic error sources.

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[Fig. 1.] Timeinterleaving digitaltoanalog converting system.

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[Fig. 2.] Operation of Kalman filter [10].

[Fig. 3.] Cosimulation setup representing nonideal degradation in time interleaving (TI) DAC systems including quantization noise, TI mismatch, and correction filter.

[Fig. 4.] Generated 64 QAM reference signal with 800 Msym/s and the uncorrected system response.

[Fig. 5.] Constellation of 64 QAM signal from cosimulation of time interleaving DAC system with (a) random and systematic error sources (b) after correction filter is applied.

[Fig. 6.] Errorvectormagnitude (EVM) of cosimulated 64 QAM with correction filter applied to time interleaving DAC.