Doctoral School of Electrical Engineering 

Department of Measurement  and Information Systems

Supervisor: Dr. Kollár István

Efficient Parameter Estimation of Sampled and Quantized Signals 

Brief presentation of the topic

The aim of my research is to estimate the parameters of sampled and quantized signals using the methods of modern estimation theory. As digital signal processing is used in almost every modern measurement  and control system, the accurate parameter estimation of these signals is fundamental to ensure reliability and good performance. The most important application – in my scope – is to test and grade analog-to-digital converters (ADCs) using the estimated parameters and the measured waveform.

Brief introduction of the research place

I participate in the PhD program of the BME Graduate School of Electrical Engineering at the Department of Measurement and Information Systems under the supervision of professor István Kollár. We have a research group containing two PhD students and a PhD candidate. We are working on strongly related topics lead by István Kollár. An important part of our work is to develop and maintain a MATLAB (and lately, LabVIEW) toolbox for ADC testing.

 

History and broader context of the research

In modern measurement and control systems signal processing is performed by digital devices (microcontrollers, CPUs, DSPs, etc.) thus these systems use sampled and quantized signals. However, the physical environment is continuous, therefore, analog-to-digital and digital-to-analog conversion is required at the boundaries of the system. Analog-to-digital conversion inherently accompanied with loss even in ideal case (assuming ideal devices), however, in non-ideal case it is fundamental to describe the type and amount of information loss. Testing and grading the ADCs is needed since the appearance of the first devices. The information loss can be described by different quality measures such as SINAD, SNR, SFDR, THD, etc. The ADC test techniques have been standardized by the IEEE and the IEC as well. IMEKO organizes a  workshop on ADC and DAC modeling and testing each year since 1997. Analog-to-digital and digital-to-analog conversion is essential in embedded systems as well.

 

Aim of the research, problems to solve

The aim of the research is to extract all the available information from signals sampled and quantized by non-ideal ADCs. Naturally, it is only feasible to estimate signals that can be described easily with a few parameters. Sine waves and periodic signals (multi-sine waves) are appropriate for parametric estimation. One of the primary applications of the sine wave estimation is to describe the quality of the conversion using standard quantities. The accuracy of the quality measures of the device under test depends on the accuracy of the estimated parameters of the analog signal. Nevertheless, the task is to estimate the parameters of the observation channel via measurements. Thus very little a priori information can be used in these methods. This little information is usually the quality of the analog signal (distortion, signal-to-noise ratio, etc.) and the algorithms can be performed based on this information (see Methods). Naturally, the accurate estimation of analog signal parameters is important not only to grade the conversion, but for several other purposes. Using an analog input either for measurement or for control, the more precise information about the input signal means better measurement results or more efficient control.

 

The most important tasks and unsolved problems for me are as follows.

 

1. Examining the theoretical boundaries of the approximate maximum likelihood estimators and comparing them with the experimental results. Calculating the Cramér-Rao lower bound and comparing them with the empirical covariance of the estimators.

2. Parameterization of the nonlinearity of the quantizes. Examination of the compressibility of the information on nonlinearity. The aim is to improve the approximation of the approximate estimator and to reduce the parameter space.

3. Investigation of the optimization problems that potentially degrade the estimation: defining appropriate termination criteria, developing fault detection and robust methods to ensure convergence even in the presence of local extrema.

Methods

The research has three main parts and different methods were used for them. The first part is the theoretical one: application of the modern estimation theory for sampled and quantized data (naturally based on existing methods). Amending the existing algorithms via e.g. extending the signal model or developing fast and robust initial estimation methods. Looking for feasible simplifications to decrease computational demand significantly. Another large part of the work is performing simulations: the developed estimation methods are tested using simulated measurements. These simulated measurements scan a particular part of the parameter space (an investigation requires multiple hundreds or even thousands of simulations). The main parameters of the measurements: number of bits, number of samples, frequency, number of periods, presence of overdrive, harmonic distortion, DC component, initial phase, etc. Usually the measurement noise shall be simulated as well. In these cases 20–50 simulated records are used with the same parameters, only the actual noise realization makes them different. This way the estimation results can be collected into populations and important properties of the estimators (variance, bias, mean squared error, normality, etc.) can be examined experimentally. The estimator populations enable to perform more complex statistical hypothesis tests. E.g. using the Kolmogorov-Smirnov test for two populations to decide whether or not the estimators come from the same distribution (practically, to decide whether or not two estimation methods give the same result). The third main research area is the validation of the algorithms using real measurement records. This stage is of primary importance, since simulations can only consider a finite set of non-ideal phenomena, whereas real measurements can produce disturbances not yet treated but shall be considered in the future. Examination of the algorithms using real measurements provide a very important feedback in the development process.

 

Results

The results relate to the parameter estimation of sampled and quantized sine waves and measurements to test and grade analog-to-digital converters.

 

1. A method to get an approximate maximum likelihood estimator for the parameters of the non-ideally quantized sine wave was suggested. The method was implemented, and its properties were examined via simulations and its feasibility was verified using real measurement records[E1].

2. The numerical properties of the cost function of the approximate maximum likelihood estimators was estimated and a numerical recipe was developed especially for this cost function based on well-known recipes [E2].

3. In cooperation with the colleagues of the Technical University of Kosice, a standardized and a novel sine wave fitting method was examined. The properties of these techniques were compared and the advantages of the novel one were presented and the cases were identified when the novel method shall be used [E3].

4. In cooperation with the researchers of the Science University of Perugia, the numerical issues related to sine wave fitting tasks were examined and multiple problems were found that can spoil the estimation. A solutions to avoid these problems was also suggested [E4].

5. Based on work of former developers, a freely available MATLAB toolbox for ADC testing was developed and maintained in standard and nonstandard ways in a unified framework. This toolbox has been already downloaded in more than 15 countries worldwide [E5], [project site].

 

Expected impact and future research

In the future the elaborated approximate maximum likelihood method will be optimized further with respect to computational demand. Parameterization of the nonlinearity of the converter shall be examined to the improve the approximation. I intend to implement the algorithms in more general and more widely used languages (e.g. C++, Python). In this way, the method will be widely applicable in the industry and can be standardized. The results of further investigations (potentially other estimation techniques) shall also be elaborated to enable standardization, e.g. to be included in the IEEE standards for ADCs and waveform recorders (IEEE-1241 and IEEE-1057, respectively).

Publications, citations, hyperlinks

Detailed list of publications (Hungarian Repository of Scientific Works)

 

Important publications:

 

1. B Renczes, I Kollár, P Carbone, A Moschitta, V Pálfi, T Virosztek: “Analyzing Numerical Optimization Problems of Finite Resolution Sine Wave Fitting Algorithms” In: Proceedings of the IEEE International Instrumentation and Measurement Technology Conference. Pisa, Italy, May 11–14, 2015. p. 1662–1667.

 

1. Virosztek Tamás, Kollár István “ADC Testing in Standard and Non-standard Ways, Executed in a Unified Framework” In: 20th IMEKO TC4 International Symposium and 18th International Workshop on ADC Modelling and Testing Research on Electric and Electronic Measurement for the Economic Upturn. Benevento, Italy, September 15–17, 2014. Paper 232. 6 p.

2. Ján Šaliga, Linus Michaeli, Ján Buša, Jozef Lipták, István Kollár, Tamás Virosztek “A Comparison of Least Squares and Maximum Likelihood Based Sine Fittings in ADC Testing” MEASUREMENT 46:(10) pp. 4362–4368. (2013)

 

1. Tamás Virosztek, István Kollár “User-Friendly Matlab Tool for Easy ADC Testing” In: anon (Ed.) 19th IMEKO TC 4 Symposium and 17th IWADC Workshop: Advances in Instrumentation and Sensors Interoperability. Barcelona, Spain, July 18–19, 2013, 2013. pp. 561–568. (ISBN:978–84–616–5438–3)

 

Important citations:

 

1. Huang C, Wang G, Yang W “A fast maximum likelihood estimation for high-resolution ADC test” In: 20th IMEKO TC4 Symposium on Measurements of Electrical Quantities: Research on Electrical and Electronic Measurement for the Economic Upturn, Together with 18th TC4 International Workshop on ADC and DCA Modeling and Testing, IWADC 2014. IMEKO-International Measurement Federation Secretariat, 2014. (ISBN 9789299007327) pp. 569–573.

2. Xu Li, Sudani Siva Kumar, Chen Degang “Efficient Spectral Testing With Clipped and Noncoherently Sampled Data” IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT (ISSN: 0018–9456) 63: (6) pp. 1451–1460. (2014)

3. Moschitta A, Schoukens J, Carbone P “Information and Statistical Efficiency When Quantizing Noisy DC Values” IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT (ISSN: 0018–9456) 64: (2) pp. 308–317. (2015)

4. Michaeli L, Saliga J, Liptak J, Godla M, Kollar I “Measurement of distorted exponential signal components using maximum likelihood estimation” MEASUREMENT (ISSN: 0263–2241) 58: pp. 503–510. (2014)

 

Hyperlinks:

 

International organizations related to this research topic:

IEEE Instrumentation & Measurement Society

IMEKO TC-4 Society

 

International journals related to this research topic:

IEEE Transactions on Instrumentation and Measurement

Measurement (Journal of the International Measurement Confederation)

 

Recently held conferences and symposia related to this topic:

17th TC-4 Workshop IWADC on ADC and DAC Modeling and Testing

18th TC-4 Workshop IWADC on ADC and DAC Modeling and Testing

2015 IEEE International Instrumentation and Measurement Technology Conference (I2MTC)