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  • 1. Baratchart, L.
    et al.
    Enqvist, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Gombani, A.
    Olivi, M.
    Minimal symmetric Darlington synthesis2007In: MCSS. Mathematics of Control, Signals and Systems, ISSN 0932-4194, E-ISSN 1435-568X, Vol. 19, no 4, p. 283-311Article in journal (Refereed)
    Abstract [en]

    We consider the symmetric Darlington synthesis of a p x p rational symmetric Schur function S with the constraint that the extension is of size 2p x 2p. Under the assumption that S is strictly contractive in at least one point of the imaginary axis, we determine the minimal McMillan degree of the extension. In particular, we show that it is generically given by the number of zeros of odd multiplicity of I (p)-SS*. A constructive characterization of all such extensions is provided in terms of a symmetric realization of S and of the outer spectral factor of I-p-SS*. The authors's motivation for the problem stems from Surface Acoustic Wave filters where physical constraints on the electro-acoustic scattering matrix naturally raise this mathematical issue.

  • 2. Baratchart, L.
    et al.
    Enqvist, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Gombani, A.
    Olivi, M.
    Minimal symmetric Darlington synthesis: A frequency domain approach2006In: Proc IEEE Conf Decis Control, 2006, p. 6732-6737Conference paper (Refereed)
    Abstract [en]

    Given a p × p Schur function S, we consider the problem of constructing a symmetric Darlington synthesis of minimal size. This amounts essentially to finding a stable all-pass square extension of S of minimal size. The characterization is done in terms of the multiplicities of the zeros. As a special case we obtain conditions for symmetric Darlington synthesis to be possible without increasing the McMillan degree for a symmetric rational contractive matrix which is strictly contractive in the right half-plane. This technique immediately extends to the case where, allowing for a higher dimension of the extension, we require no increase in the McMillan degree. Also in this case we obtain sharper results than those existing in the literature (see [1]).

  • 3.
    Byrnes, Christopher
    et al.
    KTH, Superseded Departments, Mathematics.
    Enqvist, Per
    KTH, Superseded Departments, Mathematics.
    Lindquist, Anders
    KTH, Superseded Departments, Mathematics.
    Cepstral coefficients, covariance lags, and pole-zero models for finite data strings2001In: IEEE Transactions on Signal Processing, ISSN 1053-587X, E-ISSN 1941-0476, Vol. 49, no 4, p. 677-693Article in journal (Refereed)
    Abstract [en]

    One of the most widely used methods of spectral estimation in signal and speech processing is linear predictive coding (LPC). LPC has some attractive features, which account for its popularity, including the properties that the resulting modeling filter i) matches a finite window of n + 1 covariance lags, ii) is rational of degree at most n, and iii) has stable zeros and poles. The only limiting factor of this methodology is that the modeling filter is "all-pole," i.e., an autoregressive (AR) model. In this paper, we present a systematic description of all autoregressive moving-average (ARMA) models of processes that have properties i)-iii) in the context of cepstral analysis and homomorphic filtering. Indeed, we show that each such ARMA model determines and is completely determined by its finite windows of cepstral coefficients and covariance lags. This characterization has an intuitively appealing interpretation of a characterization by using measures of the transient and the steady-state behaviors of the signal, respectively. More precisely, we show that these nth-order windows form local coordinates for all ARMA models of degree n and that the pole-zero model can be determined from the windows as the unique minimum of a convex objective function. We refine this optimization method by first noting that the maximum entropy design of an LPC filter is obtained by maximizing the zeroth cepstral coefficient, subject to the constraint i). More generally, we modify this scheme to a more well-posed optimization problem where the covariance data enters as a constraint and the linear weights of the cepstral coefficients are "positive"-in a sense that a certain pseudo-polynomial is positive-rather succinctly generalizing the maximum entropy method. This new problem is a homomorphic filter generalization of the maximum entropy method, providing a procedure for the design of any stable, minimum-phase modeling filter of degree less or equal to n that interpolates the given covariance window We conclude the paper by presenting an algorithm for realizing these biters in a lattice-ladder form, given the covariance window and the moving average part of the model. While we also show how to determine the moving average part using cepstral smoothing, one can make use of any good a priori estimate for the system zeros to initialize the algorithm. Indeed, we conclude the paper with an example of this method, incorporating an example from the literature on ARMA modeling.

  • 4.
    Byrnes, Christopher
    et al.
    KTH, Superseded Departments, Mathematics.
    Enqvist, Per
    KTH, Superseded Departments, Mathematics.
    Lindquist, Anders
    KTH, Superseded Departments, Mathematics.
    Identifiability and well-posedness of shaping-filter parameterizations: A global analysis approach2002In: SIAM Journal of Control and Optimization, ISSN 0363-0129, E-ISSN 1095-7138, Vol. 41, no 1, p. 23-59Article in journal (Refereed)
    Abstract [en]

    In this paper, we study the well-posedness of the problems of determining shaping filters from combinations of finite windows of cepstral coefficients, covariance lags, or Markov parameters. For example, we determine whether there exists a shaping filter with a prescribed window of Markov parameters and a prescribed window of covariance lags. We show that several such problems are well-posed in the sense of Hadamard; that is, one can prove existence, uniqueness (identifiability), and continuous dependence of the model on the measurements. Our starting point is the global analysis of linear systems, where one studies an entire class of systems or models as a whole, and where one views measurements, such as covariance lags and cepstral coefficients or Markov parameters, from data as functions on the entire class. This enables one to pose such problems in a way that tools from calculus, optimization, geometry, and modern nonlinear analysis can be used to give a rigorous answer to such problems in an algorithm-independent fashion. In this language, we prove that a window of cepstral coefficients and a window of covariance coefficients yield a bona de coordinate system on the space of shaping filters, thereby establishing existence, uniqueness, and smooth dependence of the model parameters on the measurements from data.

  • 5.
    Enqvist, Per
    KTH, Superseded Departments, Mathematics.
    A convex optimization approach to arma(n,m) model design from covariance and cepstral data2004In: SIAM Journal of Control and Optimization, ISSN 0363-0129, E-ISSN 1095-7138, Vol. 43, no 3, p. 1011-1036Article in journal (Refereed)
    Abstract [en]

    Methods for determining ARMA(n, m) filters from covariance and cepstral estimates are proposed. In [C. I. Byrnes, P. Enqvist, and A. Lindquist, SIAM J. Control Optim., 41 ( 2002), pp. 23-59], we have shown that an ARMA( n, n) model determines and is uniquely determined by a window r(0), r(1),..., r(n) of covariance lags and c(1), c(2),..., c(n) of cepstral lags. This unique model can be determined from a convex optimization problem which was shown to be the dual of a maximum entropy problem. In this paper, generalizations of this problem are analyzed. Problems with covariance lags r(0), r(1),..., r(n) and cepstral lags c(1), c(2),..., c(m) of different lengths are considered, and by considering different combinations of covariances, cepstral parameters, poles, and zeros, it is shown that only zeros and covariances give a parameterization that is consistent with generic data. However, the main contribution of this paper is a regularization of the optimization problems that is proposed in order to handle generic data. For the covariance and cepstral problem, if the data does not correspond to a system of desired order, solutions with zeros on the boundary occur and the cepstral coefficients are not interpolated exactly. In order to achieve strictly minimum phase filters for estimated covariance and cepstral data, a barrier-like term is introduced to the optimization problem. This term is chosen so that convexity is maintained and so that the unique solution will still interpolate the covariances but only approximate the cepstral lags. Furthermore, the solution will depend analytically on the covariance and cepstral data, which provides robustness, and the barrier term increases the entropy of the solution.

  • 6.
    Enqvist, Per
    KTH, Superseded Departments, Mathematics.
    A homotopy approach to rational covariance extension with degree constraint2001In: International journal of mathematics and computer science, ISSN 1641-876X, Vol. 11, no 5, p. 1173-1201Article in journal (Refereed)
    Abstract [en]

    The solutions to the Rational Covariance Extension Problem (RCEP) are parameterized by the spectral zeros. The rational filter with a specied numerator solving the RCEP can be determined from a known convex optimization problem. However, this optimization problem may become ill-conditioned for some parameter values. A modication of the optimization problem to avoid the illconditioning is proposed and the modified problem is solved effciently by a continuation method.

  • 7.
    Enqvist, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Covariance interpolation and geometry of power spectral densities2015In: 2009 European Control Conference, ECC 2009, 2015, p. 4505-4510Conference paper (Refereed)
    Abstract [en]

    When methods of moments are used for identification of power spectral densities, a model is matched to estimated second order statistics such as, e.g., covariance estimates. There is an infinite family of power spectra consistent with such an estimate and in applications, such as identification, we want to single out the most representative spectrum. Here, we choose a prior spectral density to represent a priori information, and the spectrum closest to it in a given quasi-distance is determined. Depending on the selected quasi-distance, the geometry of the space of power spectral densities varies, and the structure of the minimizing spectral density changes with it. Recently, the Kullback-Leibler divergence, the Itakura-Saito divergence and Hellinger distances has been shown to determine power spectral densities of rational form and with tractable properties. Here, starting instead with the structure of the power spectral density, different (quasi-)distances and geometries for power spectral densities are derived.

  • 8.
    Enqvist, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    On the simultaneous realization problem - Markov-parameter and covariance interpolation2006In: Signal Processing, ISSN 0165-1684, E-ISSN 1872-7557, Vol. 86, no 10, p. 3043-3054Article in journal (Refereed)
    Abstract [en]

    An efficient algorithm for determining the unique minimal and stable realization of a window of Markov parameters and covariances is derived. The main difference compared to the Q-Markov COVER theory is that here we let the variance of the input noise be a variable, thus avoiding a certain data consistency criterion. First, it is shown that maximizing the input variance of the realization over all interpolants yields a minimal degree solution-a result closely related to maximum entropy. Secondly, the state space approach of the Q-Markov COVER theory is used for analyzing the stability and structure of the realization by straightforward application of familiar realization theory concepts, in particular the occurrence of singular spectral measures is characterized.

  • 9.
    Enqvist, Per
    KTH, Superseded Departments, Mathematics.
    Spectral Estimation by Geometric, Topological and Optimization Methods2001Doctoral thesis, comprehensive summary (Other scientific)
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  • 10. Enqvist, Per
    Spectrum estimation by interpolation of covariances and cepstrum parameters in an exponential class of spectral densities2006In: PROCEEDINGS OF THE 45TH IEEE CONFERENCE ON DECISION AND CONTROL, VOLS 1-14, 2006, p. 799-804Conference paper (Refereed)
    Abstract [en]

    Given output data of a stationary stochastic process estimates of the covariances and cepstrum parameters can be obtained. Methods of moments have been applied to these parameters for designing ARMA processes, and it has been shown that these two sets of parameters in fact form local coordinates for the set of ARMA processes, but that some combinations of cepstrum parameters and covariances cannot be matched exactly within this class of processes. Therefore, another class of processes is considered in this paper in order to be able to match any combination of covariances and cepstrum parameters. The main result is that a process with spectral density of the form phi(z) = exp {Sigma(m)(k=0) p(k)(z(k) + z(-k))}/Sigma(n)(k=0) q(k)(z(k) + z(-k))/2 can always match given covariances and cepstrum parameters. This is proven using a fixed-point argument, and a non-linear least-squares problem is proposed for determining a solution.

  • 11.
    Enqvist, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Spectrum estimation by interpolation of covariances and cepstrum parameters in an exponentional class of spectral densities2006Conference paper (Refereed)
    Abstract [en]

    Given output data of a stationary stochastic process estimates of the covariances and cepstrum parameters can be obtained. Methods of moments have been applied to these parameters for designing ARMA processes, and it has been shown that these two sets of parameters in fact form local coordinates for the set of ARMA processes, but that some combinations of cepstrum parameters and covariances cannot be matched exactly within this class of processes. Therefore, another class of processes is considered in this paper in order to be able to match any combination of covariances and cepstrum parameters. The main result is that a process with spectral density of the form Φ exP{Σ k=0mpk(zk + z-k)}/Σ k=0nqk(zk + z-k)/2 can always match given covariances and cepstrum parameters. This is proven using a fixed-point argument, and a non-linear least-squares problem is proposed for determining a solution.

  • 12.
    Enqvist, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Avventi, Enrico
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Approximative covariance interpolation with a quadratic penalty2007In: Proceedings Of The 46th IEEE Conference On Decision And Control, Vols 1-14, 2007, p. 4489-4494Conference paper (Refereed)
    Abstract [en]

    Given output data of a stationary stochastic process estimates of the covariances parameters can be obtained. These estimates can be used to determine ARMA models to approximatly fit the data by matching the covariances exactly. However, the estimates of the covariances may contain large errors, especially if they are determined from short data sequences, and thus it makes sense to match the covariances only in an approximative way. Here we consider a convex method for solving an approximative covariance interpolation problem while maximizing the entropy and penalize the quadratic deviation from the nominal covariances.

  • 13.
    Enqvist, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Avventi, Enrico
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Approximative Linear and Logarithmic Interpolation of Spectra2009Report (Other academic)
    Abstract [en]

    Given output data of a stationary stochastic process estimates of covariance and cepstrum parameters can be obtained. These estimates can be used to determine ARMA models to approximately fit the data by matching the parameters exactly. However, the estimates of the parameters may contain large errors, especially if they are determined from short data sequences, and thus it makes sense to match the parameters in an approximate way. Here we consider a convex method for solving an approximate linear and logarithmic spectrum interpolation problem while maximizing the entropy and penalize the quadratic deviation from the nominal parameters.

    Download full text (pdf)
    fulltext
  • 14.
    Enqvist, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Minimal Itakura-Saito distance and Covariance interpolation2008In: 47TH IEEE CONFERENCE ON DECISION AND CONTROL, 2008 (CDC 2008), 2008, p. 137-142Conference paper (Refereed)
    Abstract [en]

    Identification of power spectral densities rely on measured second order statistics such as, e.g. covariance estimates. In the family of power spectra consistent with such an estimate a representative spectra is singled out; examples of such choices are the Maximum entropy spectrum and the Correlogram. Here, we choose a prior spectral density to represent a priori information, and the spectrum closest to the prior in the Itakura-Saito distance is selected. It is known that this can be seen as the limit case when the cross-entropy principle is applied to a gaussian process. This work provides a quantitative measure of how close a finite covariance sequence is to a spectral density in the Itakura-Saito distance. It is given by a convex optimization problem and by considering its dual the structure of the optimal spectrum is obtained. Furthermore, it is shown that strong duality holds and that a covariance matching coercive spectral density always exists. The methods presented here provides tools for discrimination between power spectrum, identification of power spectrum, and for incorporating given data in this process.

  • 15.
    Enqvist, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Svensson, Göran
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    A Marginal Allocation Approach to Resource Management for a System of Multiclass Multiserver Queues Using Abandonment and CVaR QoS Measures2019In: 7th International Conference on Operations Research and Enterprise Systems, ICORES 2018, Springer Verlag , 2019, p. 119-133Conference paper (Refereed)
    Abstract [en]

    A class of resource allocation problems is considered where some quality of service measure is set against the agent related costs. Three multiobjective minimization problems are posed, one for a system of Erlang-C queues and two for systems of Erlang-A queues. In the case of the Erlang-C systems we introduce a quality of service measure based on the Conditional Value-at-Risk with waiting time as the loss function. This is a risk coherent measure and is well established in the field of finance. An algebraic proof ensures that this quality of service measure is integer convex in the number of servers. In the case of the Erlang-A systems we introduce two different quality of service measures. The first is a weighted sum of fractions of abandoning customers and the second is Conditional Value-at-Risk, with the waiting time in queue for a customer conditioned on eventually receiving service. Finally, numerical experiments on the two system types with the given quality of service measures, are presented and the optimal solutions are compared.

  • 16.
    Enqvist, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Svensson, Göran
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    A state dependent chat system model2019In: ICORES 2019 - Proceedings of the 8th International Conference on Operations Research and Enterprise Systems, SciTePress , 2019, p. 121-132Conference paper (Refereed)
    Abstract [en]

    The main purpose of this paper is to introduce a model of a chat based communication system, as well as developing the necessary tools to enable resource optimization with regards to a measure of the service quality. The system is modeled by a Markov process in continuous time and with a countable state space. The construction of the intensity matrix corresponding to this system is outlined and proofs of a stationary state distribution and an efficient way of calculating it are introduced. A numerical example for system optimization when the service measure is the average sojourn time is included as well as a heuristic algorithm for quicker solution generation. 

  • 17.
    Enqvist, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Svensson, Göran
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Chat based contact center modeling system modeling, parameter estimation and missing data sampling2017In: ICORES 2017 - Proceedings of the 6th International Conference on Operations Research and Enterprise Systems, SciTePress , 2017, p. 464-469Conference paper (Refereed)
    Abstract [en]

    A Markovian system model for a contact center chat function is considered and partially validated. A hypothesis test on real chat data shows that it is reasonable to model the arrival process as a Poisson process. The arrival rate can be estimated using Maximum likelihood. The service process is more involved and the estimation of the service rate depends on the number of simultaneous chats handled by an agent. The estimation is made more difficult by the low level of detail in the given data-sets. A missing data approach with Gibbs sampling is used to obtain estimates for the service rates. Finally, we try to capture the generalized behaviour of the service-process and propose to use generalized functions to describe it when little information is available about the system at hand. 

  • 18.
    Enqvist, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Svensson, Göran
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. Teleopti WFM, Teleopti AB, Sweden.
    Multi-server marginal allocation With CVaR and abandonment based QoS measures2018In: ICORES 2018 - Proceedings of the 7th International Conference on Operations Research and Enterprise Systems, SciTePress, 2018, p. 297-303Conference paper (Refereed)
    Abstract [en]

    Two multi-objective minimization problems are posed, one for Erlang-C queues and one for Erlang-A queues. The objectives are to minimize the cost of added agents while also trying to optimize a quality of service measure. For the Erlang-C system we propose using the Conditional Value-at-Risk measure with waiting time as the loss function. We prove that this quality of service measure is integer convex in the number of servers. For the Erlang-A system we use the fraction of abandoning customers and some rate based weighting function as the service measure. Finally, a numerical comparison of the two system types is performed. The numerical results show the similarities between the two systems in terms of optimal points.

  • 19.
    Karlsson, Johan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Enqvist, Per
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Gattami, A.
    Confidence assessment for spectral estimation based on estimated covariances2016In: ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings, Institute of Electrical and Electronics Engineers (IEEE), 2016, p. 4343-4347Conference paper (Refereed)
    Abstract [en]

    In probability theory, time series analysis, and signal processing, many identification and estimation methods rely on covariance estimates as an intermediate statistics. Errors in estimated covariances propagate and degrade the quality of the estimation result. In particular, in large network systems where each system node of the network gather and pass on results, it is important to know the reliability of the information so that informed decisions can be made. In this work, we design confidence regions based on covariance estimates and study how these can be used for spectral estimation. In particular, we consider three different confidence regions based on sets of unitarily invariant matrices and bound the eigenvalue distribution based on three principles: uniform bounds; arithmetic and harmonic means; and the Marcenko-Pastur Law eigenvalue distribution for random matrices. Using these methodologies we robustly bound the energy in a selected frequency band, and compare the resulting spectral bound from the respective confidence regions.

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