Fast least squares solution of Vandermonde systems of equations

Demeure, C.J.; Scharf, Louis L.

doi:10.1109/icassp.1989.266900

Cited by 8 publications

(7 citation statements)

References 5 publications

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“…The estimate of the polynomial coefficients vector is given after a few mathematical steps as (44) where . Using the matrix inversion lemma we obtain that (44) can be written more compactly as (45) where is the projection matrix (46) We note that the above LS solution can also be obtained by exploiting the Vandermonde structure of [23]- [25] which leads to computational savings. This completes the derivation of the proposed estimator.…”

Section: The Proposed Sees Methodsmentioning

confidence: 99%

“…The AMLE is based on the assumption that PPSs tend to be orthogonal to each other, unless their parameters are nearly identical, that is (23) Using (23) we get that (24) where in the second passing we use the result that . Substituting (24) into (21) yields that (21) is equivalent to (25) Note that in the second passing we used the expression for given in (15), where instead of , we substitute the definition given in (4), and each element of is expressed using (6) and (7).…”

Section: B the Amlementioning

confidence: 99%

“…However, in practice the sampling rate may change due to carrier offset and timing inaccuracy and we need to recompute the matrix . In this case we can exploit its Vandermonde structure to obtain the LS solution [25]. We first perform a QR decomposition on as suggested in [25] which results in an equation set given by where is an matrix and is an matrix where is a upper triangle matrix.…”

Section: Computational Complexitymentioning

confidence: 99%

“…In this case we can exploit its Vandermonde structure to obtain the LS solution [25]. We first perform a QR decomposition on as suggested in [25] which results in an equation set given by where is an matrix and is an matrix where is a upper triangle matrix. The complexity of this step is and it is done once.…”

Section: Computational Complexitymentioning

confidence: 99%

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A Low Complexity Blind Estimator of Narrowband Polynomial Phase Signals

Amar

Leshem

Veen

2010

IEEE Trans. Signal Process.

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Abstract-Consider the problem of estimating the parameters of multiple polynomial phase signals observed by a sensor array. In practice, it is difficult to maintain a precisely calibrated array. The array manifold is then assumed to be unknown, and the estimation is referred to as blind estimation. To date, only an approximated maximum likelihood estimator (AMLE) was suggested for blindly estimating the polynomial coefficients of each signal. However, this estimator requires a multidimensional search over the entire coefficient space. Instead, we propose an estimation approach which is based on two steps. First, the signals are separated using a blind source separation technique, which exploits the constant modulus property of the signals. Then, the coefficients of each polynomial are estimated using a least squares method applied to the unwrapped phase of the estimated signal. This estimator does not involve any search in the coefficient spaces. The computational complexity of the proposed estimator increases linearly with respect to the polynomial order, whereas that of the AMLE increases exponentially. Simulation results show that the proposed estimator achieves the Cramér-Rao lower bound at moderate or high signal to noise ratio.Index Terms-Algebraic constant modulus algorithm, blind estimation, maximum likelihood estimation, polynomial phase signals, synthetic aperture radar.

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Section: The Proposed Sees Methodsmentioning

confidence: 99%

Section: B the Amlementioning

confidence: 99%

Section: Computational Complexitymentioning

confidence: 99%

Section: Computational Complexitymentioning

confidence: 99%

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A Low Complexity Blind Estimator of Narrowband Polynomial Phase Signals

Amar

Leshem

Veen

2010

IEEE Trans. Signal Process.

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“…We note that the Vandermonde structure of U can also be exploited in the above LS solution [14] to save computations.…”

Section: The Proposed Sees Methodsmentioning

confidence: 99%

A computationally efficient blind estimator of polynomial phase signals observed by a sensor array

Amar

Leshem

Veen

2010

2010 IEEE Sensor Array and Multichannel Signal Processing Workshop

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Consider estimating the parameters of polynomial phase signals observed by an antenna array given that the array manifold is unknown (e.g., uncalibrated array). To date, only an approximated maximum likelihood estimator (AMLE) was suggested, however, it involves a multidimensional search over the entire coefficient space. Instead, we propose a two-step estimation approach, termed as SEparateEStimate (SEES): First, the signals are separated with a blind source separation technique by exploiting the constant modulus property; Then, the coefficients of each polynomial are estimated using a least squares method from the unwrapped phase of the estimated signal. This estimator does not involve any search in the coefficient spaces and its computational complexity increases linearly with respect to the polynomial order, whereas that of the AMLE increases exponentially. Simulations show that the proposed estimator achieves the Cramér-Rao lower bound (CRLB) at moderate or high signal to noise ratio (SNR).

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A Fast Reduced Kernel Extreme Learning Machine

2016

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In this paper, we present a fast and accurate kernel-based supervised algorithm referred to as the Reduced Kernel Extreme Learning Machine (RKELM). In contrast to the work on Support Vector Machine (SVM) or Least Square SVM (LS-SVM), which identifies the support vectors or weight vectors iteratively, the proposed RKELM randomly selects a subset of the available data samples as support vectors (or mapping samples). By avoiding the iterative steps of SVM, significant cost savings in the training process can be readily attained, especially on Big datasets. RKELM is established based on the rigorous proof of universal learning involving reduced kernel-based SLFN. In particular, we prove that RKELM can approximate any nonlinear functions accurately under the condition of support vectors sufficiency. Experimental results on a wide variety of real world small instance size and large instance size applications in the context of binary classification, multi-class problem and regression are then reported to show that RKELM can perform at competitive level of generalized performance as the SVM/LS-SVM at only a fraction of the computational effort incurred.

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Fast least squares solution of Vandermonde systems of equations

Cited by 8 publications

References 5 publications

A Low Complexity Blind Estimator of Narrowband Polynomial Phase Signals

A Low Complexity Blind Estimator of Narrowband Polynomial Phase Signals

A computationally efficient blind estimator of polynomial phase signals observed by a sensor array

A Fast Reduced Kernel Extreme Learning Machine

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