P. Sherman

Bio: P. Sherman is an academic researcher from Purdue University. The author has contributed to research in topics: Toeplitz matrix & Estimator. The author has an hindex of 3, co-authored 3 publications receiving 33 citations.

Circulant approximations of the inverses of Toeplitz matrices and related quantities with applications to stationary random processes

Abstract: The circulant approximation of vector and quadratic forms involving the inverse of a Toeplitz covariance matrix R is addressed. First, a result is presented which increases the rate of convergence of the average matrix error under certain conditions on \undertilde{r} , the vector which defines R. Concerning vector and quadratic operations using R-1, it is noted that if \undertilde{x} is AR(p), then the p-banded, near-Toeplitz structure of R-1results in an O(1/N)-type mean convergence of associated errors.

Optimal estimation of an unknown deterministic signal vector using a time-invariant filter

Abstract: The problem of estimating a deterministic signal vector \undertilde{theta} from \undertilde{x} = \undertilde{theta} + \undertilde{n} is considered using quadratic loss. It is assumed that the noise \undertilde{n} is weakly stationary, and that the vector size is large. These assumptions along with a time-invariant filter constraint allow the use of Fourier transforms and a filtering approach. It is noted that in the class of time-invariant data-independent filters, given spectral knowledge of the unknown deterministic signal vector \undertilde{theta} , the best performance is achieved by a form similar to the classical Wiener filter form. This provides the motivation for a simple empirical Wiener estimator, wherein the signal spectral information is estimated from the data. This estimator is shown to dominate the MLE at least in the case where the spectral signal-to-noise ratio is uniformly \lsim 0.65.

Frequency domain analysis of a family of minimax signal estimators

Abstract: A minimax estimator of the mean of x approximately N( theta ,R/sub n/) is evaluated, where R/sub n/ is Toeplitz. The problem can be viewed as estimating a signal theta that has been corrupted by additive wide-sense stationary noise. The evaluation is carried out using a frequency-domain approach that yields advantages in design and analysis. In particular, counterintuitive performance of an entire class of minimax forms is discovered when R/sub n/ is not diagonal. A numerical example is presented to demonstrate the performance. >

Conjugate Gradient Methods for Toeplitz Systems

Abstract: In this expository paper, we survey some of the latest developments in using preconditioned conjugate gradient methods for solving Toeplitz systems. One of the main results is that the complexity of solving a large class of $n$-by-$n$ Toeplitz systems is reduced to $O(n \log n)$ operations as compared to $O(n \log ^2 n)$ operations required by fast direct Toeplitz solvers. Different preconditioners proposed for Toeplitz systems are reviewed. Applications to Toeplitz-related systems arising from partial differential equations, queueing networks, signal and image processing, integral equations, and time series analysis are given.

Asymptotic Quantum Algorithm for the Toeplitz Systems

Abstract: Solving the Toeplitz systems, which involves finding the vector $x$ such that ${T}_{n}x=b$ given an $n\ifmmode\times\else\texttimes\fi{}n$ Toeplitz matrix ${T}_{n}$ and a vector $b$, has a variety of applications in mathematics and engineering In this paper, we present a quantum algorithm for solving the linear equations of Toeplitz matrices, in which the Toeplitz matrices are generated by discretizing a continuous function It is shown that our algorithm's complexity is nearly $O(\ensuremath{\kappa}\mathrm{poly}(logn))$, where $\ensuremath{\kappa}$ and $n$ are the condition number and the dimension of ${T}_{n}$, respectively This implies our algorithm is exponentially faster than its classical counterpart if $\ensuremath{\kappa}=O(\mathrm{poly}(logn))$ Since no assumption on the sparseness of ${T}_{n}$ is demanded in our algorithm, it can serve as an example of quantum algorithms for solving nonsparse linear systems

On the performance limits of data-aided synchronization

Abstract: This paper addresses data-aided (DA) synchronization, in which the reference parameter acquisition is aided by a training sequence known to the receiver. The Cramer-Rao lower bound (CRB) for the DA timing and/or carrier phase recovery is presented. For DA parameter estimation, the CRB typically varies with the training sequence. This indicates that different training sequences offer fundamentally different performance. In the literature, the widely cited closed-form CRB for timing and carrier phase recovery was derived under the assumption that the training sequence is independent and identically distributed (i.i.d.) and sufficiently long. We derive a closed-form CRB for timing and carrier phase recovery with respect to an arbitrary training sequence and pulse shaping function for the over and under sampling cases. It turns out that the CRB is a weighted summation of the aperiodic correlation of the training sequence and the weighting factor is determined by the pulse shaping filter. Therefore, this paper reveals the fundamental link between a training sequence and its corresponding performance limit.

On the convergence of the inverses of Toeplitz matrices and its applications

Abstract: Many issues in signal processing involve the inverses of Toeplitz matrices. One widely used technique is to replace Toeplitz matrices with their associated circulant matrices, based on the well-known fact that Toeplitz matrices asymptotically converge to their associated circulant matrices in the weak sense. This often leads to considerable simplification. However, it is well known that such a weak convergence cannot be strengthened into strong convergence. It is this fact that severely limits the usefulness of the close relation between Toeplitz matrices and circulant matrices. Observing that communication receiver design often needs to seek optimality in regard to a data sequence transmitted within finite duration, we define the finite-term strong convergence regarding two families of matrices. We present a condition under which the inverses of a Toeplitz matrix converges in the strong sense to a circulant matrix for finite-term quadratic forms. This builds a critical link in the application of the convergence theorems for the inverses of Toeplitz matrices since the weak convergence generally finds its usefulness in issues associated with minimum mean squared error and the finite-term strong convergence is useful in issues associated with the maximum-likelihood or maximum a posteriori principles.