Discrete Prolate Spheroidal Wave Functions Research Articles

Non-asymptotic bounds for discrete prolate spheroidal wave functions analogous with prolate spheroidal wave function bounds

Non-asymptotic bounds for discrete prolate spheroidal wave functions analogous with prolate spheroidal wave function bounds

Discrete Prolate Spheroidal Wave Functions: Further Spectral Analysis and Some Related Applications

For fixed $W\in \big(0,\frac{1}{2}\big)$ and positive integer $N\geq 1,$ the discrete prolate spheroidal wave functions (DPSWFs), denoted by $U_{k,W}^N,$ $0\leq k\leq N-1$ form the set of the eigenfunctions of the positive and finite rank integral operator $\widetilde Q_{N,W},$ defined on $L^2(-1/2,1/2),$ with kernel $K_N(x,y)=\frac{\sin(N\pi(x-y))}{\sin(\pi(x-y))}\, \mathbf 1_{[-W,W]}(y).$ It is well known that the DPSWF's have a wide range of classical as well as recent signal processing applications. These applications rely heavily on the properties of the DPSWFs as well as the behaviour of their eigenvalues $\widetilde \lambda_{k,N}(W).$ In his pioneer work \cite{Slepian}, D. Slepian has given the properties of the DPSWFs, their asymptotic approximations as well as the asymptotic behaviour and asymptotic decay rate of these eigenvalues. In this work, we give further properties as well as new non-asymptotic decay rates of the spectrum of the operator $\widetilde Q_{N,W}.$ In particular, we show that each eigenvalue $\widetilde \lambda_{k,N}(W)$ is up to a small constant bounded above by the corresponding eigenvalue, associated with the classical prolate spheroidal wave functions (PSWFs). Then, based on the well established results concerning the distribution and the decay rates of the eigenvalues associated with the PSWFs, we extend these results to the eigenvalues $\widetilde \lambda_{k,N}(W)$. Also, we show that the DPSWFs can be used for the approximation of classical band-limited functions and they are well adapted for the approximation of functions from periodic Sobolev spaces. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.

Channel interpolation for LTE uplink systems with high mobility using Slepian sequences

Single Carrier Frequency Division Multiple Access (SC-FDMA) has been adopted for 3rd Generation Partnership Project (3GPP) Long Term Evolution (LTE) uplink communication since it combines the low peak to average power ratio feature of single-carrier transmission systems with the multipath resistance and flexible frequency allocation of orthogonal frequency-division multiple access (OFDMA). Unfortunately, SC-FDMA transmission over time-varying multipath fading channels for very high speed environments the subcarrier orthogonality is destroyed giving rise to intercarrier interference (ICI) due to channel variation within an SC-FDMA symbol. This paper is, therefore, focused on the use of time-domain channel interpolator for LTE uplink systems in large Doppler spread scenarios to improve channel tracking performance. The resulting interpolation algorithm is based on the discrete prolate spheroidal wave functions (DPSWF) which is particularly well suited to represent the rapidly time-varying fading channel due to optimum finite time and frequency support feature. The variations of the channel taps are tracked first by a Kalman filter in time domain during training symbols. Then, a DPSWF based channel interpolator is applied to recover the time variation of channel taps between training symbols within two consecutive slots in a single subframe. The performance of the proposed interpolator is compared with a polynomial interpolator whose order is adjusted adaptively to achieve the best fit under particular user mobilities. Moreover, the analytical mean square error (MSE) performance of DPSWF is derived and compared with the simulations. The results show that the DPSWF represents the time-varying fading channel very effectively and the proposed algorithm has excellent channel estimation performance in high SNR even with a very small number of channel interpolation parameters employed in the algorithm.

Causal discrete‐time system approximation of non‐bandlimited continuous‐time systems by means of discrete prolate spheroidal wave functions

AbstractIn general, linear time‐invariant (LTI) continuous‐time (CT) systems can be implemented by means of LTI discrete‐time (DT) systems, at least for a certain frequency band. If a causal CT system is not bandlimited, the equivalent DT system may has be to non‐causal for perfectly implementing the CT system within a certain frequency band. This paper studies the question to which degree a causal DT system can approximate the CT system. By reducing the approximation frequency band, the approximation accuracy can be increased‐‐‐at the expense of a higher energy of the impulse response of the DT system. It turns out, that there exists a strict trade‐off between approximation accuracy, measured in the squared integral error, and energy of the impulse response of the DT system. The theoretically optimal trade‐off can be achieved by approximations based on a weighted linear combination of the discrete prolate spheroidal wave functions (DPSWFs). The results are not limited to the case of approximating a CT system by means of a causal DT system, but they generally hold for the approximation of an arbitrary spectrum by means of a spectrum of an indexlimited time sequence. Copyright © 2008 John Wiley & Sons, Ltd.

Optimal all-pole filter design based on discrete prolate spheroidal sequences

A new constrained optimization method for all-pole digital filter design is presented. The design philosophy employed minimizes a function of the area between the ideal low-pass filter response in the passband and the actual filter response subject to a quadratic constraint which ensures filter realizability. It is shown that unique solutions are obtained which are related to eigenvectors of a Toeplitz matrix whose elements form a scaled discrete prolate spheroidal wave sequence (DPSS). It is therefore possible to exploit the properties of DPSS and discrete prolate spheroidal wave functions (DPSWF's) to reduce the filter design effort. The ratio of passband energy to total energy over the entire filter bandwidth is optimal for these designs.

The Ill-Conditioned Nature of the Limited Angle Tomography Problem

The instability of inverting the limited angle Radon transform is studied by constructing the singular value decomposition of this operator. Connections are established with previously known series expansions and with the discrete prolate spheroidal wave functions. Mollification methods are shown to delay, but not prevent, the onset of instability as the angular range of projections is decreased.