Discrete-time Periodic Signals Research Articles

Periodicity-Aware Signal Denoising Using Capon-Optimized Ramanujan Filter Banks and Pruned Ramanujan Dictionaries

We propose a ‘periodicity-aware’ hybrid analysis-synthesis framework for denoising discrete-time periodic signals. Our method uses Ramanujan filter banks (RFB) for analysis and dictionaries for synthesis. The synthesis dictionary retains appropriate subspaces for signal reconstruction, by pruning the Ramanujan dictionary based on the outputs of the RFB. Unlike the other existing denoising methods, a unique advantage of the proposed method is that the denoised output is guaranteed to be composed of integer-periodic components with periods smaller than a pre-selected value. Our method works well even when the signal length is small, and has a high SNR gain over a wide range of input signal SNRs. Furthermore, we propose to adapt each filter in the analysis bank to the incoming data, by optimizing the filter coefficients through a multi-band Capon formulation. This helps in suppressing the spurious energy peaks generated from higher period filters in the analysis bank, further improving the denoising performance. Implementing multiband Capon filters requires inverses of several autocorrelation matrices. To reduce computations, a way to recursively compute these inverses based on Levinson's recursion is discussed. Next, we prove several multirate properties of Ramanujan subspace signals. An important property among these is that after decimation, a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -th Ramanujan subspace signal still remains in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -th Ramanujan space, if and only if the decimation rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> is coprime to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> . This is helpful to further reduce computations required in the analysis part of the denoising framework by downsampling the filter outputs. Extensive Monte-Carlo simulations comparing different variants of the proposed method and several existing denoising methods are also provided.

Modified Repetitive Control for Odd-Harmonic Current Suppression in Magnetically Suspended Rotor Systems

In a magnetically suspended rotor (MSR) system, harmonic current mainly caused by mass imbalance and sensor runout would produce harmonic vibration. Repetitive control (RC) is one remarkable approach for eliminating such harmonics. Conventional RC (CRC) attempts to compensate for all frequency components simultaneously. However, in the practical MSR system, it is usual that the current comprises only odd-harmonic components. This paper investigates a modified odd-harmonic RC (MORC) scheme to respond to this issue. The signal generator of arbitrary-order discrete-time periodic signal is established, by which the general forms of CRC and ORC can be obtained. Besides, it is proved that ORC with halved data memory possesses a faster convergence rate, higher repetitive gain, wider frequency bandwidth at the targeted frequency and greater robustness to frequency fluctuation than CRC. The stability criterion for the MORC-controlled MSR system is proved as well. Simulation and experimental results have highlighted the effectiveness of the proposed approach.

Minimum Data Length for Integer Period Estimation

Detecting periodicity in a short sequence is an important problem, with many applications across science and engineering. Several efficient algorithms have been proposed for this over the years. There is a wide choice available today in terms of the tradeoff between algorithmic complexity and estimation accuracy. In spite of such a rich history, one particular aspect of period estimation has received very little attention from a fundamental perspective. Namely, given a discrete time periodic signal and a list of candidate integer periods, what is the absolute minimum datalength required to estimate its integer period? Notice that the answer we seek must be a fundamental bound, i.e., independent of any particular period estimation technique. Common intuition suggests the minimum datalength as twice the largest expected period. However, this is true only under some special contexts. This paper derives the exact necessary and sufficient bounds to this problem. The above-mentioned question is also extended to the case of mixtures of periodic signals. First, a careful mathematical formulation discussing the unique identifiability of the component periods (hidden integer periods) is presented. Once again, a rigorous theoretical framework in this regard is missing in the existing literature but is a necessary platform to derive precise bounds on the minimum necessary datalength. The bounds given here are generic, that is, independent of the algorithms used. Specific algorithm-dependent bounds are also presented in the end for the case of dictionary-based integer period estimation reported in recent years.

Periodic spline-based frames for image restoration

We present a design scheme that generates tight and semi-tightframes in discrete-time periodic signals space originated fromfour-channel perfect reconstruction periodic filter banks. Filter banksare derived from interpolating and quasi-interpolating polynomial and discrete splines. Each filter bank comprises one linear phaselow-pass filter (in most cases interpolating) and one high-passfilter, whose magnitude's response mirrors that of a low-pass filter.These filter banks comprise two band-pass filters. We introduce local discrete vanishing moments (LDVM). When the frame is tight, analysis framelets coincidewith their synthesis counterparts. However, for semi-tight frames,we swap LDVM between synthesis and analysis framelets. The designscheme is generic and it enables us to design framelets with anynumber of LDVM. The computational complexity of the the framelet transforms,which consists of calculating the forward and the inverse FFTs, doesnot depend on the number of LDVM and does depend on the size of the the impulse response fi lters. The designed frames are used for image restorationtasks, which were degraded by blurring, random noise and missingpixels. The images were restored by the application of the SplitBregman Iterations method. The frames performances are evaluated. Apotential application of this methodology is the design of asnapshot hyperspectral imager that is based on a regular digitalcamera. All these imaging applications are described.

Synthesis of discrete-time discrete-frequency Wigner distribution

A recursive algorithm is proposed for synthesizing a discrete-time periodic signal from a specified discrete-time discrete-frequency Wigner distribution by minimizing the error in the cyclic outer product. Equating the gradient of the error with respect to signal samples to zero results in a set of nonlinear simultaneous equations, which are solved using Newton-Raphson's method. A simulation example is given showing the synthesized signal over the first few iterations.

Wavelet transforms for discrete-time periodic signals

Wavelet transforms for discrete-time periodic signals are developed. In this finite-dimensional context, key ideas from the continuous-time papers of Daubechies and of Cohen, Daubechies and Feauveau are isolated to give a concise, rigorous derivation of the discrete-time periodic analogs of orthonormal and symmetric biorthogonal bases of compactly supported wavelets. These discrete-time periodic wavelets are expressed in terms of circular FIR filters, and thus lead to fast wavelet transforms whose complexity is order N.

Wigner-space synthesis of discrete-time periodic signals

A procedure that facilitates at least-squares synthesis of periodic, discrete-time signals from Wigner quasi-distributions is proposed. The scheme is based on expanding the desired time sequence on a generally nonorthogonal, Gabor-type basis whose associated biorthogonal function presumably exists. The specific basis selection may crucially affect the efficiency and quality of the ensuing synthesis procedure. The cited basis type constitutes a considerable generalization over the standardly used orthogonal variety, thus creating previously unavailable degrees of freedom. Of primary significance is the acquired capability of generating time-frequency basis functions that are well localized. Localization is a highly desirable property that can advantageously serve in various applications. It is shown and numerically demonstrated that benefits of localization as well as the fact that achieving effective time-frequency basis localization renders a certain degree of oversampling unavoidable. >