Regularized recurrent nonuniform sampling formulations in the linear canonical transform domain

Recurrent nonuniform discrete-time signal samples can be regarded as a combination of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> mutual delayed sequences of uniform discrete-time signal samples taken at one <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> th of the Nyquist sampling rate. This paper introduces a new alternative discrete-time analysis model of the recurrent nonuniform sampling scenario. This model can be described by the analysis part of a uniform discrete Fourier transform (DFT) modulated filterbank from which the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> uniformly distributed and down sampled frequency bands are mixed in a very specific way. This description gives a clear relationship between uniform and recurrent nonuniform discrete-time sampling schemes. A side benefit of this model is an efficient structure with which one can reconstruct uniform discrete-time Nyquist signal samples from recurrent nonuniform samples with known mutual delays between the nonuniform distributed samples. This reconstruction structure can be viewed as a natural extension of the synthesis part of an uniform DFT modulated filterbank.

The main purpose of this paper is to analyze the instability in waveform reconstruction from non-uniformly sampled signals generated by recurrent non-uniform discrete-time sampling scheme. From the applied equation, the matrix of conversion, [A], associates the uniformly sampled signals to non-uniformly sampled ones, and plays a key role in the reconstruction. To recover the uniformly sampled signals, we need to compute the inversion of [A], [A]−1. Although [A]−1 is known to be numerically unstable in some occasions, the fundamental reason and premise for this instability have not been discussed. Thus, by obtaining the closed form of det[A], this paper shows that the root cause of the instability of [A]−1 is related to the singularity of [A], which has det[A] = 0 resulting from specific patterns of timing offsets in the recurrent nonuniform discrete-time sampling scheme. Numerical results confirm the analysis.

The uniform and recurrent nonuniform higher-order derivative sampling problems associated with the fractional Fourier transform are investigated in this paper. The reconstruction formulas of a bandlimited signal from the uniform and recurrent nonuniform derivative sampling points are obtained. It is shown that if a bandlimited function f(t) has $$n - 1$$ order derivative in fractional Fourier transform domain, then f(t) is determined by its uniform sampling points $$f^{(l)}(knT)(l=0,1,\ldots ,n-1)$$ or recurrent nonuniform sampling points $$f^{(l)}(n(t_{p}+kNT))(l=0,1,\ldots ,n-1;p=1,2,\ldots ,N)$$ , the related sampling rate is also reduced by n times. The examples and simulations are also performed to verify the derived results.

The azimuth signal undersampled from the multi-channel SAR system for high resolution and wide swath imaging can be considerded as recurrent nonuniform sampling, which is a special case of multichannel sampling. Then from the point of view of reconstructing a bandlimited signal from recurrent nonuniform samples, we propose an new reconstruction algorithm based on filter-bank framework for suppressing azimuth ambiguities and therefore restoring the unambiguous azimuth full spectrum. The proposed algorithm has the advantage of acquiring the filter weights without matrix inversion over other reconstruction algorithms, which reduces tremendously the computation load. Finally, the performance of our algorithm is validated by numerical experiments.

Spectrum Reconstruction from Recurrent Nonuniform Sampling with Known Nonuniform Sampling Ratios

Reconstruction of N-th order nonuniformly sampled bandlimited signals using digital filter banks

Along a roughly chronological order, the azimuth signals undersampled from the multichannel synthetic aperture radar (SAR) for high-resolution wide-swath (HRWS) imaging correspond to recurrent nonuniform sampling. Only a finite-duration sequence of samples of the azimuth signal can be obtained in practical applications. A new recurrent nonuniform sampling scheme can be generated by extending these samples periodically across the boundaries provided that the azimuth signal is bandlimited. Thus, from the perspective of reconstructing recurrent nonuniform sampling, an innovative reconstruction algorithm for suppressing azimuth ambiguities of multichannel HRWS SAR is proposed, which is suitable to be implemented on digital computers. Furthermore, the presented algorithm acquires the filter weights without matrix inversion reducing tremendously the computational load.

The aim of the multichannel sampling is the reconstruction of a band-limited signal f(t), from the samples of the responses of M linear time invariant systems, each sampled by the 1/Mth Nyquist rate. As the offset linear canonical transform (OLCT) has been found wide applications in signal processing and optics fields, it is necessary to consider the multichannel sampling based on offset linear canonical transform. In this paper, we develop a multichannel sampling theorem for signals band-limited in offset linear canonical transform domains. Moreover, by designing different OLCT filters, reconstruction formulas for uniform sampling from the signal, from the signal and its first derivative or its generalized Hilbert transform are obtained based on the derived multichannel sampling theorem. Since recurrent nonuniform sampling for the signal has valuable applications, reconstruction expression for recurrent nonuniform samples of the signal band-limited in the offset linear canonical transform domain is also obtained by using the derived multichannel sampling theorem and the properties of the offset linear canonical transform.

There have been numerous interesting and useful generalizations of the sampling theorem. Some are straightforward variations on the fundamental cardinal series. Oversampling, for example, results in dependent samples and allows much greater flexibility in the choice of interpolation functions. In Chapter 7, we will see that it can also result in better performance in the presence of sample data noise. Bandlimited signal restoration from samples of various filtered versions of the signal is the topic addressed in Papoulis’ generalization [1086, 1087] of the sampling theorem. Included as special cases are recurrent nonuniform sampling and simultaneously sampling a signal and one or more of its derivatives. Kramer [772] generalized the sampling theorem to signals that were bandlimited in other than the Fourier sense. We also demonstrate that the cardinal series is a special case of Lagrangian polynomial interpolation. Sampling in two or more dimensions is the topic of Section 8.9. There are a number of functions other than the sinc which can be used to weight a signal’s samples in such a manner as to uniquely characterize the signal. Use of these generalized interpolation functions allows greater flexibility in dealing with sampling theorem type characterizations. If a bandlimited signal has bandwidth B, then it can also be considered to have bandwidthW ≥ B.

This paper considers interleaved, multichannel measurements as arise for example in time-interleaved analog-to-digital (A/D) converters and in distributed sensor networks. Such systems take the form of either uniform or recurrent nonuniform sampling, depending on the relative timing between the channels. Uniform (i.e., linear) quantization in each channel results in an effective overall signal-to-quantization-error ratio (SQNR) in the reconstructed output which is dependent on the quantizer step size in each channel, the relative timing between the channels, and the oversampling ratio. It is shown that in the multichannel sampling system when the quantization step size is not restricted to be the same in each channel and the channel timing is not constrained to correspond to uniform sampling, it is often possible to increase the SQNR relative to the uniform case. Appropriate choice of these parameters together with the design of appropriate compensation filtering is developed.

This paper considers the environment of interleaved, multi-channel measurements as arises for example in time-interleaved A/D converters and in distributed sensor networks. Such systems take the form of either uniform or recurrent nonuniform sampling, depending on the timing offset between the channels. Quantization in each channel results in an effective overall signal to noise ratio in the reconstructed output which is dependent on the quantizer step sizes, the timing offsets between the channels and the oversampling ratio. Appropriate choice of these parameters together with the design of appropriate compensation filtering is discussed.

Sampling is one of the fundamental topics in the signal processing community. Theorems proposed under this topic form the bridge between the continuous-time signals and discrete-time signals. Several sampling theorems, which aid in the reconstruction of signals in the linear canonical transform (LCT) domain, have been proposed in the literature. However, two main practical issues associated with the sampling of the LCT still remain unresolved. The first one relates to the reconstruction of the original signal from nonuniform samples and the other issue relates to the fact that only a finite number of samples are available practically. Focusing on these issues, this paper seeks to address the above from the LCT point of view. First, we extend several previously developed theorems for signals band-limited in the Fourier domain to signals band-limited in the LCT domain, followed by the derivation of the reconstruction formulas for finite uniform or recurrent nonuniform sampling points associated with the LCT. Simulation results and the potential applications of the theorem are also proposed.

Multi-channel sampling expansion for band-pass signals without channels constraints