Signal Reconstruction from Nonuniformly Spaced Samples Using Evolutionary Slepian Transform-Based POCS
© Jinsung Oh et al. 2010
Received: 29 January 2010
Accepted: 15 June 2010
Published: 7 July 2010
We consider the reconstruction of signals from nonuniformly spaced samples using a projection onto convex sets (POCSs) implemented with the evolutionary time-frequency transform. Signals of practical interest have finite time support and are nearly band-limited, and as such can be better represented by Slepian functions than by sinc functions. The evolutionary spectral theory provides a time-frequency representation of nonstationary signals, and for deterministic signals the kernel of the evolutionary representation can be derived from a Slepian projection of the signal. The representation of low pass and band pass signals is thus efficiently done by means of the Slepian functions. Assuming the given nonuniformly spaced samples are from a signal satisfying the finite time support and the essential band-limitedness conditions with a known center frequency, imposing time and frequency limitations in the evolutionary transformation permit us to reconstruct the signal iteratively. Restricting the signal to a known finite time and frequency support, a closed convex set, the projection generated by the time-frequency transformation converges into a close approximation to the original signal. Simulation results illustrate the evolutionary Slepian-based transform in the representation and reconstruction of signals from irregularly-spaced and contiguous lost samples.
The problem of signal reconstruction from Nonuniformly spaced samples is central in many practical problems in image and signal processing [1–11]. Nonuniform sampling is a common result of Nyquist-Shannon sampling caused by jittering in the sampler, but it is also the case when samples are missing, either according to some distribution or in segments.
Reconstruction of Nonuniformly sampled signals can be approached numerically by considering sinc basis as frames . Unfortunately, it is an ill-posed problem given the characteristics of the basis. Signals in practice have finite supports and can be approximated as nearly band-limited signals, based on this in [13, 14] it is shown that a more appropriate basis for signal interpolation is the Prolate Spheroidal Wave or Slepian functions . It is thus possible to develop the projection of signals of finite time support that are nearly band-limited. The representation of bandpass signals, as the modulation of baseband components, can be obtained using modulated Slepian basis. The discrete prolate spheroidal sequences (DPSSs) are the discrete form of the prolate spheroidal wave functions (PSWFs)  and can be used as the basis for the projection of the sampled signal. Using DPSS as an orthogonal basis [14, 17], it is shown to reduce the sampling rate and reconstruction error.
The reconstruction of finite energy signals can thus be viewed as an interpolation or an estimation problem in which projection of the observed signal minimizes an error criteria. Constraining the solution to satisfy time and frequency conditions iteratively, a close approximation to the signal, with the given samples, is obtained. This is the basic idea of the projection onto convex sets (POCSs) method. This method was introduced by [1, 2] as an iterative algorithm for signal restoration. Since then, the POCS method has been successfully used in many signal and image recovery problems [3–11]. Time-frequency signal representations using short-time  and fractional Fourier transform [10, 11] have been recently used to implement this type of reconstruction.
To obtain the POCS iterative solution, we consider that the signals of interest have a finite time support and an approximately finite frequency support. As such, the Slepian projection are used for this class of signals. To jointly consider time and frequency constraints, we develop a time-frequency representation from the Slepian projection. This can be done using the evolutionary spectral theory , where a signal can be represented in terms of a kernel which in turn can be obtained from the windowed signal. The magnitude square of the kernel is associated with the way the energy of the signal is distributed in time and frequency. It is also possible to obtain a similar representation, the discrete evolutionary transform (DET), for deterministic signals having components with time-varying frequency components . Imposing time and frequency limitations in the DET permits us to reconstruct the signal iteratively, that is, the iterative projection generated by the time-frequency transformation converges into a close approximation to the original signal with the given Nonuniform samples.
The rest of the paper is organized as follows. In Section 2, we consider the reconstruction of Nonuniformly sampled signals. In this section we show why the PSWF basis is more appropriate than the sinc basis for the reconstruction from Nonuniform samples when the signal is of finite time support and essentially band-limited. In Section 3 we propose the time-frequency discrete evolutionary transform (DET) as the projection operator for the implementation of a recursive projection onto complex sets (POCSs) to recover missing samples. Assuming that the baseband components of a bandpass signal has finite support in time and frequency, a DET based in PSWF or Slepian functions is possible. This will be presented in Section 4. In Section 5 we illustrate the Slepian-based DET, and its application in the reconstruction of signals missing samples randomly and in blocks. Conclusions follow.
2. Reconstruction of Nonuniformly Sampled Signals
where is the sinc function. Several issues of practical interest arise when numerically implementing this interpolation. Besides the infinite dimension of the problem, uniform sampling is not realistic. Moreover, band-limitedness is just an approximation to reality.
where is a matrix containing shifted sinc functions, is a vector with entries the samples of the signal, and are the expansion coefficients of the projection in terms of the sinc functions. The infinite dimension of this problem makes it unsolvable, and when the dimension is reduced the problem becomes ill-posed .
The problem in part is due to using shifted sinc functions as basis: these functions are not appropriate for the interpolation given that time-limited signals are of infinite frequency support according to the uncertainty principle. A more appropriate basis is given by the Prolate Spheroidal Wave functions (PSWFs) .
Thus, if is time-limited and essentially band-limited, (5) and (6) provide a reconstruction of the signal from Nonuniformly sampled signals. In , it is shown that in the case of jittering sampling, that is, a subset of the uniform samples is available, the signal can be reconstructed.
3. Evolutionary Spectral Representation and POCS
where is the window which can be expressed in nonorthogonal functions such as Gabor's, or orthogonal functions such as Malvar's .
where is the reconstructed signal after iterations. Assuming that the signal of interest is square summable and that the DET projects a signal into another square summable signal, under joint time-frequency constraints we develop an iterative POCS algorithm to recover the signal from partial information of it.
4. Evolutionary Slepian Transform-Based POCS
that is, in terms of modulated Slepian functions.
the bandpass DPSS provide an efficient representation of passband signals and accurate channel estimation [21, 22]. Allowing the center frequency to vary from zero to infinity (or zero to in the discrete domain) we then have a general representation for any signal.
The missing samples are recovered if the above equation can be solved for , or if is invertible. Given the many possible ways the missing samples could be distributed, this might not be possible. However, as indicated in [12, 23] there are cases where reconstruction is possible with the sinc interpolation, and we will show later that it is also the case when we are using the Slepian POCS.
It is important to understand that when the signal under consideration is modulated, that is, , and we use the bandpass Slepian functions, we can obtain the spectrum of or . For a signal with bandpass characteristics, the signal can be represented by a small number of bandpass DPSS coefficients and then restored by small number of projection iteration compared to baseband DPSS based DET, which will be shown in next section.
5.1. Slepian-Based DET
5.2. Reconstruction of Irregularly Sampled Signals
In this section, we perform three different simulations to illustrate the effectiveness of DET-based POCS. We use the POCS methodology for reconstruction of nonuniformly sampled and band-limited signals.
5.2.1. Nonuniform Jittering Sampling with Known Distribution
5.2.2. Nonuniform Jittering Sampling with Unknown Distribution
5.2.3. Block or Contiguous Sample Losses
In this paper, we have introduced a new discrete evolutionary Slepian transform capable of efficient representation of band-limited signals. For the evolutionary kernel window, baseband and bandpass DPSS are used for the representation of baseband and bandpass signals, respectively. The evolutionary Slepian spectrum provides an accurate representation of time-and-band limited signal in the time-frequency domain. For the reconstruction, the DET-based POCS algorithm is applied in the area of signal recovery from nonuniformly spaced subsamples. For a signal that has bandpass characteristics, the signal can be represented by a small number of bandpass DPSS coefficients with the same accuracy obtained from baseband DPSS, and then restored by small number of projection iteration with the same MAE performance compared to baseband DPSS-based DET. The DET-based POCS algorithm is shown to provide fast and accurate technique for recovering the band-limited samples from the irregularly spaced subsamples. Although there are remaining issues that need further study, for instance, the upper error bound by the number and distribution of missing samples, the proposed method shows very promising results, that is, capable of signal recovery from randomly spaced subsamples and continuous lost samples.
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