Periodic Nonuniform Sampling and Reconstruction of Modulated Signals


Student thesis: Doctoral Thesis

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Award date20 Jan 2021


The sampling and reconstruction of continuous-time signals are commonly required in many areas, including wireless communications. The Nyquist sampling theorem states that any bandlimited signal can be uniquely recovered from its uniform samples obtained with a sampling frequency which is at least twice the highest frequency component in the signal. While sampling a bandpass signal at the Nyquist rate will usually result in a higher rate than necessary, periodic nonuniform sampling (PNS) can address this issue. PNS involves N undersampled sequences of the signal with different time offsets taken at the same sampling rate. The total sampling rate of PNS is defined as the sum of these N undersampling rates. The order of PNS, namely N, should be small enough that the total sampling rate is lower than the Nyquist rate. However, the corresponding sampling and reconstruction systems are generally too complicated for practical implementation. In this thesis, we study reconstruction methods by applying the PNS of bandlimited real-valued signals at a minimal sampling rate, which can be used in data converters and image compression.

First, we present a direct reconstruction approach of the second-order sampled real-valued bandpass signal onto the baseband. Frequency-shifting interpolants for signal reconstruction are designed. Second-order sampling consists of N = 2 undersampled sequences of the signal at the same sampling rate with a time offset between the two sequences. Feasible time offsets are determined analytically to enable second-order sampling in practice. The simulation results are included to confirm the theoretical calculations. The results also demonstrate the superiority of the proposed method over several existing methods.

PNS with Nth-order (N > 2) can also be applied to bandpass signals, which allows more freedom in choosing the sample locations. Nth-order PNS is required particularly for more general signals, such as multi-band signals. We have therefore extended the second-order PNS method to apply to the Nth-order PNS. Specifically, we investigated the exact method of recovering the continuous-time real-valued dual-band signal from its periodic nonuniform samples at the optimal sampling rate and with a simple sampling and reconstruction system. The optimal sampling rate is as close to the Nyquist-Landau rate as possible. Constraints on the aliasing-free sampling frequency with first-order sampling are developed, and conditions to reconstruct a uniformly-spaced sequence from its periodic nonuniform samples at the optimal sampling rate are given. The complexity of the proposed method is far less than that of conventional schemes. The proposed method has potential applications in the fields of software defined radio design and orthogonal frequency-division multiplexing systems.

The above algorithms consider the tradeoff between the sampling rate and complexity of a PNS system. The position selection of samples is also important because the sample distribution affects the quality of the reconstruction, especially in the presence of noise. In theory, the formulation of these methods requires infinitely long data, but it is impossible to have infinite-duration samplers in practice. We therefore derived a fast, L2-optimal uniformly-sampled sequence reconstruction scheme from finite-length periodic nonuniform samples with unknown time offset information between samples. Compared to direct computation involving a pseudo-inverse matrix, matrix partition can reduce complexity. A very large condition number will result in an unreliable solution. The sampling pattern is related to the formation of the matrix, and it influences the quality of the reconstruction. Sequential forward selection is one of the search algorithms used to solve the sampling pattern selection problem. In our case, we developed a modified sequential forward selection algorithm to estimate the sampling pattern for a more complicated matrix. The stability problem associated with the sampling pattern is thus alleviated. Numerical examples comparing different reconstruction algorithms are presented in this study.