Sampling Theorem for Fractional Bandlimited Signals: A Self-Contained Proof. Application to Digital Holography

2006 ◽  
Vol 13 (11) ◽  
pp. 676-679 ◽  
Author(s):  
R. Torres ◽  
P. Pellat-Finet ◽  
Y. Torres
Keyword(s):  
Digital Holography ◽  
Sampling Theorem ◽  
Bandlimited Signals

Signal Recovery

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Interpolation Problem ◽  
Super Resolution ◽  
Sampling Theorem ◽  
Signal Recovery ◽  
Continuous Sampling ◽  
Bandlimited Signals ◽  
Special Cases ◽  
Extrapolation Problem ◽  
Definition Of

The literature on the recovery of signals and images is vast (e.g., [23, 110, 112, 257, 391, 439, 791, 795, 933, 934, 937, 945, 956, 1104, 1324, 1494, 1495, 1551]). In this Chapter, the specific problem of recovering lost signal intervals from the remaining known portion of the signal is considered. Signal recovery is also a topic of Chapter 11 on POCS. To this point, sampling has been discrete. Bandlimited signals, we will show, can also be recovered from continuous samples. Our definition of continuous sampling is best presented by illustration.Asignal, f (t), is shown in Figure 10.1a, along with some possible continuous samples. Regaining f (t) from knowledge of ge(t) = f (t)Π(t/T) in Figure 10.1b is the extrapolation problem which has applications in a number of fields. In optics, for example, extrapolation in the frequency domain is termed super resolution [2, 40, 367, 444, 500, 523, 641, 720, 864, 1016, 1099, 1117]. Reconstructing f (t) from its tails [i.e., gi(t) = f (t){1 − Π(t/T)}] is the interval interpolation problem. Prediction, shown in Figure 10.1d, is the problem of recovering a signal with knowledge of that signal only for negative time. Lastly, illustrated in Figure 10.1e, is periodic continuous sampling. Here, the signal is known in sections periodically spaced at intervals of T. The duty cycle is α. Reconstruction of f (t) from this data includes a number of important reconstruction problems as special cases. (a) By keeping αT constant, we can approach the extrapolation problem by letting T go to ∞. (b) Redefine the origin in Figure 10.1e to be centered in a zero interval. Under the same assumption as (a), we can similarly approach the interpolation problem. (c) Redefine the origin as in (b). Then the interpolation problem can be solved by discarding data to make it periodically sampled. (d) Keep T constant and let α → 0. The result is reconstructing f (t) from discrete samples as discussed in Chapter 5. Indeed, this model has been used to derive the sampling theorem [246]. Figures 10.1b-e all illustrate continuously sampled versions of f (t).

SAMPLING AND OVERSAMPLING IN SHIFT-INVARIANT AND MULTIRESOLUTION SPACES I: VALIDATION OF SAMPLING SCHEMES

2005 ◽  
Vol 03 (02) ◽  
pp. 257-281 ◽  
Author(s):  
J. A. HOGAN ◽  
J. D. LAKEY
Keyword(s):  
Summation Formula ◽  
Sampling Theorem ◽  
Poisson Summation Formula ◽  
Zak Transform ◽  
Self Similarity ◽  
Critical Rate ◽  
Sampling Schemes ◽  
Bandlimited Signals ◽  
Shift Invariant Spaces ◽  
Invariant Spaces

We ask what conditions can be placed on generators φ of principal shift invariant spaces to ensure the validity of analogues of the classical sampling theorem for bandlimited signals. Critical rate sampling schemes lead to expansion formulas in terms of samples, while oversampling schemes can lead to expansions in which function values depend only on nearby samples. The basic techniques for validating such schemes are built on the Zak transform and the Poisson summation formula. Validation conditions are phrased in terms of orthogonality, smoothness, and self-similarity, as well as bandlimitedness or compact support of the generator. Effective sampling rates which depend on the length of support of the generator or its Fourier transform are derived.

Sampling of bandlimited signals in the offset linear canonical transform domain based on reproducing kernel Hilbert space

2019 ◽  
Vol 18 (02) ◽  
pp. 1950054 ◽  
Author(s):  
Shuiqing Xu ◽  
Zhiwei Chen ◽  
Yi Chai ◽  
Yigang He ◽  
Xiang Li
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Sampling Theorem ◽  
Orthogonal Basis ◽  
Transform Domain ◽  
Linear Canonical Transform ◽  
Uniform Sampling ◽  
Bandlimited Signals ◽  
Offset Linear Canonical Transform

The offset linear canonical transform (OLCT) has proven to be a novel and effective method in signal processing and optics. Many important properties and results of the OLCT have been well studied and published. In this work, the sampling theorem of the OLCT bandlimited signals based on reproducing kernel Hilbert space has been proposed. First, we show that the bandlimited signals in the OLCT domain form a reproducing kernel Hilbert space. Then, an orthogonal basis for the OLCT bandlimited signals has been obtained based on the reproducing kernel Hilbert space. By using the orthogonal basis, the uniform sampling theory for bandlimited signals associated with the OLCT has been obtained. Furthermore, the nonuniform sampling of the OLCT bandlimited signals also has been attained. Finally, the simulations are provided to prove the usefulness and correctness of the derived results.

Shannon Wavelet Approach to Sub-Band Coding

2003 ◽  
Vol 01 (02) ◽  
pp. 233-242 ◽  
Author(s):  
Charles K. Chui ◽  
Jianzhong Wang
Keyword(s):  
Continuous Time ◽  
Wavelet Packet ◽  
Sampling Rate ◽  
Sampling Theorem ◽  
Wavelet Theory ◽  
Bandlimited Signals ◽  
Bandlimited Signal ◽  
Time Signals ◽  
Shannon Sampling Theorem ◽  
Shannon Wavelet

It is well known that the Shannon Sampling Theorem allows us to fully recover a continuous-time bandlimited signal from its digital samples, as long as the sampling rate to be chosen is not smaller than the Nyquist frequency. This theory applies to all bandlimited signals, which may or may not occupy the entire frequency band. Hence, it is intuitively convincing that for continuous-time signals, such as those in speech, that do not fully utilize the entire frequency intervals, less digital samples are required for their full recovery. Current techniques in sub-band coding are used for achieving this goal. The objective of this paper is to present a wavelet theory for establishing the mathematical foundation of this sub-band coding approach. A wavelet packet decomposition of the signal provides the optimal sub-band coding bit-rate by using the Shannon wavelet library introduced in this paper.

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