Abstract
This chapter deals with the minimum sampling interval \(T_s\) needed to correctly represent an analog signal by samples extracted periodically from it, so as to be able to reconstruct the continuous-time signal from its discrete-time version. The sampling theorem prescribes this lower limit and highlights the fact that a representative sampling is possible if, and only if, the analog signal does not contain frequencies higher than the Nyquist frequency \(1/(2 T_s)\): no finite-rate sampling can capture the variations of an analog signal which is not bandlimited. Other issues related to analog signals, such as the signal’s concentrations in the time and frequency domains and their mutual inverse dependence (uncertainty principle), as well as the definition of bounded support in both domains, are also discussed. An appendix provides a summary of the relations among the variables used to express the concept of frequency in the continuous-time and discrete-time cases.
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Notes
- 1.
This statement can be understood as follows:
$$\begin{aligned} X_s(f)= & {} \int _{-\infty }^{+\infty } x_s(t)\mathrm {e}^{-\mathrm {j}2 \pi f t} \mathrm {d}t= \int _{-\infty }^{+\infty } \left[ x(t) \sum _{n=-\infty }^{+\infty } \delta (t-nT_s)\right] \mathrm {e}^{-\mathrm {j}2 \pi f t} \mathrm {d}t= \\= & {} \sum _{n=-\infty }^{+\infty }\int _{-\infty }^{+\infty }x(t)\delta (t-nT_s)\mathrm {e}^{-\mathrm {j}2 \pi f t} \mathrm {d}t = \sum _{n=-\infty }^{+\infty }x(nT_s)\mathrm {e}^{-\mathrm {j}2 \pi f (n T_s)} = \sum _{n=-\infty }^{+\infty }x[n] \mathrm {e}^{-\mathrm {j}\omega n} \equiv X(\mathrm {e}^{\mathrm {j}\omega }). \end{aligned}$$ - 2.
The sampling theorem actually was originally demonstrated neither by Nyquist, nor by Shannon. The original proof of the sampling theorem is due to Cauchy (1841), even if Cauchy’s paper does not explicitly contain the statement of the sampling theorem. Shannon himself recognized it (Shannon 1949): “If a function f(t) contains no frequencies higher than W cycles per second, it is completely determined by giving its ordinates at a series of points spaced 1 / 2W seconds apart. This is a fact which is common knowledge in the communication art.”
- 3.
Of course the precise amount of advisable oversampling is dictated by the shape of the designed frequency response of the realizable analog filter.
- 4.
We should call this a theorem and not a principle, since it can be demonstrated. However, this is the name by which this result is usually referred to.
- 5.
The proof of the uncertainty principle for Fourier transforms actually assumes that the signal vanishes faster that \(1/\sqrt{t}\) as \(t \rightarrow \pm \infty \), so that \(\lim _{t \rightarrow \pm \infty } t x^2(t)\) is zero. In mathematical literature, this constraint is often expressed by saying that the uncertainty principle holds for Schwartz functions on the real line—a class of functions that can be thought of as smooth functions that decay rapidly towards infinity.
- 6.
Since the analog window \(w_T(t)\) is non-causal and its support is symmetrical around the origin of the time axis, this transform is real.
References
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Alessio, S.M. (2016). Sampling of Continuous-Time Signals. In: Digital Signal Processing and Spectral Analysis for Scientists. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-25468-5_4
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DOI: https://doi.org/10.1007/978-3-319-25468-5_4
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