Digital signal processing (DSP) in the real world often begins with sampling. Sampling takes measurements of a continuous waveform at regular intervals and generates a sequence of measured values.

Sampling a sine waveform.

Data Analysis and Sampling

Sampling is one of many ways to perform signal analysis. Like other analysis methods, it takes apart data. More specifically, it takes apart a continuous waveform into components or “samples.”

Data Synthesis and Sampling

After the data are separated into samples, is it possible to put them back together? That is, is synthesis possible? The answer is yes. This answer is one of the most surprising results in mathematics and is fundamental to DSP.

According to the sampling theorem, if the sample rate is at least two times the highest frequency component of the sampled waveform, perfect reconstruction is possible.

Mathematically, the basis for a band-limited function x(t) is a series of sinc functions called a Cardinal series. The coefficients for the projection of x(t) onto the basis are the samples of x(t).

Nyquist Frequency

The highest frequency component is called the Nyquist frequency. Sampling above this value is known as oversampling.

Undersampling occurs when the sample rate is less than two times this value. It can cause multiple copies of the signal in the frequency domain to overlap and result in aliasing. Aliasing occurs when a high-frequency signal masquerades as a low-frequency signal.

References

Greenhoe, Daniel J., A Book Concerning Digital Signal Processing. Self-published, 2019.

Greenhoe, Daniel J., Wavelet Structure and Design. Self-published, 2019.

Hardy, Godfrey H., “Notes on special systems of orthogonal functions (iv): the orthogonal functions of whittaker’s cardinal series.” Mathematical Proceedings of the Cambridge Philosophical Society, 37, no. 4 (1941): 331-348. DOI: 10.1017/S0305004100017977.

Higgins, John R., “Five short stories about the cardinal series.” Bulletin of the American Mathematical Society, 12, no. 1 (1985): 45-89. DOI: 10.1090/S0273-0979-1985-15293-0. See page 56, historical notes.

Higgins, John R., Sampling Theory in Fourier and Signal Analysis: Foundations. Oxford: Clarendon Press, 1996.

Kotelnikov, V. A. 1933. “On the capacity of the ‘ether’ and of cables in electrical communication.” In: Proceedings of the first All-Union Conference on the technological reconstruction of the communications sector and low-current engineering, Moscow.

Nashed, M. Zuhair and Gilbert G. Walter, “General sampling theorems for functions in reproducing kernel hilbert spaces.” Mathematics of Control, Signals and Systems, 4, no. 4 (1991): 363-390. DOI: 10.1007/BF02570568.

Shannon, Claude E., “A mathematical theory of communication.” The Bell System Technical Journal, 27, no. 3 (1948): 379-423. DOI: 10.1002/j.1538-7305.1948.tb01338.x. See Theorem 13.

Shannon, Claude E., “Communication in the presence of noise.” Proceedings of the IRE, 37, no. 1 (1949): 10-21. DOI: 10.1109/JRPROC.1949.232969. See page 1.

Whittaker, Edmund Taylor., “On the functions which are represented by the expansions of the interpolation theory.” Proceedings of the Royal Society Edinburgh, 35, (1915): 181-194. DOI: 10.1017/S0370164600017806.

Whittaker, John Macnaghten, Interpolatory Function Theory. The University Press, 1935.

Young, Robert M., An Introduction to Non-Harmonic Fourier Series, Revised Edition, 93. San Diego: Academic Press, 2001.