Introduction to Nyquist’s Theorem

Nyquist's Theorem, also called the Nyquist-Shannon Sampling Theorem, is a theorem in the field of digital signal processing. Nyquist's Theorem is credited to Harry Nyquist and Claude Shannon, though its fundamental discovery was made by E. T. Whittaker [1]. Signals are electromagnetic waves created by impulses in electromagnetic equipment. Signals are the backbone of several fields, including communications, radio broadcasting, image processing, music production, and more. Signals are often referred to as sinusoids, because they are made up of sine and cosine waves. These waves have peaks and valleys, and the heights of the peaks are called amplitudes. In signal processing, there are discrete-time signals and continuous-time signals. When time is viewed as a discrete variable, then any function of time is thought of as having distinct values at separate points in time. In this view, the variable being measured is measured once in each time period between measurements. In many applications, such as tracking the number of shipments each day, time can be considered a discrete variable. However, the physical world presents a different view: time as a continuous variable. When time is a continuous variable, the variable measured over time only has its value for an infinitely small moment in time. The infinite time values cannot all be exactly measured, because instruments of measurement only can make measurements periodically. Therefore, there has to be a jump from the continuous form of time that is apparent in the real world to the discrete form of time used in measurement. Nyquist's Theorem makes this jump.

Harry Nyquist, one of the creators of Nyquist's Theorem [2].

Nyquist's Theorem enables scientists and engineers to model a continuous-time signal with discrete time measurements. Essentially, the theorem seeks to find the minimum frequency of measurements needed to completely capture all of the information in a signal. In practical cases, one can use Nyquist's Theorem to ensure that all essential signal information is collected, while not wasting effort measuring with too much detail.

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