Frequency Domain Representation of Biomedical Signals

Fourier’s findings provide a general theory for approximating complex waveforms with simpler functions that has numerous applications in mathematics, physics, and engineering.

This section summarizes the Fourier transform and variants of this technique that play an important conceptual role in the analysis and interpretation of biological signals

Periodic Signal representation: The Trigonometric Fourier Series In 1807, Fourier showed that an arbitrary periodic signal of period T can be represented mathematically as a sum of trigonometric functions

this is achieved by summing or mixing sinusoids while simultaneously adjusting their amplitudes and frequency as illustrated for a square wave function in the Figure

If the amplitudes and frequencies are chosen appropriately, the trigonometric signals add constructively, thus recreating an arbitrary periodic signal. This is akin to combining prime colors in precise ratios to recreate an signal. This is akin to combining prime colors in precise ratios to recreate an arbitrary color and shade.

Because….. RGB are the building blocks for more elaborate colors much as sinusoids of different frequencies serve as the building blocks for more complex signals.

For the example, a first-order approximation of the square wave is achieved by fitting the square wave to a single sinusoid of appropriate frequency and amplitude. Successive improvements in the approximation are obtained by adding higher-frequency sinusoid components, or harmonics, to the first-order approximation. If this procedure is repeated indefinitely, it is possible to approximate the square wave signal with infinite accuracy.

The Fourier series summarizes this result

Compact Fourier Series The most widely used counterparts for approximating and modeling biological signals are the exponential and compact Fourier series.

The compact Fourier series is a close cousin of the standard Fourier series. This version of the Fourier series is obtained by noting that the sum of sinusoids and cosines can be rewritten by a single cosine term with the addition of a phase constant

Which lead to the compact form of the series:

Exponential Fourier Series An alternative and somewhat more convenient form of this result is obtained by noting that complex exponential functions are directly related to sinusoids and cosines through Euler’s identities

an arbitrary periodic signal can be expressed as a sum of complex exponential functions:

Where:

Fourier Transform The Fourier integral, also referred to as the Fourier transform, is used to decompose a continuous a periodic signal into its constituent frequency components

Properties of the Fourier Transform   Linearity   Time Shifting/Delay   Frequency Shifting   Convolution Theorem

Discrete Fourier Transform The DFT is essentially the digital version of the Fourier transform. The index m represents the digital frequency index, x(k) is the sampled approximation of x(t), k is the discrete time variable, N is an even number that represents the number of samples for x(k), and X(m) is the DFTof x(k)

The Z Transform The z transform provides an alternative tool for analyzing discrete signals in the frequency domain. This transform is essentially a variant of the DFT, which converts a discrete sequence into its z domain representation

The z transform plays a similar role for digital signals as the Laplace transform does for the analysis of continuous signals If a discrete sequence x(k) is represented by xk, the (one-sided) z transform of the discrete sequence is expressed by

Properties of the Z transform   Linearity   Delay   Convolution