Definition of the Fourier Series

We assume that the signal x(t)x(t) to be analyzed is well described by a real or complex valued function of a real variable tt defined over a finite interval {0≤t≤T}{0≤t≤T}. The trigonometric series expansion of x(t)x(t) is given by

where xk(t)=cos(2πkt/T)xk(t)=cos(2πkt/T) and yk(t)=sin(2πkt/T)yk(t)=sin(2πkt/T) are the basis functions for the expansion. The energy or power in an electrical, mechanical, etc. system is a function of the square of voltage, current, velocity, pressure, etc. For this reason, the natural setting for a representation of signals is the Hilbert space of L2[0,T]L2[0,T]. This modern formulation of the problem is developed in Entry 6, Entry 11. The sinusoidal basis functions in the trigonometric expansion form a complete orthogonal set in L2[0,T]L2[0,T]. The orthogonality is easily seen from inner products

and

where δ(t)δ(t) is the Kronecker delta function with δ(0)=1δ(0)=1 and δ(k≠0)=0δ(k≠0)=0. Because of this, the kkth coefficients in the series can be found by taking the inner product of x(t)x(t) with the kkth basis functions. This gives for the coefficients

and

where TT is the time interval of interest or the period of a periodic signal. Because of the orthogonality of the basis functions, a finite Fourier series formed by truncating the infinite series is an optimal least squared error approximation to x(t)x(t). If the finite series is defined by

the squared error is

which is minimized over all a(k)a(k) and b(k)b(k) by (Equation 4) and (Equation 5). This is an extraordinarily important property.

It follows that if x(t)∈L2[0,T]x(t)∈L2[0,T], then the series converges to x(t)x(t) in the sense that ε→0ε→0 as N→∞N→∞Entry 6, Entry 11. The question of point-wise convergence is more difficult. A sufficient condition that is adequate for most application states: If f(x)f(x) is bounded, is piece-wise continuous, and has no more than a finite number of maxima over an interval, the Fourier series converges point-wise to f(x)f(x) at all points of continuity and to the arithmetic mean at points of discontinuities. If f(x)f(x) is continuous, the series converges uniformly at all points Entry 11, Entry 8, Entry 3.

A useful condition Entry 6, Entry 11 states that if x(t)x(t) and its derivatives through the qqth derivative are defined and have bounded variation, the Fourier coefficients a(k)a(k) and b(k)b(k) asymptotically drop off at least as fast as 1kq+11kq+1 as k→∞k→∞. This ties global rates of convergence of the coefficients to local smoothness conditions of the function.

The form of the Fourier series using both sines and cosines makes determination of the peak value or of the location of a particular frequency term difficult. A different form that explicitly gives the peak value of the sinusoid of that frequency and the location or phase shift of that sinusoid is given by

and, using Euler's relation and the usual electrical engineering notation of j=-1j=-1,

the complex exponential form is obtained as

where

The coefficient equation is

The coefficients in these three forms are related by

and

It is easier to evaluate a signal in terms of c(k)c(k) or d(k)d(k) and θ(k)θ(k) than in terms of a(k)a(k) and b(k)b(k). The first two are polar representation of a complex value and the last is rectangular. The exponential form is easier to work with mathematically.

Although the function to be expanded is defined only over a specific finite region, the series converges to a function that is defined over the real line and is periodic. It is equal to the original function over the region of definition and is a periodic extension outside of the region. Indeed, one could artificially extend the given function at the outset and then the expansion would converge everywhere.

A Geometric View

It can be very helpful to develop a geometric view of the Fourier series where x(t)x(t) is considered to be a vector and the basis functions are the coordinate or basis vectors. The coefficients become the projections of x(t)x(t) on the coordinates. The ideas of a measure of distance, size, and orthogonality are important and the definition of error is easy to picture. This is done in Entry 6, Entry 11, Entry 17 using Hilbert space methods.

Properties of the Fourier Series

The properties of the Fourier series are important in applying it to signal analysis and to interpreting it. The main properties are given here using the notation that the Fourier series of a real valued function x(t)x(t) over {0≤t≤T}{0≤t≤T} is given by F{x(t)}=c(k)F{x(t)}=c(k) and x˜(t)x˜(t) denotes the periodic extensions of x(t)x(t).

Examples

Note the derivative of a triangle wave is a square wave. Examine the series coefficients to see this. There are many books and web sites on the Fourier series that give insight through examples and demos.

Theorems on the Fourier Series

Four of the most important theorems in the theory of Fourier analysis are the inversion theorem, the convolution theorem, the differentiation theorem, and Parseval's theorem Entry 4.

All of these are based on the orthogonality of the basis function of the Fourier series and integral and all require knowledge of the convergence of the sums and integrals. The practical and theoretical use of Fourier analysis is greatly expanded if use is made of distributions or generalized functions (e.g. Dirac delta functions, δ(t)δ(t)) Entry 14, Entry 2. Because energy is an important measure of a function in signal processing applications, the Hilbert space of L2L2 functions is a proper setting for the basic theory and a geometric view can be especially useful Entry 6, Entry 4.

The following theorems and results concern the existence and convergence of the Fourier series and the discrete-time Fourier transform Entry 13. Details, discussions and proofs can be found in the cited references.