Exponential Fourier Series

The Exponential Fourier Series uses, instead of the bases of the sines and cosines of the Trigonometric Fourier Series, an equivalent bases of exponential functions. This bases may look like

where, as before, w₀ is the base frequency of the signal and j = √-1 (often seen elsewhere as i )

The relationship between this bases and the previous trigonometric bases you just looked at is due to the very important identity often called Euler's Equation:

e^iθ = cos(θ) + i sin(θ) (where i= √-1)

or e^jθ = cos(θ) + j sin(θ) (where j= √-1) (version familiar to electrical engineers)

One does not intuitively associate exponentials with fluctuating behavior as one does with trig functions. Yet when complex numbers are involved and Euler's Equation is brought up, they become equivalent. For example, this complex exponential does not decay over time as the real version does, but rather oscillates continually with both real and imaginary parts doing so. If you are trying to envision this, think of a unit vector from the origin in the two dimensional plane, with its tip at the point (cos(θ), sin(θ) ) As θ increases in time, the tip of the vector goes around and around the origin.

The complex and trigonometric forms of Fourier Series are actually equivalent. This can be seen with a little algebra. Using trig identities cos(-θ) = cos(θ), sin(- θ) = - sin(θ) one gets that

e^-jθ = cos(θ) - j sin(θ) from e^jθ = cos(θ) + j sin(θ)

adding these two equations together and dividing by 2 yields cos(θ) = (e^jθ+ e^-jθ)/2

while subtracting them and dividing by 2j yields sin(θ) = (e^jθ- e^-jθ)/2j. Thus complex exponentials can be expressed as trig functions while trig functions can be expressed as complex exponentials. Electrical engineers prefer the complex version while physicists and mechanical engineers prefer the trigonometric.

Next, for any signal f(t) over [0,T₀], or any periodic f(t) with period T₀ we can compute the Exponential Fourier Series. We begin by writing

since our basis set is now . We must find the D₀ and D_n coefficients to find the EFS of the signal f.

As before with trigonometric Fourier Series and

We have everything we need to carry out an example now. That web page will fully explain the Exponential Fourier Series and conclude with a worked example.

We have completed our Exponential Fourier Series discussion, please Return to the Fourier Series Page.

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