Introduction to the Fourier Transform



The Fourier Transform is a mathematical technique that transforms a function of time, x(t), to a function of frequency, X(ω). It is closely related to the Fourier Series. If you are familiar with the Fourier Series, the following derivation may be helpful. If you are only interested in the mathematical statement of transform, please skip ahead to Definition of Fourier Transform.

The Fourier Transform of a function can be derived as a special case of the Fourier Series when the period, T→∞ (Note: this derivation is performed in more detail elsewhere). Start with the Fourier Series synthesis equation

$$x(t) = \sum\limits_{n = - \infty }^{ + \infty } {{c_n}{e^{jn{\omega _o}t}}} $$

where cn is given by the Fourier Series analysis equation,

$$c_n = {1 \over T}\int_T {x(t){e^{ - jn\omega_0t}}dt} $$

which can be rewritten

$$T{c_n} = \int_T {x(t){e^{ - jn\omega_0t}}dt} $$

As T→∞ the fundamental frequency, ω0=2π/T, becomes extremely small and the quantity 0 becomes a continuous quantity that can take on any value (since n has a range of ±∞) so we define a new variable ω=nω0; we also let X(ω)=Tcn. Making these substitutions in the previous equation yields the analysis equation for the Fourier Transform (also called the Forward Fourier Transform).

$$X(\omega ) = \int\limits_{ - \infty }^{ + \infty } {x(t){e^{ - j\omega t}}dt} $$

Likewise, we can derive the Inverse Fourier Transform (i.e., the synthesis equation) by starting with the synthesis equation for the Fourier Series (and multiply and divide by T).

$$x(t) = \sum\limits_{n = - \infty }^{ + \infty } {{c_n}{e^{jn{\omega _o}t}}} = \sum\limits_{n = - \infty }^{ + \infty } {T{c_n}{e^{jn{\omega _o}t}}} {1 \over T}$$

As T→∞, 1/T=ω0/2π. Since ω0 is very small (as T gets large, replace it by the quantity ). As before, we write ω=nω0 and X(ω)=Tcn. A little work (and replacing the sum by an integral) yields the synthesis equation of the Fourier Transform.

$$x(t) = \sum\limits_{n = - \infty }^{ + \infty } {X(\omega ){e^{j\omega t}}} {{d\omega } \over {2\pi }} = {1 \over {2\pi }}\int\limits_{ - \infty }^{ + \infty } {X(\omega ){e^{j\omega t}}d\omega } $$

Definition of the Fourier (and Inverse) Transform (synthesis and analysis).

Key Concept: Forward and Inverse Fourier Transforms

Forward Fourier Transform: Analysis Equation

$$X(\omega ) = \int\limits_{ - \infty }^{ + \infty } {x(t){e^{ - j\omega t}}dt} $$

Inverse Fourier Transform: Synthesis Equation

$$x(t) = {1 \over {2\pi }}\int\limits_{ - \infty }^{ + \infty } {X(\omega ){e^{j\omega t}}d\omega }$$

Alternate Forms of the Fourier Transform

There are alternate forms of the Fourier Transform that you may see in different references. Different forms of the Transform result in slightly different transform pairs (i.e., x(t) and X(ω)), so if you use other references, make sure that the same definition of forward and inverse transform are used.

Symmetric Form: Radian Frequency

& X(\omega ) = {1 \over {\sqrt {2\pi } }}\int\limits_{ - \infty }^{ + \infty } {x(t){e^{ - j\omega t}}dt} \cr
& x(t) = {1 \over {\sqrt {2\pi } }}\int\limits_{ - \infty }^{ + \infty } {X(\omega ){e^{j\omega t}}d\omega } \cr} $$

Symmetric Form: Hertz Frequency

& X(f) = \int\limits_{ - \infty }^{ + \infty } {x(t){e^{ - j2\pi ft}}dt} \cr
& x(t) = \int\limits_{ - \infty }^{ + \infty } {X(f){e^{j2\pi ft}}df } \cr} $$

Mathematical Niceties

There are several fine points of math that have been skipped over here, but in general won't create any significant difficulties for us.

Existence of the Fourier Transform requires that the x(t) be absolutely integrable,

$$\int\limits_{ - \infty }^{ + \infty } {\left| {x(t)} \right|dt} < \infty $$

Note: This would seem to present a problem, because common signals such as the sine and cosine are not absolutely integrable. We will finesse this problem, later, by considering impulse functions, δ(α), which are not functions in the strict sense since the value isn't defined at α=0.

Existence of the Fourier Transform requires that discontinuities in x(t) must be finite (i.e.,|x(α+)-x(α-)|<∞). This presents no difficulty for the kinds of functions we will consider (i.e., functions that we can produce in the lab).

Moving Ahead