The Fourier Transform is a mathematical technique that transforms a function of tim*e, 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

where *c _{n}* is given by the Fourier Series analysis equation,

which can be rewritten

$$T{c_n} = \int_T {x(t){e^{ - jn\omega_0t}}dt} $$As *T→∞* the fundamental frequency, *ω _{0}=2π/T*, becomes extremely small and the quantity

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*).

As* T→∞*, *1/T=ω0/2π*. Since *ω _{0}* is very small (as

Forward Fourier Transform: **Analysis Equation**

Inverse Fourier Transform: **Synthesis Equation**

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.

& 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} $$

& 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} $$

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,

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).

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