Consider two vectors y and Y which are related by a transformation matrix A. Then the relationship between components takes the familiar form
except for the apparently unorthodox limits. Now let the transformation matrix have the special form
where Dw is the difference between consecutive values. Then Eqn (1) can be written with a change in notation as
In the limit of an infinite number of infinitesimally spaced values this can be written
The relationship expressed in Eqn(4) is a special case of a general class of such relationships referred to as integral transforms, and in this case is the Fourier transform. The functions y(t) and Y(w) are referred to as Fourier transform pairs, either one being the Fourier transform of the other. Of course the above is anything but a rigorous exposition, but is meant to emphasize that the functions can in some ways be considered infinite dimensional vectors, related by a coordinate transformation. For the purposes of signal analysis the variables t and w are identified as time and frequency respectively. The integrand in Eqn(4) then represents a harmonic signal so the equation can be interpreted to imply that any signal is the linear superposition of harmonic signals. The quantity Y(w) can be looked upon as a measure of the amount of that frequency component present in the signal. It is important to realize that both functions y(t) and Y(w) are equally valid descriptions of the signal. True enough, our experience and mode of experimental observation through oscilloscopes makes the time domain description intuitively more acceptable than that in the frequency domain, but the latter has equivalent status and can be extremely powerful in signal analysis. A complication which arises is that the quantity Y(w) is in general complex. If written in polar form, the magnitude represents the amplitude of the frequency component, while the angle represents the phase.
The transformation described by Eqn(4) is from the frequency domain to the time domain. The inverse transform is
While there are many properties connected with the Fourier transform, two will be useful here. If the transform is decomposed into real and imaginary parts the integral transform may be written
Since y(t) is a real signal the imaginary portion must vanish. Thus
This can only be unconditionally satisfied if Yr(-w) = Yr(w) while at the same time Yi(-w) = -Yi(w), or equivalently Y(-w) = Y*(w).
Consider
Substituting w' = -w gives
Thus the frequency domain representation of y(-t) is Y*(w).