A comprehensive review of complex numbers and complex exponentials is given in Parker's textbook. Euler's formula relates the complex exponential to the complex number whose real part is the cosine function and whose imaginary part is the sine function. So the complex exponential is a convenient way to deal with both the sine and the cosine of an argument simultaneously. Trigonometry tells us that a cosine (or sine) function with a phase angle is equivalent to a linear combination of the sine and cosine function. The complex exponential allows us to deal with the sinusoidal function and phase angles in a functional form which is often easy to manipulate - the exponential form.
We can represent a time-dependent signal as an expansion of its frequency components. In other words, a time signal contains a certain amount of low frequencies, another amount of medium frequencies, and yet a different amount of high frequencies. In fact, the time signal can be expanded into an infinitely divisible spectrum of frequencies in the following mathematical format:
The scale factor F(w) for each of the frequencies w is known as the "Fourier Transform" of the original signal - it gives the amount of each frequency found in the signal f(t). If there is part of f(t) containing signals of frequency wo, then F(wo) = 0.
The Fourier transform F(w) is determined from the original signal f(t) in the following manner:
It is because of this relationship between F(w) and f(t) that we refer to the original time signal as the "Inverse" Fourier transform of F(w).
The symbol for the relationship between the inverse Fourier transform f(t) and the Fourier transform F(w) is:
These two functions are referred to as a Fourier Transform Pair.
What is the Fourier transform of a simple sinusoidal function? Obviously this is one frequency only - a pure tone. It turns out that the F.T. contains only real delta functions at + and - wo.
It f(t) were a sine function instead of a cosine function, then the F.T. would contain only imaginary delta functions, again at + and - w
So frequency components show up in the Fourier transform. For a simple sine wave oscillating at a given frequency, the Fourier transform is a delta function at that frequency and zero at all other frequencies.
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The Fourier transform pair is two EQUIVALENT representations of the same entity. The time domain is not superior to the frequency domain, it is just a different way of looking at or expressing the information contained in the entity. The entity is referred to as a "signal" in the time domain and a "spectrum" in the frequency domain.
Spatial Frequency
We have introduced the concept of the Fourier transform using the familiar "time" and "frequency" duality. Of course this is image processing, and we ultimately wish to take the Fourier transform of two dimensional images. What we end up with is a two dimesional complex spectra of "spatial frequency". The units of time-frequency are often expressed as Hz, or sec-1. Likewise, the units of spatial frequency are often expressed as cm-1. We can convert to radians but multiplying the frequency values by the factor 2 pi radians per cycle. This is true for both time-frequency and spatial frequency.