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What is a Fourier Transform?

The Fourier transform and Fourier analysis are name after Joseph Fourier, a mathematician and physicist of the early 1800s, who introduced the idea that an arbitrary function could be decomposed into sinusoidal components. If the input is a periodic function, the resultant representation will probably be an infinite sum. For arbitrary inputs, the decomposition is described via an integral. Fourier transforms and Fourier series are normally introduced in the second or third year of a college math curriculum. However, the description is accessible from the mathematics of the first year of calculus. The Fourier decomposition has broad applications in many areas of science, engineering, and mathematics.

Focusing on our sound problem, we note that “sound” will be a pressure signal, transmitted (normally, for us) through the air. We represent that pressure signal (at the ear) as a function of time, denoted x(t). The Fourier transform takes this function of time t and creates a new, related function of frequency, f:

X(f)=\int^\infty_{-\infty} x(t)e^{-i2\pi f\ t} dt,

(1)

where i=\sqrt{-1} means that we are computing with complex numbers. Note that because we are integrating using the variable , the only variable remaining in the expression is f . For each value of
f , we can use the right hand side of (1) to compute a value. Hence, expression (1) really is a function of f . If t is measured in seconds, then frequency f would be measured in Hertz. The function X is called the transform of x. It contains the same information as the signal, but in a different form. It is said to represent the signal in the frequency domain. It has lost none of the information about the signal and we can always recover the function of time from the transform by using the inverse transform (IFT).

x(t)=\int^\infty_{-\infty} X(f)e^{i2\pi f\ t} df.

(2)

The function X is said to describe the spectrum of the signal. To highlight this aspect, we would remind the reader (or introduce them to the notion) that a complex number can be represented in two forms – Cartesian or polar:

z=x+iy=Ae^{i\phi},

where the polar form emphasizes the amplitude and phase angle of that complex number. So the complex valued transform X(f) can be written in polar form

X(f)=A(f)e^{i\phi(f)},

Where A(f) represents and amplitude at frequency f and \phi(f) is a phase angle for that frequency. In terms of the Fourier decomposition, this form tells us the amplitude and phase shift of the frequency f component of the underlying signal.