THE FOURIER TRANSFORM
REPRESENTING SIGNALS AS SUMS OF SINES

We saw in the previous section that any point on a two-dimensional surface can be expressed using a pair of Cartesian or polar coordinates. We also showed that there are transforms which allow us to switch between these two representations at will. A similar situation exists for signals. We may choose to represent signals using time-based coordinates or frequency-based coordinates. The Fourier Transform is our tool for switching between these two representations.

I find it helpful to think of the frequency-domain representation as a list of phasors. The Discrete Fourier Transform takes your time-domain signal and produces a list of phasors which, when summed together, will reproduce your signal. In very broad strokes, the two representations can be thought of as looking something like this,

We combine or sum phasors by stringing them together into a chain. The center of the first phasor is placed at the origin, and the center of each subsequent phasor is “attached” to the tip of the previous phasor. Once the chain of phasors is constructed, we allow each phasor to begin rotating. We can reconstruct the time domain signal by tracing the vertical distance from the origin to the tip of the last phasor. The following visualization allows you to play with the relative magnitudes of five phasors which are linked together in a “chain”.1

You should find it relatively easy to create different wave shapes like the saw and square. This is because the phasors are harmonically related. In other words, the second phasor spins twice as quickly as the first, the third spins three times as quickly, the fourth four times, and so on. The Discrete Fourier Transform always produces a set of phasors which are harmonically related. In the next few sections we’ll dig deeply into the inner workings of the Discrete Fourier Transform. We’ll see exactly how this list of phasors is generated from the time domain signal.

1. Which is all a rather wordy and roundabout way to say that you just sum them up. The symbolic expression is straightforward, and it looks something like,

A1sin(f1 * phase) +
A2sin(f2 * phase) ...

Where Ai is the Amplitude of the ith sine and fi is the frequency of the ith sine.