Performing Fourier Transforms

 

In this section we use DADiSP to investigate some of the principles behind Fourier analysis. The central concept of Fourier analysis is that virtually any waveform can be expressed as a discrete or continuous sum of sine and cosine functions, or equivalently, as a sum of complex (i.e., real and imaginary) exponential functions.

 

This simple idea proves extremely useful because we often gain more information by viewing a signal as a sum of functions rather than as a single phenomenon.

 

Create the following Worksheet:

 

W1: gsin(128, 1/128, 1)

W2: gsin(128, 1/128, 3) /3

W3: gsin(128, 1/128, 5) /5

W4: gsin(128, 1/128, 7) /7

W5: gsin(128, 1/128, 9) /9

W6: gsin(128, 1/128, 11) /11

W7: W1 + W2

W8: W1+W2+W3+W4

W9: W1+W2+W3+W4+W5+W6

 

As we add more sine waves, the resulting waveform begins to look more and more like a square wave. If we summed an infinite number of sine waves with the proper amplitudes and frequencies, we could reproduce a square wave exactly. This square wave is a Fourier Series: a summation of discrete sine and cosine waves.

 

Now, in W10, take the XSPECTRUM of W9.

 

W10: spectrum(w9);sticks

 

The "spectrum" of a waveform is a form of Fourier transform, plotting the amplitudes of its sinusoidal components against the frequency of the sinusoid. In other words, the Fourier transform shows the amplitudes and frequencies of the sine waves that comprise our waveform.

 

Fourier transform plot. The peaks of a spectrum plot are equal to the amplitudes of the component sine waves, and the peaks are located at the frequencies of those sine waves. The spectrum graph is a shorthand way of expressing the amplitude/frequency relationship of a data series. Spectrum plots can reveal distinct components within series that appear almost random.

 

The description of a Fourier transform as a plot of sinusoidal amplitudes and frequencies that comprise a series is somewhat simplified. A signal can be continuous or discrete along the x-axis. (Although time is a common unit for the x-axis, the x-axis can be any unit of measure.)

 

The Dow Jones stock market index or the salary of a professional football player by weight are examples of discrete series. Note that a discrete series is defined only for discrete x values. Signals that are continuous in time are called time series, while discrete series are referred to as time sequences.

 

Of course, continuous signals are often converted to discrete form for manipulation on a computer. An analog-to-digital (A/D) converter transforms a continuously occurring voltage to a voltage sampled (i.e., made discrete) in time.