In this section, we employ DADiSP to examine foundational principles of Fourier analysis. The central tenet is that virtually any time-domain signal can be decomposed into a sum—either discrete or continuous—of sinusoidal components. These components may be expressed as sine and cosine functions or, equivalently, as complex exponentials comprising real and imaginary parts.
This decomposition is analytically powerful: representing a signal in terms of its frequency-domain constituents often reveals structural features and behaviors that are difficult to observe in the time domain.
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
To reveal the amplitude differences, we'll use SETY to scale all windows identically.
sety(W1..W9, -1, 1);
Windows 1 through 6 contain sine waves of 1, 3, 5, 7, 9 and 11 Hertz with amplitudes of 1, 1/3, 1/5, 1/7, 1/9 and 1/11. Windows 7 through 9 show the progressive sum of the sine waves. 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 summation of sinusoids to produce a square wave depicts the Fourier Series: considering a waveform as a summation of discrete sine and cosine component waves.
Now, in W10, we take the SPECTRUM of W9.
W10: spectrum(w9);sticks;setx(0, 12)
The spectrum of a waveform is a normalized Discrete Fourier Transform, plotting the amplitudes of its sinusoidal components against the frequency of the sinusoid. In this case, the heights of the non-zero sticks are 1, 1/3, 1/5, 1/7, 1/9 and 1/11 located at respective frequencies 1, 3, 5, 7, 9 and 11 Hertz. The Fourier transform shows the amplitudes and frequencies of the sine waves that comprise our waveform. Window 10 is an amplitude-frequency summary of the sum of windows 1 through 6. It is also an amplitude-frequency summary of the waveform in Window 9.
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.