Fourier Analysis - introduction from a practical point of viewIn calculus or mathematics, Fourier analysis is a topic which grew from the study of Fourier series. This field began with the study of the way periodic or general functions might be represented by summations of simpler trigonometric functions (sine or cosine series).
Fourier analysis has many applications in science, physics, digital signal processing, imaging, numerical analysis, acoustics... and in many other areas.
We are going to comment on some electronic applications, where there are two emphasized domains: analysis in time domain, to find the voltages or currents at specific moments, and the analysis in the frequency domain, to study (in order to possibly manipulate) the frequencies present in a given signal.
This figure shows a sine wave with a peak value Vp and a period T.
When AC analysis is presented in circuit theory courses, they usually focus on the sine wave because it helps understand many effects. Based on those effects or behaviors of electronic components we can start the study of non-sinusoidal signals.
The shapes below (a half-sine, a sawtooth and a square wave) complete their basic form over a period T, and each cycle is an exact copy of the first period or cycle. This kind of waveform with repetitive cycles is called a periodic wave with period T.
The addition of sine waves with appropriate amplitudes and phases can represent triangular waves, or a different combination of sine series and waves can produce a sawtooth signal, and another different combination may produce a half-wave signal. In other words, any periodic wave is a superposition of sine waves.
The sine waves are harmonically related, which means that the frequencies are harmonics (or multiples) of a fundamental (lowest) frequency. Given a periodic wave, you can find the period T when measured with an oscilloscope. The reciprocal of T is equal to the fundamental frequency.
The fundamental frequency f1 = 1 / T
The second harmonic has a frequency f2 = 2f1
The third harmonic has a frequency f3 = 3f1
In general, the nth harmonic has a frequency
fn = nf1
Formulas for the Fourier series
Expressed in words, the figure above means:
periodic wave = DC component +
first harmonic +
second harmonic +
In mathematical terms:
v = V0 + V1 sin(wt + Ф1) + V2 sin(2wt + Ф2) + ... + Vn sin(nwt + Фn)
This famous equation is called the Fourier series, and it means that a periodic wave is a superposition of harmonically related sine waves.
Voltage v is the value of the periodic wave at any instant, this value can be calculated by adding the DC component and the instant values of all the the harmonics. The first term in the Fourier series is V0, which is a constant and represents the DC component. The coefficients V1, V2... Vn are the peak values of the harmonics. The angular frequency w is equal to 2πf, so you can see that each term in the Fourier series represents the next higher harmonic.
Harmonic waves theoretically continue to infinity. However, five to ten terms are often enough to represent a periodic signal within an accuracy of 5%. With the right combination of magnitudes, angles and phases, it is possible to reproduce any periodic waveform.
Now, let’s do a brief experiment with our numerical software. You can see how to produce a square wave, here.
We can implement this code:
% Define and plot one cycle of a square wave
x = 0 : .01 : 2*pi;
y = square(x);
axis([min(x) max(x) min(y)-.5 max(y)+.5])
% Plot f1 and f3 and add them up
y1 = sin(x);
y2 = .2*sin(3*x);
plot(x,y1,'b', x,y2,'b:', x, y1+y2, 'g', 'linewidth', 2)
axis([min(x) max(x) min(y)-.5 max(y)+.5])
We get two figures:
The figure on the left is a plain square wave. We want to find an equivalent of that periodic wave by utilizing only sines. The figure on the right represents our first attempt to do it. We use the first and third harmonics of the original wave, we add them up and show the results. We can clearly see that the green line is getting closer to a square wave. In order to produce a better approximation, we’d need to adjust our amplitudes and include more terms in the series.
In the next article, we’re going to explore three very common periodic waves and the specific Fourier Series to produce them.
Malvino, A. P.; Electronic Principles; 2nd. edition; McGraw-Hill Co., 1979.
From 'Fourier Analysis ' to home
From 'Fourier Analysis ' to Calculus Problems
TAKING THE WAVEFORM APART
The waveform of a sound is a graph of the way the pressure changes between thewavefronts. This is usually a very convoluted pattern and the actual sound ofthe wave is not apparent from looking at the waveform. As a matter of fact, thewaveform does not usually repeat exactly from one cycle to another.
At this point I am going to have to digress into some mathematics. (SeeThe Mathematics of Electronic Music ifthis does not seem simple to you.)
Fig 1. Sine and cosine functions
The waveform produced by simple harmonic motion is the SINE WAVE. We graph asine wave by plotting the function:
To do this we divide up our graph paper horizontally into equal chunks torepresent a time scale, and for each time t we want to plot, we multiply t by2[pi]f (f=frequency) and look up the sine of the result. That sine value is whatgets used for the vertical part of the graph.
There is also a function called a cosine wave. The expression is
and it looks just like the sine wave. The difference is that the cosine of anangle is equal to the sine of an angle 90 degrees bigger. When we have twowaveforms which have the same shape and frequency but are offset in time, we saythey are out of phase by the amount of angle you have to add to the 2[pi]ft termof the first to move them together. In other words the wave defined bysin(2[pi]ft) is out of phase with the wave defined as sin(2[pi]ft+p) by theangle p.
The second simplest waveform is probably the combination of two sine waves. Anycombination of waves is interpreted by the ear as a single waveform, and thatwaveform is merely the sum of all of the waves passing that spot. Here are a fewrules about the addition of two sine waves:
- If both have the same frequency and phase, the result is a sine waveof amplitude equal to the sum of the two amplitudes.
- If both have the same frequency and amplitude but are 180 degrees out ofphase, the result is zero. Any other combinations of amplitude produce a resultof amplitude equal to the difference in the two original amplitudes.
- If both are the same frequency and amplitude but are out of phase a valueother 180 degrees, you get a sine wave of amplitude less than the sum of the twoand of intermediate phase.
- If the two sine waves are not the same frequency, the result is complex. Infact, the waveform will not be the same for each cycle unless the frequency ofone sine wave is an exact multiple of the frequency of the other.
If you explore combinations of more than two sine waves you find that thewaveforms become very complex indeed, and depend on the amplitude, frequency andphase of each component. Every stable waveform you discover will be made up ofsine waves with frequencies that are some whole number multiple of the frequencyof the composite wave.
The reverse process has been shown mathematically to be true: Any waveform canbe analyzed as a combination of sine waves of various amplitude, frequency andphase. The method of analysis was developed by Fourier in 1807 and is calledFourier Analysis.
The actual procedure for Fourier analysis is too complex to get into here, butthe result (with stable waveforms) is an expression of the form:
and so forth. The omega (looks like a w) represents the frequency in radians persecond, also known as angular frequency. The inclusion of cosine waves as wellas sine waves takes care of phase, and the letters represent the amplitude ofeach component. This result is easily translated into a bar graph with one barper component. Since the ear is apparently not sensitive to phase, we oftensimplify the graph into a sine waves only form. Such a graph is called aspectral plot:
Fig. 2 A typical spectral plot
The lowest component of the waveform is known as the FUNDAMENTAL, andthe others are HARMONICS, with a number corresponding to the multiple of thefundamental frequency. The second harmonic is twice the fundamental frequency,the third harmonic is three times the fundamental frequency, and so forth. It isimportant to recognize that the harmonic number is not the same as theequivalent musical interval name, although the early harmonics do approximatesome of the intervals. The most important relationship is that the harmonicsnumbered by powers of two are various octaves.
Non-repeating waveforms may be disassembled by Fourier means also, but theresult is a complex integral that is not useful as a visual aid. However, if wedisregard phase, these waveforms may also be represented on a spectral plot aslong as we remember that the components are not necessarily whole numbermultiples of the fundamental frequency and therefore do not qualify asharmonics. We should not say that a non-harmonic waveform is not pitched, but itis true that the worse the spectral plot fits the harmonic model the moredifficult it is to perceive pitch in a sound.
There are sounds whose waveforms are so complex that the Fourier process gives astatistical answer. (These waveforms are the sounds commonly called noise.) Youcan express the likelihood of finding a particular frequency as a component overa large enough time but you cannot assign any component a constant amplitude. Todescribe such sounds on a spectral plot, we plot the probability curve. A verynarrow band of noise will sound like a pitched tone, but as the curve widens, welose the impression of pitch, aware only of a vague highness or lowness of thesound.
Noise that spreads across the entire range of hearing is called WHITE NOISE ifit has equal probability of all frequencies being represented. Such noise soundshigh pitched because of the logarithmic response of the ear to frequency. (Ourears consider the octave 100 hz to 200 hz to be equal to the octave 1000 hz to2000 hz, even though the higher one has a much wider frequency spread, andtherefore more power.) Noise with emphasis added to the low end to compensatefor this is called PINK NOISE.
A sound event is only partially described by its spectral plot. For a completedescription, we need to graph the way the sound changes over time. There are twoways in which such graphs are presented. In the Sonogram, the horizontal axis istime, the vertical axis is frequency, and the amplitude is represented by thedarkness of the mark. There is a machine that produces this kind of chart bymechanical means.
Fig 3. A Sonogram
Computer analysis usually produces a three dimensional graph:
Fig 4. Spectragraph
The three dimensional graph gives a clearer sense of how the amplitudes ofvarious components of a sound change. This really shows amplitude envelopes foreach partial of the sound. In this case, frequency is represented by theapparent depth into the screen. Most analysis programs allow you to either showhigh frequency behind low, as this one does, or low behind high. This and theability to swap parameters among the three axes allows you to pick a view withthe least information hidden.