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Chapter 3: The Frequency DomainSection 3.3: Fourier and the Sum of Sines


In this section, we’ll try to really explain the notion of a Fourier expansion by building on the ideas of phasors, partials, and sinusoidal components that we introduced in the previous section. A long time ago, French scientist and mathematician Jean Baptiste Fourier (1768–1830) proved the mathematical fact that any periodic waveform can be expressed as the sum of an infinite set of sine waves. The frequencies of these sine waves must be integer multiples of some fundamental frequency. In other words, if we have a trumpet sound at middle A (440 Hz), we know by Fourier’s theorem that we can express this sound as a summation of sine waves: 440 Hz, 880Hz, 1,320Hz, 1,760 Hz..., or 1, 2, 3, 4... times the fundamental, each at various amplitudes. This is rather amazing, since it says that for every periodic waveform (one, by the way, that has pitch), we basically know everything about its partials except their amplitudes. Fourier SeriesWhat exactly is a Fourier series, and how does it relate to phasors? We use phasors to represent our basic tones. The amazing fact is that any sound can be represented as a combination of phaseshifted, amplitudemodulated tones of differing frequencies. Remember that we got a hint of this concept when we discussed adding phasors in Section 3.2. A phasor is essentially a way of representing a sinusoidal function. What this means, mathematically, is that any sound can be represented as a sum of sinusoids. This sum is called a Fourier series. Note that we haven’t limited these sounds to periodic sounds (if we did, we’d have to add that last qualifier about integer multiples of a fundamental frequency). Nonperiodic, or aperiodic, sounds are just as interesting—maybe even more interesting—than periodic ones, but we have to do some special computer tricks to get a nice "harmonic" series out of them for the purposes of analysis and synthesis. But let’s get down to the nittygritty. First, let’s take a look at what happens when we add two sinusoids of the same frequency. Adding a sine and cosine of the same frequency gives a phaseshifted sine of the same frequency: In fact, the amplitude of the sum, C, is given by: The phase shift is given by the angle whose tangent is equal to A/B. The shorthand for this is: We can visualize this with a phasor. Remember that the cosine is just a phaseshifted sine. Since the sine and cosine are moving at the same frequency, they are always "out of sync" by /2, so when we add them it looks like this: And we get another sinusoid of that frequency. Any periodic function of period 1 can be written as follows: Notice that these sums can be infinite!We have a nice shorthand for those possibly infinite sums (also called an infinite series): The two expressions after the Σ signs are called the Fourier coefficients of the function f(t). The Fourier coefficient A_{0} has a special name: it is called the DC term, or the DC offset. It tells you the average value of the function. The Fourier coefficients make up a set of numbers called the spectrum of the sound. Now, when you think of the word "spectrum," you might think of colors, like the spectrum of colors of the rainbow. In a way it’s the same: the spectrum tells you how much of each frequency (color) is in the sound. The values of A_{n} and B_{n} for "small" values of n make up the lowfrequency information, and we call these the loworder Fourier coefficients. Similarly, the big values of n index the highfrequency information. Since most sounds are made up of a lot of lowfrequency information, the lowfrequency Fourier coefficients have larger absolute value than the highfrequency Fourier coefficients. What this means is that it is theoretically possible to take a complex sound, like a person’s voice, and decompose it into a bunch of sine waves, each at a different frequency, amplitude, and phase. These are called the sinusoidal or spectral components of a sound. To find them, we do a Fourier analysis. Fourier synthesis is the inverse process, where we take varying amounts of a bunch of sine waves and add them together (play them at the same time) to reconstruct a sound. Sounds a bit fantastic, doesn’t it? But it works. This process of analyzing or synthesizing a sound based on its component sine waves is called performing a Fourier transform on the sound. When the computer does it, it uses a very efficient technique called the fast Fourier transform (or FFT) for analysis and the inverse FFT (IFFT) for synthesis. 



The advantage of representing a sound in terms of its Fourier series is that it allows us to manipulate the frequency content directly. If we want to accentuate the highfrequency effects in a sound (make a sound brighter), we could just make all the highfrequency Fourier coefficients bigger in amplitude. If we wanted to turn a sawtooth wave into a square wave, we could just set to zero the Fourier coefficients of the even partials. 

In fact, we often
modify sounds by removing certain frequencies. This corresponds to making
a new function where certain Fourier coefficients are set equal to zero
while all others are left alone. When we do this we say that we filter
the function or sound. These sorts of filters are called bandpass filters,
and the frequencies that we leave unaltered in this sort of situation
are said to be in the passband. A lowpass filter puts all the
low frequencies (up to some bandwidth) in the passband, while a highpass
filter puts all high frequencies (down to some cutoff) in the passband.
When we do this, we talk about highpassing and lowpassing the sound.
In the following soundfiles, we listen to a sound and its highpassed
and lowpassed versions. We’ll talk a lot more about filters in Chapter
4. 










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