Fourier Analysis

Playing a musical instrument, such as a guitar or an organ, generates a set of standing waves that cause a sympathetic oscillation in the surrounding air. Such an oscillation consists of a fundamental harmonic, whose frequency determines the pitch of the musical note heard by the listener, accompanied by a set of overtone harmonics that determine the timbre of the note. By definition, the oscillation frequencies of the overtone harmonics are integer multiples of that of the fundamental. Thus, we expect the pressure perturbation generated in a listener's ear to have the general form

$$\delta p(t) = \sum_{n=1,\infty} A_{n}\,\cos(n\,\omega\,t-\phi_{n}),$$ (5.47)

where $\omega$ is the angular frequency of the fundamental (i.e., $n=1$) harmonic, and the $A_n$ and $\phi_n$ are the amplitudes and phases of the various harmonics. The preceding expression can also be written

$$\delta p(t) = \sum_{n=1,\infty} \left[C_{n}\,\cos(n\,\omega\,t)+ S_{n}\,\sin(n\,\omega\,t)\right],$$ (5.48)

where $C_n=A_n\,\cos\phi_n$ and $S_n=A_n\,\sin\phi_n$. The function $\delta p(t)$ is periodic in time with period $\tau=2\pi/\omega$. In other words, $\delta p(t+\tau)=\delta p(t)$ for all $t$. This follows because of the mathematical identities $\cos(\theta+n\,2\pi)\equiv \cos\theta$ and $\sin(\theta+n\,2\pi)\equiv \sin\theta$, where $n$ is an integer. [Moreover, there is no $\tau'<\tau$ for which $\delta p(t+\tau')=\delta p(t)$ for all $t$.] Can any periodic waveform be represented as a linear superposition of sine and cosine waveforms, whose periods are integer subdivisions of that of the waveform, such as that specified in Equation (5.48)? To put it another way, given an arbitrary periodic waveform $\delta p(t)$, can we uniquely determine the constants $C_n$ and $S_n$ appearing in Equation (5.48)? It turns out that we can. Incidentally, the decomposition of a periodic waveform into a linear superposition of sinusoidal waveforms is commonly known as Fourier analysis.

The problem under investigation is as follows. Given a periodic waveform $y(t)$, where $y(t+\tau)=y(t)$ for all $t$, we need to determine the constants $C_n$ and $S_n$ in the expansion

$$y(t) = \sum_{n'=1,\infty} \left[C_{n'}\,\cos(n'\,\omega\,t)+ S_{n'}\,\sin(n'\,\omega\,t)\right],$$ (5.49)

where

$$\omega=\frac{2\pi}{\tau}.$$ (5.50)

It can be demonstrated that [cf., Equation (4.53)]

$$\frac{2}{\tau} \int_0^\tau \cos(n\,\omega\,t)\,\cos(n'\,\omega\,t)\,dt = \delta_{n,n'},$$ (5.51)

$$\frac{2}{\tau} \int_0^\tau \sin(n\,\omega\,t)\,\sin(n'\,\omega\,t)\,dt = \delta_{n,n'},$$ (5.52)

$$\frac{2}{\tau} \int_0^\tau \cos(n\,\omega\,t)\,\sin(n'\,\omega\,t)\,dt =0,$$ (5.53)

where $n$ and $n'$ are positive integers. Thus, multiplying Equation (5.49) by $(2/\tau)\,\cos(n\,\omega\,t)$, and then integrating over $t$ from 0 to $\tau$, we obtain

$$C_n = \frac{2}{\tau}\int_0^\tau y(t)\,\cos(n\,\omega\,t)\,dt,$$ (5.54)

where use has been made of Equation (5.51)–(5.53), as well as Equation (4.54). Likewise, multiplying Equation (5.49) by $(2/\tau)\,\sin(n\,\omega\,t)$, and then integrating over $t$ from 0 to $\tau$, we obtain

$$S_n = \frac{2}{\tau}\int_0^\tau y(t)\,\sin(n\,\omega\,t)\,dt.$$ (5.55)

Hence, we have uniquely determined the constants $C_n$ and $S_n$ in the expansion (5.49). These constants are generally known as Fourier coefficients, whereas the expansion itself is known as either a Fourier expansion or a Fourier series.

Figure 5.6: Fourier reconstruction of a periodic sawtooth waveform. The top-left, top-right, bottom-left, and bottom-right panels correspond to reconstructions using 4, 8, 32, and 64 terms, respectively, in the Fourier series.

In principle, there is no restriction on the waveform $y(t)$ in the previous analysis, other than the requirement that it be periodic in time. In other words, we ought to be able to Fourier analyze any periodic waveform. Let us see how this works. Consider the periodic sawtooth waveform

$$y(t) = A\,(2\,t/\tau-1) \mbox{\hspace{0.5cm}$0\leq t/\tau \leq 1$} ,$$ (5.56)

with $y(t+\tau)=y(t)$ for all $t$. See Figure 5.6. This waveform rises linearly from an initial value $-A$ at $t=0$ to a final value $+A$ at $t=\tau$, discontinuously jumps back to its initial value, and then repeats ad infinitum. According to Equations (5.54) and (5.55), the Fourier harmonics of the waveform are

$$C_n = \frac{2}{\tau}\int_0^\tau A\,(2\,t/\tau-1)\,\cos(n\,\omega\,t)\,dt = \frac{A}{\pi^{\,2}}\int_0^{2\pi} (\theta-\pi)\,\cos(n\,\theta)\,d\theta,$$ (5.57)

$$S_n = \frac{2}{\tau}\int_0^\tau A\,(2\,t/\tau-1)\,\sin(n\,\omega\,t)\,dt = \frac{A}{\pi^{\,2}}\int_0^{2\pi} (\theta-\pi)\,\sin(n\,\theta)\,d\theta,$$ (5.58)

where $\theta=\omega\,t$. Integration by parts (Riley 1974) yields

$$C_n =0,$$ (5.59)

$$S_n = - \frac{2\,A}{n\,\pi}.$$ (5.60)

Hence, the Fourier reconstruction of the waveform is written

$$y(t) = - \frac{2\,A}{\pi}\sum_{n=1,\infty} \frac{\sin(n\,2\pi\,t/\tau)}{n}.$$ (5.61)

Given that the Fourier coefficients fall off like $1/n$, as $n$ increases, it seems plausible that the preceding series can be truncated after a finite number of terms without unduly affecting the reconstructed waveform. Figure 5.6 shows the result of truncating the series after 4, 8, 16, and 32 terms. It can be seen that the reconstruction becomes increasingly accurate as the number of terms retained in the series increases. The annoying oscillations in the reconstructed waveform at $t=0$, $\tau$, and $2\,\tau$ are known as Gibbs' phenomena, and are the inevitable consequence of trying to represent a discontinuous waveform as a Fourier series (Riley 1974). In fact, it can be demonstrated mathematically that, no matter how many terms are retained in the series, the Gibbs' phenomena never entirely go away (Zygmund 1955).

We can slightly generalize the Fourier series (5.49) by including an $n=0$ term. In other words,

$$y(t) = C_0+ \sum_{n'=1,\infty} \left[C_{n'}\,\cos(n'\,\omega\,t)+ S_{n'}\,\sin(n'\,\omega\,t)\right],$$ (5.62)

which allows the waveform to have a non-zero average. There is no term involving $S_0$, because $\sin (n\,\omega\,t)=0$ when $n=0$. It can be demonstrated that

$$\frac{2}{\tau} \int_0^\tau \cos(n\,\omega\,t)\,dt =0,$$ (5.63)

$$\frac{2}{\tau} \int_0^\tau \sin(n\,\omega\,t)\,\,dt =0,$$ (5.64)

where $\omega=2\pi/\tau$, and $n$ is a positive integer. Making use of the preceding expressions, as well as Equations (5.51)–(5.53), we can show that

$$C_0 = \frac{1}{\tau}\int_0^\tau y(t)\,dt,$$ (5.65)

and also that Equations (5.54) and (5.55) still hold for $n>0$.

Figure 5.7: Fourier reconstruction of a periodic “tent” waveform. The top-left, top-right, bottom-left, and bottom-right panels correspond to reconstruction using the first 1, 2, 4, and 8 terms, respectively, in the Fourier series (in addition to the $C_0$ term).

As an example, consider the periodic “tent” waveform

$$y(t) = 2\,A\left\{ \begin{array}{lcl} t/\tau&\mbox{\hspace{0.... ...1/2\\ [0.5ex] 1-t/\tau &&1/2< t/\tau \leq 1 \end{array}\right.,$$ (5.66)

where $y(t+\tau)=y(t)$ for all $t$. See Figure 5.7. This waveform rises linearly from zero at $t=0$, reaches a peak value $A$ at $t=\tau/2$, falls linearly, becomes zero again at $t=\tau$, and repeats ad infinitum. Moreover, the waveform has a non-zero average. It can be demonstrated, from Equations (5.54), (5.55), (5.65), and (5.66), that

$$C_0 = \frac{A}{2},$$ (5.67)

and

$$C_n = - A\,\frac{\sin^2(n\,\pi/2)}{(n\,\pi/2)^{\,2}}$$ (5.68)

for $n>1$, with $S_n=0$ for $n>1$. In fact, only the odd-$n$ Fourier harmonics are non-zero. Thus,

$$y(t) = \frac{A}{2} - \frac{4\,A}{\pi^{\,2}}\sum_{n=1,3,5,\cdots} \frac{\cos(n\,2\pi\,t/\tau)}{n^{\,2}}.$$ (5.69)

Figure 5.7 shows a Fourier reconstruction of the “tent” waveform using the first 1, 2, 4, and 8 terms (in addition to the $C_0$ term) in the Fourier series The reconstruction becomes increasingly accurate as the number of terms in the series increases. Moreover, in this example, there is no sign of Gibbs' phenomena, because the tent waveform is completely continuous.

In our first example—that is, the sawtooth waveform—all of the $C_n$ Fourier coefficients are zero, whereas in our second example—that is, the tent waveform—all of the $S_n$ coefficients are zero. This occurs because the sawtooth waveform is odd in $t$—that is, $y(-t)=-y(t)$ for all $t$—whereas the tent waveform is even—that is, $y(-t)=y(t)$ for all $t$. It is a general rule that waveforms that are even in $t$ only have cosines in their Fourier series, whereas waveforms that are odd only have sines (Riley1974). Waveforms that are neither even nor odd in $t$ have both cosines and sines in their Fourier series.

Fourier series arise quite naturally in the theory of standing waves, because the normal modes of oscillation of any uniform continuous system possessing linear equations of motion (e.g., a uniform string, an elastic rod, an ideal gas) take the form of spatial cosine and sine waves whose wavelengths are rational fractions of one another. Thus, the instantaneous spatial waveform of such a system can always be represented as a linear superposition of cosine and sine waves; that is, a Fourier series in space, rather than in time. In fact, the process of determining the amplitudes and phases of the normal modes of oscillation from the initial conditions is essentially equivalent to Fourier analyzing the initial conditions in space. (See Sections 4.4 and 5.3.)