[6.2 EXISTENCE AND](#page-12-0) CONVERGENCE OF THE FOURIER SERIES

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6.2 EXISTENCE AND CONVERGENCE OF THE FOURIER SERIES

For the existence of the Fourier series, coefficients a0,an, and bn in Eq. (6.8) must be finite. It follows from Eq. (6.8) that the existence of these coefficients is guaranteed if x(t) is absolutely integrable over one period; that is,

$$\int_{T_0} |x(t)| \, dt < \infty \tag{6.16}$$

However, existence, by itself, does not inform us about the nature and the manner in which the series converges. We shall first discuss the notion of convergence.

6.2-1 Convergence of a Series

The key to many puzzles lies in the nature of the convergence of the Fourier series. Convergence of infinite series is a complex problem. It took mathematicians several decades to understand the convergence aspect of the Fourier series. We shall barely scratch the surface here.

Nothing annoys a student more than the discussion of convergence. “Have we not proved,” they ask, “that a periodic signal x(t) can be expressed as a Fourier series”? Then why spoil the fun by this annoying discussion? All we have shown so far is that a signal represented by a Fourier series in Eq. (6.1) is periodic. We have not proved the converse, that every periodic signal can be expressed as a Fourier series. This issue will be tackled later, in Sec. 6.5-4, where it will be shown that a periodic signal can be represented by a Fourier series, as in Eq. (6.1), where the equality of the two sides of the equation is not in the ordinary sense, but in the mean-square sense (explained later in this discussion). But the astute reader should have been skeptical of the claims of the Fourier series to represent discontinuous functions in Figs. 6.2a and 6.6a. If x(t) has a jump discontinuity, say, at t = 0, then x(0+), x(0), and x(0−) are generally different. How could a series consisting of the sum of continuous functions of the smoothest type (sinusoids) add to one value at t = 0⁻ and a different value at t = 0 and yet another value at t = 0+? The demand is impossible to satisfy unless the math involved executes some spectacular acrobatics. How does a Fourier series act under such conditions? Precisely for this reason, the great mathematicians Lagrange and Laplace, two of the judges examining Fourier’s paper, were skeptical of Fourier’s claims and voted against publication of the paper that later became a classic.

There are also other issues. In any practical application, we can use only a finite number of terms in a series. If, with a fixed number of terms, the series guarantees convergence within an arbitrarily small error at every value of t, such a series is highly desirable and is called a uniformly convergent series. If a series converges at every value of t, but to guarantee convergence within a given error requires a different number of terms at different t, then the series is still convergent, but less desirable. It goes under the name pointwise convergent series.

Finally, we have the case of a series that refuses to converge at some t, no matter how many terms are added. But the series may converge in the mean; that is, the energy of the difference between x(t) and the corresponding finite term series approaches zero as the number of terms approaches infinity.† To explain this concept, let us consider representation of a function x(t) by an infinite series

$$x(t) = \sum_{n=1}^{\infty} z_n(t)$$

Let the partial sum of the first N terms of the series on the right-hand side be denoted by xN(t), that is,

$$x_N(t) = \sum_{n=1}^N z_n(t)$$

^† The behavior is called “convergence in the mean” because minimizing the error energy over a certain interval is equivalent to minimizing the mean-square value of the error over the same interval.

614 CHAPTER 6 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER SERIES

If we approximate x(t) by xN(t) (the partial sum of the first N terms of the series), the error in the approximation is the difference x(t) − xN(t). The series converges in the mean to x(t) in the interval (0, T0) if

$$\int_0^{T_0} |x(t) - x_N(t)|^2 dt \to 0 \quad \text{as} \quad N \to \infty$$

Hence, the energy of the error x(t)−xN(t) approaches zero as N → ∞. This form of convergence does not require the series to be equal to x(t) for all t. It just requires the energy of the difference (area under |x(t)−xN(t)| 2) to vanish as N → ∞. Superficially it may appear that if the energy of a signal over an interval is zero, the signal (the error) must be zero everywhere. This is not true. The signal energy can be zero even if there are nonzero values at a finite number of isolated points. This is because although the signal is nonzero at a point (and zero everywhere else), the area under its square is still zero. Thus, a series that converges in the mean to x(t) need not converge to x(t) at a finite number of points. This is precisely what happens to the Fourier series when x(t) has jump discontinuities. This is also what makes Fourier series convergence compatible with the Gibbs phenomenon, to be discussed later in this section.

There is a simple criterion for ensuring that a periodic signal x(t) has a Fourier series that converges in the mean. The Fourier series for x(t) converges to x(t) in the mean if x(t) has a finite energy over one period, that is,

$$\int_{T_0} |x(t)|^2 dt < \infty \tag{6.17}$$

Thus, the periodic signal x(t), having a finite energy over one period, guarantees the convergence in the mean of its Fourier series. In all the examples discussed so far, Eq. (6.17) is satisfied; hence the corresponding Fourier series converges in the mean. Equation (6.17), like Eq. (6.16), guarantees that the Fourier coefficients are finite.

We shall now discuss an alternate set of criteria, due to Dirichlet, for convergence of the Fourier series.

DIRICHLET CONDITIONS

Dirichlet showed that if x(t) satisfies certain conditions (Dirichlet conditions), its Fourier series is guaranteed to converge pointwise at all points where x(t) is continuous. Moreover, at the points of discontinuities, x(t) converges to the value midway between the two values of x(t) on either side of the discontinuity. These conditions are:

• 1. The function x(t) must be absolutely integrable; that is, it must satisfy Eq. (6.16).
• 2. The function x(t) must have only a finite number of finite discontinuities in one period.
• 3. The function x(t) must contain only a finite number of maxima and minima in one period.

All practical signals, including those in Exs. 6.1, 6.2, 6.3, and 6.4, satisfy these conditions.

6.2-2 The Role of Amplitude and Phase Spectra in Waveshaping

The trigonometric Fourier series of a signal x(t) shows explicitly the sinusoidal components of x(t). We can synthesize x(t) by adding the sinusoids in the spectrum of x(t). Let us synthesize the square-pulse periodic signal x(t) of Fig. 6.6a by adding successive harmonics in its spectrum step by step and observing the similarity of the resulting signal to x(t). The Fourier series for this function as found in Eq. (6.13) is

$$x(t) = \frac{1}{2} + \frac{2}{\pi} \left( \cos t - \frac{1}{3} \cos 3t + \frac{1}{5} \cos 5t - \frac{1}{7} \cos 7t + \cdots \right)$$

We start the synthesis with only the first term in the series (n = 0), a constant 1/2 (dc); this is a gross approximation of the square wave, as shown in Fig. 6.8a. In the next step we add the dc (n = 0) and the first harmonic (fundamental), which results in a signal shown in Fig. 6.8b. Observe that the synthesized signal somewhat resembles x(t). It is a smoothed-out version of x(t). The sharp corners in x(t) are not reproduced in this signal because sharp corners mean rapid changes, and their reproduction requires rapidly varying (i.e., higher-frequency) components, which are excluded. Figure 6.8c shows the sum of dc, first, and third harmonics (even harmonics are absent). As we increase the number of harmonics progressively, as shown in Figs. 6.8d (sum up to the fifth harmonic) and 6.8e (sum up to the nineteenth harmonic), the edges of the pulses become sharper and the signal resembles x(t) more closely.

ASYMPTOTIC RATE OF AMPLITUDE SPECTRUM DECAY

Figure 6.8 brings out one interesting aspect of the Fourier series. Lower frequencies in the Fourier series affect the large-scale behavior of x(t), whereas the higher frequencies determine the fine structure such as rapid wiggling. Hence, sharp changes in x(t), being a part of fine structure, necessitate higher frequencies in the Fourier series. The sharper the change [the higher the time derivative x˙(t)], the higher are the frequencies needed in the series.

The amplitude spectrum indicates the amounts (amplitudes) of various frequency components of x(t). If x(t) is a smooth function, its variations are less rapid. Synthesis of such a function requires predominantly lower-frequency sinusoids and relatively small amounts of rapidly varying (higher-frequency) sinusoids. The amplitude spectrum of such a function would decay swiftly with frequency. To synthesize such a function, we require fewer terms in the Fourier series for a good approximation. On the other hand, a signal with sharp changes, such as jump discontinuities, contains rapid variations, and its synthesis requires a relatively large amount of high-frequency components. The amplitude spectrum of such a signal would decay slowly with frequency, and to synthesize such a function, we require many terms in its Fourier series for a good approximation. The square wave x(t) is a discontinuous function with jump discontinuities, and therefore its amplitude spectrum decays rather slowly, as 1/n [see Eq. (6.13)]. On the other hand, the triangular-pulse periodic signal in Fig. 6.4a is smoother because it is a continuous function (no jump discontinuities). Its spectrum decays rapidly with frequency as 1/n² [see Eq. (6.12)].

We can show that if the first k − 1 derivatives of a periodic signal x(t) are continuous and the kth derivative is discontinuous, then its amplitude spectrum Cn decays with frequency at least as rapidly as 1/nk+¹ [6]. This result provides a simple and useful means for predicting the asymptotic

Figure 6.8 Synthesis of a square-pulse periodic signal by successive addition of its harmonics.

rate of convergence of the Fourier series. In the case of the square-wave signal (Fig. 6.6a), the zeroth derivative of the signal (the signal itself) is discontinuous so that k = 0. For the triangular periodic signal in Fig. 6.4a, the first derivative is discontinuous; that is, k = 1. For this reason, the spectra of these signals decay as 1/n and 1/n2, respectively.

EXAMPLE 6.5 Square-Wave Synthesis by Truncated Fourier Series Using MATLAB

Use MATLAB to synthesize and plot the square wave of Fig. 6.8a using a Fourier series that is truncated to the 19th harmonic. The result should match Fig. 6.8e.

To synthesize the waveform, we use the Fourier series of Eq. (6.13).

>> x = @(t) 1.0*(mod(t+pi/2,2*pi)<=pi);
>> t = linspace(-2*pi,2*pi,10001);
>> x19 = 0.5*ones(size(t));
>> for n=1:19, x19 = x19+2/(pi*n)*sin(pi*n/2)*cos(n*t); end
>> plot(t,x19,'k-'); axis([-2*pi 2*pi -0.2 1.2]);
>> xlabel('t'); ylabel('x_{19}(t)');

As expected, the result of Fig. 6.9 matches Fig. 6.8e.

PHASE SPECTRUM: THE WOMAN BEHIND A SUCCESSFUL MAN

The role of the amplitude spectrum in shaping the waveform x(t) is quite clear. However, the role of the phase spectrum in shaping this waveform is less obvious. Yet, the phase spectrum, like the woman behind a successful man,† plays an equally important role in waveshaping. We can explain this role by considering a signal x(t) that has rapid changes, such as jump discontinuities. To synthesize an instantaneous change at a jump discontinuity, the phases of the various sinusoidal components in its spectrum must be such that all (or most) of the harmonic components will have

^† Or, to keep up with the times, the man behind a successful woman.

618 CHAPTER 6 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER SERIES

one sign before the discontinuity and the opposite sign after the discontinuity. This will result in a sharp change in x(t) at the point of discontinuity. We can verify this fact in any waveform with jump discontinuity. Consider, for example, the sawtooth waveform in Fig. 6.7b. This waveform has a discontinuity at t = 1. The Fourier series for this waveform, as given in Drill 6.1b, is

$$x(t) = \frac{2A}{\pi} \left[ \cos(\pi t - 90^\circ) + \frac{1}{2} \cos(2\pi t + 90^\circ) + \frac{1}{3} \cos(3\pi t - 90^\circ) + \frac{1}{4} \cos(4\pi t + 90^\circ) + \cdots \right]$$

Figure 6.10 shows the first three components of this series. The phases of all the (infinite) components are such that all the components are positive just before t = 1 and turn negative just after t = 1, the point of discontinuity. The same behavior is also observed at t = −1, where a similar discontinuity occurs. This sign change in all the harmonics adds up to produce very nearly a jump discontinuity. The role of the phase spectrum is crucial in achieving a sharp change in the waveform. If we ignore the phase spectrum when trying to reconstruct this signal, the result will be a smeared and spread-out waveform. In general, the phase spectrum is just as crucial as the amplitude spectrum in determining the waveform. The synthesis of any signal x(t) is achieved by using a proper combination of amplitudes and phases of various sinusoids. This unique combination is the Fourier spectrum of x(t).

Figure 6.10 Role of the phase spectrum in shaping a periodic signal.

FOURIER SYNTHESIS OF DISCONTINUOUS FUNCTIONS: THE GIBBS PHENOMENON

Figure 6.8 showed the square function x(t) and its approximation by a truncated trigonometric Fourier series that includes only the first N harmonics for N = 1, 3, 5, and 19. The plot of the truncated series approximates closely the function x(t) as N increases, and we expect that the series will converge exactly to x(t) as N → ∞. Yet the curious fact, as seen from Fig. 6.8, is that even for large N, the truncated series exhibits an oscillatory behavior and an overshoot approaching a value of about 9% in the vicinity of the discontinuity at the nearest peak of oscillation.† Regardless of the value of N, the overshoot remains at about 9%. Such strange behavior certainly would undermine anyone’s faith in the Fourier series. In fact, this behavior puzzled many scholars at the turn of the century. Josiah Willard Gibbs, an eminent mathematical physicist who was the inventor of vector analysis, gave a mathematical explanation of this behavior (now called the Gibbs phenomenon).

We can reconcile the apparent aberration in the behavior of the Fourier series by observing from Fig. 6.8 that the frequency of oscillation of the synthesized signal is Nf0, so the width of the spike with 9% overshoot is approximately 1/2Nf0. As we increase N, the frequency of oscillation increases and the spike width 1/2Nf⁰ diminishes. As N → ∞, the error power → 0 because the error consists mostly of the spikes, whose widths → 0. Therefore, as N → ∞, the corresponding Fourier series differs from x(t) by about 9% at the immediate left and right of the points of discontinuity, and yet the error power →0. The reason for all this confusion is that in this case, the Fourier series converges in the mean. When this happens, all we promise is that the error energy (over one period) → 0 as N → ∞. Thus, the series may differ from x(t) at some points and yet have the error signal power zero, as verified earlier. Note that the series, in this case, also converges pointwise at all points except the points of discontinuity. It is precisely at the discontinuities that the series differs from x(t) by 9%.‡

When we use only the first N terms in the Fourier series to synthesize a signal, we are abruptly terminating the series, giving a unit weight to the first N harmonics and zero weight to all the remaining harmonics beyond N. This abrupt termination of the series causes the Gibbs phenomenon in synthesis of discontinuous functions. Section 7.8 offers more discussion on the Gibbs phenomenon, its ramifications, and cure.

The Gibbs phenomenon is present only when there is a jump discontinuity in x(t). When a continuous function x(t) is synthesized by using the first N terms of the Fourier series, the synthesized function approaches x(t) for all t as N → ∞. No Gibbs phenomenon appears. This can be seen in Fig. 6.11, which shows one cycle of a continuous periodic signal being synthesized from the first 19 harmonics. Compare the similar situation for a discontinuous signal in Fig. 6.8.

DR ILL 6.3 Rate of Spectral Decay

By inspection of signals in Figs. 6.2a, 6.7a, and 6.7b, determine the asymptotic rate of decay of their amplitude spectra.

^† There is also an undershoot of 9% at the other side [at ^{t =} (π/2)+] of the discontinuity.

^‡ Actually, at discontinuities, the series converges to a value midway between the values on either side of the discontinuity. The 9% overshoot occurs at t = (π/2)⁻ and 9% undershoot occurs at t = (π/2)+.

Figure 6.11 Fourier synthesis of a continuous signal using first 19 harmonics.

ANSWERS

1/n, 1/n2, and 1/n, respectively.

A HISTORICAL NOTE ON THE GIBBS PHENOMENON

Normally speaking, troublesome functions with strange behavior are invented by mathematicians; we rarely see such oddities in practice. In the case of the Gibbs phenomenon, however, the tables were turned. A rather puzzling behavior was observed in a mundane object, a mechanical wave synthesizer, and then well-known mathematicians of the day were dispatched on the scent of it to discover its hideout.

Albert Michelson (of Michelson–Morley fame) was an intense, practical man who developed ingenious physical instruments of extraordinary precision, mostly in the field of optics. His harmonic analyzer, developed in 1898, could compute the first 80 coefficients of the Fourier series of a signal x(t) specified by any graphical description. The instrument could also be used as a harmonic synthesizer, which could plot a function x(t) generated by summing the first 80 harmonics (Fourier components) of arbitrary amplitudes and phases. This analyzer, therefore, had the ability of self-checking its operation by analyzing a signal x(t) and then adding the resulting 80 components to see whether the sum yielded a close approximation of x(t).

Michelson found that the instrument checked very well with most of signals analyzed. However, when he tried a discontinuous function, such as a square wave,† a curious behavior was observed. The sum of 80 components showed oscillatory behavior (ringing), with an overshoot of 9% in the vicinity of the points of discontinuity. Moreover, this behavior was a constant feature regardless of the number of terms added. A larger number of terms made the oscillations proportionately faster, but regardless of the number of terms added, the overshoot remained 9%. This puzzling behavior caused Michelson to suspect some mechanical defect in his synthesizer. He wrote about his observation in a letter to Nature (December 1898). Josiah Willard Gibbs, who was a professor at Yale, investigated and clarified this behavior for a sawtooth periodic signal in a letter to Nature [7]. Later, in 1906, Bôcher generalized the result for any function with discontinuity [8].

^† Actually, it was a periodic sawtooth signal.

Albert Michelson and Josiah Willard Gibbs

It was Bôcher who gave the name Gibbs phenomenon to this behavior. Gibbs showed that the peculiar behavior in the synthesis of a square wave was inherent in the behavior of the Fourier series because of nonuniform convergence at the points of discontinuity.

This, however, is not the end of the story. Both Bôcher and Gibbs were under the impression that this property had remained undiscovered until Gibbs’s work published in 1899. It is now known that what is called the Gibbs phenomenon had been observed in 1848 by Wilbraham of Trinity College, Cambridge, who clearly saw the behavior of the sum of the Fourier series components in the periodic sawtooth signal later investigated by Gibbs [9]. Apparently, this work was not known to most people, including Gibbs and Bôcher.

6.3 EXPONENTIAL FOURIER SERIES

By using Euler’s equality, we can express cos nω0t and sin nω0t in terms of exponentials ejn^ω0*^t* and e−jnω0*^t* . Clearly, we should be able to express the trigonometric Fourier series in Eq. (6.7) in terms of exponentials of the form ejn^ω0*^t* with the index n taking on all integer values from −∞ to ∞, including zero. Derivation of the exponential Fourier series from the results already derived for the trigonometric Fourier series is straightforward, involving conversion of sinusoids to exponentials. We shall, however, derive them here independently, without using the prior results of the trigonometric series.

This discussion shows that the exponential Fourier series for a periodic signal x(t) can be expressed as

$$x(t) = \sum_{n = -\infty}^{\infty} D_n e^{jn\omega_0 t}$$

622 CHAPTER 6 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER SERIES

To derive the coefficients Dn, we multiply both sides of this equation by e−jmω0*^t* (m integer) and integrate over one period. This yields

$$\int_{T_0} x(t) e^{-jm\omega_0 t} dt = \sum_{n=-\infty}^{\infty} D_n \int_{T_0} e^{j(n-m)\omega_0 t} dt$$

To simplify this expression, we use the orthogonality property of exponentials, which states that†

$$\int_{T_0} e^{jn\omega_0 t} e^{-jm\omega_0 t} dt = \begin{cases} 0 & m \neq n \\ T_0 & m = n \end{cases}$$

\n(6.18)

Thus,

$$\int_{T_0} x(t) e^{-jm\omega_0 t} dt = D_m T_0$$

from which we obtain

$$D_m = \frac{1}{T_0} \int_{T_0} x(t) e^{-jm\omega_0 t} dt.$$

To summarize, the exponential Fourier series can be expressed as

$$x(t) = \sum_{n = -\infty}^{\infty} D_n e^{jn\omega_0 t} \qquad \text{where} \qquad D_n = \frac{1}{T_0} \int_{T_0} x(t) e^{-jn\omega_0 t} dt \tag{6.19}$$

Observe the compactness of Eq. (6.19) and compare it with the trigonometric Fourier series expression. Such a comparison demonstrates very clearly the principal virtue of the exponential Fourier series. First, the form of the series is most compact. Second, the mathematical expression for deriving the coefficients of the series is also compact. It is much more convenient to handle the exponential series than the trigonometric one. For these reasons we shall use the exponential (rather than trigonometric) representation of signals in the rest of the book.

We can now relate Dn to trigonometric series coefficients an and bn. Setting n=0 in Eq. (6.19), we obtain

$$D_0=a_0$$

Moreover, for n = 0,

$$D_n = \frac{1}{T_0} \int_{T_0} x(t) \cos n\omega_0 t \, dt - \frac{j}{T_0} \int_{T_0} x(t) \sin n\omega_0 t \, dt = \frac{1}{2} (a_n - jb_n) \tag{6.20}$$ $$\int_{T_0} e^{j(n-m)\omega_0 t} dt = \int_{T_0} \cos{(n-m)\omega_0 t} dt + j \int_{T_0} \sin{(n-m)\omega_0 t} dt$$

Both the integrals on the right-hand side represent area under n − m number of cycles. Because n − m is an integer, both the areas are zero. Hence, Eq. (6.18) follows.

^† We can readily prove this property as follows. For the case of ^{m = n}, the integrand in Eq. (6.18) is unity and the integral is T0. When m = n, the integral on the left-hand side of Eq. (6.18) can be expressed as

and

$$D_{-n} = \frac{1}{T_0} \int_{T_0} x(t) \cos n\omega_0 t \, dt + \frac{j}{T_0} \int_{T_0} x(t) \sin n\omega_0 t \, dt = \frac{1}{2} (a_n + jb_n) \tag{6.21}$$

These results are valid for general x(t), real or complex. When x(t) is real, an and bn are real, and Eqs. (6.20) and (6.21) show that Dn and D−ⁿ are conjugates.

$$D_{-n}=D_n^*$$

Moreover, from Eq. (6.10), we observe that

$$a_n - jb_n = \sqrt{a_n^2 + b_n^2} \, e^{j \tan^{-1} \left( \frac{-b_n}{a_n} \right)} = C_n e^{j \theta_n}$$

Hence,

$$D_0=a_0=C_0$$

and

$$D_n = \frac{1}{2} C_n e^{j\theta_n} \qquad D_{-n} = \frac{1}{2} C_n e^{-j\theta_n}$$

Therefore, for n = 0,

$$|D_n| = |D_{-n}| = \frac{1}{2}C_n, \quad \angle D_n = \theta_n, \quad \text{and} \angle D_{-n} = -\theta_n \tag{6.22}$$

Note that |Dn| are the amplitudes and Dn are the angles of various exponential components. From Eq. (6.22) it follows that when x(t) is real, the amplitude spectrum (|Dn| versus ω) is an even function of ω and the angle spectrum ( Dn versus ω) is an odd function of ω. For complex x(t), Dn and D−ⁿ are generally not conjugates.