FourierSeries. In 1807, the French mathematician and physicist Joseph Fourier submitted a paper C H A P T E R

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1 C H A P T E R FourierSeries In 87, the French mathematician and physicist Joseph Fourier submitted a paper on heat conduction to the Academy of Sciences of Paris. In this paper Fourier made the claim that any function f º» can be epanded into an infinite sum of trigonometric functions, f º» a ½ k Ìa k cosºk» b k sinºk»í for 7 The paper was rejected after it was read by some of the leading mathematicians of his day. They objected to the fact that Fourier had not presented much in the way of proof for this statement, and most of them did not believe it. In spite of its less than glorious start, Fourier s paper was the impetus for major developments in mathematics and in the application of mathematics. His ideas forced mathematicians to come to grips with the definition of a function. This, together with other metamathematical questions, caused nineteenth-century mathematicians to rethink completely the foundations of their subject, and to put it on a more rigorous foundation. Fourier s ideas gave rise to a new part of mathematics, called harmonic analysis or Fourier analysis. This, in turn, fostered the introduction at the end of the nineteenth century of a completely new theory of integration, now called the ebesgue integral. The applications of Fourier analysis outside of mathematics continue to multiply. One important application pertains to signal analysis. Here, f º» could represent the amplitude of a sound wave, such as a musical note, or an electrical signal from a CD player or some other device (in this case represents time and is usually replaced by t). The Fourier series representation of a signal represents a decomposition of this signal into its various frequency components. The terms sin k and cos k

2 . Computation of Fourier Series 73 oscillate with numerical frequency of k. Signals are often corrupted by noise, which usually involves the high-frequency components (when k is large). Noise can sometimes be filtered out by setting the high-frequency coefficients (the a k and b k when k is large) equal to zero. Data compression is another increasingly important problem. One way to accomplish data compression uses Fourier series. Here the goal is to be able to store or transmit the essential parts of a signal using as few bits of information as possible. The Fourier series approach to the problem is to store (or transmit) only those a k and b k that are larger than some specified tolerance and discard the rest. Fortunately, an important theorem (the Riemann ebesgue emma, which is our Theorem.) assures us that only a small number of Fourier coefficients are significant, and therefore the aforementioned approach can lead to significant data compression.. Computation of Fourier Series The problem that we wish to address is the one faced by Fourier. Suppose that f º» is a given function on the interval Ì Í. Can we find coefficients, a n and b n, so that f º» a ½ n [a n cos n b n sin n] for? (.) Notice that, ecept for the term a, the series is an infinite linear combination of the basic terms sin n and cos n for n a positive integer. These functions are periodic with period n, so their graphs trace through n periods over the interval Ì Í. Figure shows the graphs of cos and cos 5, and Figure shows the graphs of sin and sin 5. Notice how the functions become more oscillatory as n increases. π π π π Figure The graphs of cos and cos 5. Figure The graphs of sin and sin 5. The orthogonality relations Our task of finding the coefficients a n and b n for which (.) is true is facilitated by the following lemma. These orthogonality relations are one of the keys to the whole theory of Fourier series. Be sure you know the difference between angular frequency, k in this case, and numerical frequency. It is eplained in Section 4..

3 74 Chapter Fourier Series EMMA. et p and q be positive integers. Then we have the following orthogonality relations. sin p d cos p d (.3) sin p cos q d (.4) cos p cos q d sin p sin q d if p q if p q if p q if p q (.5) (.6) We will leave the proof of these identities for the eercises. Computation of the coefficients The orthogonality relations enable us to find the coefficients a n and b n in (.). Suppose we are given a function f that can be epressed as f º» a ½ k Ìa k cos k b k sin kí (.7) on the interval Ì Í. To find a, we simply integrate the series (.7) term by term. Using the orthogonality relation (.3), we see that f º» d a (.8) To find a n for n, we multiply both sides of (.7) by cos n and integrate term by term, getting f º» cos n d a a ½ k ½ k ½ k cos n d a k b k Ìa k cos k b k sin kí cos k cos n d sin k cos n d cos n d (.9) Using the orthogonality relations in emma., we see that all the terms on the right-hand side of (.9) are equal to zero, ecept for a n cos n cos n d a n

4 . Computation of Fourier Series 75 Hence, equation (.9) becomes f º» cos n d a n for n so, including equation (.8), a n f º» cos n d for n (.) To find b n, we multiply equation (.7) by sin n and then integrate. By reasoning similar to the computation of a n, we obtain b n f º» sin n d for n. (.) Definition of Fourier series If f is a piecewise continuous function on the interval Ì Í, we can compute the coefficients a n and b n using (.) and (.). Thus we can define the Fourier series for any such function. DEFINITION. Suppose that f is a piecewise continuous function on the interval Ì Í. With the coefficients computed using (.) and (.), we define the Fourier series associated to f by f º» a ½ n [a n cos n b n sin n] (.3) The finite sum S N º» a N [a n cos n b n sin n] (.4) n is called the partial sum of order N for the Fourier series in (.3). We say that the Fourier series converges at if the sequence of partial sums converges at as N ½ We use the symbol in (.3) because we cannot be sure that the series converges. We will eplore the question of convergence in the net section, and we will see in Theorem.3 that for functions that are minimally well behaved, the can be replaced by an equals sign for most values of. EXAMPE.5 Find the Fourier series associated with the function for, f º» for We used the epression a instead of a for the constant term in the Fourier series (.7) so formulas like equation (.) would be true for n as well as for larger n.

5 76 Chapter Fourier Series We compute the coefficient a using (.8) or (.). We have a f º» d º» d For n, we use (.), and integrate by parts to get a n n f º» cos n d º» d sin n n º» sin n º cos n n» n º» cos n d sin n d Thus, since cos n º» n, the even numbered coefficients are a n, and the odd numbered coefficients are a n ̺n» Í for n We compute b n using (.). Again we integrate by parts to get a n and b n b n n f º» sin n d º» d cos n n º» cos n n n º» sin n d cos n d 3 Figure 3 The Fourier coefficients for the function in Eample.5. n The magnitude of the coefficients is plotted in Figure 3, with a n in black and b n in blue. Notice how the coefficients decay to. The Fourier series for f is f º» 4 ½ n cosºn» ºn» ½ n sin n (.6) n et s eamine the eperimental evidence for convergence of the Fourier series in Eample.5. The partial sums of orders 3, 3, and 3 for the Fourier series in Eample.5 are shown in Figures 4, 5, and 6, respectively. In these figures the function f is plotted in black and the partial sum in blue. The evidence of these figures is that the Fourier series converges to f º», at least away from the discontinuity of the function at

6 . Computation of Fourier Series 77 π π π π π π π π π Figure 4 The partial sum of order 3 for the function in Eample.5. Figure 5 The partial sum of order 3 for the function in Eample.5. Figure 6 The partial sum of order 3 for the function in Eample.5. Fourier series on a more general interval It is very natural to consider functions defined on Ì Í when studying Fourier series because in applications the argument is frequently an angle. However, in other applications (such as heat transfer and the vibrating string) the argument represents a length. In such a case it is more natural to assume that is in an interval of the form Ì Í. It is a matter of a simple change of variable to go from Ì Í to a more general integral. Suppose that f º» is defined for Then the function Fºy» f ºy» is defined for y. For F we have the Fourier series defined in Definition.. Using the formula y, the coefficients a n are given by a n Fºy» cos ny dy y f cos ny dy f º» cos n The formula for b n is derived similarly. Thus equations (.) and (.) are the special case for of the following more general result. d THEOREM.7 If f º» a È ½ n Ìa n cosºn» b n sinºn»í for, then a n b n f º» cos n f º» sin n d for n (.8) d for n (.9) Keep in mind that Theorem.7 only shows that if f can be epressed as a Fourier series, then the coefficients a n and b n must be given by the formulas in (.8) and (.9). The theorem does not say that an arbitrary function can be epanded into a convergent Fourier series.

7 78 Chapter Fourier Series The special case when n in (.8) deserves special attention. Since cos, it says a f º» d Thus a is the average of f over the interval Ì Í We will also etend Definition. to functions defined on the interval Ì Í. DEFINITION. Suppose that f is a piecewise continuous function on the interval Ì Í. With the coefficients computed using (.8) and (.9), we define the Fourier series associated to f by n f º» a ½ n a n cos n bn sin (.) Even and odd functions The computation of the Fourier coefficients can often be facilitated by taking note of the symmetries of the function f. DEFINITION. A function f º» defined on an interval is said to be even if f º» f º» for, and odd if f º» f º» for. The graph of an even function is symmetric about the y-ais as shown in Figure 7. Eamples include f º» and f º» cos. The graph of an odd function is symmetric about the origin as shown in Figure 8. Eamples include f º» 3 and f º» sin. f() f() f( a) = f(a) a a a a f( a) = f(a) Figure 7 The graph of an even function. Figure 8 The graph of an odd function. The following properties follow from the definition. PROPOSITION.3 Suppose that f and g are defined on the interval.. If both f and g are even, then f g is even.

8 . If both f and g are odd, then f g is even. 3. If f is even and g is odd, then f g is odd. 4. If f is even, then f º» d 5. If f is odd, then. Computation of Fourier Series 79 f º» d f º» d We will leave the proof for the eercises. If we remember that the integral of f computes the algebraic area under the graph of f, parts 4 and 5 of Proposition.3 can be seen in Figures 7 and 8. The Fourier coefficients of even and odd functions Parts 4 and 5 of Proposition.3 simplify the computation of the Fourier coefficients of a function that is either even or odd. For eample, if f is even, then, since sinºn» is odd, f º» sinºn» is odd by part 3 of Proposition.3, and by part 5, b n f º» sin n d Consequently, no computations are necessary to find b n. Using similar reasoning, we see that f º» cosºn» is even, and therefore a n f º» cos n d f º» cos n d Frequently integrating from to is simpler than integrating from to. Just the opposite occurs for an odd function. In this case the a n are zero and the b n can be epressed as an integral from to. We will leave this as an eercise. We summarize the preceding discussion in the following theorem. THEOREM.4 Suppose that f is piecewise continuous on the interval Ì Í.. If f º» is an even function, then its associated Fourier series will involve only the cosine terms. That is, f º» a È ½ n a n cosºn» with a n f º» cos n d for n. If f º» is an odd function, then its associated Fourier series will involve only the sine terms. That is, f º» È ½ n b n sinºn» with b n f º» sin n d for n

9 7 Chapter Fourier Series et s look at another eample of a Fourier series. EXAMPE.5 Find the Fourier series associated to the function f º» on. The function f is odd, so according to Theorem.4 its Fourier series will involve only the sine terms. The coefficients are b n n 3 Figure 9 The Fourier coefficients for the function in Eample.5. b n Using integration by parts, we obtain b n cos n n n sinºn» d cos n d º»n n Thus, the Fourier series of f º» on the interval Ì Í is f º» ½ n º» n n sin n (.6) The magnitude of the coefficients is plotted in Figure 9. The a n are not shown, since they are all equal to. The partial sums of orders 5,, and 5 for the Fourier series in (.6) are shown in Figures,, and respectively. In these figures the function f º» is plotted in black and the partial sum in blue. These figures provide evidence that the Fourier series converges to f º», at least on the open interval º». At every term in the series is equal to. Therefore the series converges to at, but not to f º». π π π π π π π π π π π π Figure The partial sum of order 5 for f() =. Figure The partial sum of order for f() =. Figure The partial sum of order 5 for f() =. EXAMPE.7 Compute the Fourier series for the saw-tooth wave f graphed in Figure 3 on the interval Ì Í. The graph in Figure 3 on the interval Ì Í consists of two lines with slope and respectively. The formula for f on the interval is given by if f º» if

10 . Computation of Fourier Series Figure 3 A saw-tooth shaped wave. a n n 3 Figure 4 The Fourier coefficients for the function in Eample.7. EXAMPE.8 The function f is even and is periodic with period. Since f is an even function, we see using Theorem.4 that only the cosine terms appear in the Fourier series, and the coefficients are given by a n º» cosºn» d for n For n we can compute the integral by observation, a º» d For n we use integration by parts to obtain a n Since cos n º» n, we see that Thus we have º» cosºn» d 4 º cos n n» 8 a n and a n ºn» for n f º» 8 ½ n cosººn»» ºn» The magnitude of the coefficients is plotted in Figure 4. The b n are not shown, since they are all equal to. Notice how fast the coefficients decay to,in comparison to those in Figures 3 and 9. The graph of the partial sum of order 3, S 3 º» 8 cos 9 cos 3 is shown in Figure 5. The sum of these two terms gives a pretty accurate approimation of the saw-tooth wave. This reflects the fact that the coefficients decay rapidly, as shown in Figure 4. The partial sum of order 9 is plotted in Figure 6. Notice that the poorest approimation occurs at the corners of the graph of the saw-tooth. This is where the function fails to be differentiable, and these facts are connected. Find the Fourier series of the function f º» sin 3 cos 4 on the interval Ì Í.

11 7 Chapter Fourier Series 3 3 Figure 5 The partial sum of order 3 for the saw-tooth function. 3 3 Figure 6 The partial sum of order 9 for the saw-tooth function. Since f is already given as a sum of sines and cosines, no work is needed. The Fourier series of f is just sin 3 cos 4. This eample illustrates an important point. According to Theorem.7, the Fourier coefficients of a function are uniquely determined by the function. Thus, by inspection, b 3, a 4 and all other coefficients are equal to. By uniqueness, these are the same values as would have been obtained by computing the integrals in Theorem.7 for the a n and b n. EXAMPE.9 Find the Fourier series of the function f º» sin on the interval Ì Í. In this eample, f is not eplicitly given as a linear combination of sines and cosines, so there is some work to do. However, if we use the trigonometric identity sin º cos» the right side is the desired Fourier series, since it is a finite linear combination of terms of the form cos n.... EXERCISES In Eercises 6 epand the given function in a Fourier series valid on the interval. Plot the function and two partial sums of your choice over the interval. Plot the same partial sums over the interval f º» sin. f º»

12 3. f º» 4. f º» sin 5. f º» cos 6. f º» sin. Computation of Fourier Series 73 In Eercises 7 6 find the Fourier series for the indicated function on the indicated interval. Plot the function and two partial sums of your choice over the interval. for, 7. f º» on Ì Í for 8. f º» 4 on Ì Í 9. f º» 3 on Ì Í. f º» sin cos on Ì Í for,. f º» on Ì Í for sin for,. f º» on Ì Í for cos for, 3. f º» on Ì Í for for, 4. f º» on Ì Í for for, 5. f º» on Ì Í for for, 6. f º» on Ì Í for 7. Epand the function f º» in a Fourier series valid on the interval. Plot both f and the partial sum S N for N Observe how the graphs of the partial sums approimate the graph of f. Plot the same graphs over the interval. 8. Epand the function f º» in a Fourier series valid on the interval. Plot both f and the partial sum S N for N Observe how the graphs of the partial sums approimate the graph of f. Plot the same graphs over the interval.

13 74 Chapter Fourier Series In Eercises 9 determine if the function f is even, odd, or neither. 9. f º» sin. f º» 3 3. f º» e. Convergence of Fourier Series. f º» 3. Use the addition formulas for sin and cos to show that cos «cos Ìcosº cosºí sin «sin Ìcosº cosºí sin «cos Ìsinº sinºí 4. Prove emma.. Hint: Use Eercise Complete the derivation of equation. for the coefficient b n 6. Prove parts,, and 3 of Proposition Prove parts 4 and 5 of Proposition Prove part of Theorem From Theorem.4, the Fourier series of an odd function consists only of sineterms. What additional symmetry conditions on f will imply that the sine coefficients with even indices will be zero? Give an eample of a function satisfying this additional condition. 3. Suppose that f is a function which is periodic with period T and differentiable. Show that f ¼ is also periodic with period T. 3. Suppose that f is a function defined on R. Show that there is an odd function f odd and an even function f even such that f º» f odd º» f even º» for all. Suppose that f is a piecewise continuous function on the interval Ì Í and that f º» a ½ n n a n cos b n sin (.) n is its associated Fourier series. Two questions arise immediately whenever an infinite series is encountered. The first question is, does the series converge? The second question arises if the series converges. Can we identify the sum of the series? In particular, does the Fourier series of a function f converge at to f º» or to something else? These are the questions we will address in this section. 3 3 Theorem.7 does not answer this question, since it assumes that f º» equals its Fourier series and then describes what the Fourier coefficients have to be.

14 Fourier Series and periodic functions The partial sums of the Fourier series in (.) have the form n S N º» a N n a n cos. Convergence of Fourier Series 75 bn sin n (.) The function S N º» is a finite linear combination of the trigonometric functions cosºn» and sinºn», each of which is periodic with period. 4 Hence for every N the partial sum S N is a function that is periodic with period. Consequently, if the partial sums converge at each point, the limit function must also be periodic with period et s consider again the function f º», which we treated in Eample.5. f º» is defined for all real numbers, and it is not periodic. We found that its Fourier series on the interval [ ] is ½ n º» n n sin n The partial sums of this series are all periodic with period. Therefore, if the Fourier series converges, the limit function will be periodic with period. Thus the limit cannot be equal to f º» everywhere. The evidence from the graphs of the partial sums in Figures,, and of the previous section indicates that the series does converge to f º» on the interval º». Since the limit must be -periodic, we epect that the limit is closely related to the periodic etension of f º» from the interval º». The situation is illustrated in Figure, which shows the partial sum of order 5 over 3 periods. The periodic etension of f º» is shown plotted in black. Since it will appear repeatedly, let s denote the periodic etension of a function f º» defined on an interval Ì Í by f p º». Usually it is easier to understand the periodic etension of a function graphically than it is to give an understandable formula for it. This is illustrated for f º» in Figure. However, for the record, the formula for the periodic etension 5 is f p º» f º k» for ºk» ºk» You will notice that f p is periodic with period To get a feeling for whether or not the Fourier series of f º» converges to its periodic etension f p, we graph both f p and the partial sum of order in Figure. Note that the graph of S (the blue curve) is close to the graph of f p ecept at the points of discontinuity of f p, which occur at the odd multiples of The accuracy of the approimation of f p º» by S º» gets worse as gets closer 4 We studied periodic functions in Section 5.5, but let s refresh our memory. A function gº» is periodic with period T if gº T» gº» for all. Notice that every integral multiple of a period is also a period. The smallest period of cosºn» and sinºn» is n, so n ºn» is also a period. Thus each function in the partial sum S N in (.) is periodic with period. 5 Notice we use less than () at the lower endpoint of each interval and less than or equal () at the upper endpoint. A choice is necessary to avoid having two values at the endpoints. This is not the only possible choice, but it is as good as any.

15 76 Chapter Fourier Series π 3π π π 3π π Figure The partial sum of order 5 for the series in Eample.5 over three periods. π 3π π π 3π π Figure The partial sum of order for the series in Eample.5 over three periods. to a point of discontinuity. This is necessary, simply because each partial sum is a continuous function, while f p is not. Furthermore, we see that S N º» for all N. Hence the Fourier series converges to at, and not to f p º» The same phenomenon occurs at ºk» for any integer k, and these are the points of discontinuity of f p. We will see these considerations reflected in our convergence theorem. Piecewise continuous functions Suppose that f is a function defined in an interval I. We define the right-hand and left-hand limits of f at a point to be f º» lim f º» and f º» lim f º» The function f is continuous at if and only if both limits eist and f º» f º» f º» In Section 5., we defined a function f to be piecewise continuous if it has only finitely many points of discontinuity in any finite interval, and if both the leftand right-hand limits eist at every point of discontinuity. Thus, for a piecewise continuous function the left- and right-hand limits eist everywhere. You will notice that the periodic etension f p of f º» on the interval Ì Í that we saw in Eample.5 is piecewise continuous. The points of discontinuity for f p are at ºk», the odd integral multiples of. We have f p ºÌºk»Í» and f p ºÌºk»Í». In fact, if f is any function that is continuous on the interval Ì Í, then the periodic etension f p is piece-

16 . Convergence of Fourier Series 77 wise continuous on all of R, and its only possible points of discontinuity are the odd multiples of. We will say that the function f has left-hand derivative at if the left-hand limit f º» eists and the limit lim f º» f º»» eists. Similarly, we will say that f has right-hand derivative at if f º» eists and the limit f º» f º lim» eists. If f is differentiable at, then f has left- and right-hand derivatives there and both are equal to f ¼ º» For an eample, we consider again the periodic etension f p of f º» on the interval Ì Í. The left- and right-hand derivatives eist at every point and are equal to, even at the points of discontinuity. Another eample is the saw-tooth wave in Eample.7. Notice that the saw-tooth function is continuous everywhere, but fails to be differentiable where is an integer. However, at these points the leftand right-hand derivatives of the saw-tooth wave eist. Convergence Since a Fourier series converges to a periodic function, we may as well assume from the beginning that the function f is already periodic. If, as was the case in Eample.5, we are given a function that is not periodic, then it is necessary to look at the periodic etension of the function. We are now in a position to state our main theorem on the convergence of Fourier series. A proof is beyond the scope of this tet, but one can be found in any advanced book on Fourier series. THEOREM.3 Suppose f º» is a piecewise continuous function that is periodic with period. If the left- and right-hand derivatives of f eist at then the Fourier series for f converges at to f º» f º» If f is continuous at, then f º» f º» f º», so assuming that the left- and right-hand derivatives of f eist at, Theorem.3 concludes that the Fourier series converges at to f º». Theorem.3 assumes the eistence of the left- and right-hand derivatives of f only at the point, and concludes that the Fourier series converges only at. This indicates that it is only the smoothness of f near that affects the convergence there. However, in most of the eamples that we will consider, the left- and righthand derivatives will eist everywhere. We will consider this special case in the net corollary.

17 78 Chapter Fourier Series COROARY.4 Suppose f º» is a piecewise continuous function that is periodic with period. Suppose in addition that f has left- and right-hand derivatives at every point.. At every point where f is continuous the Fourier series for f converges to f º».. At every point where f is not continuous the Fourier series for f converges to f º» f º» (.5) EXAMPE.6 We have verified that the hypothesis of Corollary.4 holds for the periodic etension f p of f º» on. Show that the conclusion of Corollary.4 holds at any point of discontinuity. The points of discontinuity are ºk», the odd integral multiples of. We have f p ºÌºk»Í» and f p ºÌºk»Í». Therefore, f p ºÌºk»Í» f p ºÌºk»Í» We have also seen that the Fourier series of f is ½ n º» n n sin n When ºk» every term in the series is equal to. Hence the series converges to, so the conclusion of Theorem.5 is valid at ºk». EXAMPE.7 et f º» on the interval. Without computing the Fourier coefficients, eplicitly describe the sum of the Fourier series of f for all. 6 Of course, f º» is not periodic. Therefore, we must consider its periodic etension, f p, graphed in black in Figure 3. Note that f p is continuous everywhere, and it is differentiable ecept at the odd integers, k. At these points the left- and right-hand derivatives eist. Thus the left- and right-hand derivatives eist everywhere, and Corollary.4 implies that its Fourier series converges to f p º» for all. EXAMPE.8 Consider the function f º» for, for, for Find the Fourier series for f, and describe the sum of its Fourier series. 6 The computation of this series is Eercise 8 in Section.

18 . Convergence of Fourier Series Figure 3 The partial sum S 5 () for the function in Eample.7. Since f is only defined over the interval Ì Í, we are really looking at the Fourier series of its periodic etension f p, shown plotted in black in Figure 4. For obvious reasons, f p is called the square wave. Since f (and f p ) is an odd function, Theorem.4 says that only the sine terms are present in the Fourier series, and that the coefficients are Hence b n f º» sin n d b n and b n The Fourier series associated to f p (and to f ) is f p º» 4 ½ n sin n d 4 ºn» n ̺»n Í sinºn» (.9) n The function f p is piecewise continuous, with discontinuities at all of the integers. In addition, its left- and right-hand derivatives eist everywhere. Thus, the Fourier series converges to f p º» if is not an integer. If k is an integer, the series converges to f p ºk» f p ºk» º» In fact, each term of the Fourier series in (.9) is equal to when k is an integer. The partial sum of the Fourier series of order is shown plotted in Figure 4. Gibb s phenomenon Suppose that the piecewise continuous function f has a discontinuity at, but that the left- and right-hand derivatives of f eist at. As Theorem.3 points out, the Fourier series of f converges at to Ì f º» f º»Í. However, if you look closely at the graphs of the partial sums near the points of discontinuity in Figures,, and 4, we see that the graph of the partial sum overreaches the graph of the function on each side of the discontinuity. This effect is called Gibb s phenomenon. To eamine Gibb s phenomenon a little more deeply, let s look at the graphs of some high order partial sums for the square wave. The partial sums S 3 and S 6

19 73 Chapter Fourier Series 3 3 Figure 4 The partial sum S () for the square wave in Eample Figure 5 The partial sum S 3 () for the square wave. Figure 6 The partial sum S 6 () for the square wave. are plotted in Figures 5 and 6, but only for so that we may see the overrun clearly. Notice that both partial sums display the overrun that is characteristic of Gibb s phenomenon. In the two cases the amount of the overrun is approimately the same, but for the higher order sum the duration is smaller. It can be proved that whenever a function f satisfies the hypotheses of Theorem.3, but has a discontinuity at, the graphs of the partial sums of the Fourier series display Gibb s phenomenon near. Furthermore, the ratio of the length of the interval between the upper peak and the lower peak of the partial sum to f º» f º» is approimately 79 in every case. The Riemann ebesgue emma Notice that in every eample we have considered, the Fourier coefficients approach as the frequency gets large. This is demonstrated, in particular, in Figures 3, 9, and 4 in Section. These eamples are typical of the behavior of Fourier coefficients, as the net theorem, known as the Riemann-ebesgue lemma, shows. THEOREM. Suppose f is a piecewise continuous function on the interval a b. Then lim k½ b a f º» cos k d lim k½ b a f º» sin k d As we will see in Section 5, this theorem has important applications. Basically, this theorem states that any given signal can be approimated very well by a few dominant Fourier coefficients because most of the Fourier coefficients are near zero.

20 . Convergence of Fourier Series 73 The intuitive reason behind this theorem is that as k gets very large, sin k and cos k oscillate much more rapidly than does f (see Figures and in Section ). If k is large, f º» is nearly constant on two adjacent periods of sin k or cos k. The integral over each period is almost zero, since the areas above and below the -ais almost cancel. Interpretation of the Fourier coefficients Suppose that f is a function that satisfies the hypotheses of Corollary.4. Then f º» is equal to its Fourier series f º» a ½ n a n cos n n bn sin (.) ecept at those points where f is not continuous. et s look more closely at the nth summand, which we can rewrite in terms of its amplitude and phase 7 as n f n º» a n cos b n sin n n A n cos n (.) Ô where A n an bn and tan n b n a n We see that f n º», defined in (.), is an oscillation with amplitude A n and frequency 8 n n We will call f n the component of f at frequency n n Notice that n n, where, so all of the frequencies are integer multiples of the fundamental frequency. We can interpret Corollary.4 as saying that any function that satisfies its hypotheses is an infinite linear combination of oscillatory components at frequencies that are integer multiples of the fundamental Ô frequency. The component of f at frequency n has amplitude A n an b n. The amplitude is a numerical measure of the importance of the component in the Fourier epansion. By the Riemann ebesgue lemma, the Fourier coefficients decay to as n increases, so the amplitudes A n do as well. As a result, the components at the smaller frequencies dominate the Fourier series in (.). This fact is illustrated by the plots of the magnitudes of the coefficients in Figures 3, 9, and 4 in Section. Fourier coefficients for periodic functions In this section we have been looking at periodic functions, since it is only such functions that can be the sums of Fourier series. It is worth pointing out that for a periodic function with period, the coefficients can be computed by an integral over any interval of length. More precisely, we have the following: 7 See Section This is an angular frequency. Remember that we are using angular frequencies instead of numerical frequencies unless otherwise stated.

21 73 Chapter Fourier Series PROPOSITION.3 Suppose that f is a piecewise continuous function that is periodic with period. Then for any c the Fourier coefficients for f are given by a n b n c c c c f º» cos n f º» sin n d for n d for n We will leave the proof to the eercises.... EXERCISES In Eercises 6 determine if the function f is periodic or not. If it is periodic, find the smallest positive period.. f º» sin. f º» cos 3 3. f º» 4. f º» sinº» cosº» 5. f º» 6. f º» e In Eercises 7 4 find the sum of the Fourier series for indicated function at every point in R without computing the series. Each of these is an eercise in Section. Although that is not very important, the reference is included in parentheses. 7. f º» on Ì Í (See Eercise.3) 8. f º» on Ì Í (See Eercise.4) sin for 9. f º» on Ì Í (See Eercise.7) for. f º» 4 on Ì Í (See Eercise.8). f º» 3 on Ì Í (See Eercise.9). f º» 3. f º» for on Ì Í (See Eercise.) for sin for, on Ì Í (See Eercise.) for

22 . Convergence of Fourier Series 733 for, 4. f º» on Ì Í (See Eercise.6) for 5. Compute the Fourier series for the function f º» on the interval Ì Í (See Eercise..) Use the result and Theorem.3 to show that ½ n ºn» 6. Compute the Fourier series for the function f º» on the interval Ì Í (See Eample.7.) Use the result and Theorem.3 to show that ½ n º» n n and 8 ½ n n 7. Compute the Fourier series for the function f º» 4 on the interval Ì Í Use the result, Theorem.3, and Eercise 6 to show that ½ n º» n n and ½ n 6 n Epand the function f º» in a Fourier series valid on the interval. Plot the graph of f and the partial sums of order N for N 5 and 4, as in Eercise 7 in Section. Notice how much slower the series converges to f in this eample than in Eercise 7 in Section. What accounts for the slow rate of convergence in this eample? 9. Epand the function f º» e r in a Fourier series valid for. For the case r, plot the partial sums of orders N and 3 of the Fourier series along with the graph of f p over the intervals and.. Use the previous eercise to compute the Fourier coefficients for the function f º» sinh ºe e» and f º» coshº» ºe e» over the interval.. Use Theorem.3 to determine the sum of the Fourier series of the function f defined in Eercise 8 for each in the interval.. Suppose that f is periodic with period T and differntiable. Show that f ¼ is also periodic with period T.

23 734 Chapter Fourier Series 3. Suppose that f is periodic with period T. Show that bt b f º» d at a f º» d for any a and b. Use this result to prove Proposition Suppose that f is periodic with period T. Define Fº» f ºy» dy.3 Fourier Cosine and Sine Series Show that if Ê T f ºy» dy, then F is periodic with period T. (Hint: Use Eercise 3.) In this section we will eamine the possibility of finding Fourier series of the forms and f º» f º» a ½ n ½ b n sin n for n a n cos n for The basic idea behind our method comes from Theorem.4. f e () Fourier cosine series According to Theorem.4, the Fourier series of an even function contains only cosine terms. If the function f º» is defined only for, we etend it to as an even function. The even etension of f is defined by f e º» f º» if, f º» if f() = e Figure The even etension of f() = e. For the function f º» e on the interval Ì Í, the even etension f e is plotted in blue in Figure. Since the function f e is an even function defined on Ì Í, Theorem.4 tells us that its Fourier series has the form f e º» a ½ n a n cos for (3.) n where a n n f e º» cos d for n

24 .3 Fourier Cosine and Sine Series 735 Since f e º» f º» for, this formula becomes a n n f º» cos d for n (3.) Furthermore, if we restrict ourselves to the interval Ì Í, where f e º» f º», we can write f º» a ½ n a n cos for (3.3) n with the coefficients given by (3.). The series in (3.3), with the coefficients given in (3.), is called the Fourier cosine series for f on the interval Ì Í. a n EXAMPE 3.4 Figure The Fourier cosine coefficients for f() = e. n Find the Fourier cosine series for f º» e on the interval Ì Í The coefficients in (3.) become a n e cos n d n º» n e This evaluation can be done by direct computation, by looking the integral up in an integral table, or by using a computer and a symbolic algebra program. The magnitude of these coefficients is plotted in Figure. Notice how quickly the coefficients decay to. The Fourier series is e ºe» ºe» ½ n º» n e n cos n ºe» ºe» cos cos 4 (3.5) on the interval Ì Í The partial sum S 3 º» is plotted in blue in Figure 3. The black curve in Figure 3 is the periodic etension of the even etension of the function f º» e. We will call this the even periodic etension of f, and we will denote it by f ep. Since, in this case, f ep is continuous and satisfies the hypotheses of Corollary.4, the Fourier series converges everywhere to f ep º». f() = e f o () Figure 4 The odd etension of f() = e. Fourier sine series In a similar manner, a function f can be epanded in a series which involves only sine terms. Again motivated by Theorem.4, we consider the odd etension of f, which is defined by f o º» f º» f º» if if if The odd etension of f º» e is plotted in blue in Figure 4. Since the function f o

25 736 Chapter Fourier Series 3 3 Figure 3 The partial sum S 3 of the Fourier cosine series for f() = e plotted over three periods. is an odd function defined on Ì Í, Theorem.4 tells us that its Fourier series has only sine terms. Proceeding as before, we find that f º» ½ n b n sin n for (3.6) where b n n f º» sin d for n (3.7) The series in (3.6), with the coefficients given in (3.7), is called the Fourier sine series for f on the interval Ì Í. EXAMPE 3.8 Find the Fourier sine series for f º» e on the interval Ì Í The coefficients in (3.7) become b n e sin n d nì º»n eí n This evaluation can be done by direct computation, by looking the integral up in an integral table, or by using a computer and a symbolic algebra program. Thus we have ½ e nì º» n eí sin n n n ºe» sin 4ºe» 6ºe» sin 4 sin 9 3 (3.9) 3 3 Figure 6 The partial sum S 3 of the Fourier sine series for f() = e plotted over three periods.

26 .3 Fourier Cosine and Sine Series 737 b n on the interval Ì Í The magnitude of the coefficients is plotted in Figure 5. Notice that the sine coefficients do not decay nearly as rapidly as do the cosine coefficients in Figure. The partial sum of order 3 is plotted in blue in Figure 6. It is interesting to compare this figure with Figure 3. The black curve in Figure 6 is the odd periodic etension of f º» e, which we denote by f op. In this case f op satisfies the hypotheses of Corollary.4 and fails to be continuous only at the integers. Consequently, the Fourier sine series converges to f op º» everywhere ecept at the integers. Figure 5 The Fourier sine coefficients for f() = e. n Note that while we used the even and odd etensions of f ( f e and f o ) to help derive the cosine and sine epansions, the formulas for a n and b n involve only the function f on the interval Ì Í. This is reflected by the fact that the cosine and sine epansions converge to f only on º». Outside this interval the cosine and sine epansions converge to f ep and f op, respectively. Eamples 3.4 and 3.8 illustrate these facts, but another eample might help to put things into perspective. EXAMPE 3. Find the complete Fourier series for f º» e on the interval Ì Í From (.) we have a n while from (.) we have e cos n d º» n e e n for n a n and b n Figure 7 The coefficients of the complete Fourier series for f() = e. n b n e n nºe e» sin n d º» n for n. Again there are several ways to verify this. Hence the complete Fourier series for e on Ì Í is µ e e ½ e º» n n [cos n n sin n ] (3.) n The magnitude of the coefficients is plotted in Figure 7, with a n in black and b n in blue. The partial sum of order 3 is plotted in blue Figure 8. The periodic etension of e satisfies the hypotheses of Corollary.4 and fails to be continuous only at the odd integers. Consequently, the Fourier series converges to the periodic etension everywhere ecept at the odd integers. Now we have three Fourier series that converge to f º» e on the interval º». The first in (3.5) contains only cosine terms. The second in (3.9) contains only sine terms. The third in (3.) contains both sine and cosine terms. It is interesting to compare graphs of the partial sums in Figures 3, 6, and 8. The difference between the three is what happens outside of the interval º». The cosine series converges to f ep, the even periodic etension of f. The sine series converges to f op, the odd periodic etension of f, ecept at the integers. And, finally, the full Fourier series converges to f p, the periodic etension of f, ecept at the odd integers. Of course, the same three series can be considered for any piecewise continuous function defined on an interval of the form Ì Í.

27 738 Chapter Fourier Series 3 3 Figure 8 The partial sum S 3 of the complete Fourier series for f() = e plotted over three periods.... EXERCISES In eercises 4 sketch the graph of the odd etension of the function f. Also sketch the graph of the odd periodic etension with period over three periods.. f º». f º» 3. f º» 4. f º» In eercises 5 8 sketch the graph of the even etension of the function f. Also sketch the graph of the even periodic etension with period over three periods. 5. f º» 6. f º» 7. f º» 8. f º» In Eercises 9 epand the given function in a Fourier cosine series valid on the interval. Plot the function and two partial sums of your choice over the interval. Plot the same partial sums and the function the series converges to over the interval f º». f º» sin. f º» cos. f º» 3. f º» 4. f º» 5. f º» 3 6. f º» 4

28 7. f º» 8. f º».3 Fourier Cosine and Sine Series f º» cos. f º» sin In Eercises 3 epand the given function in a Fourier sine series valid on the interval. Plot the function and two partial sums of your choice over the interval. Plot the same partial sums and the function the series converges to over the interval Same as Eercise 9. Same as Eercise 3. Same as Eercise 4. Same as Eercise 5. Same as Eercise 3 6. Same as Eercise 4 7. Same as Eercise 5 8. Same as Eercise 6 9. Same as Eercise 7 3. Same as Eercise 8 3. Same as Eercise 9 3. Same as Eercise 33. Show that the functions cosºn», n are orthogonal on the interval Ì Í. This means that Hint: Use Eercise 3. cosºn» cosºp» d if p n 34. Show that the functions sinºn», n 3 are orthogonal on the interval Ì Í. This means that Hint: Use Eercise Show that sinºn» sinºp» d if p n cosºn» sinºk» d

29 74 Chapter Fourier Series Hint: Use Eercise 3..4 The Comple Form of a Fourier Series 36. If f º» is continuous on the interval, show that its even periodic etension is continuous everywhere. Does this statement hold for the odd periodic etension? What additional condition(s) is (are) necessary to ensure that the odd periodic etension is everywhere continuous? If the piecewise continuous function f is periodic with period, then its Fourier series is f º» a ½ n n a n cos bn sin (4.) n where the coefficients are given by a n n f º» cos b n n f º» sin d d for n and for n (4.) Sometimes it is useful to epress the Fourier series in comple form using the comple eponentials, e in for n. This is possible because of the close connection between the comple eponentials and the trigonometric functions. We eplored this connection in the appendi to this book and in Section 4.3. The most important facts to know about the comple eponential are Euler s formula e iy cos y i sin y (4.3) which defines the eponential, and that all of the familiar properties of the real eponential remain true for the comple eponential. If we write down Euler s formula with y replaced by y, we get Solving (4.3) and (4.4) for cos y and sin y, we see that cos y eiy e iy e iy cos y i sin y (4.4) and sin y eiy e iy (4.5) i et s substitute these epressions into the Fourier series (4.). The nth term in the sum is the component of f at the frequency n n, and it becomes n n f n º» a n cos b n sin a n e ein in b n i a n ib n e in a n ib n e «n e in «ne in e in e in in (4.6)

30 .4 The Comple Form of a Fourier Series 74 where we have substituted «n a n ib n and «n a n ib n for n. (4.7) We will also write the constant term as f º» «a Separating the positive and negative terms, the Fourier series can be written as f º» ½ n ½ «n e in (4.8) Notice that by (4.6), the component of f at frequency n n is given by f n º» «n e i n «ne i n As a result, when we talk in terms of low frequency components we have to consider the coefficients «n and «n for small values of n. We can use (4.) to epress the coefficients «n in terms of the function f. For eample, for n we have «n a n ib n n f º» cos f º»e in d n i sin d The corresponding formulas for n and for n can be computed in the same way, and we discover that «n f º»e in d for all n. (4.9) It is important to notice that while «n is the coefficient of e in in the Fourier series (4.8), it is e in which appears in the integral in (4.9). The series (4.8), with the coefficients computed using (4.9), is called the comple Fourier series for the function f. There are several differences between the Fourier series involving cosines and sines, given in Definition., and the Fourier series using comple eponentials presented here. First, the comple Fourier series involves a sum from n ½ to n ½, rather than a sum from n to n ½. Net, for the comple Fourier series, there is one succinct formula (4.9) for the Fourier coefficients, rather than the two separate formulas for a n and b n in (.8) and (.9). For this reason, and also because computations using eponentials are easier than those using trigonometric functions, many scientists and engineers prefer to use the comple version of the Fourier series. EXAMPE 4. Find the comple Fourier series for the function f º» e on the interval Ì Í This is the function we eamined in Eamples 3.4, 3.8, and 3.. For this function it is much easier to compute the comple Fourier coefficients than the real

31 74 Chapter Fourier Series ones. The nth coefficient is α n «n e e in d e º in» d e in e in º in» n º»n ºe e» º in» Figure The coefficients of the comple Fourier series for f() = e. The last identity follows since e in e in º» n The magnitude of the coefficients is plotted in Figure. Notice that we included negative indices. The comple Fourier series is e e e ½ n ½ º» n in ein for Relation between the real and comple Fourier series We derived the comple Fourier series from the real series. In doing so we found that the comple coefficients can be computed from the real coefficients using (4.7). In turn, we can solve these relationships for the real coefficients in terms of the comple coefficients, getting a «a n «n «n and b n i º«n «n» for n (4.) These equations simplify somewhat if the function f is real valued. In that case f º» f º» so «n f º»e in d f º»e in d «n f º» e in d Consequently, if f is real valued, a n «n «n Re «n and b n i º«n «n» Im «n EXAMPE 4. Compute the coefficients of the real Fourier series for the function f º» e on the interval Ì Í We computed the comple coefficients in Eample 4. and found that «n º»n ºe e» º in»

32 .5 The Discrete Fourier Transform and the FFT 743 Since the function is real valued, we can use (4.) to find that a n Re «n º»n ºe e» n and b n Im «n º»n nºe e» n... EXERCISES. Show that the comple Fourier coefficients for an even, real-valued function are real. Show that the comple Fourier coefficients for an odd, real-valued function are purely imaginary (i.e., their real parts are zero). In Eercises find the comple Fourier series for the given function on the interval Ì Í.. f º» 3. f º» 4. f º» 5. f º» 6. f º» 7. f º» e b 8. f º» 3 9. f º». f º» cos. f º» sin. Two comple Ê valued function f and g are said to be orthogonal on the interval Ìa bí if b a f º»gº» d Show that The functions eip and e iq are orthogonal on Ì Í if p and q are different integers. 3. Use the method of proof of Theorem.7, and Eercise to show that if f º» È ½ n ½ «ne in for, then «n f º»e in d

33 744 Chapter Fourier Series.5 The Discrete Fourier Transform and the FFT Suppose that f ºt» is piecewise continuous for t. 9 Then f ºt» ½ k ½ «k e ikt where «k f ºt»e ikt dt (5.) The Fourier coefficients «k are often too difficult to compute eactly. In such a case it is useful to approimate the coefficients using a numerical integration technique such as the trapezoid rule. We remind you that for a function Ê F defined on the interval Ì Í, the trapezoid rule for approimating the integral Fºt» dt with step size h N is Fºt» dt h Fº» Fºh» Fºh» FººN»h» FºNh» If Fºt» is -periodic, then Fº» Fº» FºNh», and the preceding formula becomes et s set Fºt» dt h N j Fº jh» N N j Fº jn» Applying this formula to the integral for the Fourier coefficient in (5.), we get «k f ºt»e ikt dt N N j y j f º jn» and Û e in Then Û e in, and the approimation becomes «k N N j f º jn»e i jkn (5.) y j Û jk (5.3) The sum on the right side of equation (5.3) involves the discrete values y j f º jn». The values of f ºt» for t jn are ignored. It is a common occurrence in the digital age to replace a time dependent function with such a discrete sample of that function. For eample, f ºt» may represent a music signal that we want to transmit over the internet. The internet, or any other computer network, allows only discrete signals, so to transmit the music we replace the continuous signal f ºt» with the discrete sample y j f º jn» for j N In many digital applications signals arise that are not represented by a continuous function at all. Instead, they arise as discrete values y j at a discrete set of times t j. Such a signal is illustrated in Figure. Here, the horizontal ais represents time, which has been divided into many small time intervals. 9 In most applications of the material in this section, the function f represents a time dependent signal. Consequently, we will use t instead of as the independent variable.

34 .5 The Discrete Fourier Transform and the FFT 745 t The discrete Fourier transform Even for discrete signals such as that illustrated in Figure, it is often useful to consider the transform we found in equation (5.3). We will assume that the signal is an infinite sequence y y j ½ j ½ that is periodic with period N, meaning that y N j y j for all j. Wherever the sequence y k comes from, the transform on the right-hand side of (5.3) is important. Figure A discrete signal. DEFINITION 5.4 et y y j be a sequence of comple numbers that is periodic with period N. The discrete Fourier transform of y is the sequence y y k, where y k N j y j e ikj N N j y j Û jk for ½ k ½ (5.5) For the last epression in (5.5) we use the notation Û e in, so that Û e in. An important property of Û e in is that Û N e i Of course, it follows that Û N. From this we see that y kn N j y j Û j ºkN» N j y j Û jk Û j N N j y j Û jk y k Thus the discrete Fourier transform is also periodic of period N. et s look back at equation (5.3), where we used the trapezoid rule to approimate «k, the kth Fourier coefficient of the function f. Using (5.5), we can now write (5.3) as «k y k N (5.6) It follows from the Riemann ebesgue lemma that «k as k ½. On the other hand, the sequence y k is periodic. This implies that (5.3) is not a good approimation for large k. In fact, The trapezoid rule algorithm used to approimate the integral in (5.) loses accuracy as the integrand f ºt»e ikt becomes more oscillatory as the frequency (and inde) k increases. Therefore we would epect equation (5.3) to provide a good approimation only for k that are relatively small compared to N. There is another, related factor to consider. We have previously talked of the importance of the low frequency components of a function. When we use the comple Fourier series, this means that we include both «k and «k for small values of k. By (5.6) and the periodicity of the sequencey, «k y k N y N k N Therefore, when considering small frequency components while using the discrete Fourier transform, we must include both y k and y N k for small nonnegative values of the inde k.