Sheng-Fang Huang

11.1 Fourier Series Fourier series are the basic tool for representing periodic functions. A function ƒ(x) is called a periodic function if there exists some positive number p, called a period of ƒ(x), such that (1) ƒ(x + p) = ƒ(x) for all real x.

Properties of Periodic Functions Periodic: constant, cosine, sine functions. Non-periodic: x, x 2, x 3, e x, cosh x, and ln x, etc. If ƒ(x) has period p, it also has the period np for any integer n = 1, 2, 3, ‥‥, (2) ƒ(x + np) = ƒ(x) for all x. If ƒ(x) and g(x) have period p, then aƒ(x) + bg(x) with any constants a and b also has the period p.

Fourier Series Suppose that ƒ(x) is a given function of period 2π and is such that it can be represented by a series y, that converges and, moreover, has the sum ƒ(x). Then, using the equality sign, we write (5) and call (5) the Fourier series of ƒ(x).

Fig Cosine and sine functions with the period 2π

a 0, a n, and b n are the called Fourier coefficients of ƒ(x), given by the Euler formulas (6) 480

Derivation of the Euler Formulas Orthogonality of the Trigonometric System (3) THEOREM 1 The trigonometric system (3) is orthogonal on the interval –π ≤ x ≤ π (hence also on 0 ≤ x ≤ 2π or any other interval of length 2π because of periodicity); that is, the integral of the product of any two functions in (3) over that interval is 0, so that for any integers n and m, (9)

Derivation of the Euler Formulas Integrating on both sides of (5) from - π to π ： Multiply (5) on both sides by cosmx with any fixed positive integer m

Example 1 Periodic Rectangular Wave Find the Fourier coefficients of the periodic function ƒ(x) in Fig. 257a. The formula is (7) Solution

Fig Eample 1

The Convergence of Fourier Series The partial sums are Fig. 257 indicates the series is convergent. As the number of terms is increased, the graph gradually approaches the shape of the original square waveform. At x = 0 and x = π, all partial sums have the value zero, which is the arithmetic mean of the limits –k and k of our function.

11.2 Functions of Any Period p = 2L Change of scale: giving a function f(x) of period 2L from a function g(v) of period 2π. From Sec. 11.1, we have

Change of Scale Let f(x) = g(v) and v = kx. Because 2π=k2L, k=π/L. Thus, v=πx/L.

Coefficients of Any Period p = 2L Replace v by πx/L to obtain the Fourier series of the function ƒ(x) of period 2L (5) (6)

Example 1 Periodic Rectangular Wave Find the Fourier series of the function

Solution. From (6a) we obtain a 0 = k/2. From (6b) we obtain Thus a n = 0 if n is even and From (6c) we find that b n = 0 for n = 1, 2, ‥‥. Hence the Fourier series is

Example 2 Periodic Rectangular Wave Find the Fourier series of the function

Solution.

Example 3 Half-Wave Rectifier A sinusoidal voltage E sin ωt, where t is time, is passed through a half-wave rectifier that clips the negative portion of the wave. Find the Fourier series of the resulting periodic function

Solution. Since u = 0 when –L < t < 0, we obtain from (6a), with t instead of x, From (6b), by using formula (11) in App. A3.1 with x = ωt and y = nωt, If n = 1, the integral on the right is zero, and if n = 2, 3, ‥‥, we readily obtain

If n is odd, this is equal to zero, and for even n we have In a similar fashion we find from (6c) that b 1 = E/2 and b n = 0 for n = 2, 3, ‥‥. Consequently,