MA 201: Differentiation and Integration of Fourier Series Applications of Fourier Series Lecture - 10

Transcription

1 MA 201: Differentiation and Integration of Fourier Series Applications of Fourier Series ecture - 10

2 Fourier Series: Orthogonal Sets We begin our treatment with some observations: For m,n = 1,2,3,... cos nπx dx = 0 & sin mπx sin mπx cos mπx sin nπx dx = 0, (1) nπx cos dx = 0 { (2) nπx sin dx = 0, m n, m = n (3) cos nπx dx = { 0, m n, m = n (4) Setting x = t, observe that {1,cos nπx nπx,sin is orthogonal on [0, 2]. : n = 1, 2 3,...},

3 Thus, if we have f(x) = a { a n cos nπx +b nsin nπx }, x [0,2] then all coefficients a n and b n can be determined as a n = 1 b n = f(x)cos nπx dx, n = 0, 1, 2,... and (5) f(x)sin nπx dx, n = 1, 2,... (6)

4 Fourier Series in [0, 2] The infinite series with a n = 1 b n = 1 a [ a n cos nπx +b nsin nπx ], (7) f(x)cos nπx dx, n = 0, 1, 2,... and (8) f(x)sin nπx dx, n = 1, 2,... (9) is called Fourier series (FS) of f(x) in [0,2]. Remark The Fourier series of a function is defined whenever the integrals in (8) and (9) have meaning.

5 Fourier Series in [c,c +2] The infinite series with a n = 1 b n = 1 a [ a n cos nπx +b nsin nπx ], (10) c+2 c c+2 c f(x)cos nπx dx, n = 0, 1, 2,... and (11) f(x)sin nπx dx, n = 1, 2,... (12) is called Fourier series (FS) of f(x) in [c,c +2]. Remark The Fourier series of a function is defined whenever the integrals in (11) and (12) have meaning.

6 Gibbs Phenomenon This is about how the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. It is named after the American mathematical physicist J. W Gibbs. Consider the following periodic function whose definition in one period is: { 1, 0 < x < 1, f(x) = (13) 0, 1 < x < 2. This function can be represented as f(x) = sin(2n 1)πx, x?. (14) π 2n 1 To see how well this infinite series represents the function, let us truncate the series after N terms. et the sum of these first N terms of the infinite series be denoted by S N : S N = π N sin(2n 1)πx. (15) 2n 1

7 Gibbs Phenomenon Graphs of equation (15) are shown for N = 2,4,8,16 and 32. In the figure the overshoot (overshoot is the occurrence of a signal or function exceeding its target) at x = 1 and the undershoot at x = 1 + are characteristics of Fourier series at the points of discontinuity. This phenomenon is known as Gibbs phenomenon This phenomenon persists even though a large number of terms are considered in the partial sum. In the approximation of functions, overshoot/undershoot is one term describing quality of approximation. Here, the convergence is in point-wise sense.

8 Gibbs Phenomenon Figure : Gibbs Phenomenon with 2 terms

9 Gibbs Phenomenon Figure : Gibbs Phenomenon with 4 terms

10 Gibbs Phenomenon Figure : Gibbs Phenomenon with 8 terms

11 Gibbs Phenomenon Figure : Gibbs Phenomenon with 16 terms

12 Gibbs Phenomenon Figure : Gibbs Phenomenon with 32 terms

13 Convergence of Fourier series for piecewise continuous functions Definition et f(t) : [,] R be a piecewise C 1 -function. Define the adjusted function g(t) as follows: g(t) = 1 2 [f(t+ )+f(t )], < t <, 1 2 [f( + )+f( )], t = ±. (16) Note: The above definition tells that g(t) coincides with f(t) at all points in (,), where f(t) is continuous, but g(t) is the average of the left-hand and right-hand limits of f(t) at points of discontinuity in (,).

14 Convergence of Fourier series for piecewise continuous functions Theorem et f : [,] R be a piecewise C 1 function and let g(t) be the adjusted function as defined in (16). Then Fourier series of f(t) = g(t), for all t [,]. Note: The existence of Fourier series depends on the evaluation of Fourier coefficients. On the other hand, convergence of the Fourier series is done via adjusted function g. Thus, a highly oscillating function may be decomposed as sum of a (possibly infinite) set of simple oscillating trigonometric functions. A discontinuous function is approximated by a continuous function.

15 Applications of Fourier Series Square wave-high frequencies One application of Fourier series, the analysis of a square wave in terms of its Fourier components, occurs in electronic circuits designed to handle sharply rising pulses. Physically, square wave contains many high-frequency components. Figure : Square wave

16 Suppose that our modified square wave is defined by { 0, π < x < 0, f(x) = h, 0 < x < π. (17) We can easily calculate the Fourier coefficients to be So, the resulting series is g is the adjusted function. A 0 = h, A n = 0, n = 1,2,3,..., (18) { 2h B n = nπ, n odd, 0, n even. (19) g(x) = h 2 + 2h ( sinx + sin3x + sin5x ) +, (20) π 1 3 5

17 Figure : Square wave

18 for Fourier Series There is a very useful identity obtained from Fourier series which has found applications in electrical engineering. But before knowing Parseval s identity, we need the following theorem which is more general: Theorem If f(t) and g(t) are continuous in (,), and provided f(t) 2 dt < and g(t) 2 dt <, and if A n,b n are Fourier coefficients of f(t) and C n,d n are Fourier coefficients of g(t), then f(t)g(t)dt = 2 A 0C 0 + (A n C n +B n D n ). Proof: We can express f(t) and g(t) in terms of Fourier series as f(t) = A [ A n cos nπt +B nsin nπt ], g(t) = C [ C n cos nπt +D nsin nπt ].

19 for Fourier Series Taking product of f(t) with g(t) we obtain f(t)g(t) = A 0 2 g(t)+ [ A n g(t)cos nπt +B n g(t)sin nπt ]. Integrating this series from to gives f(t)g(t)dt = A [ 0 g(t)dt+ A n 2 Putting back the values of the Fourier coefficients C n and D n : 1 f(t)g(t)dt = A 0C g(t)cos nπt ] dt +B n g(t)sin nπt dt [A n C n +B n D n ].

20 for Fourier Series Theorem () If f(t) is continuous in the range (,) and is square integrable (i.e. f(t) 2 dt < ) and has Fourier coefficients A n and B n, then 1 [f(t)] 2 dt = A [A 2 n +Bn]. 2 This result can obtained easily from the previous theorem by taking g(t) = f(t). The left-hand side represents the mean square value of f(t). It can, therefore, be thought of in terms of energy if f(t) represents a signal. What Parseval s theorem states therefore is that the energy of a signal expressed as a waveform is proportional to the sum of the squares of its Fourier coefficients.

21 for Fourier Series Parseval s identity can be used to determine the power delivered by an electric current, I(t), flowing under a voltage, E(t), through a resistor of resistance R: P = EI = RI 2. In most applications I(t) is a periodic function. Average Power = P av = 1 2 RI 2 (t) dt = R I 2 (t) dt 2 [ ] A 2 0 = R (A 2 n +B 2 2 n), Here we have made use of the Fourier expansion of I(t): I(t) = A [ A n cos nπt +B nsin nπt ].

22 for Fourier Series Mean square of the current is: I av = 1 I 2 (t)dt = A (A 2 n +B2 n ). Root mean square of the current is: I rms = A (A 2 2 n +Bn). 2

23 Application of Example: Given the Fourier series t 2 = π deduce that 1 n 4 = π4 90. Here A 0 = 2π2 3, A n = 4( 1)n n 2, B n = 0. Parseval s identity is so that here = π. We get giving us 1 Hence the result follows. [f(t)] 2 dt = A [A 2 n +B2 n ]. 1 π t 4 dx = 2π4 π π 2π 4 5 = 2π n 4 ( 1) n n 2 cosnt, π < t < π, 16 n 4

24 Finite Vibrating String Problem The IBVP under consideration consists of the following: The governing equation: u tt = c 2 u xx, (x,t) (0,) (0, ). (21) The boundary conditions for all t > 0: u(0,t) = 0, u(,t) = 0. (22) The initial conditions for 0 x are u(x,0) = φ(x), u t (x,0) = ψ(x). (23)

25 Bernoulli s Solution In [0,π], Bernoulli gave the solution of (21) as a series of the form u = b 1 sinx cosct +b 2 sin2xcos2ct +... (24) When t = 0, we should have φ(x) = b 1 sinx +b 2 sin2x +... (25) This is possible and can be treated as the Fourier sine series of φ(x), and which converges to φ(x). Few Facts: As we are expecting u(x,t) as a solution of (21), term-wise differentiation of the series given by (24) should exist. In other sense, the limit function u of the infinite series (24) should be differentiable. Term-wise differentiation of an infinite series is not always possible. In fact, infinite sum of continuous functions may have discontinuous limit x x2 (1+x = ) 2 x 2.

26 Differentiation and integration of Fourier series The term-by-term differentiation of a Fourier series is not always permissible. Example Recall that Fourier series for f(x) = x, π < x < π is ( 1) n+1sinnx, n which converges to f(x) for all x ( π,π), that is x = ( 1) n+1sinnx. n Term-by-term differentiation leads to 1 = ( 1) n+1 cosnx, x ( π,π). which fails at x = 0. In fact, the RHS series diverges for all x(?)

27 Differentiation of Fourier series Theorem (Differentiation of Fourier series) et f(x) : R R be continuous and f(x +2) = f(x). et f (x) and f (x) be piecewise continuous on [,]. Then, The Fourier series of f (x) can be obtained from the Fourier series for f(x) by termwise differentiation. In particular, if f(x) = A { A n cos nπx +B nsin nπx }, then f (x) = nπ { A n sin nπx +B ncos nπx }.

28 Integration of Fourier series Termwise integration of a Fourier series is permissible under much weaker conditions. Theorem (Integration of Fourier series) et f(x) : [,] R be piecewise continuous function with Fourier series f(x) = A Then, for any x [,], we have { A n cos nπx +B nsin nπx }. x f(τ)dτ = x A 0 2 dτ + x { A n cos nπτ +B nsin nπτ } dτ.