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Engineering Mathematics Class # 15 Fourier Series, Integrals, and Transforms (Part 3)

Engineering Mathematics Class # 15 Fourier Series, Integrals, and Transforms (Part 3). Sheng-Fang Huang. 11.7 Fourier Integral. Consider the periodic rectangular wave ƒ L ( x ) of period 2 L > 2 given by

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Engineering Mathematics Class # 15 Fourier Series, Integrals, and Transforms (Part 3)

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  1. Engineering Mathematics Class #15Fourier Series, Integrals, and Transforms (Part 3) Sheng-Fang Huang

  2. 11.7 Fourier Integral • Consider the periodic rectangular wave ƒL(x) of period 2L > 2 given by The left part of Fig. 277 shows this function for 2L = 4, 8, … as well as the nonperiodic function ƒ(x), which we obtain from ƒLif we let L → ∞,

  3. Amplitude Spectrum • Consider the Fourier coefficients of ƒLas L increases. Since ƒLis even, bn = 0 for all n. For an, • This sequence of Fourier coefficients is called the amplitude spectrum of ƒLbecause ｜an｜ is the maximum amplitude of the ancos (nπx/L).

  4. Fig. 277. Waveforms and amplitude spectra

  5. (See Fig. 277) For increasing L these amplitudes become more and more dense on the positive wn-axis, where wn =nπ/L. • For 2L = 4, 8, 16 we have 1, 3, 7 amplitudes per “half-wave” of the function (2 sin wn)/(Lwn). • Hence, for 2L = 2kwe have 2k-1 – 1 amplitudes per half-wave. • These amplitudes will eventually be everywhere dense on the positive wn-axis (and will decrease to zero).

  6. From Fourier Series to Fourier Integral • Consider any periodic function ƒL(x) of period 2L that is represented by a Fourier series • what happens if we let L → ∞? • We should expect an integral (instead of a series) involving cos wx and sin wx with w no longer restricted to integer multiples w =wn =nπ/L of π/L but taking all values.

  7. If we insert anand bn, and denote the variable of integration by v, the Fourier series of ƒL(x) becomes We now set

  8. Then 1/L =Δw/π, and we may write the Fourier series in the form (1) • Let L → ∞ and assume that the resulting nonperiodic function is absolutely integrable on the x-axis; that is, the following limits exist:

  9. 1/L → 0, and the value of the first term on the right side of (1) → zero. Also Δw =π/L →dw. The infinite series in (1) becomes an integral from 0 to ∞, which represents ƒ(x), namely, (3) If we introduce the notations (4)

  10. Fourier integral we can write this in the form (5) This is called a representation of ƒ(x) by a Fourier integral.

  11. THEOREM 1 Fourier Integral If ƒ(x) is piecewise continuous in every finite interval and has a right-hand derivative and a left-hand derivative at every point and if the integral exists, then ƒ(x) can be represented by a Fourier integral with A and B given by (4). At a point where ƒ(x) is discontinuous the value of the Fourier integral equals the average of the left- and right-hand limits of ƒ(x) at that point. Fourier Integral

  12. Applications of Fourier IntegralsExample 2: Single Pulse, Sine Integral • Find the Fourier integral representation of the function

  13. Solution.

  14. Sine Integral • The case x = 0 is of particular interest. If x = 0, then (7) gives (8*) We see that this integral is the limit of the so-called sine integral (8) as u → ∞. The graphs of Si(u) and of the integrand are shown in Fig. 279.

  15. Fig. 279. Sine integral Si(u) and integrand

  16. In the case of the Fourier integral, approximations are obtained by replacing ∞ by numbers a. Hence the integral (9) which approximates ƒ(x).

  17. Gibbs Phenomenon • We might expect that these oscillations disappear as a → ∞. However, with increasing a, they are shifted closer to the points x = ±1. • This unexpected behavior is known as the Gibbs phenomenon.

  18. Fourier Cosine Integral and Fourier Sine Integral • If ƒ(x) is an even function, then B(w) = 0 and (10) The Fourier integral (5) then reduces to the Fourier cosine integral

  19. Fourier Cosine Integral and Fourier Sine Integral • If ƒ(x) is an odd function, then A(w) = 0 and (12) The Fourier integral (5) then reduces to the Fourier cosine integral

  20. 11.8 Fourier Cosine and Sine TransformsFourier Cosine Transform • For an even function ƒ(x), the Fourier integral is the Fourier cosine integral (1) We now set A(w)= , where c suggests “cosine.” Then from (1b), writing v =x, we have (2) and from (1a), (3)

  21. Fourier Sine Transform • Similarly, for an odd function ƒ(x), the Fourier integral is the Fourier sine integral (4) We now set B(w)= , where s suggests “sine.” From (4b), writing v =x, we have (5) This is called the Fourier sine transform of ƒ(x). From (4a) (6)

  22. Fourier Sine Transform • Equation (6) is called the inverse Fourier sine transform of . The process of obtaining from ƒ(x) is also called the Fourier sine transform or the Fourier sine transform method. • Other notations are and and for the inverses of and , respectively.

  23. Example 1: Fourier Cosine and Fourier Sine Transforms • Find the Fourier cosine and Fourier sine transforms of the function Solution:

  24. Example 2: Fourier Cosine Transform of the Exponential Function • Find Solution.

  25. Linearity, Transforms of Derivatives • The Fourier cosine and sine transforms are linear operations, (7)

  26. THEOREM 1 Let ƒ(x) be continuous and absolutely integrable on the x-axis, let ƒ'(x) be piecewise continuous on every finite interval, and let let ƒ(x) → 0 as x → ∞. Then (8) Cosine and Sine Transforms of Derivatives

  27. Formula (8a) with ƒ' instead of ƒ gives (when ƒ', ƒ'' satisfy the respective assumptions for ƒ, ƒ' in Theorem 1) hence by (8b) (9a) Similarly, (9b)

  28. Example 3:An Application of the Operational Formula (9) • Find the Fourier cosine transform (e-ax) of ƒ(x) = e-ax, where a > 0. Solution.

  29. 11.9 Fourier Transform. Discrete and Fast Fourier Transforms • The complex Fourier integral is (4)

  30. Fourier Transform and Its Inverse • Writing the exponential function in (4) as a product of exponential functions, we have (5) The expression in brackets is a function of w, is denoted by , and is called the Fourier transform of ƒ; writing v =x, we have (6)

  31. With this, (5) becomes (7) and is called the inverse Fourier transform of . • Another notationfor the Fourier transform is so that • The process of obtaining the Fourier transform (ƒ) =from a given ƒ is also called the Fourier transform or the Fourier transform method.

  32. Example 1: Fourier Transform • Find the Fourier transform of ƒ(x) = 1 if ︱x︱ < 1 and ƒ(x) = 0 otherwise. • Solution.Using (6) and integrating, we obtain As in (3) we have eiw = cos w +i sin w, e-iw = cos w –i sin w, and by subtraction eiw –e-iw = 2i sin w. Substituting this in the previous formula, we see that i drops out and we obtain the answer

  33. Example 2: Fourier Transform • Find the Fourier transform (e-ax) of ƒ(x) = e-axif x > 0 and ƒ(x) = 0 if x < 0; here a > 0. Solution.

  34. THEOREM 2 Linearity of the Fourier Transform The Fourier transform is a linear operation; that is, for any functions ƒ(x) and g(x) whose Fourier transforms exist and any constants a and b, the Fourier transform of aƒ + bg exists, and (8) Linearity. Fourier Transform of Derivatives

  35. THEOREM 3 Fourier Transform of the Derivative of ƒ(x) Let ƒ(x) be continuous on the x-axis and ƒ(x) →0 as ︱x︱→ ∞. Furthermore, let ƒ'(x) be absolutely integrable on the x-axis. Then (9)

  36. Convolution • The convolution ƒ * g of functions ƒ and g is defined by (11) Taking the convolution of two functions and then taking the transform of the convolution is the same as multiplying the transforms of these functions (and multiplying them by ):

  37. THEOREM 4 Convolution Theorem Suppose that ƒ(x) and g(x) are piecewise continuous, bounded, and absolutely integrable on the x-axis. Then (12) Convolution Theorem

  38. Convolution Theorem • By taking the inverse Fourier transform on both sides of (12), writing and as before, and noting that and 1/ in (12) and (7) cancel each other, we obtain (13)

  39. Discrete Fourier Transform (DFT) • The function f(x) is given only in terms of values at finitely many points. • Dealing with sampled values, we can replace Fourier transform by the so-called discrete Fourier transform. • Let f(x) be periodic with the period 2π. Assume N measurements are taken over the interval 0≦ x≦ 2π at regular spaced points

  40. Discrete Fourier Transform (DFT) • We now to determine a complex trigonometric polynomial that interpolates f(x) at the nodes. That is, • Hence, we must determine the coefficients c0, …, cN-1 • Multiply by and sum over k from 0 to N-1 Denote […] by r. For n = m, r = e0 = 1. The sum of these terms over k equals N.

  41. Discrete Fourier Transform (DFT) • For n≠m we have r ≠1 and by the formula for a geometric sum: Because This shows that the right side of (17) equals cmN. Thus, we obtain the desired coefficient formula:

  42. Discrete Fourier Transform (DFT) • It is practical to drop the factor 1/N from cn and define the discrete Fourier transform of the given signal to be the vector with components • This is the frequency spectrum of the signal.

  43. Fourier Matrix • In vector notation, , where the N×N Fourier matrix FN=[enk] has the entries [given in (18)]:

  44. Example 4 • Let N = 4 measurements (sample values) be given. Then w =e-2πi/N =e-πi/2 = –i and thus wnk = (–i)nk. Let the sample values be, say f = [0 1 4 9]T. Then by (18) and (19), (20)

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