Chapter 10. Classical Fourier Series

Transcription

1 Math 344, Male ab Manual Chater : Classical Fourier Series Real and Comle Chater. Classical Fourier Series Fourier Series in PS K, Classical Fourier Series in PS K, are aroimations obtained using orthogonal rojections to subsaces of the form San cos n w, sin n w n = where w = =. That is, to San, cos w, sin w,..., cos w, sin w, w = /. Because cos n w, sin n w n = is an orthogonal family, see the last age in Chater 9, and K d =, K cos n d = K sin n d =, the rojection of f is S = a C n > a n cos n w C b n sin n w, where = a n = K f cos n w d and b n = f sin n w d. K ote that the constant term a is the average value of f on K,. Eamle. et f d iecewise!,, / : in PS K,, lotted below. lot f, =K.. =. K With these definitions, d : w d / :, the n th Fourier coefficients are an, bn d f cos n w d, f sin n w d K K sin n n C sin n n C cos n K n, cos n K age 54 n K cos n n K sin n n Telling Male to assume that n is an integer yields the following simlified formulas for an and bn. simlify an, bn assuming n T integer = K C K n n, K n Observe in outut () that the constant a is indeterminate. By insection of f 's grah, its average value is C / = 3 4, so a = 3. This value can also be obtained as the limit of an as n/. a d lim n / an = 3 Imortant: This works only if an has not been simlified. ()

2 Math 344, Male ab Manual Chater : Classical Fourier Series Real and Comle The net entry defines the th order aroimation as the function S,. S, d a C n > an cos n w C bn sin n w : = Dislayed below is the order 3 aroimation, followed by its grah, along with the grah of f. S 3, = 3 4 K cos K sin K sin K 9 cos 3 K 3 sin 3 lot f, S 3,, =K.. =. K As increases the aroimations get better; see f and the order 6, 9, and 8 aroimations below. lot f, S 6,, S 9,, S 8,, =K...8. K The overshoot and undershoot in the aroimations at a jum discontinuity are called Gibbs Phenomenon. Each one is roughly % of the jum. There's no Gibbs henomenon at the endoints here, because the eriodic etension of f is continuous at those oints. See the order 3 aroimation on K, below. lot S 3,, =K..,....5 K K8 K6 K4 K The eriodic etension of f is very closely aroimated by S 3,. The overshoot and undershoot are still about % of the jum. The Amlitude and Power Sectra of a Periodic Signal The (one-sided) Amlitude Sectrum of the eriodic signal dislayed above is the lot of lines of length A n = a n C b n, the amlitude of the n th harmonic: a n cos n w C b n sin n w, in its Fourier series reresentation. The n th line is lotted at the frequency n w. A line of length a / is lotted at the origin. An d simlify an C bn assuming n T integer = n C C K Cn n. Aines d,,, a /, n w,, n w, An $ n =.. : age 55

3 Math 344, Male ab Manual Chater : Classical Fourier Series Real and Comle lot Aines, =..6, style = line, thickness = 3, tickmarks = sacing w The eriodic signal's one-sided Amlitude Sectrum. The signal's (one-sided) Power Sectrum is the lot of lines of length A n and a /. Pines d,,, a /, n w,, n w, An $ n =.. : lot Pines, =..6, style = line, thickness = 3, tickmarks = sacing w The Comle Fourier Series The comle Fourier series > The signal's one-sided Power Sectrum. It shows that ractically all of the signal's ower is carried by the constant term and its first 3 harmonics. That is, by S 3,, lotted above for K % %. n =K c n e i n w for f can be obtained from the real series by substituting cos n w = ei n w C e Ki n w and sin n w = ei n w Ke Ki n w i, and then simlifying. (To enter the comle constant K tye i [esc] [return].) The family e i n w n =K to the comle inner roduct IP u, v = of comle-valued functions is also orthogonal in PS K,, but with resect K u$v d. Therefore, the coefficient c n can be obtained with the integral formula: c n = f e Ki n w d. Just one formula works for all n, ositive, negative, or. K Alying this to the function in Eamle, age 56

4 Math 344, Male ab Manual Chater : Classical Fourier Series Real and Comle cn d f e Ki n w I ei n K d = K K n Assuming that n is an integer, this simlifies as follows. simlify cn assuming n T integer C K C I n C K n n I n K e I n C e KI n n. To get cn into standard comle form aly the evalc rocedure to outut (), assuming that n is an integer. K C K n I evalc () assuming n T integer = C n n ote. In outut, Male denotes i as I (a caital i). If you wish, I can also be used in Male inut. The real Fourier coefficients, an and bn, are times the real art of cn and K times the imaginary art of cn resectively. Comare outut (3) below to the simlified eressions for an and bn following outut (). The symbol R for real art is entered by tying Re [esc] [return]. To get I, for imaginary art, tye Im [esc] [return]. evalc R cn, evalc K I cn : simlify % assuming n T integer () K C K n n, K n (3) The Two-Sided Amlitude Sectrum The two-sided Amlitude Sectrum for the eriodic etension of f is the lot of a line of height c n at the oint n w for n =, G,G,.... For n = use c = a /. TSAines d,,, a /, n w,, n w, cn $ n =.., n w,, n w, cn $ n =K..K : lot TSAines, =K6..6, style = line, thickness = 3, tickmarks = sacing w K5 K4 K3 K K K The two-sided amlitude sectrum for the eriodic etension of f. With the ecetion of the center line, all lines are half the height of the lines in the one-sided sectrum. age 57

5 Math 344, Male ab Manual Chater : Classical Fourier Series Real and Comle Chater Procedures Math Entry (Eamle) Fourier Aroimations in PS a, b et g be in PS K,, g d iecewise!,k,k / : Then d : w d / : and an, bn d K g cos n w d, K g sin n w d : a d lim an = The grah of g confirms the constant is a/ =. n / lot g, =K.. =.5 K.5 K.5 The real coefficients can be simlified like this. g ave = simlify an, bn assuming n T integer : an, bn d o % K C K n 3 K n, K n K Calculation Define the th Fourier aroimation. S, d a C n > an cos n w C = bn sin n w : Dislay S,, then lot g and S 5,. S, = K cos K sin lot g, S 5,, =K.. C. K K..5 K sin The Comle Fourier Series The comle coefficients are cn d g e Ki n w d : K Simlify like this. cn d evalc simlify cn assuming n T integer K K K n C 4 I K n C 3 K n n 4 n K n Get the real coefficients from cn as follows. simlify evalc R cn, evalc K I cn assuming n T integer K C K n 3 K n, n K n S,, as comuted directly from comle series: a C n > R cn e i n w : evalc % = K cos K sin C sin Sectral Analysis The eriodic etension of g looks like this. Gibbs haens. lot S 3,, =K5..5, tickmarks = sacing K5 K4 K3 K K The one-sided amlitude sectrum for the etension follows. An d an C bn : Aines d,,, a/, n w,, n w, An $n =.. : lot Aines, =..3, style = line There is no line at = The one-sided ower sectrum is derived net. Pines d,,, a/, n w,, n w, An $n =.. : lot Pines, =..3, style = line o line at = S 5, carries almost all of the ower in the signal. lot S 5,, =K5..5, tickmarks = sacing.8 K5 K4 K3 K K age 58