Appendix A.l Fourier Series [Benedetto 1996 B, Bracewell1986 B, Carslaw 1930 B] A .1.1 General Rules period: x(t) = x(t + T) basic circular frequency: OJ OJ 2n / T T = 2n / OJ = 2n / OJ1 = 1 = complex Fourier series: complex coefficients: m=oo 1 T x(t) = x(t+ T) = IJmexp(-imOJt) Fm = - f x(t) exp(+imOJt)dt m=-oo T o real periodic functions: coefficients: x(t) = x(t+ T) = x*(t) Fo = Fo* = llo = Ao Fm =F~m =~(am +ibm) =.!. ~exp(iam} 2 real Fourier series I: real coefficients: x(t) = x(t+ T) = x*(t) 1 T 00 00 ao = - f x(t)dt T o =ao+ Lamcos(mOJt) + Lbmsin(mOJt) m=1 m=1 2 T am = - f x(t)cos(mOJt)dt T o 2 T bm =-f x(t)sin(mOJt)dt T o real Fourier series TI: real parameters: x(t) = x{t+ T) = x*{t) Ao =ao 00 [2 2]112 = Ao + L~cos(mOJt-am} Am = am +bm ~O m=1 am = arc tan(bm / am} 480 real even periodic functions: coefficients: x{t) = x{t + T) = x * (t) = x{ -t) Fo = Fo* = ao = Ao 1 1 Fm =F-m =Fm *=-a 2 m =-A2 m um =O;bm =0 real odd periodie functions: coefficients: x{t) = x{t+ T) = x*{t) = -x{-t) Fo =ao =Ao = 0 Fm =-F-m =-Fm *=..!..ib 2 m um = ±tr 12;am = 0 bm = Am sign( Um) A .1. 2 Real Periodie Functions Norrnalized parameters: Cü =Cü, = 1 and T =2tr a) Square wave: x{t) = x{t + 2tr) = sign(sin t) = i[sint + ..!..sin 3t + ..!..sin 5t + ... ] tr 3 5 • • 1 -T .2'Tr -2T .1T t -1 b) Asymmetrie sawtooth: x{t)=x{t+2tr)=xltr for _tr:s;x+tr=~(sint _ sin2t + sin3t _+ ... ) tr 1 2 3 481 e) Symmetrie sawtooth: 2X / n for - n / 2 ~ x ~ + n / 2 } x{t) = x{t + 2n) = { 2 - (2x / n) for + n /2 ~ x ~ +3n /2 =-28(.1'3 smt-zsm t+zsm1 '5 t-+ ... ) n 3 5 I d) Modulus of sine: 2 4 (COS2t cos4t cos6t ) x{t)=x{t+2n)= Ismtl=--- . --+--+--+ ... n n 1·3 3·5 5·7 I e) Reetified sine: . ). 1 1. 2 (cos 2 t cos 4 t cos 6 t ) x{t)=x{t+2n)=H( smt ·smt=-+-smt-- --+--+--+... n 2 n 1·3 3·5 5·7 482 f) Rectangle: I für Itl < 'C I 2} x( t) = x( t + T) = II( t I 'C) = {1/ 2 for Itl = 'C I 2 o für Itl > 'C I 2 =~+3.. fm-1sin(m'C/2)cosmt with 'C<2n 2n n m;1 x g) Simple Fourier se ries a) Sums valid für 0 < t < 2n: ~LJ m -I.sm mt = -1 ( n - t ) m;1 2 fm-1cos mt = -..!.Cn[2(1- cost)] m;1 2 483 ß) Sums valid for r2 < 1: i rmsinmt = rSint[ 1-2reost+ r2 rl m=\ I m-Irmsinmt = are tan{r sint(1- reos trI} m=1 = -I ~>meos mt = (1- reos t)[ 1-2reos t + r2 ] m=1 Im-1rmeos mt = -2. en{ 1-2reost+ r2 } m=l 2 484 A • 2 Fourier Transformation [Benedetto 1996 B, Bracewell 1986 B, Campbell & Foster 1948 B, Carslaw 1930 B, Champeney 1973 B, Erdelyi et al. 1954 B, Poularikas 1995 B, Zayed 1996 B] A .2. 1 General Rules function: Fourier transforrn: 1 +~ +~ x( t) = - f F( w) exp( -irot )dw F(w) = f x(t)exp(+irot)dt 27r _~ -~ real function: x(t) = x *(t) F( -w) = F * (w) even function: 1 ~ ~ x(t) = x( -t) = - f F( w) cos wt· dw F( w) = F( - w) = 2 f x( t) cos wt . dt 7r o 0 odd function: .~ ~ x(t) = -x(-tl = -~ f F( w) sin rot· dw F( w) = -F(-w) = 2i f x(t) sin rot· dt 7r o 0 x(-t) F(-w) x(t / -r) -r F(w-r) x(t- -r) exp(iw-r) F(w) t n x(t) (_i)n d n F( w) / dwn co rIx(t) i f F( w') dw' -~ dnx(t)/ dtn (-iw)n F(w) t f x(t')dt' (i / w)F(w) -~ product: convolution: +~ 27r Xl (t) X2 (t) f FI(Q)F2(w-Q)dQ -~ convolution: product: +~ f XI( -r)X2(t- -r)d-r FI (w ) F2(w) -~ autocorrelation: power spectrum: +~ Wiener-Khintchine theorem x(-r)x(-r+t)d-r f IF(w)12 ~ x(t / 1")cosQt ~[F( 1"(w +Q)) + F( 1"(w - Q))] 2 x(t / -r)sinQt ;i [F( 1"(w + Q)) - F( 1"(w -Q))] 485 A. 2.2 Real Functions funetion: Fourier transforrn: I +- +- x( t) = - f F( OJ) exp( -iOJt )dOJ F(OJ) = x(t)exp(+iOJt)dt 27r f I -- 27rÖ(OJ)-- cosnt 7r[ Ö(OJ - n) + Ö(OJ + n)] sinn-r i7r[ ö( OJ - n) - ö( OJ + n)] Dirae delta function Ö: ö(t) I ö(t --r) exp(iOJ'Z') dßö('Z')/ dtß [-iOJ t sign function: 2i - signt OJ Heavyside unit step H: i H(t) = .!.[I + signt] 7rÖ(OJ)+- 2 OJ H(t) exp(-at) [a -/OJ. r l rectangle function ll: fO'IX 1<l/2} ll(t)= r1/2forlxl=1/2 sinc( OJ / 27r) = ~sin( OJ / 2) OJ o for lxi> 1/ 2 triangle function A: { 1 -lxi for lxi :5 1} A(t) = sinc2(OJ / 27r) = ~sin2 (OJ / 2) o for lxi ~ I OJ sin 7rt sinct=-- ll( OJ / 27r) 7rt sinc2t A(OJ / 27r) 2i sin2( OJ / 2) ~ll(t-~) - ll(t+~) OJ ll(t)cos7rt -1[.sznc (OJ--- I) +sznc. (OJ-+- 1)] 2 27r 2 27r 2 ll(t)sin27rt ~ [sinc( 2: -1) - sinc( 2: + 1) ] exp{ -(t / 'Z')2} 7r1/2 'Z' exp{ -( OJ'Z' / 2)2 } t exp( -7rt2) i(27r)-1 OJexp{ _OJ 2 / 47r} Itl- 1I2 127r / OJII/2 486 1I2 Itl- sign t I'1 2 n I m1" 2slgnm . exp(-Itl) 2[I+ m2 t exp( -I tl) sign t inm[l+ m2t exp( -Itl)sinc(x In) arctan(21 ( 2) exp(-t)H(t) [1-imrl t·exp(-t)H(t) [1-imr2 sech 1rX = [cosh 1rX t sech( m I 2) = [cosh ( ml 2) t 2 sech21rX = [cosh 1rX r m[ n sinh ( m I 2) t tanht i[ sinh ( m / 2) t ll(t)[ 1- 4t2] ;2 [( m / 2r l sin( m 12) - cos( m / 2)] nSinc(2:)+ (~}inc(2: -1) ll(t)cos2m + n sinc( ~+ 1) 2 2n ln[ 1+ Cl' / t)2 ] 2n m-I [1- exp( -mr)] 2 2 ln[ r +t ] 2n m-I [exp( -mT) - exp( -mr)] T2 +t2 2[t2 + r2t (n / r)exp[ -Tim!] Bessel function Jo: Jo(t / r) 2ll( mr 12)[ r-2 - m2t /2 Bessel functions: }zn: Chebyshev polynomials T2n : 2 ll(mr/2)T2n (mr) J2n (t/r) [-2r -m2f2 Bessel functions Jm: Legendre polynomials Pn: r Il2J dt) in(2ntl/2 ·ll(m/2)·Pn(m) n+-2 Legendre polynomials Pn: Bessel functions Jm: ll(t/2)Pn(t) in (2n Im )112 J I (m) n+-2 Chebyshev polynomials Tn: Bessel functions Jn: ll(t / 2) Tn (t) ninJn(m) [ I-t2f2 487 A • 3 Laplace Transformation [Abramowitz & Stegun 1965 B, Bracewell 1986 B, Doetsch 1970 B, Doetsch 1971/73 B, Erdelyi et al. 1954 B, Pöschl 1956 B, Poularikas 1995 B, Zayed 1996 B] A.3.1 General Rules For the definition of the convolution see (3.2 - 24a&b). If F(t) and F(t) are continuous at t = 0, then F(t = +0) and F(t = +0) can be replaced by F(O) and F(O). ~ L{F(t)} = f F(t)exp(-pt)dt o L{ aFI (t) + bF2 (t)} = aF1(p)+bF2 (p) L{F(at)} = F(p/ a) L{H(t-to) F(t-to)} = F(p) exp( -top), to > 0 L{(-tt F(t)} = dnF(p)/ dpn L{ exp( +Pot) F(t)} = F(p- Po) L{F(t)} = pF(p)-F(t=+o) L{F(t)} = p2F(p)-pF(t=+O)-F(t=+o) LU F(t)dt} = p-1F(p) T L{F(t) = F(t+T)} = [1- e-Tp r f F(t)e-tPdt, F(t) periodic o L{F1(t)* F2 (t)} = F1 (p) F2 (p), convolution where F1(t < 0) = F2 (t < 0) = 0 488 A. 3.2 Heaviside and Dirac Functions This list eontains Laplaee transforms of the Heaviside step funetion H(t), the Dirae delta funetion &,.t) and its derivatives 8n)(t). In these equations t = +0 indieates a time just after t =O. L{c5(t-to)} exp( -to p), to > 0 L{ :r: c5(t - to)} = L{ c5(n)(t - to)} = pn exp( -top) L{ c5(t - (+Ü))} =1 L{:r: c5(t-{+Ü))} =L{c5(n)(t-{+ü))} L{c5{t)} =112 L{H(t-to)} =p-1exp(-top), to>O L{H{t)} =l/p L{~{ _1)n H(t - ~nr)} 489 A.3.3 Real Functions L{(;::)!} L{exp( -at)} =(p+arl L{texp(-at)} =(p+ar2 L{ (;::)! exp (-at)} =(p+ar o L{ exp( -at) - exp( -bt)} = [(p+a)(p+b)t b-a L{sin at} = a[p2 + a2 t L{eos at} = p[p2 +a2t L{ sin at exp( -bt)} =a [(p+b) 2+a 2]-1 L{eos at exp( -bt)} =(p+b)[(p+b)2 +a2( L{sinh at} =a[p2 - a2 rl L{eosh at} = p[p2 -a2t Il2 -1/2 L{( 1ftr } =p 0 -(o+.!.) L{ 4 n! t(O-1I2)} =p 2 (2n)!-fii L{Jo(at)} =[p2 +a2r1/2 L{t sin at} = 2ap[p2 + a2 r2 L{t eos at} =(p2 _a2)[p2 +a2t L{t-Isin at} = are tan ( a / p) L{sin2( ~)} =(a2 /2p)[p2 + a2 t L{lsinatl} = a[p2 + a2 t tanh(2a /rep) L{+fn t} =-p-I[fnp-C] C=O.5772 ..