transform summary

This is the Title PhL 3.26.05 This has been added as Appendix E to spectral doc. Appendix E: Table of Transforms Fourier Integral Transform: x(t) aperiodic and continuous, X(ω) continuous, no image spectra X(ω) = !Syntax Error, Idt x(t) e-iωt projection = transform (1.1) x(t) = (1/2π)!Syntax Error, Idω X(ω) e+iωt expansion = inverse transform (1.2) Generalized Fourier Integral Transform: recovery contour passes below all singularities of X(ω) X(ω) = !Syntax Error, Idt x(t) e-iωt projection = transform (6.4) x(t) = (1/2π) !Syntax Error, Idω X(ω) e+iωt expansion = inverse transform (6.5) Fourier Cosine Transform Xc(ω) = 2 !Syntax Error, Idt x(t) cos(ωt) projection = transform x(t) = (1/π)!Syntax Error, Idω Xc(ω) cos(ωt) expansion = inverse transform (1.7) Fourier Sine Transform Xs(ω) = 2 !Syntax Error, Idt x(t) sin(ωt) projection = transform x(t) = (1/π)!Syntax Error, Idω Xs(ω) sin(ωt) expansion = inverse transform (1.8) Fourier Integral Transform in Appendix C notation: k is an arbitrary convention constant x^(ω) = k!Syntax Error, Idt x(t) e-iωt projection = transform x(t) = (1/2πk) !Syntax Error, Idω x^(ω) e+iωt expansion = inverse transform f^(ω) = k!Syntax Error, Idu f(u) e-iωu Fourier transform of f(u) f^-1(t) = (1/2πk) !Syntax Error, Idu f(u) e+iut inverse Fourier transform of f(u) (C.1) Fourier Series Transform: x(t) periodic and continuous with period T1, spectrum is discrete x(t) = !Syntax Error, I xpulse(t - nT1) (14.1) Complex form: cm ≡ (1/T1) !Syntax Error, I dt xpulse(t) e-imωt = (1/T1) !Syntax Error, I dt x(t) e-imωt (14.16) x(t) =!Syntax Error, Icm e+imωt (15.1) cm = c(mω1) = (1/T1)Xpulse(mω1) (14.8) and (14.10) Real form: am ≡(2/T1) !Syntax Error, Idt xpulse(t) cos(mω1t) = (2/T1) !Syntax Error, I dt x(t) cos(mω1t) (15.6) bm ≡ (2/T1) !Syntax Error, Idt xpulse(t) sin(mω1t) = (2/T1) !Syntax Error, I dt x(t) sin(mω1t) (15.7) x(t) = a0/2 + !Syntax Error, I am cos(mω1t) + !Syntax Error, I bm sin(mω1t) (15.9) Laplace Transform: Right-sided (causal) x(t) vanishes for t < 0, x(t) and X(s) are continuous X(s) = !Syntax Error, Idt x(t) e-st projection = transform (6.9) x(t) = (1/2πi) !Syntax Error, Ids X(s) e+is expansion = inverse transform (6.10) Relation to the Fourier Integral Transform: X(s) = X(s/i) X(ω) = X(iω) (6.8) Digital Fourier Transform: x(tn) aperiodic, spectrum X'(ω) is continuous and contains image spectra X'(ω) ≡ !Syntax Error, I∆t x(tn) e-iωt projection = transform (22.2) x(tn) = !Syntax Error, Idω X'(ω) e+iωt expansion = inversion (22.4) or X'(ω) ≡ T1!Syntax Error, Ixn e-iωnT projection = transform xn = x(tn). xn = !Syntax Error, Idω X'(ω) e+iωnT expansion = inversion where: T1 = ∆t ω1 = 2π/T1 = 2π/∆t tn = n ∆t = n T1 . Relation to the Fourier Integral Transform: X'(ω) = !Syntax Error, IX(ω - mω1) = [ X(ω) + !Syntax Error, IX(ω - mω1) ] // image spectra (23.1) Z Transform: x(tn) is aperiodic, spectrum X"(z) is continuous and contains image spectra X"(z) = !Syntax Error, Ixn z-n projection = transform (24.3) xn = !Syntax Error, Idz X"(z) zn-1 expansion = inversion (24.4) where contour C goes once counterclockwise around the unit circle in the z-plane. Relation to the Digital Fourier Transform and the Fourier Integral Transform: X"(z) ≡ X'(ω) = !Syntax Error, IX(ω - mω1) z = eiωT Discrete Fourier Transform of a Pulse Train : x(tn) periodic, spectrum discrete, m integer x(tm) = !Syntax Error, I xpulse(tm - nT1) sampled pulse train, tm = (m/N)T1 c'm ≡ (1/N) !Syntax Error, Ixpulse(tn) e-imn(2π/N) projection = transform (27.9) x(tn) = !Syntax Error, Ic'm e+imn(2π/N) expansion = inverse transform (27.11), (27.12) Discrete Fourier Transform : A is an arbitrary convention constant c'm ≡ (A/N) !Syntax Error, Ixn e-imn(2π/N) m = 0,1...N-1 projection = transform xn = (1/A) !Syntax Error, Ic'm e+imn(2π/N) n = 0,1,...N-1 expansion = inverse transform (27.22) Hilbert Transform: Xh(ω) = (1/π) dω' projection = transform X(ω) = - (1/π) dω' expansion = inverse transform (C.41)