Here we have the polar (or magnitude-phase) form of FT/ W(f) /=[X2(f)+Y2(f)]1/2:magnitude spectrume(f)=arctg[Y(f)/X(f)]:phase spectrumInverse Fourier transform:w(t)=F-1[W(f]= - W(f)exp[j2元ft]dfEx. Spectrum of an exponential pulse:w(t)=e-t, t>0W(f)=0J e-t exp[j2元ft]dt=1/(1+j2元f)X(f)=Y(f)=/ W(f) / =[X2(f)+Y2(f)]1/2, 0(f)=arctg[Y(f)/X(f)]
Here we have the polar (or magnitude-phase) form of FT: │W(f)│=[X2 (f)+Y2 (f)]1/2:magnitude spectrum θ(f)=arctg[Y(f)/X(f)]:phase spectrum Inverse Fourier transform: w(t)=F -1 [W(f)]= -∞∫ ∞W(f)exp[j2πft]df Ex. Spectrum of an exponential pulse: w(t)=e-t , t>o W(f)=0 ∫ ∞ e -t exp[j2πft]dt=1/(1+j2πf) X(f)= Y(f)= │W(f)│=[X2 (f)+Y2 (f)]1/2, θ(f)=arctg[Y(f)/X(f)]
Properties of Fourier TransformsTheorem:Spectral symmetry of real signalsW(t) is real, then W(-f)=W*(f)Proof. See textDeduction: / w(-f) |= / W(f) | :The magnitude spectrum iseven function of f0(-f)= - 0(f): the phase spectrum is oddSummary: f,frequency (Hz),an FT's parameter that specifies w(t)'sinterested frequency.FT looks for frequency f in w(t) over all timeW(f) is complex in generalw(t) real,then W(-f)-W*(f)
• Theorem:Spectral symmetry of real signals. W(t) is real, then W(-f)=W* (f) Proof. See text Deduction: │W(-f)│=│W(f)│:The magnitude spectrum is even function of f θ(-f)= - θ(f): the phase spectrum is odd Summary: • f,frequency (Hz),an FT’s parameter that specifies w(t)’s interested frequency. • FT looks for frequency f in w(t) over all time • W(f) is complex in general • w(t) real,then W(-f)=W* (f) Properties of Fourier Transforms
Parseval's theoremParseval's theorem wi(t) w2*(t) dt-.. Wi(f) W2*(f)dfif wi(t) =w2(t) =w(t),we haveE= - ow2(t)dt- Jo0/ w(f) / 2dfProof: directly from FTEnergy spectral density(ESD): Definition:The ESD is defined for energy waveforms by:E(f)= / W(f) |2 J/HzBy using Parseval's theorem we haveE=-JE(f)df
• Parseval’s theorem: -∞∫ ∞w1(t) w2 * (t) dt=-∞∫ ∞ W1(f) W2 * (f)df if w1(t) =w2(t) =w(t),we have E= -∞∫ ∞w2 (t)dt=-∞∫ ∞ │W(f)│2df Proof: directly from FT Energy spectral density(ESD) • Definition:The ESD is defined for energy waveforms by: E(f)= │W(f)│2 J/Hz By using Parseval’s theorem we have E =-∞∫ ∞ E(f)df Parseval’s theorem
Power spectral density(PSD): For power waveforms,we have a similar function calledPSD.(see later)Another properties of FT:W(f) is real if w(t) is even. W(f) is complex if w(t) is oddWe have some basic and important FT's theorems at page50Most important theorems:W(f)e-j oTd:w(t-Ta)time delayfrequency translation:+/2[ W(f-f)ejo+W(f+ f.)e-jo]w(t)cos(wct+0)
• For power waveforms,we have a similar function called PSD.(see later) Another properties of FT: • W(f) is real if w(t) is even • W(f) is complex if w(t) is odd We have some basic and important FT’s theorems at page50. Most important theorems: time delay :w(t-Td) W(f)e-j ωTd frequency translation: w(t)cos(ωc t+θ) 1/2[ W(f-fc)ejθ+W(f+ fc)e-jθ] Power spectral density(PSD)
Convolution:Wi(t)*w2(t)Wi(f)W2(f)j2元fW(f)Differentiator: dw(t)/dtIntegrator: - Jtw(t)dt(f)/ (j2元f)+1/2W(0)8(f)Freguency translation:(w(t) is real)F[w(t)coso,t]W(f)ffwe can use the FT's definition to prove these theorem.(Homeworks)
Convolution:w1(t)*w2(t) W1(f)W2(f) Differentiator: dw(t)/dt j2πfW(f) Integrator: -∞∫ tw(t)dt W(f)/ (j2πf)+1/2W(0)δ(f) Frequency translation:(w(t) is real) we can use the FT’s definition to prove these theorem.(Home works) W(f) f f F[w(t)cosωct] fc