Dirac delta function and unit step function: Dirac delta function is very useful (perhaps the mostuseful)in communication system's analysisDefinition: S(x) is defined byJw(x)8(x) dx=w(0)where w(x) is any function that is continuous at x-0Or we can equally define the (x) as:.J∞8(x) dx=1ands(x)=o0 when x=08(x)=0 when x#0So we can use two delta functions’ definitions withoutdifference
• Dirac delta function is very useful (perhaps the most useful)in communication system’s analysis. • Definition: δ(x) is defined by -∞∫ ∞w(x)δ(x) dx=w(0) where w(x) is any function that is continuous at x=0. Or we can equally define the δ(x) as: -∞∫ ∞δ(x) dx=1 and δ(x)=∞ when x=0 δ(x)=0 when x≠0 So we can use two delta functions’ definitions without difference. Dirac delta function and unit step function
Delta function's properties The sifting property: Jw(x)8(x-xo) dx=w(xo).An useful delta function's expression:8(x) =- Je±j2xy dyProof: we have delta function's FTJe-j2元ft 8(t) dt=e0=1and take the inverse Fourier transform of above equation.then8(t) =- Je+j2rft df: (x) is even: (x) = S(-x)
• The sifting property: -∞∫ ∞w(x)δ(x-x0) dx=w(x0) • An useful delta function’s expression: δ(x) =-∞∫ ∞e±j2πxy dy Proof: we have delta function’s FT: -∞∫ ∞e -j2πft δ(t) dt=e0=1 and take the inverse Fourier transform of above equation, then δ(t) =-∞∫ ∞e +j2πft df • δ(x) is even: δ(x) = δ(-x) Delta function’s properties
Unit step function:closely related with o(x)Definition: u(t) is defined by:u(t)=1 for t>0 and u(t)=0 for t<0Jo 8(x) dx=u(x)Properties:anddu(t)/dt= S(t)Ex. Spectrum of a sinusoidv(t)=Asin0ot, o=2元fofrom FT ,we have:V(f) = -Jo(A/2j)(ej2元fot - e-j2fot )e-j2nft dt= j(A/2)[8(f+fo) -8(f-fo)]/ V(f) / =(A/2) 8(f-fo)+ (A/2) 8(f+fo),0(f)=元/2 for f>0 and 0(f)= -元/2 for f<0
• Unit step function:closely related with δ(x) • Definition: u(t) is defined by: u(t)=1 for t>0 and u(t)=0 for t<0 Properties: -∞∫ ∞ δ(x) dx=u(x) and du(t)/dt= δ(t) Ex. Spectrum of a sinusoid v(t)=Asinω0t, ω0=2πf0 from FT ,we have: V(f) = -∞∫ ∞(A/2j)(ej2πf 0 t - e -j2πf 0 t )e-j2πft dt = j(A/2)[δ(f+f0) -δ(f-f0)] │V(f)│=(A/2) δ(f-f0)+ (A/2) δ(f+f0), θ(f)=π/2 for f>0 and θ(f)= -π/2 for f<0
/ v(f) /e(f)A/2fffo- foMagnitude spectrumPhase spectrum
│V(f)│ f - f0 f0 A/2 f θ(f) Magnitude spectrum Phase spectrum
Conclusion:A sinusoid waveform has mathematically twofrequency components( at f-±fo) and his magnitudespectrum is a line spectraRectangular pulse:Definition:The rectangular pulse II() is defined byII(t/T)=1 for / t|≤T/2II(t/T)=0 for I t|≥T/2Definition:The function sinc() is defined bysinc(x)=(sinx)/(πx)or the function Sa () is difined bySa (x)=sinc(x/元)=sin x/xTwo very important functions in digital communicationsystem's analysis
• Conclusion:A sinusoid waveform has mathematically two frequency components( at f=±f0) and his magnitude spectrum is a line spectra. Rectangular pulse: • Definition:The rectangular pulse Π(·) is defined by: Π(t/T)=1 for │t│≤T/2 Π(t/T)=0 for │t│≥T/2 • Definition:The function sinc(·) is defined by: sinc(x)=(sinπx)/(πx) or the function Sa (·) is difined by Sa (x)=sinc(x/π)=sin x/x Two very important functions in digital communication system’s analysis