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Chap.2 Signals and Spectra

Chap.2 Signals and Spectra. Object: mathematical tools to describe and analyze the signals Fourier series and transform Important function:Dirac delta function,rectangular function, periodic function and sinc function and their Fourier transforms

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Chap.2 Signals and Spectra

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  1. Chap.2 Signals and Spectra Object:mathematical tools to describe and analyze the signals Fourier series and transform Important function:Dirac delta function,rectangular function, periodic function and sinc function and their Fourier transforms frequency analyze(time function and his spectrum) some properties of signal (DC value ,root mean square value,…) power spectral density and autocorrelation function linear systems:linear time-invariant systems,impulse response,transfer function,distortionless transmission bandwidth concept:baseband,passband and bandlimited signals and noise *sampling theorem (dimensionality theorem) summary

  2. 2-1. Properties of Signals and Noise • Signal:desired part of waveforms; Noise:undesired part • Electric signal’s form:voltage v(t) or current i(t) (time function) • In this chapter,all signals are deterministic. • But in communication systems,we will be face the stochastic waveforms Deterministic results stochastic results by analogy Signal analysis:first importance

  3. Physically realizable waveforms • Non zero values over a finite time interval • non zero values over a finite frequency interval • a continuous time function • a finite peak value • only real values In general,the waveform is denoted by w(t) When t→±∞,we have w(t) →0,but w(t) is defined over (+∞,-∞) The math model of waveform can violate some or all above conditions. Ex. w(t)=sinωt,physically this waveform can not be existed.

  4. The classifications of waveforms Waveforms: • signal or noise • digital or analog • deterministic or nondeterministic(stochastic) • physically realizable or nonphysically realizable • power type or energy type • periodic or nonperiodic Power type:the average power of the waveform is finite(math model) Energy type:the average energy of the waveform is finite(all physically realizable signal)

  5. Some important math operations • Time average operator:dc(direct current) value of time function Definition: the time average operation is given by: 〈[·]〉=lim1/T-T/2∫ T/2[·]dt 〈[·]〉is time average operator. The operator is linear.(Why?) Definition : w(t) is periodic with period T0 if w(t)=w(t+ T0) for all t where T0 is smallest positive number that satisfies above relationship. Theorem:if w(t) is periodic,the time average operation can be reduced to 〈[·]〉=1/T-T/2-a∫ T/2+a[·]dt where T is period of w(t)

  6. DC value • Definition:the dc value of w(t) is given by its time average, 〈w(t)〉. Wdc=〈w(t)〉=lim1/T-T/2∫ T/2w(t)dt or Wdc=〈w(t)〉=1/(t2-t1)∫w(t)dt Power • Definition:the instantaneous power is given by: p(t) = v(t)i(t) and the average power is : P=<p(t)>=<v(t)i(t)> i(t) + v(t) - circuit

  7. Rms Value and Normalized Power • Definition:the root mean square (rms) value of w(t)is given by: Wrms=[<w2(t)>]1/2 • Theorem:if a load (R) is resistive,the average power is: P= <v2(t)>/R= <i2(t)>R= V2rms/R=I 2rmsR • Definition:if R=1Ω,the average power is called normalized power. Then i(t) = v(t) = w(t) and P= <w2(t)> Energy and Power Waveforms: • Definition:w(t) is a power waveform if and only if the normalized power P is finite and nonzero(0<P<∞). • Definition:the total normalized energy is given by:

  8. Definition:w(t) is an energy waveform if and only if the total normalized energy is finite and nonzero (0<E<∞). Waveform: power signal or energy signal Energy finite Average Power=0 Power finite Energy=∞ Physically realizable waveform:Energy waveform Periodic waveform:Power waveform Decibel • Definition:the decibel gain of a circuit is given by dB=10log(average power out/average power in) =10log(Pout/Pin)

  9. For normalized power case(R=1Ω),we have: dB=20log(Vrms out /Vrms in)= 20log(Irms out /Irms in)

  10. Fourier Transform and Spectra w(t),voltage or current,time function analysis in time domain. Their fluctuation as a function of time is an important characteristic to analyze the signal’s comportment when they present in the transmission channel or other communication’s units. Frequency analysis of signal. Tool to realize the frequency domain analysis of signal Fourier Transformation • Definition:The Fourier Transform (FT) of w(t) is : W(f)=F[w(t)]= -∞∫∞w(t)exp[-j2πft]dt f :frequency (unit:Hz if t is in sec) In general,W(f) is called a two-sided spectrum of w(t) Some properties: W(f) is a complex function so W(f)=X(f)+jY(f)=│W(f)│exp[jθ(f)]

  11. 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)]

  12. Properties of Fourier Transforms • 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)

  13. Parseval’s theorem • 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

  14. Power spectral density(PSD) • 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 附录A. Most important theorems: time delay :w(t-Td) W(f)e-j ωTd frequency translation: w(t)cos(ωct+θ) 1/2[ W(f-fc)ejθ+W(f+ fc)e-jθ]

  15. 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) F[w(t)cosωct] W(f) f f fc

  16. Dirac delta function and unit step function • 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.

  17. Delta function’s properties • 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)

  18. 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: -∞∫xδ(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)(ej2πf0t - e-j2πf0t )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

  19. │V(f)│ θ(f) A/2 f f - f0 f0 Magnitude spectrum Phase spectrum

  20. 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

  21. Ex. Spectrum of a rectangular pulse W(f)= -∞∫∞ Π(t/T)e-j2πft dt =-T/2∫ T/21e-j2πft dt =Tsin(2πfT/2)/(2πfT/2)=TSa(πfT) so we have: Π(t/T) TSa(πfT) Time domain Frequency domain TSa(πfT) Π(t/T) 1 T t -T/2 T/2 1/T 2/T f

  22. For an ideal low pass filter, we have its time response: Π(f/2W) 2WSa(2πWt) • Conclusion:ideal LPF physically unrealizable • The equivalent LPF plays a special role in digital comm. • For triangular pulse,we have : Λ(t/T) TSa2(πTf) Π(f/2W) 2WSa(2πWt) 2W 1 t -W W f -1/2W 1/2W

  23. Convolution • Definition:the convolution of waveforms w1(t) and w2(t) gives a third function w3(t) defined by: w3(t)= w1(t)*w2(t)= -∞∫∞ w1(λ)w2(t-λ)dλ Ex.We have: Λ(t/T) = Π(t/T) * Π(t/T) then F[Λ(t/T) ]=F[Π(t/T)]F[Π(t/T)] = TSa2(πTf) Ex.Spectrum of a switched sinusoid: w(t)= Π(t/T) Asin(ω0t)= Π(t/T) Acos(ω0t-π/2) we have :Π(t/T) TSa(πfT) Asin(ω0t) j(A/2)[δ(f+f0) -δ(f-f0)] So W(f)= [TSa(πfT)] *{j(A/2)[δ(f+f0) -δ(f-f0)] } = j(A/2){Sa[πT(f+f0)] - Sa[πT(f-f0)]}

  24. Time domain: Frequency domain W(t) t -T/2 T/2 │W(f)│ AT/2 f -f0 f0

  25. Power spectral density and autocorrelation function • Power spectral density w(t) and its truncated waveform wT(t) : wT(t)=w(t) -T/2<t<T/2 and wT(t) =0 t elsewhere wT(t)’s average normalized power P: P=lim 1/T -T/2∫ T/2wT2(t)dt = lim 1/T -∞∫∞ wT2(t)dt By Parseval’s theorem, we have P=lim1/T -T/2∫ T/2│WT(f)│2df =-∞∫∞ lim[│WT(f)│2/T]df where WT(f) is the wT(t)’s FT. Units:P Watts │WT(f)│2/T:Watts/Hz PSD:definition

  26. Definition:the PSD for a deterministic power waveform is Pw(f)= lim[│WT(f)│2/T]Watts/Hz Pw(f) :always a positive function of frequency. Not sensitive to the phase spectrum of w(t). So with Pw(f),the normalized average power of w(t) can be given by: P= -∞∫∞ Pw(f) df Ex. Pw(f) f

  27. Autocorrelation function Definition:the autocorrelation of a real (physical) waveform is: Rw(τ) =<w(t)w(t+ τ)> = lim 1/T -T/2∫ T/2w(t)w(t+ τ)dt Wiener-Khintchine theorem: Rw(τ) Pw(f) For P ,we have: P =<w2(t)> =W2rms=-∞∫∞ Pw(f) df = Rw(0) Ex. PSD of a Sinusoid w(t)=Asinωct the autocorrelation function: Rw(τ) =<w(t)w(t+ τ)> =A2/2cosωc τ so the PSD Pw(f) : Pw(f) = A2/4[δ(f+fc) +δ(f-fc)]

  28. And the average power P: P= -∞∫∞ Pw(f) df = A2/2 Pw(f) A2/4 A2/4 -fc fc f

  29. Fourier series • Complex Fourier series w(t)=Σcnej2πf0t a<t<T0 and cn=1/T0 a∫a+T0w(t)e-j2πf0tdt • Quadrature Fourier series (see p.70) • Polar Fourier Series(see p.70)

  30. Line spectra for periodic waveforms • Theorem: If a waveform w(t) is periodic with period T0,the spectrum of w(t) is W(f)= Σcn δ(f-nf0) f0=1/ T0 and cn is the coefficients of w(t)’s complex Fourier series. Proof. (delta function’s properties) W(f) c-1 c1 c-2 c2 c-3 c0 c3 f -3f0 -2f0 -f0 0 f0 2f0 3f0

  31. Conclusion: periodic function line spectra no dc value c0=0 no periodic components continuous spectra • Power spectral density for periodic waveforms Theorem:for a periodic waveform,the PSD is given by: Pw(f)=Σ│cn │2δ(f-nf0) where{ cn} are the corresponding Fourier coefficients.

  32. Review of linear systems • Linear system superposition law y(t)=L[a1x1(t)+ a2x2(t)]= a1L[x1(t)]+a2L[x2(t)] input output X(f) spectrum Y(f) Rx(τ) autocorrelation Ry(τ) Px(f) PSD Py(f) h(t) H(f) x(t) y(t)

  33. Impulse response Input Output Dirac delta function δ(t) Impulse response h(t) x(t) y(t)=x(t)*h(t) Transfer function H(f)=F[h(t)] Py(f)=│H(f)│2Px(f) so power transfer function of a linear network: Gh(f) = Py(f) /Px(f) = │H(f)│2

  34. Ex. RC low-pass filter • Sol. Using Kirchhoff’s voltage law,we have: • Where • FT of this differential equation: i(t) x(t) y(t)

  35. The transfer function: • And the impulse response of RC LPF: • The power transfer function: • Where

  36. Distortionless transmission • Distortionless communication systems(channels): the output of the channel is just proportional to a delayed version of the input • so we have: y(t)=Ax(t-Td) A:gain;Td:time delay we can represent the transfer function of such system by: Y(f)=AX(f)e-j2πf Td and H(f)=Y(f)/X(f)=A e-j2πf Td Implication:(a linear time-invariant and distortionless system) 1.│H(f)│=constant =A :the magnitude spectrum is constant. 2. Phase response is a linear function of frequency

  37. When │H(f)│=constant =A no amplitude distortion • When θ(f)= -2πf Td no phase distortion • Time delay of the system is defined by: Td(f)= -(1/2πf) θ(f) so no phase distortion is equivalent to: Td(f)=constant

  38. Ex.Distortion caused by a RC LP filter H(f)=Y(f)/X(f)=1/[1+j(2πRC)f] the amplitude response is │H(f)│=1/[1+(f/f0)]1/2 Where f0= 1/(2πRC) and the phase response is θ(f)= -arctg(f/f0) the corresponding time delay function is Td(f)=1/(2πf) arctg(f/f0) Effect? i(t) x(t) y(t)

  39. Bandlimited signals and noise • In communication systems,often we deal with signals and noise called bandlimited. • Definition:a waveform w(t) is said to be (absolutely) bandlimited to B Hz if W(f)=F[w(t)]=0 for │f│≥B • Definition: a waveform w(t) is (absolutely) timelimited if w(t)=0 for │t│≥T • Theorem:an absolutely bandlimited waveform cannot be absolutely timelimited and vice versa. • Dilemma :physically realizable waveform timelimited so not bandlimited. How to do?

  40. Hilbert Transform • Definition:The Hilbert transform of f(t) is defined as: • Where f(t) is real. • The inverse Hilbert transform is: • We have:

  41. Spectra: • So H(f) can be calculated by: f(t)

  42. Properties of Hilbert transform: If f(t) is even，then is odd. If f(t) is odd，then is even .

  43. Analytic signals • Definition:实信号f(t)，则定义复信号Z(t): 为f(t)的解析信号。 性质：

  44. 性质（续）：

  45. Ex.解析信号的确定 • 方法：时域：虚部是实部的Hilbert变换 频域：z(t)的Fourier变换是其实部 的Fourier变换的两倍 若复信号的Fourier在f<0时恒为0 故可验证 是解析信号

  46. 已知实函数f(t)，求其解析信号 • Hilbert变换 • 求F(f) • Ex.求 的解析信号。 • 解： • 因此

  47. 频带信号与带通系统

  48. Bandpass signaling: • Definition:A baseband waveform has a spectral magnitude that is nonzero for frequencies in the vicinity of the origin and negligible elsewhere. • Definition:A bandpass waveform has a spectral magnitude that is nonzero for frequencies in some band concentrated about a frequency f= ±fc ,where fc>>0.The spectral magnitude is negligible elsewhere. fc is called the carrier frequency. • fc may be arbitrarily assigned. • Definition:Modulation is the process of imparting the source information onto a bandpass signal with a carrier frequency fc by the introduction of amplitude and/or phase perturbation.This bandpass signal is called the modulated signal s(t),and the baseband source signal is called the modulating signal m(t).

  49. Complex envelope representation • All banpass waveforms can be represented by their complex envelope forms. • Theorem:Any physical banpass waveform can be represented by: v(t)=Re{g(t)ejωct} Re{.}:real part of {.}.g(t) is called the complex envelope of v(t),and fc is the associated carrier frequency.Two other equivalent representations are: v(t)=R(t)cos[ωct+θ(t)] and v(t)=x(t)cos ωct-y(t)sin ωct where g(t)=x(t)+jy(t)=R(t) ejθ(t)

  50. Representation of modulated signals • The modulated signals a special type of bandpass waveform • So we have s(t)=Re{g(t)ejωct} the complex envelope is function of the modulating signal m(t): g(t)=g[m(t)] g[.]: mapping function All type of modulations can be represented by a special mapping function g[.]. See pages231-232 for g[.]

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