fourier representation of signal and systems

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fourier representation of signal and systems

  1. 1. Introduction to Analog And DigitalCommunications Second Edition Simon Haykin, Michael Moher
  2. 2. Chapter 2 Fourier Representation of Signalsand Systems 2.1 The Fourier Transform 2.2 Properties of the Fourier Transform 2.3 The Inverse Relationship Between Time and Frequency 2.4 Dirac Delta Function 2.5 Fourier Transforms of Periodic Signals 2.6 Transmission of Signals Through Linear Systems : Convolution Revisited 2.7 Ideal Low-pass Filters 2.8 Correlation and Spectral Density : Energy Signals 2.9 Power Spectral Density 2.10 Numerical Computation of the Fourier Transform 2.11 Theme Example : Twisted Pairs for Telephony 2.12 Summary and Discussion
  3. 3. 2.1 The Fourier Transform Definitions  Fourier Transform of the signal g(t) ∞ G( f ) = ∫ −∞ g (t ) exp(− j 2πft )dt (2.1) exp(− j 2πft ) : the kernel of thr formula defining the Fourier transform ∞ g (t ) = ∫ −∞ G ( f ) exp( j 2πft )df (2.2) exp( j 2πft ) : the kernel of thr formula defining the Inverse Fourier transfom  A lowercase letter to denote the time function and an uppercase letter to denote the corresponding frequency function  Basic advantage of transforming : resolution into eternal sinusoids presents the behavior as the superposition of steady-state effects  Eq.(2.2) is synthesis equation : we can reconstruct the original time-domain behavior of the system without any loss of information. 3
  4. 4.  Dirichlet’s conditions 1. The function g(t) is single-valued, with a finite number of maxima and minima in any finite time interval. 2. The function g(t) has a finite number of discontinuities in any finite time interval. 3. The function g(t) is absolutely integrable ∞ ∫−∞ g (t ) dt < ∞  For physical realizability of a signal g(t), the energy of the signal defined by ∞ ∫ 2 g (t ) dt −∞ must satisfy the condition ∞ ∫ g (t ) dt < ∞ 2 −∞  Such a signal is referred to as an energy signal.  All energy signals are Fourier transformable. 4
  5. 5.  Notations  The frequency f is related to the angular frequency w as w = 2πf [rad / s ]  Shorthand notation for the transform relations G ( f ) = F[ g (t )] (2.3) g (t ) = F −1[G ( f )] (2.4) g (t ) ⇔ G ( f ) (2.5) 5
  6. 6.  Continuous Spectrum  A pulse signal g(t) of finite energy is expressed as a continuous sum of exponential functions with frequencies in the interval -∞ to ∞.  We may express the function g(t) in terms of the continuous sum infinitesimal components, ∞ g (t ) = ∫ −∞ G ( f ) exp( j 2πft )df  The signal in terms of its time-domain representation by specifying the function g(t) at each instant of time t.  The signal is uniquely defined by either representation.  The Fourier transform G(f) is a complex function of frequency f, G ( f ) = G ( f ) exp[ jθ ( f )] (2.6) G ( f ) : continuous amplitude spectrum of g(t) θ ( f ) : continuous phase spectrum of g(t) 6
  7. 7.  For the special case of a real-valued function g(t) G (− f ) = G * ( f ) G (− f ) = G ( f ) θ (− f ) = −θ ( f ) * : complex conjugation The spectrum of a real-valued signal 1. The amplitude spectrum of the signal is an even function of the frequency, the amplitude spectrum is symmetric with respect to the origin f=0. 2. The phase spectrum of the signal is an odd function of the frequency, the phase spectrum is antisymmetric with respect to the origin f=0. 7
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  10. 10. Fig. 2.3 10
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  16. 16. 2.2 Properties of the Fourier Transrom Property 1 : Linearity (Superposition) Let g1 (t ) ⇔ G1 ( f ) and g 2 (t ) ⇔ G2 ( f ) then for all constants c1 and c2, c1 g1 (t ) + c2 g 2 (t ) ⇔ c1G1 ( f ) + c2G2 ( f ) (2.14) Property 2 : Dialation Let g (t ) ⇔ G ( f ) 1 f g (at ) ⇔ G   (2.20) a a The dilation factor (a) is real number ∞ 1 ∞  f  F [ g (at )] = ∫ −∞ g (at ) exp(− j 2πft )dt F [ g (at )] = −∞ a ∫ g (τ ) exp − j 2π  τ dτ  a  1 f = G  a a 16
  17. 17.  The compression of a function g(t) in the time domain is equivalent to the expansion of ite Fourier transform G(f) in the frequency domain by the same factor, or vice versa. Reflection property  For the special case when a=-1 g (−t ) ⇔ G (− f ) (2.21) 17
  18. 18. Fig. 2.6 18
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  20. 20. ` 20
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  22. 22.  Property 3 : Conjugation Rule Let g (t ) ⇔ G ( f ) then for a complex-valued time function g(t), g * (t ) ⇔ G * (− f ) (2.22) Prove this ; ∞ g (t ) = ∫ −∞ G ( f ) exp( j 2πft )df ∞ g (t ) = ∫ G * ( f ) exp(− j 2πft )df * −∞ ∞ g (t ) = − ∫ G * (− f ) exp( j 2πft )df * −∞ ∞ = ∫ −∞ G * (− f ) exp( j 2πft )df g * (−t ) ⇔ G * ( f ) (2.23) 22
  23. 23.  Property 4 : Duality If g (t ) ⇔ G ( f ) G (t ) ⇔ g (− f ) (2.24) ∞ g (−t ) = ∫−∞ G ( f ) exp(− j 2πft )df  Which is the expanded part of Eq.(2.24) in going from the time domain to the frequency domain. ∞ g (− f ) = ∫ −∞ G (t ) exp(− j 2πft )dt 23
  24. 24. Fig. 2.8 24
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  26. 26.  Property 5 : Time Shifting If g (t ) ⇔ G ( f ) g (t − t0 ) ⇔ G ( f ) exp(− j 2πft0 ) (2.26)  If a function g(t) is shifted along the time axis by an amount t0, the effect is equivalent to multiplying its Fourier transform G(f) by the factor exp(-j2πft0). Prove this ; ∞ F [ g (t − t0 )] = exp(− j 2πft0 ) ∫ −∞ g (τ ) exp(− j 2πτ )dτ = exp(− j 2πft0 )G ( f ) Property 6 : Frequency Shifting If g (t ) ⇔ G ( f ) exp( j 2πf c t ) g (t ) ⇔ G ( f − f c ) (2.27) 26
  27. 27.  This property is a special case of the modulation theorem  A shift of the range of frequencies in a signal is accomplished by using the process of modulation. ∞ F [exp( j 2πf c t ) g (t )] = ∫ g (t ) exp[− j 2πt ( f − f c )]dt −∞ = G( f − fc ) Property 7 : Area Under g(t) If g (t ) ⇔ G ( f ) ∞ ∫ −∞ g (t )dt = G (0) (2.31)  The area under a function g(t) is equal to the value of its Fourier transform G(f) at f=0. 27
  28. 28. Fig. 2.9 28
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  30. 30. Fig. 2.9 Fig. 2.2 30
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  32. 32.  Property 8 : Area under G(f) If g (t ) ⇔ G ( f ) ∞ g ( 0) = ∫ −∞ G ( f )df (2.32)  The value of a function g(t) at t=0 is equal to the area under its Fourier transform G(f). Property 9 : Differentiation in the Time Domain Let g (t ) ⇔ G ( f ) d g (t ) ⇔ j 2πfG ( f ) (2.33) dt  Differentiation of a time function g(t) has the effect of multiplying its Fourier transform G(f) by the purely imaginary factor j2πf. dn g (t ) ⇔ ( j 2πf ) n G ( f ) (2.34) dt n 32
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  34. 34. Fig. 2.10 34
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  36. 36.  Property 10 : Integration in the Time Domain Let g (t ) ⇔ G ( f ) , G(0) = 0, 1 ∫ t g (τ )dτ ⇔ G (t ) (2.41) −∞ j 2πf  Integration of a time function g(t) has the effect of dividing its Fourier transform G(f) by the factor j2πf, provided that G(0) is zero. verify this ; g (t ) =  g (τ )dτ  d ∫ t dt   −∞    F  t g (τ )dτ   G ( f ) = ( j 2πf ) ∫   −∞    36
  37. 37. Fig. 2.11 37
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  41. 41.  Property 11 : Modulation Theorem Let g1 (t ) ⇔ G1 ( f ) and g 2 (t ) ⇔ G2 ( f ) ∞ g1 (t ) g 2 (t ) ⇔ ∫ −∞ G1 (λ )G2 ( f − λ )dλ (2.49)  We first denote the Fourier transform of the product g1(t)g2(t) by G12(f) g1 (t ) g 2 (t ) ⇔ G12 ( f ) ∞ G12 ( f ) =∫ −∞ g1 (t )g 2 (t ) exp(− j 2πft )dt ∞ g (t ) = ∫ 2 G2 ( f ) exp(− j 2πf t )df −∞ ∞ ∞ G12 ( f ) = ∫ ∫ g1 (t )G2 ( f ) exp[− j 2π ( f − f )t ]df dt −∞ −∞ G12 ( f ) = G2 ( f − λ )  g1 (t ) exp(− j 2πλt )dt dλ ∞ ∞ −∞∫   ∫−∞   the inner integral is recognized simply as G1 (λ ) 41
  42. 42. ∞ G12 ( f ) = ∫ −∞ G1 (λ )G2 ( f − λ )dλ This integral is known as the convolution integral Modulation theorem  The multiplication of two signals in the time domain is transformed into the convolution of their individual Fourier transforms in the frequency domain.  Shorthand notation G12 ( f ) = G1 ( f ) ∗ G2 ( f ) g1 (t ) g 2 (t ) ⇔ G1 ( f ) ∗ G2 ( f ) (2.50) G1 ( f ) ∗ G2 ( f ) = G2 ( f ) ∗ G1 ( f ) 42
  43. 43.  Property 12 : Convolution Theorem Let g1 (t ) ⇔ G1 ( f ) and g 2 (t ) ⇔ G2 ( f ) ∞ ∫−∞ g1 (τ ) g 2 (t − τ )dτ ⇔G1 ( f )G2 ( f ) (2.51)  Convolution of two signals in the time domain is transformed into the multiplication of their individual Fourier transforms in the frequency domain. g1 (t ) ∗ g 2 (t ) = G1 ( f )G2 ( f ) (2.52) Property 13 : Correlation Theorem Let g1 (t ) ⇔ G1 ( f ) and g 2 (t ) ⇔ G2 ( f ) ∞ ∫ g1 (t ) g 2 (t − τ )dt ⇔G1 ( f )G2* ( f ) (2.53) * −∞  The integral on the left-hand side of Eq.(2.53) defines a measure of the similarity that may exist between a pair of complex-valued signals ∞ ∫−∞ g1 (t ) g 2 (τ − t )dt ⇔G1 ( f )G2 ( f ) (2.54) 43
  44. 44.  Property 14 : Rayleigh’s Energy Theorem Let g (t ) ⇔ G ( f ) ∞ ∞ ∫ g (t ) dt = ∫ 2 2 G ( f ) df (2.55) −∞ −∞  Total energy of a Fourier-transformable signal equals the total area under the curve of squared amplitude spectrum of this signal. ∞ ∫ g (t ) g * (t − τ )dt ⇔G ( f )G ( f ) = G ( f ) * 2 −∞ ∞ ∞ 2 ∫ g (t ) g (t − τ )dt = ∫ G ( f ) exp( j 2πfτ )df * (2.56) −∞ −∞ 44
  45. 45. 45
  46. 46. 2.3 The Inverse Relationship Between Time and Frequency1. If the time-domain description of a signal is changed, the frequency-domain description of the signal is changed in an inverse manner, and vice versa.2. If a signal is strictly limited in frequency, the time-domain description of the signal will trail on indefinitely, even though its amplitude may assume a progressively smaller value. – a signal cannot be strictly limited in both time and frequency. Bandwidth  A measure of extent of the significant spectral content of the signal for positive frequencies. 46
  47. 47.  Commonly used three definitions 1. Null-to-null bandwidth  When the spectrum of a signal is symmetric with a main lobe bounded by well-defined nulls – we may use the main lobe as the basis for defining the bandwidth of the signal 2. 3-dB bandwidth  Low-pass type : The separation between zero frequency and the positive frequency at which the amplitude spectrum drops to 1/√2 of its peak value.  Band-pass type : the separation between the two frequencies at which the amplitude spectrum of the signal drops to 1/√2 of the peak value at fc. 3. Root mean-square (rms) bandwidth  The square root of the second moment of a properly normalized form of the squared amplitude spectrum of the signal about a suitably chosen point. 47
  48. 48.  The rms bandwidth of a low-pass signal g(g) with Fourier transform G(f) as follows : 1/ 2  f G ( f ) df  ∞ ∫ 2 2   Wrms =  −∞ ∞  (2.58)  ∫ G ( f ) df  2  It lends itself more readily − ∞ mathematical   to evaluation than the other two definitions of bandwidth  Although it is not as easily measured in the lab. Time-Bandwidth Product  The produce of the signal’s duration and its bandwidth is always a constant (duration) × (bandwidth) = constant  Whatever definition we use for the bandwidth of a signal, the time-bandwidth product remains constant over certain classes of pulse signals 48
  49. 49.  Consider the Eq.(2.58), the corresponding definition for the rms duration of the signal g(t) is 1/ 2  ∫ t g (t ) dt  ∞ 2 2   Trms = −∞ ∞  (2.59)  ∫ g (t ) dt  2  −∞  The time-bandwidth product has the following form 1 TrmsWrms ≥ (2.60) 4π 49
  50. 50. 2.4 Dirac Delta Function The theory of the Fourier transform is applicable only to time functions that satisfy the Dirichlet conditions ∞ ∫ g (t ) dt < ∞ 2 −∞ 1. To combine the theory of Fourier series and Fourier transform into a unified framework, so that the Fourier series may be treated as a special case of the Fourier transform 2. To expand applicability of the Fourier transform to include power signals-that is, signals for which the condition holds. 1 T ∫ g (t ) dt < ∞ 2 lim 2T T →∞ −T Dirac delta function  Having zero amplitude everywhere except at t=0, where it is infinitely large in such a way that it contains unit area under its curve. 50
  51. 51. δ (t ) = 0, t ≠ 0 (2.61) ∞ ∫−∞ δ (t )dt = 1 (2.62) We may express the integral of the product g(t)δ(t-t0) with respect to time t as follows : ∞ ∫ −∞ g (t )δ (t − t0 )dt = g (t0 ) (2.63) Sifting property of the delta function ∞ ∫−∞ g (τ )δ (t − τ )dτ = g (t ) (2.64) g (t ) ∗ δ (t ) = g (t ) The convolution of any time function g(t) with the delta function δ(t) leaves that function completely unchanged. – replication property of delta function. The Fourier transform of the delta function is ∞ F [δ (t )] = ∫ −∞ δ (t ) exp(− j 2πft )dt 51
  52. 52.  The Fourier transform pair for the Delta function F [δ (t )] = 1 δ (t ) ⇔ 1 (2.65) The delta function as the limiting form of a pulse of unit area as the duration of the pulse approaches zero. A rather intuitive treatment of the function along the lines described herein often gives the correct answer. Fig. 2.12 52
  53. 53. Back Next 53
  54. 54. Fig. 2.13(a)Fig. 2.13(b) 54
  55. 55. Back NextFig.2.13(A) 55
  56. 56. Back NextFig2.13(b) 56
  57. 57.  Applications of the Delta Function 1. Dc signal  By applying the duality property to the Fourier transform pair of Eq.(2.65) 1 ⇔ δ ( f ) (2.67)  A dc signal is transformed in the frequency domain into a delta function occurring at zero frequency ∞ ∫ −∞ exp(− j 2πft )dt = δ ( f ) ∞ ∫−∞ cos(2πft )dt = δ ( f ) (2.68) 2. Complex Exponential Function  By applying the frequency-shifting property to Eq. (2.67) exp( j 2πf c t ) ⇔ δ ( f − f c ) (2.69) Fig. 2.14 57
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  59. 59. 3. Sinusoidal Functions  The Fourier transform of the cosine function 1 cos(2πf c t ) = [exp( j 2πf c t ) + exp(− j 2πf c t )] (2.70) 2 1 cos(2πf c t ) ⇔ [δ ( f − f c ) + δ ( f + f c )] (2.71) 2 Fig. 2.15  The spectrum of the cosine function consists of a pair of delta functions occurring at f=±fc, each of which is weighted by the factor ½ 1 sin( 2πf c t ) ⇔ [δ ( f − f c ) − δ ( f + f c )] (2.72) 2j4. Signum Function + 1, t > 0  sgn(t ) = 0, t = 0 − 1, t < 0  Fig. 2.16 59
  60. 60. Back NextFig2.15 60
  61. 61. Back NextFig 2.16 61
  62. 62.  This signum function does not satisfy the Dirichelt conditions and therefore, strictly speaking, it does not have a Fourier transform exp(− at ), t >0  g (t ) = 0, t = 0 (2.73) − exp(− at ), t < 0  Its Fourier transform was derived in Example 2.3; the result is given by − j 4πf G( f ) = a 2 + (2πf ) 2 The amplitude spectrum |G(f)| is shown as the dashed curve in Fig. 2.17(b). In the limit as a approaches zero, − j 4πf F[sgn(t )] = lim a →0 a 2 + (2πf ) 2 sgn(t ) ⇔ 1 (2.74) 1 jπf = jπf At the origin, the spectrum of the approximating function g(t) is zero for a>0, whereas the spectrum of the signum function goes to infinity. Fig. 2.17 62
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  64. 64. 5. Unit Step Function  The unit step function u(t) equals +1 for positive time and zero for negative time. 1, t > 0 1  u (t ) =  , t = 0 2 0, t < 0   The unit step function and signum function are related by 1 u (t ) = [sgn((t ) + 1)] (2.75) 2  Unit step function is represented by the Fourier-transform pair • The spectrum of the unit step function contains a delta function weighted by a factor of ½ and occurring at zero frequency 1 1 u (t ) ⇔ + δ ( f ) (2.76) j 2πf 2 Fig. 2.18 64
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  66. 66. 6. Integration in the time Domain (Revisited)  The effect of integration on the Fourier transform of a signal g(t), assuming that G(0) is zero. ∫ t y (t ) = g (τ )dτ (2.77) −∞  The integrated signal y(t) can be viewed as the convolution of the original signal g(t) and the unit step function u(t) ∞ y (t ) = ∫ −∞ g (τ )u (t − τ )dτ 1, τ < t 1  u (t − τ ) =  , τ = t 2 0, τ > t   The Fourier transform of y(t) is  1 1  Y ( f ) = G( f ) + δ ( f ) (2.78)  j 2πf 2  66
  67. 67.  The effect of integrating the signal g(t) is G ( f )δ ( f ) = G (0)δ ( f ) 1 1 Y( f ) = G ( f ) + G (0)δ ( f ) j 2πf 2 1 1 ∫ t g (τ )dτ ⇔ G ( f ) + G (0)δ ( f ) (2.79) −∞ j 2πf 2 67
  68. 68. 2.5 Fourier Transform of Periodic Signals  A periodic signal can be represented as a sum of complex exponentials  Fourier transforms can be defined for complex exponentials Consider a periodic signal gT0(t) ∞ gT0 (t ) = ∑ c exp( j 2πnf t ) n = −∞ n 0 (2.80) 1 T /2 cn = ∫ gT0 (t ) exp(− j 2πnf 0t )dt (2.81) 0 T0 −T / 2 0 Complex Fourier coefficient 1 f0 = (2.82) T0 f0 : fundamental frequency 68
  69. 69.  Let g(t) be a pulselike function  T0 T0  gT (t ), − ≤ t ≤ g (t ) =  0 2 2 (2.83) 0,  elsewhere ∞ gT0 (t ) = ∑ g (t − mT ) m = −∞ 0 (2.84) ∞ cn = f 0 ∫ −∞ g (t ) exp(− j 2πnf 0t )dt = f 0G (nf 0 ) (2.85) ∞ gT0 (t ) = f 0 ∑ G (nf 0 ) exp( j 2πnf 0t ) (2.86) n = −∞ 69
  70. 70.  One form of Possisson’s sum formula and Fourier-transform pair ∞ ∞ ∑ g (t − mT ) = f ∑ G(nf ) exp( j 2πnf t ) m = −∞ 0 0 n = −∞ 0 0 (2.87) ∞ ∞ ∑ g (t − mT ) ⇔ f ∑ G(nf )δ ( f − nf ) m = −∞ 0 0 n = −∞ 0 0 (2.88) Fourier transform of a periodic signal consists of delta functions occurring at integer multiples of the fundamental frequency f0 and that each delta function is weighted by a factor equal to the corresponding value of G(nf0). Periodicity in the time domain has the effect of changing the spectrum of a pulse-like signal into a discrete form defined at integer multiples of the fundamental frequency, and vice versa. 70
  71. 71. Fig. 2.19 71
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  74. 74. 2.6 Transmission of Signal Through Linear Systems : Convolution Revisited In a linear system,  The response of a linear system to a number of excitations applied simultaneously is equal to the sum of the responses of the system when each excitation is applied individually. Time Response  Impulse response  The response of the system to a unit impulse or delta function applied to the input of the system.  Summing the various infinitesimal responses due to the various input pulses,  Convolution integral  The present value of the response of a linear time-invariant system is a weighted integral over the past history of the input signal, weighted according to the impulse response of the system ∞ y (t ) = ∫ −∞ x(τ )h(t − τ )dτ (2.93) ∞ y (t ) = ∫ h(τ )x(t − τ )dτ (2.94) −∞ 74
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  77. 77. Fig. 2.21 77
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  79. 79.  Causality and Stability  Causality  It does not respond before the excitation is applied h(t ) = 0, t < 0 (2.98)  Stability  The output signal is bounded for all bounded input signals (BIBO) x(t ) < M for all t ∞ y (t ) = ∫−∞ h(τ )x(t − τ )dτ (2.99)  Absolute value of an integral is bounded by the integral of the absolute value of the integrand ∞ ∞ ∫ h(τ )x(t − τ ) dτ ≤ ∫ h(τ ) x(t − τ ) dτ ∞ ∫ −∞ −∞ ∞ y(t) ≤ M h(τ ) dτ = M ∫ h(τ ) dτ −∞ −∞ 79
  80. 80.  A linear time-invariant system to be stable  The impulse response h(t) must be absolutely integrable  The necessary and sufficient condition for BIBO stability of a linear time-invariant system ∞ ∫−∞ h(t ) dt < ∞ (2.100) Frequency Response  Impulse response of linear time-invariant system h(t),  Input and output signal x(t ) = exp( j 2πft ) (2.101) ∞ y (t ) = ∫ h(τ ) exp[ j 2πf (t − τ )]dτ −∞ ∞ = exp( j 2πft ) ∫ h(τ ) exp(− j 2πfτ )dτ (2.102) −∞ 80
  81. 81. ∞H( f ) = ∫ h(t ) exp(− j 2πft )dt (2.103) y (t ) = H ( f ) exp( j 2πft ) (2.104) −∞  Eq. (2.104) states that  The response of a linear time-variant system to a complex exponential function of frequency f is the same complex exponential function multiplied by a constant coefficient H(f)  An alternative definition of the transfer function ∞ H( f ) = y (t ) x(t ) x ( t ) =exp( j 2πft ) (2.105) x(t ) = ∫ −∞ X ( f ) exp( j 2πft )df (2.106) ∞ x(t ) = ∆f →0 lim ∑ X ( f ) exp( j 2πft )∆f f = k∆f k = −∞ (2.107) ∞ y (t ) = ∆f →0 ∑ H ( f ) X ( f ) exp( j 2πft )∆f lim f = k∆f k = −∞ ∞ = ∫−∞ H ( f ) X ( f ) exp( j 2πft )df (2.108) 81
  82. 82.  The Fourier transform of the output signal y(t) Y ( f ) = H ( f ) X ( f ) (2.109) The Fourier transform of the output is equal to the product of the frequency response of the system and the Fourier transform of the input  The response y(t) of a linear time-invariant system of impulse response h(t) to an arbitrary input x(t) is obtained by convolving x(t) with h(t), in accordance with Eq. (2.93)  The convolution of time functions is transformed into the multiplication of their Fourier transforms H ( f ) = H ( f ) exp[ jβ ( f )] (2.110) Amplitude response or magnitude response Phase or phase response H ( f ) = H (− f ) β ( f ) = − β (− f ) 82
  83. 83.  In some applications it is preferable to work with the logarithm of H(f) ln H ( f ) = α ( f ) + jβ ( f ) (2.111) α ( f ) = ln H ( f ) (2.112) α ( f ) = 20 log10 H ( f ) (2.113) The gain in decible [dB] α ( f ) = 8.69α ( f ) (2.114) 83
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  85. 85.  Paley-Wiener Criterion  The frequency-domain equivalent of the causality requirement  α( f )  ∞ ∫  −∞ 1 + f  2 df < ∞ (2.115)  85
  86. 86. 2.7 Ideal Low-Pass Filters Filter  A frequency-selective system that is used to limit the spectrum of a signal to some specified band of frequencies The frequency response of an ideal low-pass filter condition  The amplitude response of the filter is a constant inside the passband -B≤f ≤B  The phase response varies linearly with frequency inside the pass band of the filter exp(− j 2πft0 ), − B ≤ f ≤ B H( f ) =  (2.116) 0, f >B 86
  87. 87.  Evaluating the inverse Fourier ∫ B transform of the transfer function h(t ) = exp[ j 2πf (t − t0 )]df (2.117) −B of Eq. (2.116) sin[ j 2πB (t − t0 )] h(t ) = π (t − t0 ) = 2 B sin c[2B(t - t 0 )] (2.118) We are able to build a causal filter that approximates an ideal low-pass filter,  with the approximation improving with increasing delay t0 sin c[2B(t - t 0 )] << 1, for t < 0 87
  88. 88.  Pulse Response of Ideal Low-Pass Filters  The impulse response of Eq.(2.118) and the response of the filter h(t ) = 2 B sin c(2 Bt ) (2.119) ∞ y (t ) = ∫ −∞ x(τ )h(t − τ )dτ ∫ T /2 = 2B sin c[2 B (t − τ )]dτ −T / 2  sin[ 2πB (t − τ )]  ∫ T /2 = 2B  dτ (2.120) −T / 2  2πB (t − τ )  λ = 2πB(t − τ ) 1  sin λ  2 πB ( t +T / 2 ) y (t ) = ∫  π 2πB ( t −T / 2 )  λ  dλ 1  2πB ( t +T / 2 )  sin λ  2 πB ( t −T / 2 ) sin λ    =  π0 ∫   λ  dλ − 0 ∫   dλ λ    1 = {Si[2πB (t + T / 2)] − Si[2πB (t − T / 2)]} (2.121) π 88
  89. 89.  Sine integral Si(u) sin x ∫ u Si(u ) = dx (2.122) 0 x  An oscillatory function of u, having odd symmetry about the origin u=0.  It has its maxima and minima at multiples of π.  It approaches the limiting value (π/2) for large positive values of u. 89
  90. 90.  The maximum value of Si(u) occurs at umax= π and is equal to π 1.8519 = (1.179) × ( ) 2 The filter response y(t) has maxima and minima at T 1 t max =± ± 2 2B 1 y (t max ) = [Si(π ) − Si(π − 2πBT )] π 1 = [Si(π ) + Si(2πBT − π )] π π Odd symmetric property of the sine integral Si(2πBT − π ) = (1 ± ∆ ) 2 1 Si(π ) = (1.179)(π / 2) y (t max ) = (1.179 + 1 ± ∆ ) 2 1 ≈ 1.09 ± ∆ (2.123) 2 90
  91. 91.  For BT>>1, the fractional deviation ∆ has a very small value  The percentage overshoot in the filter response is approximately 9 percent  The overshoot is practically independent of the filter bandwidth B 91
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  96. 96.  When using an ideal low-pass filter  We must use a time-bandwidth product BT>> 1 to ensure that the waveform of the filter input is recognizable from the resulting output.  A value of BT greater than unity tends to reduce the rise time as well as decay time of the filter pulse response. Approximation of Ideal Low-Pass Filters  The two basic steps involved in the design of filter 1. The approximation of a prescribed frequency response by a realizable transfer function 2. The realization of the approximating transfer function by a physical device.  the approximating transfer function H’(s) is a rational function H ( s ) = H ( f ) j 2πf = s ( s − z1 )( s − z 2 ) ⋅ ⋅ ⋅ ( s − z m ) Re([ pi ]) < 0, for all i =K ( s − p1 )( s − p2 ) ⋅ ⋅ ⋅ ( s − pn ) 96
  97. 97.  Minimum-phase systems  A transfer function whose poles and zeros are all restricted to lie inside the left hand of the s-plane.  Nonminimum-phase systems  Transfer functions are permitted to have zeros on the imaginary axis as well as the right half of the s-plane. Basic options to realization  Analog filter  With inductors and capacitors  With capacitors, resistors, and operational amplifiers  Digital filter  These filters are built using digital hardware  Programmable ; offering a high degree of flexibility in design 97
  98. 98. 2.8 Correlation and Spectral Density : Energy Signals Autocorrelation Function  Autocorrelation function of the energy signal x(t) for a large τ as ∞ Rx (τ ) = ∫ −∞ x(t ) x * (t − τ )dτ (2.124)  The energy of the signal x(t)  The value of the autocorrelation function Rx(τ) for τ=0 ∞ R x ( 0) = ∫ 2 x(t ) dt −∞ 98
  99. 99.  Energy Spectral Density  The energy spectral density is a nonnegative real-valued quantity for all f, even though the signal x(t) may itself be complex valued. Ψx ( f ) = X ( f ) 2 (2.125) Wiener-Khitchine Relations for Energy Signals  The autocorrelation function and energy spectral density form a Fourier- transform pair ∞ Ψx ( f ) = ∫ −∞ Rx (τ ) exp(− j 2πfτ )dτ (2.126) ∞ Rx (τ ) = ∫ −∞ Ψx ( f ) exp( j 2πfτ )df (2.127) 99
  100. 100. 1. By setting f=0  The total area under the curve of the complex-valued autocorrelation function of a complex-valued energy signal is equal to the real-valued energy spectral at zero frequency ∞ ∫−∞ Rx (τ )dτ = Ψx (0)2. By setting τ=0  The total area under the curve of the real-valued energy spectral density of an erergy signal is equal to the total energy of the signal. ∞ ∫−∞ Ψx ( f )d f = Rx (0) 100
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  102. 102.  Effect of Filtering on Energy Spectral Density  When an energy signal is transmitted through a linear time-invariant filter,  The energy spectral density of the resulting output equals the energy spectral density of the input multiplied by the squared amplitude response of the filter. Y( f ) = H ( f )X ( f ) Ψy ( f ) = H ( f ) Ψx ( f ) (2.129) 2  An indirect method for evaluating the effect of linear time-invariant filtering on the autocorrelation function of an energy signal 1. Determine the Fourier transforms of x(t) and h(t), obtaining X(f) and H(f), respectively. 2. Use Eq. (2.129) to determine the energy spectral density Ψy(f) of the output y(t). 3. Determine Ry(τ) by applying the inverse Fourier transform to Ψy(f) obtained under point 2. 102
  103. 103. Fig. 2.30 103
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  105. 105. Fig. 2.31 105
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  107. 107. Fig. 2.32 107
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  109. 109.  Interpretation of the Energy Spectral Density  The filter is a band-pass filter whose amplitude response is  ∆f ∆f 1, f c − ≤ f ≤ fc + H( f ) =  2 2 0, otherwise  (2.134)  The amplitude spectrum of the filter output Y( f ) = H( f ) X ( f )  ∆f ∆f  X ( fc ) , fc − ≤ f ≤ fc + = 2 2 0,  otherwise (2.135)  The energy spectral density of the filter output  ∆f ∆f  Ψx ( f c ), f c − ≤ f ≤ fc + Ψy ( f ) =  2 2 0,  otherwise (2.136) Fig. 2.33 109
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  111. 111.  The energy of the filter output is ∞ ∞ E y = ∫ Ψy ( f )df = 2 ∫ Ψy ( f )df −∞ 0 The energy spectral density of an energy signal for any frequency f  The energy per unit bandwidth, which is contributed by frequency components of the signal around the frequency f E y ≈ 2Ψx ( f c )∆f (2.137) Ey Ψx ( f c ) ≈ (2.138) 2∆f Fig. 2.33 111
  112. 112.  Cross-Correlation of Energy Signals  The cross-correlation function of the pair ∞ Rxy (τ ) = ∫ −∞ x(t ) y * (t − τ )dt (2.139)  The energy signals x(t) and y(t) are said to be orthogonal over the entire time domain  If Rxy(0) is zero ∞ ∫−∞ x(t ) y * (t )dt = 0 (2.140)  The second cross-correlation function ∞ Ryx (τ ) = ∫−∞ y (t ) x * (t − τ )dt (2.141) Rxy (τ ) = Ryx (−τ ) (2.142) * 112
  113. 113.  The respective Fourier transforms of the cross-correlation functions Rxy(τ) and Ryx(τ) ∞ ∫ Ψxy ( f ) = −∞ Rxy (τ ) exp(− j 2πfτ )dτ (2.143) ∞ Ψyx (f)=∫ Ryx (τ ) exp(− j 2πfτ )dτ (2.144) −∞ With the correlation theorem Ψxy ( f ) = X ( f )Y * ( f ) (2.145) Ψyx ( f ) = Y ( f ) X * ( f ) (2.146) The properties of the cross-spectral density 1. Unlike the energy spectral density, cross-spectral density is complex valued in general. 2. Ψxy(f)= Ψ*yx(f) from which it follows that, in general, Ψxy(f)≠ Ψyx(f) 113
  114. 114. 2.9 Power Spectral Density The average power of a signal is 1 T P = lim ∫ x(t ) dt (2.147) 2 T →∞ 2T −T P<∞  Truncated version of the signal x(t)  t  xT (t ) = x(t )rect   2T   x(t ), − T ≤ t ≤ T = 0, otherwise (2.148) xT (t ) ⇔ X T ( f ) 114
  115. 115.  The average power P in terms of xT(t) 1 ∞ P = lim ∫ 2 xT (t ) dt (2.149) T →∞ −∞ 2T ∞ ∞ ∫ xT (t ) dt = ∫ 2 2 X T ( f ) df −∞ −∞ 1 ∞ P = lim ∫ 2 X T ( f ) df (2.150) T →∞ −∞ 2T  ∞ 1 2 P= ∫  lim  − ∞ T → ∞ 2T X T ( f ) df  (2.151) Power spectral density or Power spectrum 1 S x ( f ) = lim 2 XT ( f ) (2.152) T →∞ 2T The total area under the curve of the power spectral density of a power signal is equal to the average power of that signal. ∞ P= ∫−∞ S x ( f )df (2.153) 115
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  118. 118. 2.10 Numerical Computation of the Fourier Transform The Fast Fourier transform algorithm  Derived from the discrete Fourier transform  Frequency resolution is defined by fs 1 1 ∆f = = = (2.160) N NTs T  Discrete Fourier transform (DFT) and inverse discrete Fourier transform of the sequence gn as g n = g (nTs ) (2.161)  j 2π  N −1 Gk = ∑ g n exp − kn , k = 0,1,..., N − 1 (2.162) n =0  N  1 N −1  j 2π  g n = ∑ Gk exp kn , n = 0,1,..., N − 1 (2.163) N k =0  N  118
  119. 119.  Interpretations of the DFT and the IDFT  For the interpretation of the IDFT process,  We may use the scheme shown in Fig. 2.34(b)  j 2π   2π   2π  exp kn  = cos kn  + j sin  kn   N  N  N    2π   2π  = cos kn , sin  kn , k = 0,1,..., N − 1 (2.164)  N   N   At each time index n, an output is formed by summing the weighted complex generator outputs Fig. 2.34 119
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  122. 122.  Fast Fourier Transform Algorithms  Be computationally efficient because they use a greatly reduced number of arithmetic operations  Defining the DFT of gn N −1 Gk = ∑ n =0 g nW nk , k = 0,1,..., N − 1 (2.165)  j 2π  W = exp −  (2.166)  N  W N = exp(− j 2π ) = 1 W N / 2 = exp(− jπ ) = −1 W ( k +lN )( n + mN ) = W kn , for m, l = 0,±1,±2,... N = 2L 122
  123. 123.  We may divide the data sequence into two parts ( N / 2 ) −1 N −1 Gk = ∑g W n =0 n nk + ∑g W n= N / 2 n nk ( N / 2 ) −1 ( N / 2 ) −1 = ∑n=0 g nW + nk ∑n =0 g n + N / 2W k ( n + N / 2 ) ( N / 2 ) −1 = ∑(g n =0 n + g n + N / 2W kN / 2 )W kn , k = 0,1,..., N − 1 (2.167) W kN / 2 = (−1) k  For the case of even k, k=2l, xn = g n + g n + N / 2 (2.168)  j 4π  W = exp − 2  ( N / 2 ) −1  N  ∑ N G2 l = xn (W 2 ) ln , l = 0,1,..., − 1 (2.169)  j 2π  n =0 2 = exp −   N /2 123
  124. 124.  For the case of odd k N k = 2l + 1, l = 0,1,..., −1 2 yn = g n − g n + N / 2 (2.170) ( N / 2 ) −1 G2 l +1 = ∑n =0 ynW ( 2 l +1) n ( N / 2 ) −1 ∑ N = [ ynW n ](W 2 ) ln , l = 0,1,..., − 1 (2.171) n =0 2 The sequences xn and yn are themselves related to the original data sequence Thus the problem of computing an N-point DFT is reduced to that of computing two (N/2)-point DFTs. Fig. 2.35 124
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  127. 127.  Important features of the FFT algorithm 1. At each stage of the computation, the new set of N complex numbers resulting from the computation can be stored in the same memory locations used to store the previous set.(in-place computation) 2. The samples of the transform sequence Gk are stored in a bit-reversed order. To illustrate the meaning of this latter terminology, consider Table 2.2 constructed for the case of N=8. Fig. 2.36 127
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  129. 129.  Computation of the IDFT  Eq. (2.163) may rewrite in terms of the complex parameter W N −1 ∑G W 1 gn = k − kn , n = 0,1,..., N − 1 (2.172) N k =0 N −1 Ng = * n ∑ k =0 Gk*W kn , 0,1,..., N − 1 (2.173) 129
  130. 130. 2.11 Theme Example : Twisted Pairs for Telephony Fig. 2.39  The typical response of a twisted pair with lengths of 2 to 8 kilometers  Twisted pairs run directly form the central office to the home with one pair dedicated to each telephone line. Consequently, the transmission lines can be quite long  The results in Fig. assume a continuous cable. In practice, there may be several splices in the cable, different gauge cables along different parts of the path, and so on. These discontinuities in the transmission medium will further affect the frequency response of the cable. Fig. 2.39 130
  131. 131.  We see that, for a 2-km cable, the frequency response is quite flat over the voice band from 300 to 3100 Hz for telephonic communication. However, for 8-km cable, the frequency response starts to fall just above 1 kHz.  The frequency response falls off at zero frequency because there is a capacitive connection at the load and the source. This capacitive connection is put to practical use by enabling dc power to be transported along the cable to power the remote telephone handset. It can be improved by adding some reactive loading  Typically 66 milli-henry (mH) approximately every 2 km.  The loading improves the frequency response of the circuit in the range corresponding to voice signals without requiring additional power. Disadvantage of reactive loading  Their degraded performance at high frequency Fig. 2.39 131
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  133. 133. 2.12 Summary and Discussion Fourier Transform  A fundamental tool for relating the time-domain and frequency-domain descriptions of a deterministic signal Inverse relationship  Time-bandwidth product of a energy signal is a constant Linear filtering  Convolution of the input signal with the impulse response of the filter  Multiplication of the Fourier transform of the input signal by the transfer function of the filter Correlation  Autocorrelation : a measure of similarity between a signal and a delayed version of itself  Cross-correlation : when the measure of similarity involves a pair of different signals 133
  134. 134.  Spectral Density  The Fourier transform of the autocorrelation function Cross-Spectral Density  The Fourier transform of the cross-correlation function Discrete Fourier Transform  Standard Fourier transform by uniformly sampling both the input signal and the output spectrum Fast Fourier transform Algorithm  A powerful computation tool for spectral analysis and linear filtering 134
  135. 135. Thank You !

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