Chapter 6 THE SAMPLING PROCESS 6.1 Introduction 6.2 Fourier Transform Revisited

Transcription

1 Chapter 6 THE SAMPLING PROCESS 6.1 Introduction 6.2 Fourier Transform Revisited Copyright c 2005 Andreas Antoniou Victoria, BC, Canada aantoniou@ieee.org July 14, 2018 Frame # 1 Slide # 1 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

2 Introduction Digital filters are often used to process discrete-time signals that have been generated by sampling continuous-time signals. Frame # 2 Slide # 2 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

3 Introduction Digital filters are often used to process discrete-time signals that have been generated by sampling continuous-time signals. Frequently digital filters are designed indirectly through the use of analog filters. Frame # 2 Slide # 3 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

4 Introduction Digital filters are often used to process discrete-time signals that have been generated by sampling continuous-time signals. Frequently digital filters are designed indirectly through the use of analog filters. In order to understand the basis of these techniques, the spectral relationships among continuous-time, impulse-modulated, and discrete-time signals must be understood. Frame # 2 Slide # 4 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

5 Introduction Digital filters are often used to process discrete-time signals that have been generated by sampling continuous-time signals. Frequently digital filters are designed indirectly through the use of analog filters. In order to understand the basis of these techniques, the spectral relationships among continuous-time, impulse-modulated, and discrete-time signals must be understood. These relationships are derived by using the Fourier transform, the Fourier series, the z transform, and Poisson s summation formula. Frame # 2 Slide # 5 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

6 Introduction Cont d Impulse-modulated signals comprise sequences of continuous-time impulse functions and to understand their significance, the properties of impulse functions must be understood. On the other hand, Poisson s summation formula is based on a relationship between the Fourier series and the Fourier transform of periodic signals. Frame # 3 Slide # 6 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

7 Introduction Cont d Impulse-modulated signals comprise sequences of continuous-time impulse functions and to understand their significance, the properties of impulse functions must be understood. On the other hand, Poisson s summation formula is based on a relationship between the Fourier series and the Fourier transform of periodic signals. This presentation begins with a review of the Fourier transform. Frame # 3 Slide # 7 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

8 Introduction Cont d Impulse-modulated signals comprise sequences of continuous-time impulse functions and to understand their significance, the properties of impulse functions must be understood. On the other hand, Poisson s summation formula is based on a relationship between the Fourier series and the Fourier transform of periodic signals. This presentation begins with a review of the Fourier transform. Then impulse functions are defined and their properties are examined. Frame # 3 Slide # 8 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

9 Introduction Cont d Impulse-modulated signals comprise sequences of continuous-time impulse functions and to understand their significance, the properties of impulse functions must be understood. On the other hand, Poisson s summation formula is based on a relationship between the Fourier series and the Fourier transform of periodic signals. This presentation begins with a review of the Fourier transform. Then impulse functions are defined and their properties are examined. Subsequently, the application of the Fourier transform to impulse functions and periodic signals is investigated. Frame # 3 Slide # 9 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

10 Review of Fourier Transform The Fourier transform of a continuous-time signal x(t) is defined as X (jω) = x(t)e jωt dt (A) Frame # 4 Slide # 10 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

11 Review of Fourier Transform The Fourier transform of a continuous-time signal x(t) is defined as X (jω) = x(t)e jωt dt (A) In general, X (jω) is complex and can be written as X (jω) = A(ω)e jφ(ω) where A(ω) = X (jω) and φ(ω) = arg X (jω) Frame # 4 Slide # 11 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

12 Review of Fourier Transform The Fourier transform of a continuous-time signal x(t) is defined as X (jω) = x(t)e jωt dt (A) In general, X (jω) is complex and can be written as X (jω) = A(ω)e jφ(ω) where A(ω) = X (jω) and φ(ω) = arg X (jω) Functions A(ω) and φ(ω) are the amplitude spectrum and phase spectrum of the signal, respectively. Frame # 4 Slide # 12 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

13 Review of Fourier Transform The Fourier transform of a continuous-time signal x(t) is defined as X (jω) = x(t)e jωt dt (A) In general, X (jω) is complex and can be written as X (jω) = A(ω)e jφ(ω) where A(ω) = X (jω) and φ(ω) = arg X (jω) Functions A(ω) and φ(ω) are the amplitude spectrum and phase spectrum of the signal, respectively. Together, the amplitude and phase spectrums constitute the frequency spectrum. Frame # 4 Slide # 13 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

14 Review of Fourier Transform Cont d X (jω) = x(t)e jωt dt (A) Function x(t) is the inverse Fourier transform of X (jω) and is given by x(t) = 1 X (jω)e jωt dω (B) 2π Frame # 5 Slide # 14 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

15 Review of Fourier Transform Cont d X (jω) = x(t)e jωt dt (A) Function x(t) is the inverse Fourier transform of X (jω) and is given by x(t) = 1 X (jω)e jωt dω (B) 2π Eqs. (A) and (B) can be written in operator format as respectively. X (jω) = Fx(t) and x(t) = F 1 X (jω) Frame # 5 Slide # 15 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

16 Review of Fourier Transform Cont d X (jω) = x(t)e jωt dt (A) Function x(t) is the inverse Fourier transform of X (jω) and is given by x(t) = 1 X (jω)e jωt dω (B) 2π Eqs. (A) and (B) can be written in operator format as respectively. X (jω) = Fx(t) and x(t) = F 1 X (jω) An alternative shorthand notation is x(t) X (jω) Frame # 5 Slide # 16 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

17 Convergence Theorem The convergence theorem of the Fourier transform states that if T lim x(t) dt < T T then the Fourier transform of x(t), X (jω), exists and its inverse can be obtained by using the equation x(t) = 1 2π X (jω)e jωt dω Frame # 6 Slide # 17 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

18 Convergence Theorem The convergence theorem of the Fourier transform states that if T lim x(t) dt < T T then the Fourier transform of x(t), X (jω), exists and its inverse can be obtained by using the equation x(t) = 1 2π X (jω)e jωt dω Many signals that are of considerable interest in practice violate the above condition, for example, impulse functions, impulse-modulated signals, and periodic signals. Frame # 6 Slide # 18 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

19 Convergence Theorem The convergence theorem of the Fourier transform states that if T lim x(t) dt < T T then the Fourier transform of x(t), X (jω), exists and its inverse can be obtained by using the equation x(t) = 1 2π X (jω)e jωt dω Many signals that are of considerable interest in practice violate the above condition, for example, impulse functions, impulse-modulated signals, and periodic signals. However, convergence problems can be circumvented by paying particular attention to the definition of impulse functions. Frame # 6 Slide # 19 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

20 Impulse Functions The unit impulse function has been defined in the past as { 1 δ(t) = lim p τ (t) = lim τ for t τ/2 τ 0 τ 0 0 otherwise Frame # 7 Slide # 20 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

21 Impulse Functions The unit impulse function has been defined in the past as { 1 δ(t) = lim p τ (t) = lim τ for t τ/2 τ 0 τ 0 0 otherwise Obviously, this is an infinitesimally thin, infinitely tall pulse whose area is equal to unity for any finite value of τ. Frame # 7 Slide # 21 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

22 Impulse Functions Cont d Pulse function p τ (t) for three values of τ: τ = 0.50 τ = 0.10 τ = p τ (t) t 1 (a) Frame # 8 Slide # 22 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

23 Mathematical Problem The Fourier transform of the unit impulse function as defined in the past should be given by the integral X (jω) = = x(t)e jωt dt lim [ p τ (t)]e jωt dt τ 0 where p τ (t) = { 1 τ for t τ/2 0 otherwise Frame # 9 Slide # 23 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

24 Mathematical Problem The Fourier transform of the unit impulse function as defined in the past should be given by the integral X (jω) = = x(t)e jωt dt lim [ p τ (t)]e jωt dt τ 0 where p τ (t) = { 1 τ for t τ/2 0 otherwise If we now attempt to evaluate the function p τ (t)e jωt at τ = 0, we find that it becomes infinite and, therefore, the above integral cannot be evaluated. Frame # 9 Slide # 24 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

25 Mathematical Problem Cont d We can write F lim τ 0 p τ (t) = τ/2 lim [ p τ (t)]e jωt dt τ 0 [ ] 1 lim dt τ τ/2 τ 0 Frame # 10 Slide # 25 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

26 Mathematical Problem Cont d We can write F lim τ 0 p τ (t) = τ/2 lim [ p τ (t)]e jωt dt τ 0 [ ] 1 lim dt τ τ/2 τ 0 Since the area of the pulse function p τ (t) is unity for any finite value of τ, we might be tempted to assume that the area is equal to unity even for τ = 0, i.e., F lim τ 0 p τ (t) = 1 Frame # 10 Slide # 26 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

27 Mathematical Problem Cont d The Fourier transform of p τ (t) for a finite τ is given by F p τ (t) = 1 τ Fp τ (t) = 2 sin ωτ/2 ωτ Frame # 11 Slide # 27 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

28 Mathematical Problem Cont d The Fourier transform of p τ (t) for a finite τ is given by F p τ (t) = 1 τ Fp τ (t) = 2 sin ωτ/2 ωτ Obviously, this is well defined and, interestingly, it has the limit lim F p τ (t) = 1 τ 0 Frame # 11 Slide # 28 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

29 Mathematical Problem Cont d The Fourier transform of p τ (t) for a finite τ is given by F p τ (t) = 1 τ Fp τ (t) = 2 sin ωτ/2 ωτ Obviously, this is well defined and, interestingly, it has the limit So far so good! lim F p τ (t) = 1 τ 0 Frame # 11 Slide # 29 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

30 Mathematical Problem Cont d If we now attempt to find the inverse Fourier transform of 1, we run into certain mathematical difficulties. Frame # 12 Slide # 30 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

31 Mathematical Problem Cont d If we now attempt to find the inverse Fourier transform of 1, we run into certain mathematical difficulties. From the definition of the inverse Fourier transform, we have F 1 1 = 1 2π [ = 1 2π e jωt dω cos ωt dω + j ] sin ωt dω Frame # 12 Slide # 31 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

32 Mathematical Problem Cont d If we now attempt to find the inverse Fourier transform of 1, we run into certain mathematical difficulties. From the definition of the inverse Fourier transform, we have F 1 1 = 1 2π [ = 1 2π e jωt dω cos ωt dω + j ] sin ωt dω However, mathematicians will tell us that these integrals do not converge or do not exist! Frame # 12 Slide # 32 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

33 Mathematical Problem Cont d Summarizing, by cheating a little bit we can get a more or less meaningful Fourier transform for the unit impulse function. Frame # 13 Slide # 33 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

34 Mathematical Problem Cont d Summarizing, by cheating a little bit we can get a more or less meaningful Fourier transform for the unit impulse function. Unfortunately, it is impossible to recover the impulse function from its Fourier transform by applying the inverse Fourier transform. Frame # 13 Slide # 34 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

35 Mathematical Problem Cont d The impulse-function problem can be circumvented in two ways, a practical and a theoretical one: Frame # 14 Slide # 35 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

36 Mathematical Problem Cont d The impulse-function problem can be circumvented in two ways, a practical and a theoretical one: The practical approach is easy to understand and apply but it lacks rigor. Frame # 14 Slide # 36 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

37 Mathematical Problem Cont d The impulse-function problem can be circumvented in two ways, a practical and a theoretical one: The practical approach is easy to understand and apply but it lacks rigor. The theoretical approach is rigorous but it is rather abstract and more difficult to understand or apply in practical situations. Frame # 14 Slide # 37 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

38 Practical Approach to Impulse Functions In the practical approach to impulse functions, a function γ(t) is said to be a unit impulse function if, for any continuous function x(t) over the range ɛ < t < ɛ, the following relation is satisfied: γ(t)x(t) dt x(0) (C) Frame # 15 Slide # 38 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

39 Practical Approach to Impulse Functions In the practical approach to impulse functions, a function γ(t) is said to be a unit impulse function if, for any continuous function x(t) over the range ɛ < t < ɛ, the following relation is satisfied: γ(t)x(t) dt x(0) (C) The special symbol is used to signify that the two sides can be made to approach one another to any desired degree of precision but cannot be made exactly equal. Frame # 15 Slide # 39 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

40 Practical Approach to Impulse Functions In the practical approach to impulse functions, a function γ(t) is said to be a unit impulse function if, for any continuous function x(t) over the range ɛ < t < ɛ, the following relation is satisfied: γ(t)x(t) dt x(0) (C) The special symbol is used to signify that the two sides can be made to approach one another to any desired degree of precision but cannot be made exactly equal. Now consider the pulse function lim p τ (t) = p ɛ (t) = τ ɛ { 1 ɛ for t ɛ/2 0 otherwise where ɛ is a small but finite constant. Frame # 15 Slide # 40 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

41 Practical Approach Cont d γ(t)x(t) dt x(0) (C) If we let γ(t) = lim p τ (t) τ ɛ in the left-hand side of Eq. (C), we obtain lim [ p τ (t)]x(t) dt = τ ɛ ɛ/2 ɛ/2 1 x(t) dt ɛ 1 ɛ x(0) ɛ/2 ɛ/2 dt x(0) Frame # 16 Slide # 41 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

42 Practical Approach Cont d γ(t)x(t) dt x(0) (C) If we let γ(t) = lim p τ (t) τ ɛ in the left-hand side of Eq. (C), we obtain lim [ p τ (t)]x(t) dt = τ ɛ ɛ/2 ɛ/2 1 x(t) dt ɛ 1 ɛ x(0) ɛ/2 ɛ/2 dt x(0) Thus we conclude that the very thin pulse function limτ ɛ p τ (t) behaves as an impulse function and, therefore, we can write δ(t) = lim τ ɛ p τ (t) Frame # 16 Slide # 42 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

43 Practical Approach Cont d δ(t) = lim τ ɛ p τ (t) Now if we apply the Fourier transform to the impulse function as defined, we get lim p 2 sin ωτ/2 τ (t) lim τ ɛ τ ɛ ωτ Frame # 17 Slide # 43 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

44 - Practical Approach Cont d lim p 2 sin ωτ/2 τ (t) lim τ ɛ τ ɛ ωτ As τ is reduced, the pulse function at the left tends to become thinner and taller whereas the sinc function at the right tends to be flattened out ǫ = 0.50 ǫ = 0.10 ǫ = δω p ǫ (t) t 1 (a) sin (ωǫ/2)/(ωǫ/2) ω ω 2 ǫ = ǫ = 0.10 ǫ = ω (b) Frame # 18 Slide # 44 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

45 Practical Approach Cont d For some small but finite ɛ, the sinc function will be equal to unity to within an error δ ω over some frequency range ω /2 < ω < ω / δω 0.8 sin (ωǫ/2)/(ωǫ/2) ω ω 2 ǫ = ǫ = 0.10 ǫ = ω (b) Frame # 19 Slide # 45 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

46 Practical Approach Cont d Therefore, we can write 2 sin ωτ/2 δ(t) = lim p τ (t) lim = i(ω) τ ɛ τ ɛ ωτ where i(ω) may be referred to as a frequency-domain unity function. Frame # 20 Slide # 46 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

47 Practical Approach Cont d Summarizing, the Fourier transform of a time-domain impulse function is a frequency-domain unity function, and the inverse Fourier transform of a frequency-domain unity function is a time-domain impulse function, Frame # 21 Slide # 47 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

48 Practical Approach Cont d Summarizing, the Fourier transform of a time-domain impulse function is a frequency-domain unity function, and the inverse Fourier transform of a frequency-domain unity function is a time-domain impulse function, i.e., δ(t) i(ω) Frame # 21 Slide # 48 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

49 Practical Approach Cont d Summarizing, the Fourier transform of a time-domain impulse function is a frequency-domain unity function, and the inverse Fourier transform of a frequency-domain unity function is a time-domain impulse function, i.e., δ(t) i(ω) Since i(ω) 1 for the frequency range of interest, we can write δ(t) 1 where the wavy double arrow signifies that the relation is approximate with the understanding that it can be made as exact as desired by making ɛ sufficiently small. Frame # 21 Slide # 49 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

50 Practical Approach Cont d The impulse and unity functions can be represented by the idealized graphs: δ(t) i(ω) 1 1 t (a) ω Frame # 22 Slide # 50 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

51 Properties of Impulse Functions Assuming that x(t) is a continuous function of t over the range ɛ < t < ɛ, the following relations apply: (a) δ(t τ)x(t) dt = δ( t + τ)x(t) dt x(τ) (b) δ(t τ)x(t) = δ( t + τ)x(t) δ(t τ)x(τ) (c) δ(t)x(t) = δ( t)x(t) δ(t)x(0) (See textbook for proofs.) Frame # 23 Slide # 51 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

52 Frequency-Domain Impulse Functions Given a transform pair δ(t) i(ω) where δ(t) = lim τ ɛ p τ (t) 2 sin ωτ/2 i(ω) = lim 1 τ ɛ ωτ the corresponding transform pair for ω < ω i(t) 2πδ(ω) where 2 sin tɛ/2 i(t) = 1 tɛ δ(ω) = p ɛ (ω) for t < t can be generated by applying the symmetry theorem of the Fourier transform. Frame # 24 Slide # 52 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

53 Frequency-Domain Impulse Functions Cont d i(t) 2πδ(ω) Function i(t) is a time-domain unity function whereas δ(ω) is a frequency-domain unit impulse function. Frame # 25 Slide # 53 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

54 Frequency-Domain Impulse Functions Cont d i(t) 2πδ(ω) Function i(t) is a time-domain unity function whereas δ(ω) is a frequency-domain unit impulse function. Since i(t) 1, we have 1 2πδ(ω) Frame # 25 Slide # 54 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

55 Frequency-Domain Impulse Functions Cont d i(t) 2πδ(ω) or 1 2πδ(ω) This transform pair can be represented by the idealized graphs shown. i(t) 1 δ(ω) 2π t (b) ω Frame # 26 Slide # 55 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

56 Properties of Frequency-Domain Impulse Functions Assuming that X (jω) is a continuous function of ω over the range ɛ < ω < ɛ, the following relations apply: (a) = δ(ω ϖ)x (jω) dt δ( ω + ϖ)x (jω) dt X (jϖ) (b) δ(ω ϖ)x (jω) = δ( ω + ϖ)x (jω) δ(t ϖ)x (jϖ) (c) δ(ω)x (jω) = δ( ω)x (jω) δ(t)x (0) (See textbook for details.) Frame # 27 Slide # 56 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

57 Fourier Transforms of Exponentials Since δ(t) i(ω) the application of the time-shifting theorem gives δ(t t 0 ) i(ω)e jωt 0 and since i(ω) 1, we get δ(t t 0 ) e jωt 0 Frame # 28 Slide # 57 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

58 Fourier Transforms of Exponentials Since δ(t) i(ω) the application of the time-shifting theorem gives and since i(ω) 1, we get δ(t t 0 ) i(ω)e jωt 0 δ(t t 0 ) e jωt 0 Now applying the frequency-shifting theorem to the frequency-domain impulse function, we obtain and since i(t) 1, we get i(t)e jω 0t 2πδ(ω ω 0 ) e jω 0t 2πδ(ω ω 0 ) Frame # 28 Slide # 58 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

59 Fourier Transforms of Sinusoidal Signals We know that and i(t)e jω 0t 2πδ(ω ω 0 ) i(t)e jω 0t 2πδ(ω + ω 0 ) Frame # 29 Slide # 59 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

60 Fourier Transforms of Sinusoidal Signals We know that and i(t)e jω 0t 2πδ(ω ω 0 ) i(t)e jω 0t 2πδ(ω + ω 0 ) If we add the two equations, we get i(t)(e jω0t + e jω0t ) = 2i(t) cos ω 0 t 2π[δ(ω + ω 0 ) + δ(ω ω 0 )] and since i(t) 1, we have cos ω 0 t π[δ(ω + ω 0 ) + δ(ω ω 0 )] Frame # 29 Slide # 60 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

61 Fourier Transforms of Sinusoidal Signals Cont d cos ω 0 t π[δ(ω + ω 0 ) + δ(ω ω 0 )] x(t) X(jω) π t ω 0 ω 0 ω Frame # 30 Slide # 61 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

62 Fourier Transforms of Sinusoidal Signals Cont d As before, and i(t)e jω 0t 2πδ(ω ω 0 ) i(t)e jω 0t 2πδ(ω + ω 0 ) Frame # 31 Slide # 62 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

63 Fourier Transforms of Sinusoidal Signals Cont d As before, and i(t)e jω 0t 2πδ(ω ω 0 ) i(t)e jω 0t 2πδ(ω + ω 0 ) If we subtract the top equation from the bottom one, we have i(t)(e jω0t e jω0t ) = 2ji(t) sin ω 0 t 2π[δ(ω + ω 0 ) δ(ω ω 0 )] and since i(t) 1, we can write sin ω 0 t jπ[δ(ω + ω 0 ) δ(ω ω 0 )] Frame # 31 Slide # 63 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

64 Fourier Transforms of Periodic Signals An arbitrary periodic signal can be represented by the Fourier series x(t) = X k e jkω 0t k= Frame # 32 Slide # 64 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

65 Fourier Transforms of Periodic Signals An arbitrary periodic signal can be represented by the Fourier series x(t) = X k e jkω 0t k= Hence F x(t) = 2πX k Fe jkω0t 2πX k δ(ω kω 0 ) k= k= or x(t) 2π X k δ(ω kω 0 ) k= Frame # 32 Slide # 65 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

66 Fourier Transforms of Periodic Signals Cont d x(t) 2π X k δ(ω kω 0 ) k= Summarizing, the frequency spectrum of a periodic signal can be represented by an infinite sequence of numbers X k for < k <, i.e., the Fourier-series coefficients as shown in Chap. 2 Frame # 33 Slide # 66 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

67 Fourier Transforms of Periodic Signals Cont d x(t) 2π X k δ(ω kω 0 ) k= Summarizing, the frequency spectrum of a periodic signal can be represented by an infinite sequence of numbers X k for < k <, i.e., the Fourier-series coefficients as shown in Chap. 2 or Frame # 33 Slide # 67 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

68 Fourier Transforms of Periodic Signals Cont d x(t) 2π X k δ(ω kω 0 ) k= Summarizing, the frequency spectrum of a periodic signal can be represented by an infinite sequence of numbers X k for < k <, i.e., the Fourier-series coefficients as shown in Chap. 2 or by an infinite sequence of frequency-domain impulse functions of strength 2πX k for < k < as shown in the previous slide. Frame # 33 Slide # 68 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

69 Theoretical Approach to Impulse Functions The Fourier transform pairs generated through the practical approach are approximate since the pulse width ɛ cannot be reduced to absolute zero. Frame # 34 Slide # 69 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

70 Theoretical Approach to Impulse Functions The Fourier transform pairs generated through the practical approach are approximate since the pulse width ɛ cannot be reduced to absolute zero. However, by defining impulse functions in terms of generalized functions, analogous, but exact, Fourier transform pairs can be generated. Frame # 34 Slide # 70 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

71 Theoretical Approach to Impulse Functions The Fourier transform pairs generated through the practical approach are approximate since the pulse width ɛ cannot be reduced to absolute zero. However, by defining impulse functions in terms of generalized functions, analogous, but exact, Fourier transform pairs can be generated. Unfortunately, impulse functions so defined are rather impractical and difficult to implement in a laboratory. Frame # 34 Slide # 71 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

72 Theoretical Approach to Impulse Functions The Fourier transform pairs generated through the practical approach are approximate since the pulse width ɛ cannot be reduced to absolute zero. However, by defining impulse functions in terms of generalized functions, analogous, but exact, Fourier transform pairs can be generated. Unfortunately, impulse functions so defined are rather impractical and difficult to implement in a laboratory. See textbook for more details and references on generalized functions. Frame # 34 Slide # 72 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

73 Summary of Fourier Transforms Derived x(t) X (jω) δ(t) 1 1 2πδ(ω) δ(t t 0 ) e jωt 0 e jω 0t 2πδ(ω ω 0 ) cos ω 0 t π[δ(ω + ω 0 ) + δ(ω ω 0 )] sin ω 0 t jπ[δ(ω + ω 0 ) δ(ω ω 0 )] Frame # 35 Slide # 73 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2

74 This slide concludes the presentation. Thank you for your attention. Frame # 36 Slide # 74 A. Antoniou Digital Signal Processing Secs. 6.1, 6.2