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1 CHAPTER 4 THE CONTINUOUS-TIME FOURIER TRANSFORM Outlines 2 Covers O & W pp. 284-309 Derivation of the CT Fourier Transform pair Examples of Fourier Transforms Fourier Transforms of Periodic Signals Properties of the CT Fourier Transform Fourier Transform 3 We have shown that Fourier series are useful in analyzing periodic signals, but many (most) signals are aperiodic. Need a more general tool –– Fourier transform. Fourier’s own derivation of the CT Fourier transform x(t) – an aperiodic signal – view it as the limit of a periodic signal as T → ∞ The harmonic components are spaced ωo = 2π/T apart, as T → ∞, and ωo → 0, then ω = kωo becomes continuous ⇓ Fourier series Fourier integral Motivating Examples: Square wave 4 So, on with the derivation of FT … 5 Derivation (continued) 6 Derivation (continued) 7 With CTFT, now the Frequency Response of an LTI System makes complete sense 8 (s = j) ejt H(j) H(j)ejt CT Frequency response: H j ht e jt dt H(j) is the FT of h(t) For what finds of signals can we do this? 9 It works also even if x(t) is infinite duration, but satisfies: a) Finite energy xt 2 dt In this case, there is zero energy in the error 1 et x t X j e d et dt 0 jt 2 Then 2 b) Dirichlet conditions (including) (i) x t dt 2 x t at points of continuity 1 X j e jt d 1 xt 0 xt 0 2 2 midpoint at discontinuity (ii) Gibb’s phenomenon For what finds of signals can we do this? (Cont.) 10 c) By allowing impulses in x(t) or in X(j) , we can represent even more signals E.g. It allows us to consider FT for periodic signals Ex #1 11 Find the Fourier Transform for the signal (a) x(t) = (t) X j t e jt dt 1 1 t e jt d - Synthesis equation for t 2 (b) x(t) = (t – t0) X j t t0 e jt dt e jt0 - Linear phase shift in Ex #2: Exponential function 12 Find the Fourier Transform for the signal x(t) = e-atu(t), a > 0 X j xt e jt dt e ate jt dt e-(a+j)t 1 a j t 1 ( )e a j 0 a j Ex # 3 13 Find the Fourier Transform for the signal xt e a t ,a 0 Solution X j e a t e jt dt 0 e at e jt dt e ate jt dt 0 Ex #3 (continued) 14 1 1 X j a j a j 2a 2 a 2 Ex #4: A square pulse in the time-domain 15 Determine the Fourier Transform of rectangular pulse signal 1 t T1 x t 0 t T1 Fourier transform of this signal is T1 X j x(t )e jt dt T1 T1 (1)e jt dt T1 sin T1 2 Ex #5 16 Determine the x(t) whose Fourier Transform is 1 W X j 0 W Solution 1 xt X j e jt d 2 W 1 sin Wt xt e d t jt 2 W The sinc function 17 Definition sin sinc This function arise frequently in Fourier Transform and in study of LTI systems. The sinc function (Cont.) 18 The signal in the ex# 4 and ex#5 can be express in the form of sinc function sin sinc Ex#4 T1 sin X j 2 sin T1 2T1 T 2T1sinc T1 1 Ex #5 Wt sin sin Wt W W Wt xt sinc t Wt Example #4: (continued) 19 Note the inverse relation between the two widths Uncertainty principle Useful facts about CTFT’s 20 From ex# 5, we see that if W increases, X(jω) becomes broader, while the main peak of x(t) at t=0 become higher and the width of the first lobe of this signal becomes narrower. If W→∞, X(jω) = 1 for all ω. The x(t) converges to impulse as in ex#1. Example #5 A Gaussian xt e at2 21 X j e at2 jt e dt 2 j 2 j 2 a t j t a e a 2a 2a dt a t j 2 2 e 2a dt e 4 a a 2 e 4a a CT Fourier Transforms of Periodic Signals 22 Suppose X(j) = ( - 0) 1 1 j0t x t 0 e d 2 e - Periodic in t with jt 2 frequency 0 That is e j0t 2 - 0 - All the energy is concentrated in More generally, if x(t) = x(t+T), then one frequency 0 xt a e k jk0t X j 2a k k 0 k k Example #6: 23 xt a e k jk0t X j 2a k k 0 k k 1 j0t 1 j0t x t cos 0t e e a(1) = ½ 2 2 a(-1) = ½ X j 0 0 Example #7 24 Find Fourier transform of the signal solution The Fourier series coefficients are sin k0T1 ak k sin k0T1 jk0t xt e k k 25 sin k0T1 jk0t xt e k k and the Fourier transform of this signal is xt a e k jk0t X j 2a k k 0 k k 2 sin k0T1 X j k0 k k For T = 4T1, In comparison with Example 3.5(a), the only differences are a proportionality factor 2π and the use of impulses rather than a bar graph. 26 FT 2 sin k0T1 X j k0 k k For T = 4T1, In comparison, the only differences are a proportionality factor 2π and the use of impulses rather than a bar graph. FS sin k0T1 jk0t xt e k k Example #8 27 Find Fourier transform of the signal x(t) = sin0t Solution The Fourier series coefficients are 1 1 a1 a1 2j 2j ak = 0, k ≠ 1 or −1 28 The Fourier transform are xt a e k jk0t X j 2a k k 0 k k X j 0 0 j j Example #9 Sampling function 29 x t t nT - Sampling function n T 2 1 1 x t ak 2xt e dt T jk0t T T 1 jk0t x t a e k jk0t e k k T 30 1 jk0t xt e xt X j T k 2 k 2 X j k T T k0 Properties of the CT Fourier Transform 1) Linearity 2) Time Shifting 3) Conjugate and conjugate Symmetry 4) Time-Scaling 5) Differentiation and Integration 6) Duality Properties of the CT Fourier Transform 32 1) Linearity ax(t) + by(t) aX(j) + bY(j) 2) Time Shifting x(t-t0) e jt0 X j Proof: x t t0 e jt dt e jt x t e jtdt 0 t FT magnitude unchanged X(j) e j0t X j X j Linear change in FT phase e j0t X j X j 0t a phase shift Linear with Example #10 33 Calculate Fourier Transform of x(t) 1 x t x1 t 2.5 x2 t 2.5 2 34 sin 2 12 X 1 j e jt dt 2 1 2 sin3 2 32 X 2 j e jt dt 2 3 2 sin 2 2 sin3 2 X j e j5t 2 35 12 1 X 1 j x t 2.5e jt dt 1 2 2 Let t = t – 2.5 12 12 1 1 j 5 2 X 1 j x t e j t 2.5 dt e 2xt e jtdt 1 2 2 2 1 j 5 2 sin 2 12 1 j 5 2 2de e jt e j 2 1 The Properties Keep on Coming… 36 3) Conjugate and conjugate Symmetry if x(t) X(j) then x*(t) X*(-j) 37 Conjugate symmetry If x(t) is real then X(jω) has conjugate symmetry x(t) real X(-j) = X*(j) If we express X(j) in rectangular form as X(j) = ℜe[X(j)] + ℑm[X(j)] 38 then if x(t) is real ℜe[ X(j)] = ℜe[X(−j)] and ℑm[X(j)] = −ℑm[X(−j)] The real part of Fourier transform is an even function of frequency and the imaginary part is an odd function of frequency 39 If we express in polar form as X(jω) = ∣X(jω)∣ej∢X(jω) |X(jω)| is an even function of ω and ∢X(jω) is an odd function of ω Thus, when computing the Fourier transform of a real signal, the real and imaginary parts or magnitude and phase of the transform need only be specified for positive frequencies, as the values for negative frequencies can be determined directly from the values for ω > 0 using the relations above. 40 If x(t) is real then it can always be expressed in terms of the sum of an even function and an odd function. x(t) = xe(t) + xo(t) From the linearity property F{x(t)} = F{xe(t)} + F{xo(t)} F{xe(t)} is a real function F{xo(t)} is purely imaginary 41 With x(t) real, we can conclude that x(t) ↔ X(j) Ev{x(t)} ↔ ℜe{X(j)} Od{x(t)} ↔ jℑm{X(j)} Example #11 42 Find The Fourier transform of the signal x(t) = e−a∣t∣ , a > 0 by using the symmetry properties solution From Example #2, we have 1 e u t at F a j 43 x(t) = e−a∣t∣ = e−atu(t) + eatu(−t) e atu t e atu t 2 2 2 Ev e atu t Since e-atu(t) is real 1 Ev e u t at e F a j It follow that 1 2a X j 2e 2 a j a 2 44 4) Time-Scaling x at 1 X j a a a = -1 x(-t) X(-j) a) x(t) real and even x(t) = x(-t) = x*(t) X(j) = X(-j) = X*(j) - Real & even b) x(t) real and odd x(t) = -x(-t) = x*(t) X(j) = -X(-j) = -X*(j) – Purely imaginary & odd 45 c) X(j) = Re{X(j)} + Im{X(j)} For real x(t) = Ev{x(t)} + Od{x(t)} 46 5) Differentiation and Integration x(t) X(j) dx t 1 jX j e jt d dt 2 dx t Differentiation jX j dt t 1 x d j X j X 0 Integration Example #12 47 Determine the Fourier transform X(j) of the unit step x(t) = u(t) g t t G j 1 t x t g d X j x t e jt dt Take FT of both sides G j e jt dt X j G 0 j 0 1 1 G 0 j j Example #13 48 Calculate FT X(j) for the signal x(t) d g t x t dt 49 2 sin G j e j e j Using integration property G j X j G0 j with G(0)= 0 2 sin 2 cos X j j 2 j 50 6) Duality 1, t T 1 2 sin T1 x1 t X 1 j 0, t T 1 sin Wt 1, W x2 t X 2 j t 0, W 51 FT of sinc function 52 jWt W sin Wt 1 e jWt e 1 jt x t e t t 2j 2tj W W 1 1 x t e d 2 jt H j e jt d 2 W x(t) W X(j) 1 -W W W W The CT Fourier Transform Pair 53 x(t) X(j) - FT X j x t e jt dt (Analysis Equation) 1 - Inverse FT x t X j e d (Synthesis Equation) jt 2 Example #14 54 Find the Fourier transform of the signal g(t) by using duality property 2 g t 1 t 2 solution We consider the signal x(t) whose Fourier transform is 1 X j 1 2 55 From example #3 , with a = 1 2 xt e X j t F 1 2 The synthesis equation for this Fourier transform pair is 1 2 jt 1 2 e d t e 2 56 Multiplying this equation by 2π and replacing t by -t, we obtain : 2 jt t 2e 2 e d 1 Interchanging the name of variables t and ω 2 jt 2e 2 e dt 1 t Right-hand side is the Fourier transform of g(t) 2 F 2 2e 1 t Parseval 's Relation 57 If x(t) and X(jω) are Fourier transform pair 1 xt dt 2 X j 2 2 d Example #15 58 Evaluate the following expression d xt D xt 2 E dt dt t 0 by using Fourier transform in the figure below 59 Evaluate E infrequency domain 1 X j 2 E d 2 which evaluate to 5/8 for figure (a) and to 1 for figure (b) Evaluate D infrequency domain 60 Noting that We conclude which evaluate to zero for figure (a) and to -1/(2√π) for figure (b) The Convolution Property 61 If Then 62 This property is important property in signals and system. As expressed in the equation, the Fourier transform maps the convolution of two signals into the product of their Fourier transforms. H(jω), is the Fourier transform of the impulse response, is the frequency response and capture the change in complex amplitude of the Fourier transform of the input at each frequency ω. The frequency response H(jω) plays as important a role in the analysis of LTI systems. 63 Many of the properties of LTI systems can be conveniently interpreted in the term of H(jω). For example, 64 The convergence of the Fourier transform is guaranteed only under certain conditions, and consequently, the frequency response cannot be defined for every LTI system. If, however, an LTI system is stable, then its impulse response is absolutely integratble; that is , Example #16 65 Consider the LTI system with impulse response H(t) = (t−t0) The frequency response is H ( j) = e− jt0 For any input x(t) with Fourier transform H(jω), the Fourier transform of the output is Example #17 66 Consider the LTI system for which the input x(t) and the output y(t) are related by From the differentiation property The frequency response of the differentiator is Example #18 67 Consider the LTI system for which the input x(t) and the output y(t) are related by t y t x d The impulse response for this system is u(t) and the frequency response of the system is 68 By using convolution property Example #19 69 Consider Ideal low pass filter which have the frequency response 1, c H j 0, c 70 From example #5, the impulse response h(t) of this ideal filter is Example #20 71 Determine the response of an LTI system with impulse response to the input signal 72 solution Transform the signal into the frequency domain. From Example #2, the Fourier transform of x(t) and h(t) are Therefore 73 Assuming that a ≠ b, the partial fraction expansion for Y(jω) takes the form We find that Therefore 74 The output can find by inverse Fourier transform If a = b Recognizing this as 75 We can use the dual of the differentiation property and the consequently Example #21 76 Determine the response of an ideal lowpass filter to an input the impulse response of the ideal lowpass filter is 77 solution The output is the convolution of 2 sinc functions 78 Therefore The inverse Fourier transform of Y(jω) is The Multiplication Property 79 The multiplication in time domain corresponds to convolution in frequency domain Sometime referred to as the Modulation property Example #22 80 Find the spectrum R(jω) of r(t) = s(t)p(t) when p(t) = cosω0t and 81 solution P j 0 0 1 R j S j P j d 2 1 1 S j 0 S j 0 2 2 82 Example #23 83 Find the spectrum G(jω) when g(t) = r(t)p(t) and r(t) and p(t) are the signals from Example 22 solution By using linearity property to the spectrum R(jω) 84 Example #24 85 Find the Fourier transform of the signal x(t) solution The key is to recognize x(t) as the product of two sinc functions 86 Applying the multiplication property The Fourier transform of each function is a rectangular pulse, we can proceed to convolution those pulses to obtain the X(jω) 87 88 Systems Characterized by Linear Constant- Coefficient Differential quations 89 A particularly important and useful class of continuous- time LTI system is those for which the input and output satisfy a linear constant coefficient differential equation of the form 90 By the convolution property Y(jω) = H(jω)X(jω) or where X(jω), Y(jω) and H(jω) are the Fourier transforms of the input x(t), output y(t) and impulse response h(t). 91 consider applying the Fourier transform to the equation in slide 89 from linear property 92 and from the differentiation property Or 93 Thus Observe that H(jω) is thus a rational function; that is, it is a ratio of polynomials in (jω) and the frequency response for the LTI system can be written directly by inspection Example #25 94 Find the impulse response of the LTI system with a>0 Solution Fourier transform of the system is jY(j) + aY(j) = X(j) 95 From Example #2, the inverse Fourier Transform of equation above is Example #26 96 Find the impulse response of the LTI system Solution The frequency response is 97 We factor the denominator of the right-hand side By using the partial-fraction expansion The inverse Fourier transform of each term Example #27 98 Consider the system of Example 26, find the output of the system when the input is x(t) = e−tu(t) Solution 99 By using the partial-fraction expansion 100 The inverse Fourier transform of each term

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Fourier transform, Time Fourier, Fourier series, Signals and Systems, Frequency response, Discrete-time Fourier transform, Periodic Signals, periodic signal, LTI system, Continuous-Time Signals, continuous time, Discrete Fourier Transform, impulse response, Discrete-Time Signals, frequency domain, Digital Signal Processing, sinc function, Discrete Time, discrete-time Fourier, sampling rate, Time domain, Discrete-Time Systems, Laplace Transform, series representation, Fourier transforms, for Real

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Fourier transform, Time Fourier, Fourier series, Signals and Systems, Frequency response, Discrete-time Fourier transform, Periodic Signals, periodic signal, LTI system, Continuous-Time Signals, continuous time, Discrete Fourier Transform, impulse response, Discrete-Time Signals, frequency domain, Digital Signal Processing, sinc function, Discrete Time, discrete-time Fourier, sampling rate, Time domain, Discrete-Time Systems, Laplace Transform, series representation, Fourier transforms, for Real,

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