Bài giảng Signal & Systems - Chapter 4: Fourier transform representation of signal

Nội dung text: Bài giảng Signal & Systems - Chapter 4: Fourier transform representation of signal

1. Ch-4: Fourier transform representation of signal P4.1 . Use the Fourier transform analysis equation to calculate the Fourier transform of the following signals: a) f(t)=e−2(t − 1) u(t− 1) b) f(t)=e −2|t − 1| d c) f(t)= δ(t+ 1) +δ(t − 1) d) f(t)= [u(-2-t)+u(t-2)] dt Sketch and label the magnitude of each Fourier transform. P4.2 . Determine the Fourier transform of each of the following periodic signals: π π π π a) f(t)=sin(2 t+4 ) b) f(t)=1+cos(6 t+8 ) P4.3 . Use the Fourier transform synthesis equation to determine the inverse Fourier transform of: a) F( ω)=2 πδ (ω)+ πδ (ω − 4π)+ πδ (ω+4 π) ω ω−1 ω + 1 b) F( )=2rect(2 )− 2rect( 2 ) Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-4: Fourier transform representation of signal P4.4 . Given that f(t) has the Fourier transform F( ω), express the Fourier transform of the signals listed below in terms of F( ω). You may use the Fourier transform properties. a) f1 (t)=f(1− t)+f( − 1 − t) b) f2 (t)=f(3t− 6) c) f (t)=d2 f(t− 1) 3 dt 2 P4.5 . For each of the following Fourier transforms, use Fourier properties to determine whether the corresponding time-domain signal is (i) real, imaginary, or neither and (ii) even, odd, or neither. Do this without evaluating the inverse of any of the given transform. ω ω−1 ω ω ω a) F1 ( )=rect(2 ) b) F2 ( )=cos(2 )sin(2 ) ω ω jB( ω) ω ω ω ω ω π b) F3 ( )=A( )e ; where A( )=(sin2 )/ , B( )=2 + 2 ∞ c) F ( ω)= 1 |n| δ(ω − nπ ) 4 ∑n=−∞ ( 2) 4 Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 1
2. Ch-4: Fourier transform representation of signal +∞ nπ n π P4.11 . Consider the signal f(t)=∑ sinc()4δ (t− 4 ) n=−∞ sint  a) Determine g(t) such that f(t)=  g(t) πt  b) Use the multiplication property of the Fourier transform to argue that F( ω) is periodic. Specify F( ω) over one period. P4.12 . Determine the continuous-time signal corresponding to each of the following transform. a) F( ω)=2sin[3( ω −2π)]/( ω − 2π) b) F( ω)=cos(4 ω+π/3) c) F( ω)=2[δω (−− 1) δω ( +1)]+3[ δω ( −− 2 π) δω ( +2 π)] d) F( ω) as given by the magnitude and ph ase plots of Figure P4.12a e) F( ω) as in Figure P4.12b Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-4: Fourier transform representation of signal P4.13 . Let F( ω) denote the Fourier transform of the signal f(t) depicted in Figure P4.13. +∞ a) Find ∠ F( ω) b) Find F(0) c) Find F( ω)d ω ∫−∞ +∞ 2sin ω +∞ d) Evaluate F( ω) ej2 ω d ω e) Evaluate |F( ω)|2 d ω ∫−∞ ω ∫−∞ f) Sketch the inverse Fourier transform of Re{F( ω)} Note: you should perform all these calculations without explicitly evaluating F( ω) Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 3
3. Ch-4: Fourier transform representation of signal P4.18 . Shown in Figure P4.18 is the frequency response H( ω) of a continuous-time filter referred to as a low-pass differentiator. For each of the input signals f(t) below, determine the filtered output signal y(t). a) f(t)=cos(2 πt+ θ) b) f(t)=cos(4 πt+ θ) c) f(t)=|sin(2 πt)| Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-4: Fourier transform representation of signal P4.19 . Shown in Figure P4.19 is |H( ω)| for a low-pass filter. Determine and sketch the impulse response of the filter for each of the following phase characteristics: a) ∠ H( ω)=0 b) ∠ H( ω)= ωT,where T is a constant π/2 ω>0 c) ∠ H( ω)=  -π/2 ω ω H( ω)=  c 0 otherwise a) Determine the impulse response h(t) for this filter b) As ωc is increased, does h(t) get more or less concentrated about the origin? c) Determine s(0)& s( ∞), where s(t) is the step response of the filter Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 5
4. Ch-4: Fourier transform representation of signal P4.23 . Suppose f(t)=sin200 πt+2sin400 πt and g(t)=f(t)sin400 πt. If the product g(t)sin400 πt is passed through an ideal low-pass filter with cutoff frequency 400 π and pass-band gain of 2, determine the signal obtained at the output of the low-pass filter. P4.24 . Suppose we wish to transmit the signal sin1000 πt f(t)= πt using a modulator that creates the signal w(t)=[f(t)+A]cos(10000 πt) Determine the largest permissible value of the modulation index m that would allow asynchronous demodulation to be use to recover f(t) from w(t). For this problem, you should assume that the maximum magnitude taken on by a side lobe of a sinc function occurs at the instant of time that is exactly halfway between the two zero-crossings enclosing the side lobe. Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-4: Fourier transform representation of signal P4.25 . An AM-SSB/SC system is applied to a signal f(t) whose Fourier transform F( ω) is zero for | ω|> ωM. the carrier frequency ωc used in the system is greater than ωM. Let g(t) denote the output of the system, assuming that only the upper sidebands are retained. Let q(t) denote the output of the system, assuming that only the lower sidebands are retained. The system in Figure P4.25 is proposed for converting g(t) into q(t). How should the parameter ω0 in the figure be related to ωc? What should be the value of pass- band gain A Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 7
5. Ch-4: Fourier transform representation of signal a) If F( ω) is given by the spectrum shown in Figure P4.27(b), sketch the spectrum of the scrambled signal y(t). b) Using amplifiers, multipliers, adders, oscillators, and whatever ideal filters you find necessary, draw the block diagram for such an ideal scrambler. c) Again using amplifiers, multipliers, adders, oscillators, and ideal filters, draw a block diagram for the associated unscrambler. Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-4: Fourier transform representation of signal P4.28 . Figure P4.28(a) the system that to perform single-sideband modulation. With F( ω) illustrated in Figure P4.28(b), sketch Y 1(ω), Y2(ω), and Y( ω) for the system in Figure P4.28(a), and demonstrate that only the upper-sidebands are retained Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 9