Fourier Transform... Reference material: Gabel and Roberts \"Signals and Systems

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Fourier Transform... Reference material: Gabel and Roberts "Signals and Systems" 2/e or 3/e chapter 5 Consider the signal x(t), which consists of a single rectangular pulse of unit height, is symmetric about the origin, and has a total width T1,. Sketch (t). Sketch (t), which is a periodic repetition of x(t) with period T0 = 3T1/2. Compute X(omega), the Fourier transform of x(t). Sketch |X(omega)| for |omega| 6 pi/T1. Compute ak, the Fourier series coefficients of (t). Sketch ak for k = 0, 1, 2, 3. Using your answers to (c) and (d), verify that, for this example, alpha k = 1/T0 X(omega)| Write a statement that indicates how the Fourier series for a periodic function can be obtained if the Fourier transform of one period of this periodic function is given. Find the Fourier transform of each of the following signals and sketch the magnitude and phase as a function of frequency, including both positive and negative frequencies. delta(t - 5) e-atu(t), a real, positive e(-1 + j2)t u(t) In this problem we explore the definition of the Fourier transform of a periodic signal. Show that if x31t) = ax1t(t) + bx2(t), then X3(omega) = aX1(omega) + bX2(omega). Verify that ej omega 0 t = ½ pi 2 pi delta (omega - omega 0)ef omega t dw From this observation, argue that the Fourier transform of ej omega 0 is 2 pi delta (omega - omega 0). Recall the synthesis equation for the Fourier series: (t) = akejk(2 pi(T)t By taking the Fourier transform of both sides and using the results to parts (a) and (b), show that (omega) = 2 pi ak delta(omega -2 pi k/T) Sketch (omega) for your answer to Problem P8.1(d) for |omega| 4 pi. Consider the often-used alternative definition of the Fourier transform, which we will call Xa(f). The forward transform is written as Xa(f) = x(t)e-j2 pi ft dt, where f is the frequency variable in hertz. Derive the inverse transform formula for this definition. Sketch Xa(f) for the signal discussed in Problem P6.1. A second, alternative definition is Find the inverse transform relation. Xa(f) = 1/ x(t)e-jvt dt Find the inverse transform relation.

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you are not allowed to ask so many questions in a single post pls split them in multiple posts thanks

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