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9 Fourier Transform Properties
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Problems
P9.
Determine the Fourier transform of x(t) = e-t/u(t) and sketch (a) IX(w)I (b) <tX(w) (c) Re{X(w)} (d) Im{X(w)}
P9.
Figure P9.2 shows real and imaginary parts of the Fourier transform of a signal x(t).
Re I x (C) ]In I(X(O))
-W W -W W
Figure P9.
(a) Sketch the magnitude and phase of the Fourier transform X(W). (b) In general, if a signal x(t) is real, then X(-w) = X*(w). Determine whether x(t) is real for the Fourier transform sketched in Figure P9.2.
P9.
Determine which of the Fourier transforms in Figures P9.3-1 and P9.3-2 correspond to real-valued time functions. (a) IX(W)| 4 X(co)
CF CO
Figure P9.3-
P9-
Signals and Systems P9-
(b) Rel (^) X()t (^) = XR (^) (W) Im IX(W)= X, (W)
(^2) lr
-W W
I
Figure P9.3-
P9.
(a) By considering the Fourier analysis equation or^ synthesis^ equation,^ show^ the validity in^ general^ of^ each^ of^ the^ following^ statements: (i) If^ x(t)^ is^ real-valued,^ then^ X(o)^ =^ X*(-c). (ii) If x(t) = x*(- t),^ then^ X(w)^ is^ real-valued. (b) Using the statements in^ part^ (a),^ show^ the validity^ of each of^ the^ following statements: (i) If^ x(t)^ is^ real^ and^ even,^ then^ X(o)^ is^ real^ and^ even. (ii) If x(t) is real and^ odd,^ then^ X(w)^ is^ imaginary^ and^ odd.
P9.
(a) In the lecture, we derived the^ transform^ of^ x(t)^ =^ e^ -'u(t). Using^ the linearity and scaling properties, derive the Fourier transform of e -"t = x(t) + x(- t). (b) Using part (a) and the duality property, determine the Fourier transform of 1/(1 + t^2 (c) If 1
r(t) = 1 + (3t)^2
find R(w). (d) x(t) is sketched in Figure P9.5. If y(t) = x(t/2), sketch^ y(t),^ Y(w),^ and^ X(w).
x(t)
A
-T T
Figure P9.
Signals and Systems P9-
x(t) I (^) N y(t)
cos (wet)
Figure P9.9-
X(W)
-W0 W 0 Figure P9.9-
P9.
Compute the Fourier transform of each of the following signals: (a) [e~a cos wotlu(t), a > 0 (b) e 31 sin 2t
(C) sin^ irt^ sin^ 2irt rt rt
P9.
Consider the following linear constant-coefficient differential equation (LCCDE): dy(t) (^) + (^) 2y(t) (^) = (^) A cos (^) wet dt Find the value of wo such that y(t) will have a maximum amplitude of A/3. Assume that the resulting system^ is^ linear^ and^ time-invariant.
P9.
Suppose an LTI system is described by the following LCCDE: d'ytt) 2dytt)^ 4dx(t) dt^2
dt
+ 3y(t) = dt
- x(t)
Fourier Transform^ Properties^ /^ Problems P9-
(a) Show that the left-hand side of the equation has^ a^ Fourier^ transform^ that^ can be expressed as
A(w)Y(w), where Y(w) =^ J{y(t)} Find A(w). (b) Similarly, show that the right-hand side^ of^ the^ equation^ has^ a^ Fourier^ transform that can be expressed as
B(w)X(w), where X(w) = {x(t)}
(c) Show^ that^ Y(w)^ can^ be^ expressed^ as^ Y(w)^ =^ H(w)X(w)^ and^ find^ H(w).
P9.
From Figure P9.13, find y(t) where
x(t) =^ sin(wot) t and^ h(t)^ =^ sin(2wot) t
x(t) (^) h(t) yt)
Figure P9.
P9. (a) Determine the energy in the signal^ x(t)^ for^ which^ the^ Fourier^ transform^ X(w)^ is given by Figure P9.14.
(b) Find the inverse Fourier transform of X(w) of^ part^ (a).
