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8 Continuous-Time
Fourier Transform
Recommended
Problems
P8. 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^ T 1. (a) Sketch x(t). (b) Sketch t(t), which^ is^ a^ periodic^ repetition^ of^ x(t)^ with^ period^ To^ =^ 3T^1 /2.
(c) Compute X(w), the Fourier transform^ of^ x(t).^ Sketch^ IX(w)^ I^ for^ I w^ s^ 6w/Ti.
(d) Compute a,, the Fourier series coefficients of t(t). Sketch ak for k = 0 +1, +2, 3. (e) Using your answers to (c) and (d), verify that, for this example, 1 ak = (^) 1-XM (^) (w) (^) / (f) 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.
P8. Find the Fourier transform of each of^ the^ following^ signals and^ sketch^ the^ magni tude and phase as a function of^ frequency,^ including^ both positive^ and^ negative frequencies. (a) b(t - 5) (b) e -a'u(t), a^ real,^ positive (c) e (-I+j^2 )tu (t)
P8. In this problem we explore the^ definition^ of^ the^ Fourier^ transform^ of^ a^ periodic signal. (a) Show that if x 3 (t) = axi(t) + bx 2 (t), then X 3 (w)^ =^ aXi(w)^ +^ bX^2 (w). (b) Verify that
e-= -^ 2,7b(w^ -^ wo)ej-t^ dw
From this observation, argue that the Fourier transform^ of^ eiwO'^ is^ 27rb(w^ -^ wo). (c) Recall the synthesis equation for^ the^ Fourier^ series:
k= -
P8-
Signals and Systems P8-
By taking the Fourier transform of both (^) sides and using the results to parts (a) and (b), show that 2irk k~)= 2ra T
(d) Sketch X(w) for your answer to Problem P8.1(d) for I oI 4ir/To.
P8. (a) Consider the often-used alternative (^) definition of the Fourier transform, which we will call X,(f). The forward transform is written (^) as
Xa(f) = f (^) x(t)e -j 2 f t^ dt,
where f is the frequency variable in hertz. Derive the inverse transform (^) formula for this definition. Sketch X(f) for the signal discussed in Problem P8.1. (b) A second, (^) alternative definition is
Xb(v) = 1 x(t)ej" dt
Find the inverse transform relation.
P8. Consider (^) the periodic signal t(t) in Figure P8.5-1, which is composed (^) solely of impulses.
X(t)
(^2) 'e (^) ' (^) t -7 -6 -S (^) -4 -3 -2 -1 0 1 (^2) 3 4 5 6 7 Figure P8.5-
(a) What^ is^ the^ fundamental^ period^ To? (b) Find the Fourier series of I(t). (c) Find^ the^ Fourier^ transform^ of^ the^ signals^ in^ Figures^ P8.5-2^ and^ P8.5-3.
(i) I I
Signals and Systems P8-
(e) IXe(Co)l 4 Xe o 1 -
-1 +
Figure P8.6-
Optional
Problems
P8. In (^) earlier lectures, the response of an LTI system to an input (^) x(t) was shown to be y(t) = x(t) __* h(t), where h(t) is the system impulse response. (a) Using the fact that
y(t) = x(t) * h(t) (^) = x(r)h(t - (^) r) dr,
show that
Y(w) =^ f^ f^ x(r-)h(t^ -^ r)e^ j^ dr^ dt
(b) By appropriate change of variables, show that Y(w) (^) = X(w)H(w), where X(w) is the Fourier transform of x(t), and H(w) is the Fourier transform of (^) h(t).
P8. Consider the impulse train
p(t) = (^ ot^ -^ kT) k=- shown in^ Figure^ P8.8-1.
Continuous-Time Fourier Transform / Problems P8-
p (t)
t -2T -T 0 T 2T
Figure P8.8-
(a) Find the Fourier series of^ p(t).
(b) Find the Fourier transform^ of^ p(t).
(c) Consider the signal x(t) shown in^ Figure^ P8.8-2,^ where^ Ti^ <^ T.
x( t)
_-T T T1 T 2 2
Figure P8.8-
Show that^ the^ periodic^ signal^ t(t),^ formed^ by^ periodically^ repeating^ x(t), satisfies
2(t) = x(t)*^ p(t)
(d) Using the result to Problem^ P8.7^ and^ parts^ (b)^ and^ (c)^ of^ this^ problem,^ find^ the Fourier transform^ of^ t(t)^ in^ terms^ of^ the^ Fourier^ transform^ of^ x(t).
