PROBLEMS 661(c)x 3 (t)={t
|t|,t=^0
0 ,t= 0*14.1.6For the following neither even nor odd signals,
find the even and odd parts of the signals.
(a)
x 4 (t)={
t, t≥ 0
0 ,t< 0(b)x 5 (t)=sint+cost14.1.7Classify the following signals into energy-type
or power-type signals, and determine the energy
or power content of the signal.
(a)x 1 (t)=e−tcostu− 1 (t)
(b)x 2 (t)=e−tcost
(c)x 3 (t)=sgn(t)=
1 ,t> 0
− 1 ,t< 0
0 ,t= 0
Note:∫
eax cos^2 xdx= 4 +^1 a 2 [(a cos^2 x+
sin 2x)+^2 a]eax.14.1.8(a) Based on Example 14.1.3, comment on
whetherx(t)=Acos 2πf 1 t+B 2 πf 2 tis
an energy- or a power-type signal.
(b) Find its energy or power content forf 1 =f 2
andf 1 =f 2.14.1.9For the power-type signals given, find the power
content in each case.
(a)x(t)=Aej(^2 πf^0 t+θ).
(b)x(t)=u− 1 (t), the unit-step signal.14.1.10Show that the product of two even or two odd
signals is even, whereas the product of an even
and an odd signal is odd.
14.1.11The triangular signal is given by
(t)=
t+ 1 , − 1 ≤t≤ 0
−t+ 1 , 0 ≤t≤ 1
0 , otherwise(a) Sketch the triangular pulse.
(b) Sketchx(t)=∑+∞
n=−∞(t−^2 n).
(c) Sketchx(t)=∑+∞
n=−∞(−^1 )
n(t−n).14.1.12For real, even, and periodic functions with period
T 0 , the Fourier-series expansion can be expressed
asx(t)=
a 0
2
+∑∞n= 1ancos(
2 π
n
T 0
t)wherean=
2
T 0∫α+T 0αx(t)cos(
2 π
n
T 0
t)
dtDetermineanfor the following signals:
(a)x(t)=|cos 2πf 0 t|, full wave rectifier out-
put.
(b)x(t)=cos 2πf 0 t+|cos 2πf 0 t|, half-wave
rectifier output.
*14.1.13For realx(t) given by Equation (14.1.11), identify
the even and odd parts ofx(t).
14.1.14Three alternative ways of representing a real peri-
odic signalx(t) in terms of Fourier-series expan-
sion are given in Problem 14.1.3. Determine the
expansion coefficientsxnof each of the periodic
signals shown in Figure P14.1.14, and for each
signal also determine the trigonometric Fourier-
series coefficientsanandbn.
14.1.15Certain waveforms can be viewed as a combi-
nation of some other waveforms for which the
Fourier coefficients are already known. Exam-
ple 14.1.4 shows some periodic waveforms for
which the coefficientsa 0 ,an, andbnare given by
Equations (14.1.12) through (14.1.14), respec-
tively. Use those to find the nonzero Fourier-
series coefficients for the waveforms given in
Figure P14.1.15.
14.1.16Determine the bandwidthWfrom two criteria, (i)
An<(An)max/10, fornf 1 >W, and (ii)An<
(An)max/20, fornf 1 >W, for the following
cases. (Note thatAstands for amplitude.)
(a) Waveform of Figure E14.1.4(a), withA=
π, D= 0. 25 μs, andT= 0. 5 μs.
(b) Waveform of Figure E14.1.4(b), withA=
π^2 andT= 2. 5 μs.
(c) Waveform of Figure E14.1.4(d), withA=π
andT=10 ms.
(d) Waveform of Figure E14.1.4(e), withA=π
andT= 800 μs.
14.1.17Consider the rectangular pulse trainx(t) of Figure
E14.1.4(a), withA=2 andD=T/2. Letv(t)=
x(t)−1.
(a) Sketchx(t) andv(t).