660 SIGNAL PROCESSINGsgn(t)=
1 ,t> 0
− 1 t< 0
0 ,t= 0
which can be expressed as the limit of the
signalxn(t) defined byxn(t)=
e−^1 /n,t> 0
−e^1 /n,t< 0
0 ,t= 0
asn→∞. Sketch the waveform as the limit
ofxn(t).*14.1.2A large number of building blocks in a commu-
nication system can be modeled by LTI (linear
time-invariant) systems, for which the impulse
response completely characterizes the system.
Consider the system described by
y(t)=∫t−∞x(τ) dτwhich is called an integrator. Investigate whether
the system is LTI by finding its response tox(t−
t 0 ).
14.1.3For a real periodic signalx(t) with periodT 0 ,
three alternative ways to represent the Fourier
series expansion are:x(t)=+∞∑−∞xnej^2 π
Tn
0 t=
a 0
2
+∑∞n= 1[
ancos(
2 π
n
T 0
t)+bnsin(
2 π
n
T 0
t)]=x 0 + 2∑∞n= 1|xn|cos(
2 πn
T 0
t+xn)where the corresponding coefficients are
obtained fromxn=1
T 0∫α+T 0αx(t)e−j^2 π
Tn
0 tdt=
an
2
−jbn
2an=
2
T 0∫α+T 0αx(t)cos(
2 π
n
T 0
t)
dtbn=
2
T 0∫α+T 0αx(t)sin(
2 π
n
T 0t)
dt|xn|=
1
2√
a^2 n+bn^2xn=−arctan(
bn
an)in which the parameterαin the limits of the
integral is arbitrarily chosen asα=0orα=
−T 0 /2, for convenience.
(a) Show that the Fourier-series representation
of an impulse train is given byx(t)=∑+∞n=−∞δ(t−nT 0 )=
1
T 0+∞∑n=−∞ej^2 π
Tn
0 tAlso sketch the impulse train.
(b) Obtain the Fourier-series expansion for the
signalx(t) sketched in Figure P14.1.3 with
T 0 =2, by choosingα=−^1 / 2.
14.1.4(a) Show that the sum of two discrete periodic
signals is periodic.
(b) Show that the sum of two continuous peri-
odic signals is not necessarily periodic; find
the condition under which the sum of two
continuous periodic signals is periodic.
14.1.5Classify the following signals into even and odd
signals:
(a)
x 1 (t)=
e−t,t> 0
−e−t,t< 0
0 ,t= 0(b)x 2 (t)=e−|t|1− 1− (^2) t
2
− 1
5 1
− 2
5
2
3
− 2
1
− 2
1
2
3
2
Figure P14.1.3