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Unformatted text preview: will be lost. So, the bandwidth of
7jJ(t) cannot be less than B Hz. Hence, it is exactly equal to B Hz. 2 t.F+X ~1"1/>'P + X
7 (4.89) where X is unknown. To d etermine X , recall t hat for t he c ase k ---> 0, we found t he
b andwidth t o b e 2 B. B ut a s Eq. (4.89) indicates, t his b andwidth is X w hen k ---> O.
T herefore, X = 2 B, a nd
B EM = 2 (t.F + B ) Hz (4.90) A m ore rigorous derivation of this result a ppears i n reference 4. N ote t hat w hen
k ---> 0, t .F ---> 0 a nd t .F « B . O n t he o ther h and w hen k is very large, t .F » B .
T he former case is k nown as t he n arrowband a ngle m odulation a nd t he l atter is
known as t he w ideband a ngle modulation.
Recall t hat for FM, .b(t) = m (t), a nd 1/>~ = m p, w here m p is t he p eak a mplitude
o f m (t). Similarly, for P M, 1/>(t) = m (t). Hence, 1/>~ = m~, w here m~ is t he p eak
a mplitude of m(t). T hus
a nd (4.91) We observe an interesting fact in angle modulation. T he b andwidth o f t he
m odulated s ignal is a djustable by choosing s uitable value of t .F o r t he c onstant k
( kj in F M o r kp i n P M). A mplitude m odulation lacks t his f eature. T he b andwidth
of each AM scheme is fixed. I t is a general principle in communication t heory
tThis assertion implies an assumption >b(t)lmax = 1>b(t)lminl 298 4 Continuous-Time Signal Analysis: T he Fourier Transform t hat widening a signal bandwidth makes t he signal more immune t o noise during
transmission. T hus, widening the transmission bandwidth makes angle modulated
signals can be m ade more immune t o noise. Moreover, this very property allows
us t o reduce t he signal power required to achieve t he same quality o f transmission.
Thus, angle m odulation allows us t o exchange signal power for bandwidth.
Also, because of its constant amplitude, angle modulation has a major advantage over a mplitude modulation. This feature makes angle modulation less susceptible to nonlinear distortion. We shall see in t he following section (Sec. 4.8-3) t hat
n o distortion results when we pass an angle modulated signal through a nonlinear
device whose o utput y(t) a nd t he i nput x (t) a re related by y(t) = x 2 (t) [in general
y (t) = L ;anxn(t)J. Such a nonlinearity can be disastrous in amplitude modulated
systems. This i s t he p rimary reason why angle modulation is used in microwave
relay systems, where nonlinear operation of amplifiers and o ther devices has thus
far been unavoidable a t t he required high power levels. In addition, t he c onstant
amplitude of F M gives it a kind of immunity against rapid fading. T he effect of
amplitude variations caused b y rapid fading can b e eliminated by using automatic
gain control a nd b andpass limiting4. Angle modulation is also less vulnerable t han
a mplitude m odulation t o small interference from adjacent channels. B ut t he price
for all these advantages is p aid in terms of increased bandwidth. We c an demonstrate t hat for t he same bandwidth, t he pulse code modulation ( PCM), discussed
in C hapter 5, is s uperior to angle modulation 4 . 4 .8-3 Generation and Demodulation o f Angle M...
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