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Unformatted text preview: t he p oints of discontinuities, where it changes momentarily by
infinite amount. I t is n ot i mmediately a pparent how a n i nstantaneous frequency can be
changed by a n infinite a mount a nd t hen c hanged back t o t he original frequency in zero
time. Let us consider t he d irect approach. F or P M
P M f or m (t) is F M for m (t). T his assertion also follows from Eq. (4.84c) o r Fig.
4.42b. Fi = Fe + ~~ m (t) = 108 + 105m(t) Fi + 105m(t) 10 1[m(t)]minl - 2 95 F or F M F or F M [see Eq. (4.85a)]
W i = We + k fm(t). D ividing t hroughout by 211", we o btain t he e quation in t erms o f
t he v ariable F (frequency in Hz). T he i nstantaneous frequency F i is Fi = Fe A ngle M odulation F M a nd P M waveforms. E xample 4 .22
S ketch F M a nd P M waves for t he d igital modulating signal m (t) d epicted in Fig. 4.44a.
T he c onstants k f a nd kp a re 211"(10 5) a nd rr / 2, respectively, a nd Fe = 100 MHz. I n this case 'PPM(t) = A sin wet when m (t) = 1 o r - 1/3. T his will certainly cause ambiguity a t t he receiver when A sin wet is received. Such a mbiguity never arises if k pm(t) is
r estricted to t he r ange (-11", rr) .
T he a mbiguity arises only when m (t) h as j ump discontinuities. In such a case, t he
phase of 'PPM (t) changes instantaneously. Because a phase 'Po + 2nrr is i ndistinguishable
from t he p hase 'Po, a mbiguities will be inherent in t he d emodulator unless the phase 296 4 Continuous-Time Signal Analysis: T he F ourier Transform variations are limited to the range (-71", 71"). For this reason kp should be small enough to
restrict the phase change kpm(t) to the range (-71", 71").
No such restriction on kp is required if m(t) is continuous. In this case the phase
change is not instantaneous, but gradual over a time, and a phase 'Po + 2n7l" will exhibit
n additional carrier cycles over the case of phase of only 'PD. This conclusion can also be
verified from Example 4.21, where the maximum phase change t..'P = 101T.
Because a bandlimited signal cannot have jump discontinuities, we can say t hat when
m (t) is bandlimited, kp has no restrictions. • 4.8-2 Bandwidth of Angle-Modulated Signals Unlike a mplitude m odulation, t here is no simple relationship between t he baseband signal waveform a nd t he c orresponding angle m odulated waveform. T he s ame
is t rue o f their s pectra. B ecause of nonlinear n ature o f angle modulation, derivation
of PEM(W), t he f requency s pectrum o f t he m odulated signal is extremely complicated a nd c an b e o btained o nly for few special cases. Generally, t he b andwidth of
a n a ngle m odulated s ignal is infinite even w hen t he b aseband signal b andwidth is
finite. However, m ost o f t he signal power (or energy) resides in a finite band. We
shall now t ry t o e stimate t his e ssential b andwidth o f an angle m odulated signal.
L et u s s tart w ith t he a ngle m odulated s ignal in Eq. (4.80), a nd c onsider first
t he case of small k (k ---> 0).
'PEM (t) 4.8 Ang...
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