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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 ContinuousTime 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. • 4.82 Bandwidth of AngleModulated 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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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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