Unformatted text preview: r i '-- A I d 293 We have seen t hat P M a nd F M a re not different kind o f m odulation, b ut t wo.
special cases o f generalized angle modulation. Such a view is very fruitful because
i t shows t he convertibility of one t ype of angle modulation (such as P M) t o a nother
(such as FM). This convertibility is q uite clear in Fig. 4.42. For instance, we show
l ater t hat t he b andwidth of F M is a pproximately 2kfm p, where m p is t he peak
amplitude of m (t). We c an derive t he equivalent result for P M by referring t o
Fig. 4.42b, which shows t hat P M is a ctually t he F M when t he m odulating signal
is m (t). Clearly, t he b andwidth of P M is approximately 2kpmp', where m p ' is t he
p eak a mplitude o f m (t). T his argument shows t hat if we analyze one type of angle
modulation (such as FM), we could readily extend those results t o a ny other kind.
Historically, the angle modulation concept began with FM. Hence, i t is c ustomary
t o analyze F M a nd t hen modify those results for o ther forms, such as P M. B ut t his
does n ot imply t hat F M is s uperior t o o ther kinds of angle modulation. O n t he
contrary, P M is superior t o F M for most analog signals such as audio a nd video.
Actually, t he o ptimum performance is realized neither by P M nor FM, b ut by some
o ther form, depending on t he n ature o f t he b aseband (message) signal. "'FM(t) ,
m (t) 4.8 Angle Modulation "'PM(t) -I / 2 X 10-4 V - ---I / \t(a) "V 20,000 - 20,000 -----'U
D I [_
(c) D Phase
Modulator F ig. 4.42 Phase and frequency modulation are inseparable. All In The Family E quations (4.84b) and (4.85c) indicate t hat P M a nd FM are n ot only very
similar b ut a re inseparable. Replacing m (t) in Eq. (4.84b) with
m (a) da changes
P M i nto FM. T hus, a signal t hat is a n F M wave corresponding to m (t) is also t he P M
wave corresponding t o m (a) da (Fig. 4.42a). Similarly, a P M wave corresponding
t o m (t) is t he F M wave corresponding t o m (t) (Fig. 4.42b).
We c onclude t hat j ust by looking a t an angle-modulated carrier, we c annot
tell whether it is F M or P M. I n fact, it is meaningless to enquire if a certain angle
m odulated wave is FM o r P M. A n analogous situation would be t o ask a person
(who is m arried, w ith children), whether he is a f ather or a son. T he person would
be puzzled because he is b oth, a father (of his child) and a son (of his father). t ( b) t (d) F ig. 4.43 FM and PM waveforms. • E xample 4.21 Sketch FM and PM waves for the modulating signal m (t) depicted in Fig. 4.43a. The
constants k f and kp are 27r (10 5 ) and 107r, respectively, and the carrier frequency F e is
100 MHz. 294 4 4 .8 C ontinuous-Time S ignal A nalysis: T he F ourier T ransform + ~~ m (t) (Fi) min = 10 8
(F;)max = 108 = 10 8 5 + 105 [m(t)]max = Because m (t) switches back a nd f orth from 1 to - 1 a nd vice versa, t he F M wave frequency
switches back a nd f orth from 99.9 MHz t o 100.1 MHz and vice versa, as shown in Fig. 4.44b.
This scheme of a carrier frequency modulation by a digital signal is known as f requencyshift k...
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