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Unformatted text preview: sibilities are p hase m odulation
( PM) a nd f requency m odulation ( FM). In P M, t he angle O(t) is varied linearly
with m (t) :
(4.84a)
where kp is a c onstant a nd We is t he c arrier frequency. T he resulting P M wave is 4 .81 T he Concept o f Instantaneous Frequency As seen above, the carrier frequency is changing continuously every i nstant in
FM. P rima facie, this does not make much sense because t o define a frequency, we
m ust have a sinusoidal signal a t least over one cycle with t he same frequency. We
c annot imagine a sinusoid whose frequency is different a t every instant. This problem reminds u s of our first encounter with t he concept of i nstantaneous v elocity
in our beginning mechanics course. Until t hat time, we were used t o t hinking of
velocity as being constant over a time interval a nd were incapable of even imagining
t hat velocity could vary a t each instant. B ut a fter some mental struggle, t he idea
gradually sinks in. We never forget, however, t he wonder a nd a mazement t hat was
caused by t he idea when it was first introduced. A similar experience awaits the
reader with t he c oncept of instantaneous frequency.
Let us consider a generalized sinusoidal signal 'P(t) given by ' PPM (t) = A cos [wet + kpm(t)] (4.84b) T he i nstantaneous frequency Wi(t) in this case is given by Wi(t) = dO = We
dt + kpm(t) (4.84c) Hence in phase modulation, t he i nstantaneous frequency Wi varies linearly with t he
derivative of t he m odulating signal. I f t he i nstantaneous frequency Wi is varied
linearly with the modulating signal, we have frequency modulation. Thus, in FM,
t he i nstantaneous frequency Wi is Wi(t) = We + k fm(t) where k f is a c onstant. From Eq. (4.83b), we find t he angle O(t) as (4.85a) 4 ContinuousTime Signal Analysis: T he Fourier Transform 292 B(t) = [oo[w e+ kfm(a)] da = wet + k f [too m(a) da (4.85b) Here we have a ssumed t he c onstant t erm in B(t) t o be zero without loss of generality.
Thus, t he F M wave is
(4.85c)
Observe t hat b oth P M a nd F M a re special cases of the exponentially modulated
signal 'PEM(t) i n E q. (4.80). I f h(t) = 6(t) in Eq. (4.81), t hen use o f t he sampling
p roperty of t he impulse in Eq. (4.81) yields ?j;(t) = m (t), a nd Eq. (4.80) reduces t o
P M in Eq. (4.84b). Similarly, if h (t) = u(t), t hen t he fact t hat u (t  a) = l over
 00 < a :s t y ields J m (a)h(t  a) da = J m (a) da, a nd Eq. (4.80) reduces to FM
in Eq. (4.85c). m (t) :
, f j m(Cl)dCl ,
Phase
Modulator This discussion also shows t hat we need not discuss methods of generation and
demodulation of each type of modulation. Figure 4.42 clearly indicates t hat t he P M
c an be generated by a n F M generator, a nd t he F M c an be generated by a P M generator. O ne of t he m ethods of generating FM in practice (the Armstrong indirectFM
system) actually integrates m (t) a nd uses i t t o p hasemodulate a carrier. Similar
remarks apply to demodulation of F M a nd P M. ; Frequency
Modulator m (t) dt m (t) i Frequency
Modulato...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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