31-03-2012, 12:35 PM
Angle Modulation
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Introduction
Consider a sinusoid, ( ) c c A f t0
cos 2π + φ , where Ac is the (constant)
amplitude, fc is the (constant) frequency in Hz and φ0 is the initial phase angle.
Let the sinusoid be written as ( ) c A t cos ⎡ ⎤ θ ⎣ ⎦ where ( ) c t f t0 θ = 2π + φ . In chapter
4, we have seen that relaxing the condition that Ac be a constant and making it a
function of the message signal m(t ) , gives rise to amplitude modulation. We
shall now examine the case where Ac is a constant but θ(t ), instead of being
equal to 2πfc t + φ0 , is a function of m(t ) . This leads to what is known as the
angle modulated signal. Two important cases of angle modulation are Frequency
Modulation (FM) and Phase modulation (PM). Our objective in this chapter is to
make a detailed study of FM and PM.
An important feature of FM and PM is that they can provide much better
protection to the message against the channel noise as compared to the linear
(amplitude) modulation schemes. Also, because of their constant amplitude
nature, they can withstand nonlinear distortion and amplitude fading. The price
paid to achieve these benefits is the increased bandwidth requirement; that is,
the transmission bandwidth of the FM or PM signal with constant amplitude and
which can provide noise immunity is much larger than 2W , where W is the
highest frequency component present in the message spectrum.
Principles of Communication Prof. V. Venkata Rao
Indian Institute of Technology Madras
5.2
Now let us define PM and FM. Consider a signal s (t ) given by
s (t ) = Ac cos ⎡⎣θi (t )⎤⎦ where θi (t ) , the instantaneous angle quantity, is a
function of m(t ) . We define the instantaneous frequency of the angle modulated
wave s (t ) , as
( ) i ( )
i
d t
f t
d t
1
2
θ
=
π
(5.1)
(The subscript i in θi (t ) or fi (t ) is indicative of our interest in the instantaneous
behavior of these quantities). If ( ) i c t f t0 θ = 2π + φ , then fi (t ) reduces to the
constant fc , which is in perfect agreement with our established notion of
frequency of a sinusoid. This is illustrated in Fig. 5.1.
Fig. 5.1: Illustration of instantaneous phase and frequency
Curve 1 in Fig. 5.1 depicts the phase behavior of a constant frequency sinusoid
with φ0 = 0. Hence, its phase, as a function of time is a straight line; that is
θi (t ) = 2πfc t . Slope of this line is a constant and is equal to the frequency of
the sinusoid. Curve 2 depicts an arbitrary phase behavior; its slope changes
with time. The instantaneous frequency (in radians per second) of this signal at
t = t1 is given by the slope of the tangent (green line) at that time.
Principles of Communication Prof. V. Venkata Rao
Indian Institute of Technology Madras
5.3
a) Phase modulation
For PM, θi (t ) is given by
θi (t ) = 2πfc t + kp m(t ) (5.2)
The term 2πfc t is the angle of the unmodulated carrier and the constant kp is
the phase sensitivity of the modulator with the units, radians per volt. (For
convenience, the initial phase angle of the unmodulated carrier is assumed to be
zero). Using Eq. 5.2, the phase modulated wave s (t ) can be written as
( ) PM c c p ( ) ⎡⎣s t ⎤⎦ = A cos ⎡⎣2πf t + k m t ⎤⎦ (5.3)
From Eq. 5.2 and 5.3, it is evident that for PM, the phase deviation of s (t ) from
that of the unmodulated carrier phase is a linear function of the base-band
message signal, m(t ) . The instantaneous frequency of a phase modulated
signal depends on dm(t ) ( )
m t
d t
= ' because i ( ) p ( )
c
d t k
f mt
d t
1 '
2 2
θ
= +
π π
.
b) Frequency Modulation
Let us now consider the case where fi (t ) is a function of m(t ) ; that is,
fi (t ) = fc + kf m(t ) (5.4)
or ( ) ( )
t
i t 2 fi d
− ∞
θ = π ∫ τ τ (5.5)
( )
t
2 fc t 2 kf m d
− ∞
= π + π ∫ τ τ (5.6)
kf is a constant, which we will identify shortly. A frequency modulated signal
s (t ) is described in the time domain by
( ) ( )
t
FM c c f s t A cos 2 f t 2 k m d
− ∞
⎡ ⎤
⎣⎡ ⎦⎤ = ⎢ π + π τ τ⎥ ⎢⎣ ⎥⎦
∫ (5.7)
kf is termed as the frequency sensitivity of the modulator with the units Hz/volt.
From Eq. 5.4 we infer that for an FM signal, the instantaneous frequency
Principles of Communication Prof. V. Venkata Rao
Indian Institute of Technology Madras
5.4
deviation of s (t ) from the (unmodulated) carrier frequency fc is a linear function
of m(t ) . Fig. 5.2 to 5.5 illustrate the experimentally generated AM, FM and PM
waveforms for three different base-band signals. From these illustrations, we
observe the following:
i) Unlike AM, the zero crossings of PM and FM waves are not uniform (zero
crossings refer to the time instants at which a waveform changes from
negative to positive and vice versa).
ii) Unlike AM, the envelope of PM or FM wave is a constant.
iii) From Fig. 5.2(b) and 5.3(b), we find that the minimum instantaneous
frequency of the FM occurs (as expected) at those instants when m(t ) is
most negative (such as t = t1) and maximum instantaneous frequency
occurs at those time instants when m(t ) attains its positive peak value, mp
(such as t = t2 ). When m(t ) is a square wave (Fig. 5.4), it can assume
only two possible values. Correspondingly, the instantaneous frequency
has only two possibilities. This is quite evident in Fig. 5.4(b).
Principles of Communication Prof. V. Venkata Rao
Indian Institute of Technology Madras
5.5
Fig 5.2: AM and FM with tone modulation
Principles of Communication Prof. V. Venkata Rao
Indian Institute of Technology Madras
5.6
Fig. 5.3: AM and FM with the triangular wave shown as m(t )
Principles of Communication Prof. V. Venkata Rao
Indian Institute of Technology Madras
5.7
Fig. 5.4: AM and FM with square wave shown as m(t )
iv) A triangular wave has only two possibilities for its slope. In Fig. 5.5(b), it has
a constant positive slope between t1 and t2 , and constant negative slope to
the right of t2 for the remaining part of the cycle shown. Correspondingly,
the PM wave has only two values for its fi (t ) , which is evident from the
figure.
v) The modulating waveform of Fig. 5.5© is a square wave. Except at time
instants such as t = t1, it has zero slope and ( )
t t
dm t
d t
= 1
is an impulse.
Principles of Communication Prof. V. Venkata Rao
Indian Institute of Technology Madras
5.8
Therefore, the modulated carrier is simply a sinusoid of frequency fc ,
except at the time instants when m(t ) changes its polarity. At t = t1, fi (t )
has to be infinity. This is justified by the fact that at t = t1, θi (t ) undergoes
sudden phase change (as can be seen in the modulated waveform) which
implies d i (t )
d t
θ
tends to become an impulse.
Eq. 5.3 and 5.7 reveal a close relationship between PM and FM. Let
( ) ( )
t
mI t m d
− ∞
= ∫ τ τ . If mI (t ) phase modulates a carrier with modulator
sensitivity kp = 2πkf , then the resulting signal is actually an FM signal as given
by Eq. 5.7. Similarly a PM signal can be obtained using frequency modulator by
differentiating m(t ) before applying it to the frequency modulator. (Because of
differentiation, m(t ) should not have any discontinuities.)
As both PM and FM have constant amplitude Ac , the average power of a
PM or FM signal is,
c
av
P A
2
2
= ,
regardless of the value of kp or kf