ANGLE MODULATION AND DEMODULATION

4.1 Nonlinear modulation:

 In AM signals, the amplitude of a carrier is modulated by a signal m(t), and,
hence, the information content of m(t) is in the amplitude variations of the carrier.
 The other two parameters of the carrier sinusoid, namely its frequency and phase,
can also be varied in proportion to the message signal as frequency-modulated
(FM) and phase modulated (PM) signals, respectively.
 In FM ω  t   ωc  km  t  , where k is an arbitrary constant.
 If the peak amplitude of m(t) is mp , then the maximum and minimum values of
the carrier frequency would be ωc  kmp and ωc  kmp , respectively. Hence, the
spectral components would remain within this band with a bandwidth 2kmp ,
centered at ω c .
 Therefore it is expected that controlling the constant parameter k can control the
modulated signal bandwidth. Therefore we can make the information bandwidth
arbitrary small by using an arbitrary small k. Unfortunately, experimental results
showed that it is wrong! It was found that the FM bandwidth is always greater
than (at best equal to) the AM bandwidth. [Will be explained later]
4.1.1 Concept of instantaneous frequency
 FM signal can vary the instantaneous frequency in proportion to the modulating
signal m(t). This means that the carrier frequency is changing continuously every
instant.
 Prima facie, this does not make much sense, since to define a frequency we must
have a sinusoid signal at least over one cycle (or half-cycle or a quarter-cycle)
with the same frequency. This leads to the concept of instantaneous frequency
(similar to instantaneous velocity in mechanics).
 Let us consider a generalized sinusoidal frequency signal φ  t  is given by
φ  t   Acos θ  t  , where θ  t  is the generalized angle and is a function of t.
 The generalized angle of θ  t  happens to be a tangential to the angle ωc t  θ0 at
some instant t.
 Around t, over a small interval t  0 , the signal φ  t   Acos θ  t  and the
sinusoid are identical; that is, φ  t   A cos  ωc t  θ0  t1  t  t 2 .
 Therefore over this small interval  t , the angular frequency of φ  t  is ω c .
Because  ωc t  θ0  is tangential to θ  t  , the angular frequency φ  t  is the slope
of this angle over this small interval θ  t  .
 We can generalize this concept at every instant and define the instantaneous
frequency ωi at any instant t is the slop of θ  t  at t.
 Thus, for φ  t  , the instantaneous angular frequency and the generalized angle
t
dθ
are related via ωi  or, θ  t    ωi  α  dα .
dt 

 We can now see the possibility of transmitting the information of m(t) by varying
the angle θ of a carrier. Such technique of modulation, where the angle of the
carrier is varied in some manner with a modulating signal m(t), are known as
angle modulation or exponential modulation. Two simple possibilities are FM and
PM.
 In PM, the angle θ  t  is varied with m(t): θ  t   ωc t  θ0  k p m  t  , where k p is a
constant and ω c is the carrier frequency.
 Assuming θ0  0 , without loss of generality, θ  t   ωc t  k p m  t  .
 The resulting PM wave is φPM  t   A cos ωc t  kpm  t  .
 The instantaneous angular frequency ωi  t  in this case is given by

ω i  t   ωc  k p m  t  .
 Hence, in PM, the instantaneous angular frequency ωi valies linearly with the
derivative of the modulating signal.
 If the instantaneous frequency ωi is varied linearly with the modulating signal,
we have FM. Thus, in FM the instantaneous angular frequency ωi is
ωi  t   ωc  k f m  t  , where kf is a constant.
t t
 The angle θ  t  is now θ  t    ωc  k f m  α  dα  ωc t   k f m α dα . Here we have
 

assumed the constant term in θ  t  to be zero without loss of generality.

 t

 The FM wave is φFM  t   A cos ωc t   k f m  α dα 
  
4.1.2 Relationship between FM and PM

 Expressions of φPM  t  and φFM  t  reveal that they are not only very similar but
are inseparable. Replacing m(t) in the expressing of φPM  t  by  m α dα changes
the PM into FM.
 Thus a signal that is an FM wave corresponding m(t) is also the PM wave
corresponding to  m  α  dα . Similarly, a PM wave corresponding to m(t) is the FM

wave corresponding to m  t  .
 Therefore, by looking only at an angle-modulated signal φ  t  , there is no way of
telling whether it is FM or PM. However there demodulation methods should be
different.
 In both the expressions of PM and FM the angle of a carrier is varied in proportion
to some measure of m(t). In PM, it is directly proportional to m(t), whereas in FM,
it is proportional to the integral of m(t).
 As shown below, a frequency modulator can be directly used to generate an FM
or the message m(t) can be processed by a filter (differentiator) with transfer
function H(s)=s to generate PM signals.

 The generalized angle-modulated carrier φEM  t  can be expressed as

 t

φEM  t   A cos ωc t  ψ  t   A cos ωc t   m  α  h  t  α  dα  .
  
Now PM and FM are just two special cases with h  t   k pδ  t  and h  t   k f u  t  ,
respectively.
  As long as H(s) is a reversible operation (or invertible), m(t) can be recovered from
ψt .
 Although the instantaneous frequency and phase of an angle-modulated wave
can vary with time, the amplitude A remains constant. Hence, the power of an
angle-modulated wave is always A 2 2 , regardless the value of k p and kf .

Example 4.1
Sketch FM and PM waves for the modulating signal m(t) shown below. The constants
k p and kf are 2π 105 and 10π , respectively, and the carrier frequency fc is 100 MHz.

Solution:

For FM:

The instantaneous frequency fi is

kf 2π  105
fi  fc  m  t   108  m  t   108  105 m  t 
2π 2π

fi,min  108  105 m  t  min  99.9 MHz

fi,max  108  105 m  t  max  100.1 MHz

Because m(t) increases and decreases linearly with time, the instantaneous frequency
increases linearly from 9909 to 100.1 MHz over a half-cycle and decreases linearly from
100.1 to 99.9 MHz over the remaining half-cycle of the modulating signal,
For PM:

PM for m(t) is FM for m  t  .

kp  10π  
fi  fc  m  t   108  m  t   108  5 m  t 
2π 2π

fi,min  108  5 m  t   108  105  99.9 MHz
min


fi,max  108  5 m  t   100.1 MHz
max


Because m  t  switches back and forth from a value of -20,000 to 20,000, the carrier

frequency switches back and forth from 99.9 to 100.1 MHz every half-cycle of m  t  , as
shown.

This indirect method of sketching PM [using m  t  to frequency-modulate a carrier] works
as long as m(t) is a continuous signal. If m(t) is discontinuous, it means that the PM signal

has sudden phase changes and , hence m  t  contains impulses. This indirect method
fails at points of discontinuity. In such case, a direct approach should be used at the point
of discontinuity to specify sudden phase change. This is demonstrated in the next
example.

Example 4.2
Sketch FM and PM waves for the digital modulating signal m(t) shown below. The
constants k p and kf are 2π 105 and π 2 , respectively, and the carrier frequency fc is
100 MHz.
Solution:
For FM:

kf 2π  105
fi  fc  m  t   108  m  t   108  105 m  t 
2π 2π

Because m(t) switches from -1 to 1 and vice versa, the FM wave frequency switches back
and forth between 99.9 and 100.1 MHz, as shown.

This scheme of carrier frequency modulation by a digital signal is called frequency shift
keying (FSK) because information digits are transmitted by keeping different frequencies.
For PM:
kp  π  1 
fi  fc  m  t   108  m  t   108  m  t 
2π 2  2π 4

The derivative of m  t  is zero except at points of discontinuity of m(t) where impulses of
strength 2 are present. This means that the frequency of the PM signal stays the same
except at these isolated points of time! It is not immediately apparent how an
instantaneous frequency can be changed by an infinite amount and then changed back
to the original frequency in zero time.

 π   A sin  ωc t  when m  t   1
φPM  t   A cos ωc t  k p m  t   A cos ωc t  m  t    
 2    A sin  ωc t  when m  t   1

This PM wave is shown above.

This scheme of carrier PM by a digital signal is called phase shift keying because
information digits are transmitted by shifting carrier phase.

The amount of phase shift discontinuity in φPM  t  at the instant where m(t) is
discontinuous is k p md , where md is the amount of discontinuity in m(t) at that instant. In
the present example, the amplitude of m(t) changes by 2 (from -1 to +1 ) at the
π
discontinuity. Hence the phase discontinuity is k p md   2  π rad, which confirms the
2
result.

When m(t) is a digital signal, φPM  t  shows a phase discontinuity where m(t) has a jump
discontinuity. To avoid ambiguity in demodulation, in such a case, the phase deviation
 3π
k p m  t  must be restricted to a range  π,π  . For example, if k p were in the present
2
 3π 
example, then φPM  t   A cos ωc t  m  t  . In this case φPM  t   A sin  ωc t  when
 2 
m(t)=1 or -1/3. This will certainly cause ambiguity at the receiver when A sin  ωc t  is
received. Specifically, the receiver cannot decide the exact value of m(t).

When m(t) has jump discontinuities, the phase of φPM  t  changes instantaneously.
Because φo  2nπ (n is integer) is indistinguishable from the phase φ o , ambiguities will be
inherent in the demodulator unless the phase variations are limited to the range  π,π  .

Nos such restriction on k p is required if m(t) is continuous. In this case phase change is
not instantaneous, but gradual over time, and a phase change φo  2nπ will exhibit n
additional carrier cycles in the case of phase of only φ o . We can detect the PM wave by
using an FM demodulator followed by the integrator. The additional n cycles will be
detected by the FM demodulator, and the subsequent integration will yield a phase 2nπ .
Hence, the phase φ o and φo  2nπ can be detected without ambiguity. This conclusion
can also be verified from the previous example where phase change is 10π .

Because a band limited signal cannot have jump discontinuities, we can also say that
when m(t) is band-limited, k p has no restrictions.

4.2 Bandwidth of angle-modulated wave

 Unlike AM, angle modulation is nonlinear and no properties of Fourier transform

can be directly applied for its bandwidth analysis.
t
 To determine the bandwidth of an FM wave, let us define a  t    m  α  dα and

jωct kf a  t  jkf a  t 
φ̂FM  t   Ae   Ae  Ae jωct such that its relation to FM signal is
φFM  t   Re φ
ˆ FM  t  .
jk f a  t 
 Expanding the exponential e in power series yields
 k 2f k fn n  jωc t
φ̂FM  t   A 1  jk f a  t   a2 t    jn a t   e and
 2! n! 
 k2 
φFM  t   Re φ
ˆ FM  t    A cos  ωc t   k f a  t  sin  ωc t   f a 2  t  cos  ωc t   
 2! 
[The power a n  t  indicates that the angle modulation is non-linear]
 Above equation reveals that the modulated wave consists of an unmodulated carrier
plus various amplitude-modulated terms, such as a  t  sin  ωc t  , a 2  t  cos  ωc t  etc.
 The signal a(t) is an integral of m(t). If M(f) is band-limited to B, A(f) is also band-
limited to B. This is because integration is a linear operation equivalent to passing a
signal through a transfer function 1/2πf . The spectrum of a 2  t  is simply A(f)*a(f)
and is band-limited to 2B. Similarly the spectrum of a n  t  is band-limited to nB.
Hence, the spectrum consists of an unmodulated carrier plus spectra of a(t), a 2  t 
,…, a n  t  ,… centered at ω c . Clearly, the modulated wave is not band-limited. It has
an infinite bandwidth and is not related to the modulating signal spectrum, as was
the case in AM.
 Although the bandwidth of an FM wave is theoretically infinite, for practical signals
with bounded |a(t)|, k f a  t  will remain finite. Because n! increases much faster
n k nfa  t 
than k f a  t  , we have 0 for large n. Hence, we shall see that most of the
2!
modulated-signal power resides in a finite bandwidth.
 This is the principal foundation of the bandwidth analysis for angle-modulations.
There are two distinct possibilities in terms of bandwidths – narrowband FM and
wideband FM.

4.2.1 Narrowband angle modulation approximation

 When kf is very small such that kf a  t  1 then all higher order terms are negligible
except for the first two. We then have a good approximation
φFM  t   A cos  ωct   kf a  t  sin  ωct  .
This approximation is a linear modulation that has an expression similar to that of
the AM signal with message signal a(t).
 Because the bandwidth of a(t) is B Hz, the bandwidth of φFM  t  is also 2B Hz
according to the frequency-shifting property due to the term a  t  sin  ωc t  . For this
reason, the FM signal for the case kf a  t  1 is called narrowband FM (NBFM).
 Similarly narrowband PM (NBPM) signal is approximated by
φPM  t   A cos  ωc t   k pm  t  sin  ωct  .
NBPM also has the approximate bandwidth of 2B Hz.
 A comparison of NBFM with AM reveals that both have the same modulated
bandwidth 2B. The sideband spectrum of FM has a phase shift of π 2 with respect
to the carrier, whereas that of AM in phase with the carrier.
 π   π 
cos  ωc t   sin  2  ωc t    sin  ωc t  2  
    
 Despite the apparent similarities, the Am and FM signals have very different
waveform. In an AM signal the oscillation frequency is constant and the amplitude
varies with time, whereas in an FM signal, the amplitude stays constant and the
frequency varies with time.
4.2.2 Wideband FM (WBFM) bandwidth analysis: The fallacy exposed
 FM signal is meaningful only if its frequency deviation is large enough. In other
words, practical FM choose the constant kf large enough that the condition
kf a  t  1 is not satisfied. We call FM signal in such case wideband FM (WBFM).
Thus in analyzing the bandwidth of WBFM, we cannot ignore all the higher order
terms.
 Consider a low-pass m(t) with bandwidth B Hz. This signal is approximated by a
staircase signal m̂(t) , as shown below. The signal m(t) is now approximated by pulses
of constant amplitude. It is relatively easier to analyze FM corresponding to m̂(t)
because the constant amplitude pulses.
 For convenience, each of these pulses will be called a ‘cell’. To ensure that m̂(t) has
all the information of m(t), the cell width in m̂(t) must be no greater than the Nyquist
interval of 1/2B second according to the sampling theorem.
 The FM signal corresponding to this cell is a sinusoid of frequency k f m  t k  and
duration T=1/2B, as shown.

 The FM signal of m̂(t) consists of a sequence of such constant frequency sinusoid

pulses of duration 1/2B corresponding to various cells of m̂(t) .
 The FM spectrum for m̂(t) consists of the sum of the Fourier transforms of these
sinusoidal pulses corresponding to all these cells.
 The Fourier transform of a sinusoidal pulses (corresponding to the kth cell) is a sinc
function, shown.
 The minimum and maximum amplitudes of the cells are mp and mp .
 Therefore the FM spectrum bandwidth is approximately (considering the main lobe
1  k f mp 
of sinc function) BFM 
2π
 2k f mp  8πB   2   2B 
 2π 
 Previously the maximum and minimum carrier frequencies were approximated as
ωc  k f mp and ωc  k f mp , respectively (section 4.1). Hence, the spectral components
was assumed to remain within this band with a bandwidth 2k f mp , centered at ω c .
The implicit assumption was that a sinusoid of frequency ω has its entire spectrum
centered at ω . This is true only of everlasting sinusoid with T   (because it turns
the sinc function into an impulse). For a sinusoid of finite duration T seconds, the
spectrum is spread out by the sinc function on either side of ω by at least the main
lobe width of 2π T .
 Given the deviation of the carrier frequency (rad/sec) by k f mp , we shall denote the
fmax  fmin mp
peak frequency deviation in Hz by f . Thus f  k f  kf . The estimated
2.2π 2π
FM bandwidth (in Hz) can then be expressed as BFM 2  f  2B  .
 The bandwidth estimates thus obtained is somehow higher than the actual value
because this is the bandwidth corresponding to the staircase approximation of m(t),
not the actual m(t), which is considerably smoother. Hence, the actual FM bandwidth
is somewhat smaller than this value. A better FM bandwidth approximation is
between  2f,2f  4B  .
 In the case of NBFM, kf is very small. Hence, given a fixed mp , f is very small (in
comparison to B) for NBFM. In this case, we can ignore the small f term with the
result BFM  4B . But we showed that for narrowband, the FM bandwidth is
approximately 2B Hz.
 k f mp 
 This indicates that a better bandwidth estimate is BFM  2  f  B   2   B .
 2π 
This formula is known as Carson’s rule.
 For a truly wideband case, where f B , above equation can be approximated as
BFM  2f .
 If we define a deviation ratio β as β  f B , Carson’s rule can be expressed in terms
of the deviation ratio as BFM  2B β  1 .
 The deviation ratio controls the amount of modulation and, consequently, plays a
role similar to the modulation index in AM. For the special case of tone-modulated
FM, the deviation ratio is called the modulation index.
4.2.3 Phase modulation

 All the result derived for FM can be directly applied to PM. Thus, for PM, the

instantaneous frequency is given by ωi  ωc  k p m  t  .
 
m t  m t
 Therefore, the peak frequency deviation f  k p min
. max
2.2π
[This equation can be applied only if m(t) is a continuous function of time. If m(t) has
jump discontinuities, its derivative does not exist. In such case, we should use the
direct approach to find φPM  t  and then determine ω from φPM  t  ].

   mp
 If we assume that mp  m  t    m t  then f  k p .
max min 2π
  
 k p mp
 Therefore, BPM  2  f  B   2  B .
 2π 
 
 
 One very interesting aspect of FM is that ω  k f mp depends only on the peak
value of m(t). it is independent of the spectrum of m(t). On the other hand, in PM,
  
ω  k p mp depends on the peak value of m  t  . But m  t  depend strongly on the
spectral composition of m(t).
 The presence of higher frequency component in m(t) implies rapid time variations,

resulting in a higher value of mp . Conversely, predominance of lower frequency

components will result in a lower value of mp . Hence, whereas the FM signal
bandwidth is practically independent of the spectral shape of m(t), the PM signal
bandwidth is strongly affected by the spectral shape of m(t). For m(t) with a
spectrum concentrated at lower frequencies, BPM will be smaller than when the
spectrum m(t) is concentrated at higher frequencies.

4.2.4 Spectral analysis of tone frequency modulation

 For an FM carrier with a generic message signal m(t), the spectral analysis
requires the use of staircase signal approximation. Tone modulation is a special
case for which precise spectral analysis is possible: that is, when m(t) is a
sinusoid.
 Let m  t   α cos  ωmt  . Therefore with the assumption that initially a     0 , we
 α 
jωc t  k f sin ωm t  
α
have a  t   sin  ωmt  and φ̂FM  t   Ae 
ωm 
.
ωm
 Moreover, ω  k f mp  αk f and the bandwidth of m(t) is 2πB  ωm rad/sec.
 f ω αk f
 The deviation ratio (or, in this case modulation index) is β    .
B 2πB ωm
jωct βsin ωmt  jβsin ωmt 
 Hence, φ̂FM  t   Ae   Ae jωcte .
jβsin ωmt 
 e is a periodic signal with period 2π ωm and can be expanded by the

jβsin ωmt 
exponential Fourier series, as usual, e   Dne jωmt where
n
m πω π
ωm jβsin ωmt   jnωmt 1 jβsin x  nx 
Dn  
2π π ω
e e dt 
2π π
e dx
m
 The integral on the right-hand side cannot be evaluated in a closed form but must
be integrated by expanding the integrand in infinite series. This integral has been
extensively tabulated and is denoted by Jn  β  , the Bessel function of the first

jβsin ωmt 
kind and the nth order (shown). Thus e   Jn β e jnωmt .
n

 Therefore,
 
φ̂FM  t   A  Jn β e 
j ωc t  nωmt 
and φFM  t   A  Jn β cos  ωc  nωm  t .
n n

 The tone modulated FM signal has a carrier component and an infinite number
of sidebands of frequencies ωc  ωm , ωc  2ωm ,….. , ωc  nωm , ….. , as shown.
This is in stark contrast to the DSB-SC spectrum of only one sideband on either
side of the carrier frequency. The strength of the nth sideband at ω  ωc  nωm
is Jn  β  .
  From the plot of Jn  β  it can be seen that for a given β , Jn  β  decreases with n,
and there are only a finite number of significant sideband spectral lines.
 It can be seen that Jn  β  is negligible for n  β  1. Hence, the number of
significant sideband impulses is β  1 .
 The bandwidth of the FM carrier is given by BFM  2 β  1 fm  2  f  B which
corroborates our previous result.
 When β 1 (NBFM), there is only one significant sideband and the bandwidth
BFM  2fm  2B .
 The method for finding the spectrum of a tone-modulated FM wave can be used
for finding the spectrum of an FM wave when m(t) is a general periodic signal. In
jkf a  t 
this case φ̂FM  t   Ae jωcte .
 Because a(t) is a periodic signal, e f   is also a periodic signal, which can be
jk a t

expressed as an exponential Fourier series. After this, it is relatively straight

forward to write φFM  t  in terms of the carrier and sidebands.

Example 4.3

Estimate BFM and BPM for the modulating signal m(t) in figure below for k f  2π  105
and k p  5π . Assume the essential bandwidth of the periodic m(t) as the frequency of
its third harmonics. Repeat the problem if the amplitude of m(t) is doubled . Also repeat
the problem if m(t) is time expanded by a factor of 2.

Solution:

The peak amplitude of m(t) is unity. Hence mp  1 .

The Fourier series for this periodic signal is given by m  t    Cn cos  nω0t  where
n

 8
2π 4  2 2 n odd
ω0  4
 10 π and Cn  n π
2  10 
0 n even

It can be seen that the harmonic amplitudes decreases rapidly with n. the third
harmonic is only 11% of the fundamental, and the fifth harmonic is only 4% of the
fundamental. This means the third and fifth harmonic powers are 1.21 and 0.16%,
respectively, of the fundamental component power. Hence, we are justified in assuming
the essential bandwidth of m(t) as the frequency of its third harmonic, that is,

B  3  104 2  15  103 Hz  15 kHz

For FM

k f mp 2π  105
f    105 Hz  100 kHz and
2π 2π

BFM  2  f  B   2 100  15   230 kHz

f 100
Alternatively, the deviation ratio β is given by β   and
B 15

 100 
BFM  2B β  1  2 15   1  230 kHz
 15 

For PM

The peak amplitude of m  t  is 2 104 . Therefore,


k p mp 5π  2 104
f    5  104 Hz  50 kHz and
2π 2π

BPM  2  f  B   2 50  15   130 kHz

f 50
Alternatively, the deviation ratio β is given by β   and
B 15

 50 
BPM  2B β  1  2 15   1  130 kHz
 15 

Doubling m(t) doubles its peak value. Hence mp  2 . But its bandwidth is unchanged
so that B = 15 kHz.

For FM

k f mp 2π  2  105
f    2  105 Hz  200 kHz and
2π 2π
BFM  2  f  B   2  200  15   430 kHz

f 200
Alternatively, the deviation ratio β is given by β   and
B 15

 200 
BFM  2B β  1  2 15   1  430 kHz
 15 

For PM

Doubling m(t) doubles its derivative so that now mp  4  104 and


k p mp 5π  4  104
f    105 Hz  100 kHz and
2π 2π

BPM  2  f  B  2 100  15   230 kHz

f 100
Alternatively, the deviation ratio β is given by β   and
B 15

 100 
BPM  2B β  1  2 15   1  230 kHz
 15 

Observed that doubling the signal amplitude roughly doubles frequency deviation of
both FM and PM waves.

If m(t) is time expanded by a factor of 2 then the time period of m(t) is 4 104 .

The time expansion of a signal by a factor of 2 reduces the signal spectral width
(bandwidth) by a factor of 2. The fundamental frequency is now 2.5 kHz and its third
harmonic is 7.5 kHz. Hence, B = 7.5 kHz, which is half the previous bandwidth.
Moreover time expansion does not affect the peak amplitude and thus mp  1 . However,
 
mp is halved, that is mp  104 .

For FM:

k f mp 2π  105
f    105 Hz  100 kHz
2π 2π

BFM  2  f  B   2 100  7.5   215 kHz

For PM

k p mp 5π  104
f    2.5  104 Hz  25 kHz
2π 2π

BPM  2  f  B   2  25  7.5   65 kHz

Note that time expansion of m(t) has very little effect on the FM bandwidth, but it halves
the PM bandwidth. This verifies our observation that the PM spectrum strongly
dependent on the spectrum m(t).

Example 4.4

An angle-modulated signal with carrier frequency ωc  2π  105 is described by the

equation φEM  t   10cos ωct  5sin 3000t   10sin 2000πt  . Find the (a) power of the
modulated signal, (b) frequency deviation f , (c) the deviation ratio β , (d) phase
deviation φ and € estimate the bandwidth of φEM  t  .

Solution
The signal bandwidth is the highest frequency in m(t) (or its derivative). In this case

2000π
B  1000 Hz
2π

102
(a) The carrier amplitude is 10, and the power is P   50
2
(b) To find the frequency deviation f , we find the instantaneous frequency ωi , given
by
d
dt
 θ  t    ωc  15000 cos  3000t   20000π cos  2000πt 
The carrier deviation is 15000cos  3000t   20000π cos  2000πt  . The two
sinusoids will add in phase at some point, and the maximum value of this
expression is 15000  20000π . This is the maximum carrier deviation ω . Hence

ω 15000  20000π
f    12387.32 Hz
2π 2π

f 12387.32
(c) β    12.387
B 1000
(d) The angle θ  t   ωt  5sin 3000t   10sin 2000πt  . The phase deviation is the
maximum value of the angle inside the parenthesis, and is given by φ  15 rad
.
 (e) BEM  2  f  B  2 12387.32  12.387   26774.65 Hz

Observe the generality of this method of estimating the bandwidth of an angle modulated
waveform. We need not know whether it is FM, PM, or some other kind of angle
modulation. It is applicable to any angle-modulated signal.

4.3 Generating FM waves

 Basically, there are two ways of generating FM waves: indirect and direct. We first
describe narrow band FM generator that is utilized in the indirect FM generation
of wideband angle modulation signals.
4.3.1 NBFM Generation

 For NBFM and NBPM signals, we have shown earlier that because of kf a  t  1
and kpm  t  1 , respectively, the modulated signal can be approximated by

φNBFM  t  A cos  ωct   kf a  t  sin  ωct 

φNBPM  t  A cos  ωc t   kpm  t  sin  ωc t 
Both approximations are linear and are similar to the expression of the AM wave.
 Above equations suggest a possible method of generating narrowband FM and
PM signals by using DSB-SC modulators, as shown below.

 It is important to point out that the NBFM generated by the circuit (Fig. (b)) has
some distortion because of the approximation. The output of this NBFM
modulator also has some amplitude variations. A nonlinear device designed to
limit the amplitude of a bandpass signal can remove most of this distortion.
4.3.2 Bandpass limiter

 The amplitude variations of an angle-modulated carrier can be eliminated by

what is known as a bandpass limiter, which consists of a hard limiter followed
by a bandpass filter.

 The input-output characteristic of a hard limiter is shown below (Fig. (b)).

Observe that the bandpass limiter output to a sinusoid will be a square wave of
unit amplitude regardless of the incoming sinusoid amplitude. Moreover, the zero
crossing of the incoming sinusoid are preserved in the output because when the
input is zero, the output is also zero.

 Thus an angle-modulated sinusoid input vi  t   A  t  cos θ  t  results in a

constant amplitude, angle –modulated square wave vo  t  , as shown (Fig.(c)). As
we have seen, such a nonlinear operation preserves the angle modulation
information.

 When vo  t  is passed through a bandpass filter centered at ω c , the output is a

angle-modulated wave, of constant amplitude.
 To show this consider the incoming angle-modulated wave vi  t   A  t  cos θ  t 
t
where θ  t   ωc t  k f  m  α  dα .

  The output vo  t  of the hard limiter is +1 or -1, depending on whether
vi  t   A  t  cos θ  t  is positive or negative.
 Because A t  0 , vo  t  can be expressed as a function of θ:

 1 cos  θ   1

vo  θ    .
 1 cos  θ   1
 Hence vo as a function of θ is a periodic square wave with period 2π (Fig. (d)),
which can be expanded by a Fourier series
4 1 1 
vo  θ   cos  θ   cos  3θ   cos 5θ   
π 3 5 

t
 At any instant t, θ  t   ωc t  k f  m  α  dα . Hence, the output vo  t  is given by


vo θ  t   vo ωc t  k f  m  α  dα 

 
4
π
 1
3
  
Or, vo θ  t   cos ωc t  k f  m  α  dα  cos 3 ωc t  k f  m α  dα  


 The output, therefore, has original FM wave plus frequency-multiplied FM waves
with multiplication factors 3, 5, 7, … .
 We can pass the output of the hard limiter through a bandpass filter with a center
frequency ω c and a bandwidth BFM (Fig. (a)). The filter output eo  t  is the desired
angle-modulated carrier with a constant amplitude,
4

eo  t   cos ωc t  k f  m  α  dα 
π  
 Although we derived results for FM, this applies to PM (angle modulation in
general) as well.
 The bandpass filter not only maintains the constant amplitude of the angle-
modulated carrier but also partially suppresses the channel noise when the noise
is small.

4.3.3 Indirect method of Armstrong

 In Armstrong’s indirect method, NBFM is generated through the circuit, shown

in section 4.3.1. The NBFM is then converted to WBFM by using additional
frequency multipliers.
 A frequency multiplier can be realized by a nonlinear device followed by a
bandpass filter.
 Let us consider a nonlinear device whose output signal y(t) to an input x(t) is
given by y  t   a 2 x 2  t  .
 If an FM signal passes through this device, then the output signal will be
y  t   a 2 cos2 ωc t  k f  m  α  dα   0.5a 2  0.5a 2 cos 2ωc t  2k f  m  α  dα 
   
 Thus, a bandpass filter centered at 2ωc would recover an FM signal with twice
the original instantaneous frequency.
 To generalize, a nonlinear device may have the characteristics of
y  t   a 0  a1x  t   a 2 x 2  t   a n x n  t 
 If x  t   A cos ωc t  k f  m  α  dα  , then by using trigonometric identities, we can
 
readily show that y(t) is of the form
y  t   c 0  c1 cos ωc t  k f  m  α  dα   c 2 cos 2ωc t  2k f  m  α  dα  
   
c n cos nωc t  nk f  m  α  dα 
 
 Hence, the output will have spectra at ω c , 2ωc ,… , nωc , with frequency
deviations f , 2f ,…., nf , respectively.
 Thus, a bandpass filter centering at nωc can recover an FM signal whose
instantaneous frequency has been multiplied by a factor of n.
 These devices, consisting of nonlinearity and bandpass filters, are known as
frequency multipliers.
 In fact, a frequency multiplier can increase both the carrier frequency and the
frequency deviation by an integer n.
 This forms the basis of the Armstrong indirect frequency modulator. First,
generate an NBFM approximately. Then multiply the NBFM frequency and limit
its amplitude variation.
 Generally we require to increase f by a very large factor n. This increases the
carrier frequency also by n. Such a large increase in the frequency may not be
needed. In this case we can apply frequency mixing to shift down the carrier
frequency to the desired value.
 A simplified diagram of a commercial FM transmitter using Armstrong’s method
is shown below. The final output is required to have a carrier frequency 91.2 MHz
and f  75 kHz .
 We begin with a carrier frequency fc1  200 kHz generated by a crystal oscillator.
This frequency is chosen because it is easy to construct stable crystal oscillators
as well as balanced modulators at this frequency.
 To maintain β 1, as required NBFM, the deviation f is chosen to be 25 Hz.
The baseband spectrum ranges from 50 Hz to 15 kHz. The choice of f  25 Hz
is reasonable because it gives β  0.5 for the worst possible case  fm  50 Hz  .
 To achieve f  75 kHz , we need a multiplication factor of 75000/25 = 3000. This
can be done by two multiplier stages, of 64 and 48, giving a total multiplication
of 3072 and f  76.8 kHz .
 The multiplication can be effected by using frequency doublers and triplers in
cascade, as needed. Thus a multiplication of 64 can be obtained by six doublers
in cascade, and a multiplication of 48 can be obtained by four doublers and a
tripler in cascade.
 Multiplication of fc1  200 kHz by 3072, however, would yield a final carrier of
about 600 MHz. This problem is solved by using a frequency translation, or
conversion, after the first multiplier, as shown.
 The first multiplication by 64 results in the carrier frequency
fc2  200 kHz  64  12.8 MHz , and the carrier deviation f2  25  64  1.6 kHz .
We now use a frequency converter (or mixer) with carrier frequency 10.9 MHz.
 This results in a new carrier frequency fc3  12.8  10.9  1.9 MHz .
 The frequency converter shifts the entire spectrum without altering f . Hence
f3  1.6 kHz .
 Further multiplication, by 48, yields
fc4  1.9  48  91.2 MHz and f4  1.6  48  76.8 kHz .
 This scheme has the advantage of frequency stability, but it suffers from inherent
noise caused by excessive multiplication and distortion at lower modulating
frequencies, where f fm is not small enough.
 Two kinds of distortion arise in this scheme: amplitude distortion and frequency
distortion.
 The NBFM wave is given by
φFM  t   A cos  ωct   kf a  t  sin ωct   AE t  cos ωct  θ t 

Where E  t   1  k 2f a 2  t  and θ  t   tan1 kf a  t  .

 Amplitude distortion occurs because the amplitude AE(t) of the modulated
waveform is not constant. This is not a serious problem because amplitude
variations can be eliminated by a bandpass limiter, as discussed earlier.
 Ideally θ  t  should be k f a  t  . Instead, the phase θ  t  in the preceding equation
is θ  t   tan1 kf a  t  and the instantaneous frequency ωi  t  is
 
 kf a t kf m t
ωi  t   θ  t     k f m  t  1  k 2f a 2  t   k f4a 4  t   

1 k 2f a 2 t 1  k 2f a 2  t 
Ideally instantaneous frequency should be k f m  t  . The remaining terms in this
equation are the distortion.
 Let us investigate the effect of this distortion in tone modulation where
m  t   α cos  ωmt  , a  t   αsin  ωmt  ωm , and the modulation index β  αk f ωm :
ωi  t   βωm cos  ωmt  1  β2 sin2  ωmt   β4 sin4  ωmt  


 It is evident from this equation that the scheme has odd-harmonic distortion, the
most important term being the third harmonic. Ignoring the remaining terms,
this equation becomes
 β2  β3ωm
ωi  t  βωm cos  ωmt  1  β2 sin2  ωmt   βωm 1   cos  ωmt   cos 3ωmt 
 4 4
distortion
desired

Example 4.5
Design an Armstrong indirect FM modulator to generate an FM signal with carrier
frequency 97.3 MHz and f  10.24 kHz . A NBFM generator fc1  20 kHz and f  5 Hz
is available. Only frequency doublers can be used as multipliers. Additionally, a local
oscillator (LO) with adjustable frequency between 400 and 500 kHz is readily available
for frequency mixing.
Solution:

The modulator is shown below. We need to determine M1 , M2 and fLO .

The NBFM generator generates fc1  20 kHz and f1  5 Hz . The final WBFM should
have fc4  97.3 MHz and f4  10.24 kHz .

We first find the total factor of frequency multiplication

f 10240
M1.M2  4   2048  211 .
f1 5

Because only frequency doublers can be used, we have three equations: M1  2n1 ,
M2  2n2 , and n1  n2  11 .
It is also clear that, fc2  2n1 fc1 and fc4  2n2 fc3 .

To find fLO , there are three possible relationships: fc3  fc2  fLO and fc3  fLO  fc2 . Each
should be tested to determine one that will fall in 400 kHz  fLO  500 kHz .

(a) Firs, we test fc3  fc2  fLo . This case leads to

 
97.3 106  2n2 2n1 fc1  fLo  2n1n2 fc1  2n2 fLo  211  20 103  2n2 fLo .


Thus, we have fLo  2n2 4.096 107  9.73 107  0 . 
This is outside the local oscillator frequency range.
(b) Next, we test fc3  fc2  fLo . This case leads to
 
97.3 106  2n2 2n1 fc1  fLo  2n1n2 fc1  2n2 fLo  211  20 103  2n2 fLo

 
Thus, we have fLo  2n2 9.73 107  4.096 107  2n2  5.634 107 .
If n2=7, then fLo  440 kHz , which is within the realizable range of the local
oscillator.
(c) If we choose fc3  fLo  fc2 , then we have
 
97.3 106  2n2 fLo  2n1 fc1  2n2 fLo  2n1n2 fc1  2n2 fLo  211  20 103

 
Thus, we have fLo  2n2 9.73 107  4.096 107  2n2 13.826 107
No integer n2 will lead to a realizable fLo .

Thus, the final design is M1  16 , M2  128 , and fLo  440 kHz .

4.3.4 Direct generation

 In a voltage-controlled oscillator (VCO), the frequency is controlled by an external
voltage. The oscillation frequency varies linearly with the control voltage. We can
generate an FM wave by using the modulating signal m(t) as a controlled voltage.
This gives ωi  t   ωc  k f m  t  .
 One can construct a VCO using an operational amplifier and a hysteresis
comparator, such as Schmitt trigger circuit.
 Another way of accomplishing the same goal is to vary one of the reactive
parameters (C or L) of the resonant circuit of an oscillator. A reverse-biased
semiconductor diode acts as a capacitor whose capacitance varies with the bias
voltage. The capacitance of these diodes, known under several names (e.g.,
varicap, varactor, voltacap), can be approximated as a linear function of the bias
voltage m(t) over a limited range.
 In Hartley or Colpitt oscillators, for instance, the frequency of oscillation is given
1
by ω0  .
LC
 If the capacitance C is varied by the modulating signal m(t), that is, if
C  Co  km  t  then
 1 1 1 1  km  t  
ω0     1  
L Co  km  t 
1
 km  t    km  t  2
LCo  2Co 
LCo 1   LCo 1 
 Co  
 Co 
km  t 
Where 1
Co
[We have applied the Taylor series approximation 1  x   1  nx for x
n
1 with
n=1/2]
 km  t   1 kωc
 Thus, ω0  ωc 1    ωc  k f m  t  where ωc  , kf 
 2Co  LCo 2Co
 Because C  Co  km  t  , the maximum capacitance deviation is
2k f Co mp
C  kmp  .
ωc
C 2k f mp 2f k f mp
 Hence,   [As f  ]
Co ωc fc 2π
 In practice, f fc is usually small, and, hence, C is a small fraction of Co , which
helps limit the harmonic distortion that arises because of the approximation used
in this derivation.
 We may also generate direct FM by using a suitable core reactor, where the
inductance of a coil is varied by a current through a second coil (also wound
around the same core). This results in a variable inductor whose inductance is
proportional to the current in the second coil.
 Direct FM generation generally produces sufficient frequency deviation and
requires little frequency multiplication. But this method has poor frequency
stability.
 In practice feedback is used to stabilize frequency. The output frequency is
compared with a constant frequency generated by a stable crystal oscillator. An
error signal (error in frequency) is detected and fed back to the oscillator to correct
the error.

4.4 Demodulation of FM signals

 The information in an FM signal resides in the instantaneous frequency
ωi  ωc  k f m  t  . Hence a frequency-selective network with a transfer function of
the form H  f   2πaf  b over the FM band would yield an output proportional
to the instantaneous frequency (provided the variations of ωi are slow in
comparison to the time constant of the network).
 There are several possible with such characteristics. The simplest among them
is an ideal differentiator with the transfer function j2πf .
 If we apply φFM  t  to an ideal differentiator, the output is
d     
t t
 
φ̂FM  t   A cos ωc t  k f  m  α  dα    A ωc  k f m  t  sin ωc t  k f  m α  dα  π 
dt 
    
   
Both the amplitude and the frequency of the signal φ̂FM  t  are modulated, the
envelope being A ωc  kf m  t  .

 Because ω  k f mp  ωc , we have ωc  k f m  t   0 for all t, and m(t) can be

obtained by envelope detection of φ̂FM  t  .
 The amplitude A of the incoming FM carrier must be constant. If the amplitude
A are not constant, but a function of time, there would be an additional term
containing dA/dt on the right-side of the above equation. Even if this term is
neglected, the envelope of φ̂FM  t  would be A  t  ωc  kf m  t  , and the envelope-
detector output would be proportional to m(t)A(t), still leading to distortion.
Hence, it is essential to maintain A constant.
  Several factors, such as channel noise and fading, cause A to vary. This variation
in A should be suppressed via the bandpass limiter before the signal is applied
to the FM detector.

4.4.1 Practical frequency demodulators

 The role of the differentiator can be replaced by any linear system whose
frequency response contains a linear segment of positive slope. One simple device
would be an RC high-pass filter, shown below.

j2πfRC
 The RC frequency response is simply H  f    j2πfRC if 2πfRC 1.
1  j2πfRC
Tus, if the parameter RC is very small such that its product with the carrier
frequency ωcRC 1, the RC filter approximates a differentiator.
 Similarly, a simple tuned RLC circuit followed by an envelope detector can also
serve as a frequency detector because its frequency response |H(f)| below the
resonance frequency ω0  1 LC approximates a linear slope. Thus such a
receiver design requires that ωc  ω0  1 LC .
 Because the operation is on the slope of |H(f)|, this method is also called slope
detection.
 Since, however, the slope of |H(f)| is linear over only a small band, there is
considerable distortion in the output. This fault can be partially corrected by a
balanced discriminator formed by two slope detectors
 Another balanced demodulator, the ratio detector, also widely used in the past,
offers a better protection against carrier amplitude variations than does the
discriminator.
 Zero-crossing detectors are also used because of advances in digital integrated
circuits. The first step to use the amplitude limiter to generate the rectangular
pulse output.
 The resulting rectangular pulse train of varying width can then be applied to
trigger a digital counter. These are frequency counters designed to measure the
instantaneous frequency from the number of zero crossings.
  The rate of zero crossings is equal to the instantaneous frequency of the input
signal.

4.4.2 FM demodulation via PLL

 Consider a PLL that is in lock with input signal sin ωct  θi  t  and output error
signal eo  t  .
t
π
 When the input signal is an FM signal, θi  t   k f  m  α  dα  2 , then

t
π
θo  t   k f  m  α  dα  2  θe  t  .


 With PLL in lock we can assume a small frequency error θe  t   0 . Thus, the loop
1 d  
t
1 π kf
filter output signal is eo  t   θo  t   k f  m  α  dα   θe  t  m t  .
c c dt   2  c
Thus, the PLL acts as an FM demodulator.

 If the incoming signal is a PM wave, then eo  t   k p m  t  c . In this case we need
to integrate eo  t  to obtain the desired signal m(t).
 To more precisely analyze PLL behavior as an FM demodulator, we consider the
case of a small error (linear model of PLL) with |H(s)|=1. For this case, feedback
AKH  s  AK
analysis of the small-error PLL becomes Θo  s   Θi  s   Θi s  .
s  AKH s  s  AK
 If Eo  s  and M(s) are the Laplace transforms of eo  t  and m(t), respectively, then
k f M s 
we have Θi  s   and sΘo  s   cEo  s  .
s
[Taking the Laplace transform of the equation θi  t  and eo  t  , shown above]
s s AK s AK k f M  s  k f AK
 Hence, Eo  s  Θo  s   Θi  s    M s  .
c c s  AK c s  AK s c s  AK
 Thus, the PLL output eo  t  is a distorted version of m(t) and is equivalent to the
output of a single-pole circuit (such as simple RC circuit) with transfer function
k f AK
to which m(t) as the input.
c s  AK
 To reduce distortion, we must choose AK well above the bandwidth of m(t), so
that eo  t  k f m  t  c .
 In the presence of small noise, the behavior of the PLL is comparable to that of a
frequency discriminator. The advantage of the PLL over a frequency discriminator
appears only when the noise is large.
4.5 Effects of nonlinear distortion and interference
4.5.1 Immunity of angle modulation to nonlinearities
 A very useful feature of angle modulation is its constant amplitude, which makes
it less susceptible to nonlinearities.
 Consider, for instance, an amplifier with second-order nonlinear distortion whose
input x(t) and output y(t) are related by y  t   a 0  a1x  t   a 2 x 2  t   a n x n  t  .
Clearly, the first term is the desired signal amplification term, while the remaining
terms are the unwanted nonlinear distortion.
 For the angle modulated signal x  t   Acos ωc t  ψ t  .
 Trigonometric identities can be applied to rewrite the non-ideal system output
y(t) as
y  t   c0  c1 cos ωc t  ψ t   c2 cos 2ωct  2ψ t    cn cos nωct  nψ t 
 Because sufficiently large ω c makes each component of y(t) separable in
frequency domain, a bandpass filter centered at ω c with bandwidth equaling BFM
(or BPM ) can extract the desired FM signal component c1 cos ωc t  ψ t  without
any distortion.
 This shows that angle-modulated signals are immune to nonlinear distortion.
 A similar nonlinearity in AM not only causes unwanted modulation with carrier
frequencies nωc but also causes distortion of the desired signal. For instance, if
a DSB-SC signal m  t  cos  ωc t  passes through a nonlinearity
y  t   ax  t   bx 3  t  , the output is
y  t   am  t  cos  ωc t   bm3  t  cos 3  ωc t 
 3b 3  b
Or, y  t   am  t   m  t  cos  ωc t   m3  t  cos  3ωc t 
 4  4
 3b 3 
Passing this through a bandpass filter still yields am  t   m  t  cos  ωc t  .
 4 
3b 3
Observe the distortion component m  t  present along with the desired signal
4
am  t  .
 Immunity from nonlinearity is the primary reason for the use of angle modulation
in microwave radio relay systems, where power levels are high. This requires
highly efficient nonlinear class C amplifiers.
 In addition, the constant amplitude of FM gives it a kind of immunity to rapid
fading. The effect of amplitude variations caused by rapid fading can be
eliminated by using automatic gain control and bandpass limiting.
 These advantages made FM attractive as the technology behind the first-
generation cellular phone system.
  The same advantage of FM also make it attractive for microwave radio relay
systems. In the legacy analog long-haul telephone systems, several channels are
multiplexed by means of SSB signals to form L-carrier signals. The multiplexed
signals are frequency-modulated and transmitted over a microwave radio relay
system with many links in tandem. In this application, however, FM is used not
to reduce noise effects but to realize other advantages of constant amplitude, and,
hence, NBFM rather than WBFM is used.

4.5.2 Interference effect

 Angle modulation is also less vulnerable than AM to small-signal interference
from adjacent channels.
 Let us consider the simple case of the interference of an unmodulated carrier
A cos  ωc t  with another sinusoid Icos  ωc  ω t  . The received signal r(t) is
r  t   Acos  ωct   Icos  ωc  ω t  A  Icos  ωt  cos  ωct   Isin  ωt  sin  ωct 
Or, r  t   Er  t  cos ωc t  ψd  t 
Isin  ωt 
where ψd  t   tan1
A  Icos  ωt 
 When the interfering signal is small in comparison to the carrier I A ,
I
ψd  t  sin  ωt  .
A
 The phase of Er  t  cos ωc t  ψd  t  is ψd  t  , and its instantaneous frequency is

ωc  ψd  t  .
 If the signal Er  t  cos ωc t  ψd  t  is applied to an ideal phase demodulator, the
output y d  t  would be ψd  t  . Similarly, the output y d  t  of an ideal frequency

demodulator would be ψd  t  . Hence
 I
 A sin  ωt  for PM
yd  t   
 Iω cos  ωt  for FM
 A
 Observe that in either case, the interference output is inversely proportional to
the carrier amplitude A. Thus, the larger the carrier amplitude A, the smaller the
interference effect.
 This behavior is very different from the AM signals, where the interference output
is independent of the carrier amplitude. Hence, angle-modulated systems are
much better than AM systems at suppressing weak interference.
 For AM signal with an interfering sinusoid Icos  ωc  ω t  is given by
r  t   A  m  t  cos  ωct   Icos  ωc  ω t 
 Or, r  t   A  m  t   Icos  ωt  cos  ωct   Isin  ωt  sin  ωct 
The envelope of this signal is
2
E  t    A  m  t   Icos  ωt    I2 sin2  ωt 
Or, E  t   A  m  t   Icos  ωt 
Thus the interference signal at the envelope detector is Icos  ωt  , which is
independent of the carrier amplitude A.
 Because of the suppression of weak interference in FM, we observe what is known
as the capture effect when listening to FM radios. For two transmitters with
carrier frequency separation less than the audio range, instead of getting
interference, we observe that the stronger carrier effectively suppresses (captures)
the weaker carrier.
 Subjective tests show that an interference level as low as 35 dB in the audio
signals can cause objectionable effects. Hence in AM, the interference level should
be kept below 35 dB. On the other hand, for FM, because of the capture effect,
the interference level need only be below 6 dB.
 The interference amplitude (I/A for PM and Iω A foe FM) vs. ω at the receiver
output is shown below. The interference amplitude is constant for all ω in PM
but increases linearly with ω in FM.


4.5.3 Interference due to channel noise:

 The channel noise acts as interference in an angle-modulated signal.

 We shall consider the most common form of noise, white noise, which has a
constant power spectral density. Such a noise may be considered as a sum of
sinusoids of all frequencies in the band. All component have the same amplitudes
(because of uniform density).
 This means I is constant for all ω , and the amplitude spectrum of the interference
at the receiver output is shown above.
 The interference amplitude spectrum is constant for PM, and increases linearly
with ω for FM.
4.5.4 Preemphasis and deemphasis in FM broadcasting
 It is shown that in FM, the interference (the noise) increases linearly with
frequency, and the noise power in the receiver output is concentrated at higher
frequencies.
 On the other hand the PSD of an audio signal m(t) is concentrated at lower
frequencies below 2.1 kHz, as shown below. Thus, the noise PSD is concentrated,
where m(t) is weakest.


 This may seem like a disaster. But actually, in this very situation there is a
hidden opportunity to reduce noise greatly. The process, shown below, works as
follows.

 At the transmitter, the weaker high-frequency components (beyond 2.1 kHz) of

the audio signal m(t) are boosted before modulation by a preemphasis filter of
transfer function Hp  f  . At the receiver, the demodulator output is passed
through a deemphasis filter of transfer function Hd  f   1 Hp  f  . Thus, the
deemphasis filter undoes the preemphasis by attenuation (deemphasizing) the
higher frequency components (beyond 2.1 kHz), and thereby restores the original
m(t).
 The noise, however, enters the channel, and therefore has not been pre-
emphasized (boosted). However, it passes through the deemphasis filter, which
attenuates its higher frequency components, where most of the noise power is
concentrated. Thus, the process of Preemphasis-deemphasis (PDE) leaves the
desired signal untouched but reduces the noise power considerably.
4.5.5 Preemphasis and deemphasis filters

 The FM has smaller interference than PM at lower frequencies, while the opposite
is true at higher frequencies. If we can make our system behave like FM at lower
frequencies and behave like PM at higher frequencies, we will have the best. This
is accomplished by a system used in commercial broadcasting with the
preemphasis (before modulation) and deemphasis (after demodulation) filters
Hp  f  and Hd  f  , shown below.

 The frequency f1 is 2.1 kHz, and f2 is typically 30 kHz or more (well beyond
range), so that f2 does not even enter into the picture. These filters can be realized
by simple RC circuits.
 The choice of f1  2.1 kHz was apparently made on an experimental basis. It was
found that this choice of f1 maintained the same amplitude mp with or without
preemphasis. This satisfied the constraint of a fixed transmission bandwidth.
j2πf  ω1
 The preemphasis transfer function is Hp  f   K where K, the gain, is set
j2πf  ω2
ω2 j2πf  ω1
a value of ω2 ω1 . Thus, Hp  f   .
ω1 j2πf  ω2
2πf
 For 2πf ω1 , Hp  f  1 . For frequencies ω1 2πf ω2 , Hp  f  .
ω1
 Thus the preemphasizer acts as a differentiator at intermediate frequencies. This
means that FM with PDS is EM over the modulating signal frequency range 0 to
2.1 kHz and is nearly PM over the range of 2.1 to 15 kHz as desired.
ω1
 The deemphasis filter Hd  f  is given by Hd  f   .
j2πf  ω1
 2πf
 Note that for 2πf ω2 , Hp  f  . Hence Hp  f  Hd  f  1 over the baseband of
ω1
0 to 15 kHz.
 It can be shown that the PDE enhances the SNR by 13.27 dB (a power ratio
21.25).
 The PDE method of noise reduction is not limited to FM broadcast. It is also used
in audiotape recording and in (analog) phonograph recording; where hissing noise
is also concentrated at the high-frequency end. A sharp, hissing sound is caused
by irregularities in the recording material.
 The Dolby noise reduction systems for audio tapes operates on the same
principle.
 We could also use PDE in AM broadcasting to improve the output SNR.
 In practice, however, this is not done for several reasons. First, the output noise
amplitude in AM is constant with frequency and does not increase linearly as in
FM. Second, introduction of PDE would necessitate modifications of receivers
already in use. Third, increasing high-frequency component amplitudes
(preemphasis) would increase interference with adjacent stations (no such
problem arises in FM). Moreover, an increase in the frequency deviation ratio β
at high frequencies would make detector design more difficult.

4.6 Super heterodyne analog AM/FM receivers

 The radio receiver used in broadcast AM and FM systems, is called
superheterodyne receiver, shown below. It consists of a radio-frequency (RF)
section, a frequency converter, an intermediate-frequency (IF) amplifier, an
envelope detector, and an audio amplifier.
 The RF section consists basically of a tunable filter and an amplifier that picks
up the desired station by tuning the filter to the right frequency band.
 The next section, the frequency mixer (converter) translates the carrier ω c to a
fixed IF frequency ωIF . For this purpose, the receiver uses a local oscillator (LO)
whose frequency fLO is exactly fIF above the incoming carrier frequency fc ; that
is fLO  fc  fIF .
 The simultaneous tuning of the local oscillator and the RF tunable filter is done
by one joint knob. Tuning capacitors in both circuits are ganged together and are
designed so that the tuning frequency of the local oscillator is always fIF Hz above
the tuning frequency fc of the RF filter. This means every station that is tuned is
translated to a fixed carrier frequency of fIF Hz by the frequency converter for
subsequent processing at IF.
 This superheterodyne receiver structure is broadly utilized in most broadcast
systems. The intermediate frequencies are chosen to be 455 kHz (AM radio), 10.7
MHz (FM radio), and 38 MHz (TV reception).
 For AM signals, the translation of all stations to a fixed intermediate frequency
(455 kHz) allows us to obtain adequate selectivity.
 It is difficult to design precise bandpass filters of bandwidth 10 kHz (the
modulated audio spectrum) if the center frequency fc is very high. This is
practically true in the case of tunable filters. Hence, the RF filter cannot provide
adequate selectivity against adjacent channels.
 When the signal is translated to an IF frequency by a converter, it is further
amplified by an IF amplifier (usually a three-stage amplifier), which does have
good selectivity. This is because the IF frequency is reasonably low; moreover its
center-frequency is fixed and factory-tuned. Hence, the IF section can effectively
suppress adjacent-channel interference because of its high selectivity. It also
amplifies the signal for envelope detection.
 In reality, the entire selectivity is practically realized in the IF section; the RF
section plays a negligible role. The main function of the RF section is image
frequency suppression.
 The output of the mixer, or converter, consists of components of the difference
between the incoming signal  fc  and the local oscillator frequencies  fLO  (i.e.,
fIF  fLO  fc ).
 Now consider the AM example. If the incoming carrier frequency fc  1000 kHz ,
then fLO  fc  fIF  1000  455  1455 kHz .
 But another carrier with fc  1455  455  1910 kHz , will also be picked up
because the difference fIF   fc  fLO  1910  1455  455 kHz .
 The station at 1910 kHz is said to be the image of the station of 1000 kHz.
 AM stations that are 2fIF  910 kHz apart are called image stations and both
would appear simultaneously at the IF output, were it not for the RF filter at
receiver input.
 The RF filter may provide poor selectivity against adjacent stations separated by
10 kHz, but it can provide reasonable selectivity against a station separated by
 910 kHz. Thus, when we wish to tune in a station at 1000 kHz, the RF filter,
tuned to 1000 kHz, provides adequate suppression of the image station at 1910
kHz.
 The receiver, shown above, converts the incoming carrier frequency to the IF by
using a local oscillator frequency fLO higher than the incoming carrier frequency
and, hence, is called a superheterodyne receiver.
 We pick fLO higher than fc because this leads to a smaller tuning ratio of the
maximum to minimum tuning frequency for the local oscillator.
 The AM broadcasting frequencies range from 530 to 1710 kHz. The
superheterodyne fLO ranges from 1005 to 2055 kHz (ratio of 2.045), whereas the
subheterodyne range of fLO would be 95 to 1145 kHz (ratio 12.05). It is much
easier to design an oscillator that is tunable over s smaller frequency ratio.
 In early days, the entire selectivity against adjacent stations was realized in the
RF filter. Because this filter often had poor selectivity, it was necessary to use
several stages (several resonant circuits) in cascade for adequate selectivity.
 In the earlier receivers each filter was tuned individually. It was very time-
consuming and cumbersome to tune a station by bringing all resonant circuits
into synchronism. This task was made easier as variable capacitors were ganged
together by mounting them on the same shaft rotated by one knob.
 But variable capacitors are bulky, and there is a limit to the number that can be
ganged together.
 These factors, in turn, limited the selectivity available from receivers.
Consequently, adjacent carrier frequencies had to be separated widely, resulting
in fewer frequency bands.
 It was the superheterodyne receiver that made it possible to accommodate many
more radio channels.

4.7 FM broadcasting system

 The FCC has assigned a frequency range of 88 to 108 MHz from FM broadcasting,
with a separation of 200 kHz between adjacent stations and a peak frequency
deviation f  75 kHz .
 A monophonic FM receiver is identical to the superheterodyne AM receiver,
shown before, except the intermediate frequency is 10.7 MHz and the envelope
detector is replaced by a PLL or a frequency discriminator followed by a
deemphasizer.
 Earlier FM broadcasts were monophonic. Stereophonic FM broadcasting, in
which two audio signals, L (left microphone) and R (right microphone), are used
for a more natural effect, was proposed later.
 The FCC ruled that the stereophonic system had to be compatible with the
original microphonic system. This means that the older microphone receivers
should be able to receive L+R, and the total transmission bandwidth for the two
signals (L and R) should still be 200 kHz, with f  75 kHz for the two combined
signals. This would ensure that the older receivers could continue to receive
 monophonic as well as stereophonic broadcast, although the stereo effect would
be absent.

 A transmitter and a receiver for a stereo broadcast are shown below. At the
transmitter, the two signals L and R are added and subtracted to obtain L+R and
L-R.
 These signals are preemphasized. The preemphasized signal  L  R  DSB-SC
modulates a carrier of 38 kHz obtained by doubling the frequency of 19-kHz
signal that is used as a pilot. The  L  R  is used directly.
 All there signals (the third being the pilot) form a composite baseband signal m(t)
ω t
m  t    L  R    L  R  cos  ωc t   cos  c 
 2 

 The reason for using a pilot of 19 kHz rather than 38 kHz is that it is easier to
separate the pilot at 19 kHz because there are no signal components within 4
kHz of that frequency.
 The receiver operation is self-explanatory. A monophonic receiver consists of only
the upper branch of the stereo receiver and, hence, receives only L+R. This is of
course the complete audio signal without the streo effect. Hence the system is
compatible. The pilot is extracted, and (after doubling its frequency) it is used to
demodulate coherently the signal  L  R  cos  ω t  .
c
 

 An interesting aspect of stereo transmission is that the peak amplitude of the

composite signal m(t) is practically the same as that of the monophonic signal (if
we ignore the pilot), and, hence, f -which is proportional to the peak signal
amplitude for stereophonic transmission-remains practically the same as for the
monophonic case. This can be explained by the so-called interleaving effect as
follows.
 The L and R signals are very similar in general. Hence, we can assume their
peak amplitude to be equal to A p .

 Under the worst possible condition, L and R will reach their peaks at the same
time, yielding m  t   2Ap  α . In the monophonic case, the peak amplitude of
max

the baseband signal  L  R  is 2A p . Hence, the peak amplitudes in the two cases
differ only by α , the pilot amplitude.
 To account for this, the peak sound amplitude in the stereo case is reduced to
90% of its full value.
 This amounts to a reduction in the signal power by  0.9  0.81 or 1 dB. Thus,
2

the effective SNR is reduced by 1 dB because of the inclusion of the pilot.

4.8 SNR of angle modulated systems

 A block diagram of an angle-modulated system is shown below. The angle-
modulated (or exponentially modulated) carrier φEM  t  can be written as
k p m  t  for PM
 t
φEM  t   Acos ωct  ψ t  where ψ  t    and m(t) is the
 k f  m  α  dα for FM
 
message signal.
  The channel noise n(t) at the demodulator input is a bandpass noise with PSD
Sn  ω  and bandwidth 2  f  B  . The noise n(t) can be expressed in terms of
quadrature components as n  t   nc  t  cos  ωc t   ns  t  sin  ωc t  where nc  t  and
ns  t  are low-pass signals of bandwidth f  B . The bandpass noise n(t) may also
be expressed in terms of the envelope n  t   En  i  cos ωc t  Θn  t  .
 In AM, the signal output can be calculated by assuming channel noise to be zero,
and the noise output can be calculated by assuming the modulating signal to be
zero. This is a consequence of linearity. The signal and noise do not form
intermodulation product.
 Unfortunately, Angle modulation (and particularly wide-band angle modulation)
is a nonlinear type of modulation. Here, superposition does not apply. (As can be
done in AM).
 We can show that because of special circumstances, however, even in angle
modulation the noise output can be calculated by assuming the modulating
signal to be zero.
 To prove this we shall first consider the case of PM and then extend those results
to FM.

4.8.1 Phase modulation

 For wideband modulation, the modulating signal m(t) bandwidth is B, and the
noise bandwidth is 2  f  B  , with f B . Hence, the phase and frequency
variations of the modulated carrier are much slower than the variations of n(t).
 The modulated carrier appears to have constant frequency and phase over several
cycles, and, hence, the carrier appears to be unmodulated. We may therefore
calculate the output noise by assuming m(t) to be zero (or a constant).
 To calculate the signal and noise powers at the output, we shall first construct a
phasor diagram of the signal y i  t  at the demodulator input, as shown below.
yi  t   Acos ωct  ψ t   n  t   Acos ωct  ψt   En t  cos ωct  Θn t 
Or, yi  t   R  t  cos ωct  ψ t   ψ t 
Where ψ  t   k p m  t  for PM.
 For small-noise case, where En A for “almost all t,” ψ  t  π 2 for “almost all
En  t 
t,” and ψ  t  sin Θn  t   ψ  t   .
A
 The demodulator detects the phase of the input y i  t  . Hence, the demodulator
output is
En  t 
y o  t   ψ  t   ψ  t   k p m  t  
sin Θn  t   ψ  t   .
A
Note that the noise term ψ  t  involves the signal ψ  t  due to the nonlinear
nature of angle.
 Because ψ  t  (baseband signal) varies much more slowly than Θn  t  (wide-band
noise), we can approximate ψ  t  by a constant ψ ,
En  t  En  t  En  t 
ψ  t  sin Θn  t   ψ  sin Θn  t   cos  ψ  cos Θn  t   sin  ψ
A A A
ns  t  nc  t 
Or, ψ  t   cos  ψ 
cos  ψ
A A
 Also, because nc  t  and ns  t  are incoherent for white noise,
cos2  ψ sin2  ψ Sns  ω 
Sψ  ω   Sns  ω   Snc  ω  
[Because Snc  ω   Sns  ω  ].
A2 A2 A2
 For a white channel noise with PSD N/2
N
 for f  f  B
Sψ  ω    A2

0 Otherwise
 The demodulator noise bandwidth is f  B . But because the useful signal
bandwidth is only B, the demodulator output is passed through a low-pass filter
of bandwidth B to remove the out-of-band noise.
 Hence the PSD of the low-pass filter output noise is
N
 for ω  2πB
Sno  ω    A2 and
0 for ω  2πB

 N  2NB
No  2B  2   2 .
A  A
  Signal output power So  k 2p m2 .
2 2 2
So A k p m
 Thus,  .
No 2NB
 These results are valid for small noise, an they apply to both WBPM and NBPM.
S A2 2 A2 S
We also have γ  i   and o  k 2p m2 γ .
NB NB 2NB No

  S  2
2 m
 Also for PM, ω  mp where mp  m  t  . Hence, o   ω   2  γ .
  max N0  mp 
 
 Note that the bandwidth of angle modulated wave is 2f (for wide-band case).
Thus, the output SNR increase with the square of the transmission bandwidth;
that is, the output SNR increases by 6 dB for each doubling of the transmission
bandwidth.
 This result is valid only when the noise power is much smaller than the carrier
power. Hence, the output SNR cannot be increased indefinitely by increasing the
transmission bandwidth because this also increases the noise power, and at
some stage the small-noise assumption is violated. When the noise power
becomes comparable to the carrier power, the threshold appears, and a further
increase in bandwidth actually reduces the output SNR instead of increasing it.
 For tome modulation, m  t   α cos  ωmt  .
 
 For this case m2  α2 2 and mp  m  t   αωm . Hence
  max
2 2
So 1  ω  1  f 
   γ    γ
No 2  ωm  2  fm 
 Above equations are valid for both NBPM and WBPM.

4.8.2 Frequency modulation

 Frequency modulation may be considered as a special case of phase modulation,

t
where the modulating signal is  m  α  dα . At the receiver, we can demodulate


FM with a PM demodulator followed by a differentiator, as shown. The PM

t
modulator output is k f  m  α  dα . The subsequent differentiator yields the


output k f m  t  , so that So  kf m2 .
 The phase demodulator output noise will be identical to that calculated earlier,
with PSD N A2 for white channel noise.
 The noise is passed through an ideal differentiator whose transfer function is jω
. Hence, the PSD Sno  ω  of the output noise is
2
jω times the PSD and is
 Nω2
 for ω  2πB
Sno  ω    A 2 .
0 for ω  2πB

 The PSD of the output noise is parabolic (shown below) and the output noise
B
N 8π2NB3
power is No  2 2  2πf  df  .
0 A 3A 2

So  k 2 m2   A 2 2   k 2 m2 
 Hence, the output SNR is  3 f     3  f γ.
No   2πB 2   NB    2πB 2 
   
 Because ω  k f mp ,

 f   m   m2 
2 2
So
 3    2  γ  3β2 γ  2  .
No  B   mp   mp 
 
 Recall that the transmission bandwidth is about 2f . Hence, for each doubling
of the bandwidth, the output SNR increases by 6 dB. Just as in the case of PM,
the output SNR does not increase indefinitely because threshold appears as the
increased bandwidth makes the channel noise power comparable to the carrier
power.
S 3
 For tone modulation m2 m2p  0.5 and o  β2 γ .
No 2
 The output SNR is plotted as a function of γ for various values of β , below. The
dotted portion of the curves indicates the threshold region. Although the curves
are valid for tone modulation only ( m2 m2p  0.5 ), they can be used for any other
 modulating signal m (t) simply by shifting them vertically by a factor

m 2
m2p  0.5  2m
2
m2p .

 We observe that for tone modulation FM is superior to PM by a factor of 3. This

does not mean that FM is superior to PM for other modulating signals as well. In
fact, we shall see that PM is superior to FM most of the practical signals.
 So No PM  2πB2 m2p
. Hence, if  2πB m2p  3mp2 , PM is
2
 It can be seen that 
 So No FM 3mp2
superior to FM.
 If PSD of m(t) is concentrated at lower frequencies, low-frequency components
predominate in m(t), and mp is small. This favors PM.
 Therefore, in general, PM is superior to FM when Sm  ω  is concentrated at lower
frequencies and FM is superior to PM when Sm  ω  is concentrated at higher
frequencies. This explains why FM is superior to PM in tone modulation where
all signal power is concentrated at the highest frequency in the band. But for
most of the practical signals, the signal power is concentrated at lower
frequencies, and PM proves superior to FM.

4.8.3 Narrow-band modulation

 The equations derived thus far are valid for both narrow-band and wide-band
modulation.
 The narrow-band exponential modulation (NBEM) is approximately and is very
similar to AM. In fact, the output SNRs for NBEM and AM are similar.
 To see this, consider the cases of NBPM and AM
φAM  t   A cos  ωc t   m  t  cos  ωc t 
φNBPM  t   A cos  ωc t   Ak p m  t  sin  ωc t   A cos  ωc t   m1  t  sin  ωc t 
Where m1  t   Ak p m  t 
 Both φAM and φNBPM contain a carrier and a DSB term. In φNBPM the carrier and
the DSB component are out of phase by π 2 rad, whereas in φAM they are in
phase. But the π 2 -rad phase difference has no effect on the power. Thus m(t)
in φAM is analogous to m1  t  in φNBPM .
 Now let us compare the output SNRs for AM and NBPM. For AM
 So  m2
   γ
 No AM A2  m2
Whereas for NBPM with m1  t   Ak p m  t 

 So  2 m12
   k p m γ  γ
 No PM A2
 Note that for NBPM, we require that kpm  t  1 , that is m1  t  A 1 . Hence,

S  m12 S 
A2 A2  m12 and  o   γ , which is of the same form as  o  .
 No PM A2  m12  No  AM
Hence NBPM is very similar to AM.
 Under the best possible conditions, however, AM outperforms NBPM because for
AM, we need only to satisfy the condition A  m  t   0 , which implies
m  t max  A . Thus for tone modulation, the modulation index for AM can be
nearly equal to unity. For NBPM, however, the narrow-band condition would be
equivalent to requiring μ 1. Hence, although AM and NBPM have identical
performance for a given value of μ , AM has the edge over NBPM from the SNR
viewpoint.
 It is interesting to look for the line (in terms of f ) that separates narrow-band
and wide-band FM.
 We may consider the dividing line to be that value of f for which the output
SNR for FM is equal to the maximum output SNR for AM when μ  1 or A  mp .
 m2  m2
Hence, 3β2 γ  2   γ
 mp  A 2  m2
 
 
1 m2
2
m2p 1 m2p 1 1 1 1 
Or, β     
3 A 2  m2 m2 3 A 2  m2 3 A 2 m2  m2 m2
p p  3 1  m2 m2 
  p 
   
1
 Because m2 m2p  1 for practical signals, and β2  or, β2  0.6 , f  0.6B .
3