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Chapter 9.
Frequency and Phase Modulation
Figure 9-1. General block diagrams for transmitters of
amplitude, frequency, and phase modulation.
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Frequency Modulation
 A frequency-modulated signal is any periodic signal whose
instantaneous frequency f is deviated from an average
value fc by an information signal m(t).
 The maximum amount by which the instantaneous
frequency is deviated from the carrier frequency fc is
called the peak deviation, Dfc(pk).
 Hence the instantaneous output frequency is expressed as
f = fc + kovm(t)
(9-1)
It is assumed that when vm(t) = 0, Dfc = 0 and f = fc.
Otherwise,
Dfc = kovm(t) and f = fc + Dfc
(9-2)
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Frequency Modulation
Figure 9-2. Frequency variations in FM are
directly proportional to the modulating voltage.
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Modulation Index
 In general a signal with angle modulation is expressed
simply as
s(t) = A.cosq(t)
(9-3)
= A.cos(wt+f),
(9-4)
where either the angle f or the angle wt is varied.
 When f is varied by the information signal, f(t) a m(t),
the result is called Phase Modulation (PM).
 If f is held constant and the angle wt is modulated by
the information signal deviating the carrier frequency,
the result is called Frequency Modulation (FM).
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Modulation Index
 Angular frequency w is the rate of change of phase,
that is,
w = dq/dt
 Since w = 2pf, then
dq/dt = 2pfc + 2pkovm(t)
(9-6)
This differential equation is solved for q(t)
(9-7)
where qo is the arbitrary starting phase at t = 0.
(9-8)
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Modulation Index

For vm(t) = Vpkcos2pfmt, the integral of the third term
is evaluated as follows:
(9-9)

The arbitrary constant phase fo at t = 0, can be
combined with qo in Equ. (9-8) and set to zero
q(t) = 2pfct + (koVpk/fm).sin2pfmt
= 2pfct + [Dfc(pk)/fm].sin2pfmt
(9-10)
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Modulation Index
 The cosine-modulated FM signal is written, using
Equ. (9-3), as
sFM(t) = Acos(2pfct + mf sin2pfmt)
(9-11)
where mf = Dfc(pk)/fm is the peak of the sinusoidally
varying carrier phase angle.
 The modulation index, in units of radians, for
sinusoidal FM is calculated from
mf = Dfc(pk)/fm
(9-12)
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Modulation Index
 EXAMPLE 9-1:
A 1-MHz VCO with a measured sensitivity of ko = 3
kHz/V is modulated with a 2-Vpk, 4-kHz sinusoid.
Determine the following:
1. The maximum frequency deviation of the carrier.
2. The peak phase deviation of the carrier and
therefore the modulation index.
3. mf if the modulation voltage is doubled.
4. mf for Vm(t) = 2cos[2p(8kHz)t] volts.
5. Express the FM signal mathematically for a cosine
carrier and the cosine-modulating signal of part 4.
The carrier amplitude is 10-Vpk.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
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Modulation Index
 Solution:
1. Dfc(pk) = koVm(pk) = (3 kHz/V)(2Vpk) = 6 kHz.
2. From Equ. (9-12) the peak phase deviation by modulating
sinusoid is Dfc(pk)/fm = 6 kHz/4 kHz = 1.5 rad = mf .
3. For a linear modulator Df a DVm so mf = 3.0 rad is double
the answer of part 2. Of course this can be computed from
Equ. (9-12). Since Vm increases to 4V, then mf = Dfc(pk)/fm
= (4V x 3kHz)/4kHz = 3 (radians).
4. The modulation is sinusoidal with 2Vpk so the carrier
frequency deviation is the same as part 1 --- Dfc(pk) = 6 kHz.
Since fm = 8kHz, the peak carrier phase deviation and
modulation index is mf = 6 kHz/8 kHz = 0.75.
5. The modulating signal is a cosine (so is the carrier). Therefore
using the value of mf = 0.75, we have
vFM = 10cos(2p106t + 0.75sin2p8x103t).
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
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Sidebands and Spectrum
 For low deviation (mf < 0.25), called narrowband FM
(NBFM), there is a carrier and a set of sidebands
much like AM.
 The two sidebands are shifted 90° relative to the
carrier (as compared to AM.). This comparison is
illustrated in Fig.9-3.
Figure 9-3. Comparison of
phasor relationships in AM
and FM/PM.
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Sidebands and Spectrum
 Figure 9-4 shows a comparison of AM and NBFM
phasor variations.
 Notice that the amplitude for the NBFM seems to
vary slightly in the illustration.
 This is because the angle drawn for the illustration
is greater than 14.3o (0.25 rad).
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Sidebands and Spectrum
Figure 9-4. Comparison of instantaneous
transmitted phasors –sAM(t) and sPM(t).
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
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Sidebands and Spectrum
 The spectrum for angle-modulated signals (FM and PM)
may be determined from
vFM = Acos(wct + mf sinwmt)
(9-13)
= Acos(mf sinwmt)coswct - Asin(mf sinwmt)sinwct. (9-14)
 For low-deviation FM (mf < 0.25) the angular variations of
the cosine and the sine in the brackets of Equ. (9-14) can be
approximated as cosDf = 1 and sinDf = Df. Therefore
narrowband FM can be approximated as
vNBFM = Acoswct – A(mf sinwmt) sinwct
(9-15)
 Also,
Amf sin(wmt).sin(wct)
= (Amf/2)cos(wc-wm)t – (Amf/2)cos(wc+wm)t.
This shows clearly the constant amplitude carrier and the
quadrature sidebands as illustrated in Figure 9-4.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
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Wideband FM
 The amplitudes of the frequency components [sinusoid
(mfsinwmt)] cannot be evaluated in closed form but are
tabulated in the table of Bessel functions of the 1st kind
of order n (Table 9-1).
 The continuous Bessel functions Jn(mf) are shown in
Figure 9-5, where
Vo(pk) = AJo(mf)
(9-16)
gives the peak voltage amplitude of the carrier for any
modulation index mf up to and including mf = 10.
 The peak voltage of sidebands on either side of the carrier
is given by
Vn(pk) = AJn(mf)
(9-17)
where n = 1, 2, …is the order of the sideband -- that is,
the number of the particular sideband pair.
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Wideband FM
Figure 9-5. Amplitudes of carrier and sidebands for
FM (and PM) relative to the unmodulated carrier.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
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Wideband FM
Table 9-1. Bessel Function of the First Kind, Jn(mf).
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Wideband FM
Fig. 9-6. Frequency spectra for FM signals with constant
modulation frequency fm but different carrier-frequency
deviations.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
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Wideband FM
 Notice that for mf < 0.25, the carrier drops by less
than 2% and only one set of sidebands, the 1st-order
side-band J1(mf), have a value exceeding 1% (0.01) of
the unmodulated carrier.
 These values justify the approximations used for
NBFM. The 1st-order sets of sidebands are separated
from the carrier by fm.
 For mf > 0.25, there are additional sets of sidebands.
This is the wideband FM case, and the sideband sets
are separated from the carrier by fm, 2fm, 3fm, …, nfm
corresponding to the order of the Bessel function
Jn(mf). This can be seen in the example FM signal
spectra of Figure 9-6.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
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Wideband FM
 EXAMPLE 9-2:
An FM signal expressed as vFM = 1000cos(2p107t +
0.5cos2p104t) is measured in a 50-W antenna.
 Determine the following:
1. Total power;
2. Modulation index;
3. Peak frequency deviation;
4. Modulation sensitivity if 200 mVpk is required to
achieve part3.
5. Spectrum;
6. Bandwidth (99%) and approximate circuit bandwidth
by Carson's rule.
7. Power in the smallest sideband (just one sideband of
the 99% bandwidth).
8. Total information power.
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Wideband FM
 Solution:
1. PT = (V(pk))2/2R = 10002/100 = 10 kW.
2. The 0.5cos2p104t term gives the peak phase variation
and mf =0.5.
3. mf = Dfc(pk)/fm , hence Dfc(pk) = 0.5 x 104 Hz = 5 kHz.
4. ko = Dfc / DVm = 5kHz/0.2V = 25 kHz/V.
5. The spectrum is determined from Table 9-1 with
mf = 0.5, A = 1000Vpk, and fm = 10 kHz.
The carrier is AJo(0.5) = 940Vpk at 10 MHz.
The 1st sets of sidebands are AJ1(0.5) = 240Vpk
at 9.990 MHz and 10.010 MHz, and
the 2nd (and least significant) sidebands are
AJ2(0.5) = 30Vpk at 9.98 MHz and 10.020 MHz.
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Wideband FM
6. From 5, the 99% information bandwidth is 2x20 kHz = 40kHz,
and from Carson's rule, BW = 2(10 kHz + 5 kHz) = 30kHz for
reasonably low circuit distortion.
7. There are two ways to arrive at the solution: From part 5
each sideband is 30Vpk, so Plsb = (30Vpk)2/100 = 9W. From
Equ. (9-18) and PT = 10kW, Plsb = (0.03)2 x 10kW = 9W.
8. The power in the information part of the signal is PT - Pc,
where the carrier power is Pc= (940Vpk)2/100 = 8.836kW.
PT - Pc = 10 kW – 8.836 kW = 1.164 kW.
Low-index modulation is not efficient (11.64% in this case).
Incidentally, the total power must add up as follows:
PT = Pc+P1+P2 = 8.836 kW + 2[(2402/100)+9W]
= 10.006 kW,
a 0.06% round-off error.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
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Linear Phase Modulation
 Linear phase modulation is expressed by
vPM = Acos[wct + kpvm(t)]
(9-23)
where the proportionality constant kp, in rad/V,
is
called the modulator sensitivity.
 For a sinusoidal modulation signal of vm(t) =
Vpkcos2pfmt, the PM signal becomes
sPM(t) = Acos[2pfct + kpVpkcos2pfmt]
(9-25)
= Acos[2pfct + mpcos2pfmt]
(9-26)
 where
mp = kpVpk = Df(pk)
(9-28)
is the peak phase deviation of the sinusoidally varying
carrier phase angle or the modulation index in units of
radians.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
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Linear Phase Modulation
 When Equ. (9-26) is expanded by Trigonometric
identity and the narrow-band approximation
(Df < 0.25 rad) applied, the result becomes
vNBPM(t) = Acoswct – A(mpcoswmt).sinwct (9-29)
Figure 9-8. NBPM using a balanced modulator
and sideband phase shift.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
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Diode Modulator for PM
 Figure 9-9 shows the most straightforward method of
producing a PM signal.
 A voltage-variable capacitance diode (called Varicap,
Varactor, and tuning diode) is used to vary the tuning of a
parallel-resonant circuit.
 The stable crystal-controlled-frequency carrier signal will
thus be phase-shifted in accordance with the modulation
voltage.
Figure 9-9. Phase modulation system.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
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Diode Modulator for PM
 The vertical scale in the diode voltage-capacitance curve
of Figure 9-10 will yield the fractional change DCd/Cd in
diode capacitance for small input voltage changes.
 The capacitance changes can be estimated by determining
the slope of the C(V) curve at the bias point V. Thus
dCd/dV = (d/dV).[Co(1+2|V|)-1/2]
= -Cd/(1+2|V|)
(9-30a)
or
DCd/Cd = -DV/(1+2|VR|)
(9-30b)
where VR is the reverse bias across Cd, and DV is the peak
value of Vm, the modulation signal voltage.
 Equ. (9-30) is then used to determine the tuned-circuit
frequency shift
Dfo/fo = -0.5(DCd/Cd)
(9-31)
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
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Diode Modulator for PM
Figure 9-10. Ratio of junction capacitance at reverse-bias
to junction capacitance at 6V reverse-bias versus applied
reverse-bias voltage.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
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Diode Modulator for PM
 For sinusoid angle modulation mp = Df(pk), and
knowing the peak input voltage swing, the modulator
sensitivity can be calculated from
kp = Df/DVm
(9-32)
 An exact formula for the phase shift is derived from
the phase-versus-frequency curve for the parallel tank
impedance, where
Z = Rp/(1+jQr)
(9-33a)
r = f/fo – fo/f
= [1+(Dfo/fo)] - [1+(Dfo/fo)]-1
(9-33b)
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Diode Modulator for PM
 Dfo is a small deviation, less than 10% from center
frequency fo.
 The phase-versus-frequency curve, f(f), follows a
tangent curve that is the phase of impedance Z;
 that is, f(f) = -tan-1Qr for which the magnitude
provides the FM modulation index given by
Df = mp = tan-1Qr
(9-34)
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Derivation of Modulator Equations
 The modulator sensitivity can be calculated from
kp = df/dVm
= (dC/dV).(df/dC).(df/df)
In general,

Cd(V) = Co(1 + 2|VR|)-p
(9-35)
(9-38)
The rate of diode capacitance change with voltage is
computed by differentiating Equ. (9-38) as
dCd/dVR = -2pCd/(1+2VR)
(9-39)
where, for various diode junction doping profiles,
0.3 < p < 0.6.
 For convenience, let the power be p = 0.5, resulting in
dCd/dVR = -Cd/(1+2|VR|)
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
(9-40)
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Derivation of Modulator Equations
 The second term in Equ. (9-35), df/dC, is
df/dC = df/dCd = -0.5fo/Cd

(9-41)
For small frequency deviations, the phase change is
also small, so that
(9-42a)
(9-42b)

which, when combined with f = -tan-1Qr
= -Qr, yields the approximation
df/df = -2Q/fo
(9-43)
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Derivation of Modulator Equations
 Substituting Eqs. (9-40), (9-41), and (9-43) into
Equ. (9-35) yields
df/dVm = [-Cd/(1+2|VR|)].[-fo/2Cd].[-2Q/fo]
= -Q/(1+2|VR|)
(9-44)

and the magnitude of Equ. (9-44) gives the
phase modulator sensitivity at the circuit bias
point VR as
kp = Q/(1 + 2|VR|)
(9-45)
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Derivation of Modulator Equations
 The modulation index mp resulting from a
change in voltage, dVm, applied to the tuning
diode is, from Equ. (9-28),
mp = QDVm(pk)/(1 + 2|VR|)
(9-46)
for the circuit of Figure 9-9.

Eqs. (9-45) and (9-46) are exact slope
equations for a simple phase modulator.
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Indirect FM from PM
Figure 9-16. High-index FM from a phase modulator
preceded an integrator (Amstrong method).
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Indirect FM from PM

Equ. (9-8) for FM, when substituted into the general angle
modulation expression (Equ. (9-3)), yields
vFM(t) = Acos[2pfct + qo + 2pko∫vm(t)dt]


(9-60)
This equation reveals the trick needed for using a phase
modulator to produce FM: integrate the modulating signal
vm(t) before modulating the phase modulator.
To achieve the high-modulation indices required by the
FCC, frequency multipliers follow the FM modulator.
 As an example, NBPM and FM have an index less than
0.25 rad; in order to produce 5.0 rad (Dfc(pk)/fm =
75kHz/15kHz = 5.0), a multiplication of at least 20 is
required.
34
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Indirect FM from PM
Fig. 9.5. (a) Generation of AM; (b) Generation
of NBPM; (c) Generation of NBFM.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
35
Indirect Generation of FM:
The Armstrong Method
 1). Integrate the modulating signal s(t) to produce
the signal
2). Phase-modulate a carrier with z(t) to produce NBFM.
3). Use a system of frequency multipliers to convert NBFM
wideband FM.
Figure 7.6-1. Method of generating indirect FM.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
36
Indirect Generation of FM:
The Armstrong Method

A block diagram of an IFM system is shown in Fig. 7.6-1.
At point A the output is
(7.6-1)
and at point B the output is proportional to
(7.6-2)

The frequency multiplier produces a signal with frequency
given by
w = N.dq1(t)/dt,
where
q1(t) = wc1t + b1sinwmt.
Hence
w = Nwc1 + Nb1wmcoswmt.
(7.6-3)

The new phase is proportional to the integral of N.dq1(t)/dt, so at C
we have
x(t) = cos[Nwc1t + Nb1sinwmt]
= cos(wct + b sinwmt),
(7.6-4)
where b is now a value typical of wideband FM.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
37
Indirect Generation of FM:
The Armstrong Method
 Example
Let fm range from 50 to 5000 Hz, and let the maximum
frequency deviation, Df, at the output be 75 kHz. Then
bmin = (75x103)/(5x103) = 15
bmax = (75x103)/50 = 1500.

If [b1]max = 0.5, then the required frequency multiplication
is
N = bmax/[b1]max = 1500/0.5 = 3000

The maximum allowed frequency deviation at the input is
Df1 = (75x103)/3000 = 25 Hz.
38
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Indirect Generation of FM:
The Armstrong Method
 If ihe maximum specified Df at the output is 75 kHz.
If the initial carrier frequency fc1 were, say, 200 kHz,
then the final frequency would be fc = 600 MHz.
 This figure is too high for standard FM broadcasting,
amd frequecncy converters are used to reduce fc to
the desired band.
 For example, if we heterodyne x(t) with a carrier
wave of frequency, fLO, the modulation index Nb1
remains unaffected but the wave is shifted to the
new carrier frequency, Nf1 - fLO.
39
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Indirect Generation of FM:
The Armstrong Method
 A block diagram of an Armstrong-type indirect FM
transmitter is shown in Fig. 7.6-2. The largest modulation
index furnished by the NBFM system is determined by the
lowest modulating frequency (≒50 Hz), and this determines
the required frequency multiplication.
 For example, if b1 = 0.5 for a 50-Hz tone, then the 75-kHz
frequency deviation requires a b = 1500. The required
multiplication is 1500/0.5 = 3000. The largest tone
frequency determines the bandwidth.
 The value of b1 for the 15-kHz signal is approximately
1.7x10-3. The extremely small initial values of b1 are
required to prevent distortion due to amplitude-modulation
effects that can occur in the generation of NBFM.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
40
Indirect Generation of FM:
The Armstrong Method
Figure 7.6-2. Amstrong-type indirect FM transmitter.
41
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Frequency Stability and AFC
 High-frequency deviation is not practical with
crystal-controlled VCOs even with multiplication.
And the stability is reduced by the multiplication
factor N.
 LC-tuned VCOs have good deviation sensitivity but
poor stability with respect to frequency drifts due to
aging effects and temperature changes.
 One solution to these dilemmas is to modulate the LC
VCO, then lock the average value of the VCO
frequency to a crystal reference through a narrowbandwidth phase-locked loop. This technique is
shown in Figure 9-20.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
42
Frequency Stability and AFC
 The frequency divider of Figure 9-20 is useful for
minimizing interference from the crystal oscillator (XO).
 The low-pass filter (LPF) prevents feedback of modulation
frequencies (FM feedback) and eliminates the possibility of
the loop locking to a sideband.
Figure 9-20. Phase-locked FM generator.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
43
Frequency Stability and AFC
Fig. 9-21. Transmitter automatic frequency control loop.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
44
Frequency Stability and AFC
 One more technique for stabilizing an oscillator frequency
is the automatic frequency control (AFC) system shown in
Figure 9-21. This system is used for stabilizing the carrier
center frequency against drift in FM transmitters.
 AFC without the multiplier and down-converter (mixer and
XO) is used in receivers for stabilizing the first local
oscillator. In many television receivers it is called AFT-automatic fine-tuning.
 The Crosby AFC loop of Figure 9-21 is an excellent system
to analyze for understanding feedback. Suppose the VCO
frequency fo drifts by an amount dfo. Then with switch S1
open (position 2), there will be no feedback and the transmit
frequency will be
fxmt = (N1 x N2 x N3)( fo + dfo)
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
45
Frequency Stability and AFC
 To meet FCC specifications,
dfxmt = (N1N2N3)dfo < 2 kHz
(9-71)
With the generator set exactly to
f* = N1N2 fo
the frequency at the frequency discriminator input is
fd = fXO – f*.
 The frequency discriminator is a special LC tunedcircuit that changes frequency variations to amplitude
variations (FM to AM) and is followed by peak AM
detectors in a balanced configuration.
 When properly aligned the circuit is tuned to fd and the
output voltage is zero volt.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
46
Frequency Stability and AFC
 The vo(t)/fd(t) transfer characteristic of the discriminator is
linear over a wide frequency range, with a slope (gain) of
Dvo/Dfd = kd
An input frequency shift of +Dfd will result in an output
voltage change of
+Dvo = kdDfd = +Vo.
 The purpose of the low-pass filter (LPF) in Figure 9-21 is to
prevent any modulation signals from being fed back (FM
feedback), thereby reducing the modulation index.
 The very slow dc changes in Vo due to Dfd-drifts detected by
the frequency discriminator are fed back to the VCO with
the correct polarity for forcing the VCO back to nearly fo.
47
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Frequency Stability and AFC
 Let us go back to switch S1. The signal at pin 1 of S1 has
an open-loop frequency error of dfol* = N1N2dfo. When
the switch is closed to pin 1, the system with feedback
will reduce this frequency discriminator input to be Dfd
= -dffb*.
 The discriminator output voltage will be (Vo)fb=
-kddffb*, so the VCO frequency will be corrected to
(dfo)fb = ko(Vo)fb = -kokddffb*.
 This frequency correction, when multiplied by N1 and
N2, is the correction signal which reduces the open-loop
error of f*. That is,
dffb* = dfol* - kokdN1N2dffb*
(9-73)
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Frequency Stability and AFC
 Solving for in Equation 9-73 yields the amount of drift
at f* for the system with feedback:
dffb* = dfol*/[1+ kokdN1N2]
(9-74)
The error in the transmitted carrier frequency will be
dfxmt(fb) = N3 x dffb*
(9-75)
 Equation (9-75) shows the expected result for a system
with feedback, the open-loop drift is reduced 1+(loop
gain), where the magnitude of the gain once-aroundthe-loop is
loop gain = kokdN1N2
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Frequency Stability and AFC
Figure 9-21. Transmitter automatic frequency
control loop.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
50
Frequency Stability and AFC
Example 9-8:
 A VCO with a long-term frequency stability of dfo/fo =
200 ppm is used in the AFC system of Figure 9-21.
 The transmit carrier frequency is to be 108 MHz,
and the circuit component gains are as follows:
ko = 10 kHz/V, N1 = 2, N2 = 2, N3 = 3, kd = 2V/kHz, also
fXO = 37 MHz. All circuits except the VCO are
considered to be perfectly stable.
 Determine the following:
1. fo, f*, and fd .
2. dfxmt open-loop; is it within FCC specs?
3. Transmitted carrier frequency and frequency
error close-loop.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
51
Frequency Stability and AFC
 Solution:
1. fo = 108MHz/12 = 9MHz; f* = 4fo = 36 MHz;
fd = fXO - f* = 1MHz.
2. Without feedback, the VCO will drift by (dfo/fo) x fo =
200 Hz/MHz x 9 MHz = 1800 Hz. At the transmitter
output this will be dfxmt(ol) = 12 x 1800 Hz = 21.6 kHz.
This is also the same as 200 ppm x 108 MHz = 21.6
kHz. In any case, it is way beyond the specified +2
kHz and fxmt(ol) = 108.0216 MHz.
3. dfol*= 4dfo = 4x1800Hz = 7200Hz.
The loop gain is (10kHz/V).(2x2).(2V/kHz) = 80.
The drift at f* is reduced by 81 so that dffb* =
7200Hz/81 = 88.9Hz, which at the transmitter output
becomes dfxmt(fb)= 88.9Hz x 3 = 266.7Hz.
The transmitted carrier, instead of being 108MHz,
is fxmt(fb) = 108.0002667MHz---well within +2kHz.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
52
FM RECEIVERS
 Receivers for FM and PM are super-heterodyne
receivers like their AM counterparts. However, the
AGC of AM receivers is not used for FM and PM
because there is no information in the amplitude of the
transmitted signal.
 Constant amplitude into the FM detector is still
desirable, so additional IF gain is used and the final
IF amplifiers are allowed to saturate.
 The IF amplifiers used in the output stages require
special design consideration for good saturation
characteristics and are called limiters, or passband
limiters when the output is filtered to preserve the
information bandwidth.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
53
FM Demodulators
 There are three general categories of FM demodulator
circuits:
1. Phase-locked loop (PLL) demodulator;
2. Slope detection/FM discriminator;
3. Quadrature detector:
They all produce an output voltage proportional to
the instantaneous input frequency.
54
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Phase-Locked Loop Demodulator
Figure 9-22. Frequency discriminator circuit symbol.
55
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Phase-Locked Loop Demodulator
Figure 9-23. Phase-locked loop FM detector.
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Phase-Locked Loop Demodulator
 As seen in Figure 9-23, the PLL consists of a phase detector (PD),
low-pass filter (LPF), and voltage-controlled oscillator (VCO).
 The PLL VCO, with voltage-to-frequency conversion sensitivity of
ko, is tuned to the FM carrier frequency with Vo = 0.
 The loop produces a voltage Vo which forces the VCO to
track the input frequency. Hence, vo will be directly
proportional to the input frequency variations of the FM
carrier:
fc(t) = ko(xmt) x vm(t)
 With limiter ahead of demodulator and a properly compensated
loop, the circuit gain will be constant, so that
vo(t) = kdfc(t) = kd[ko(xmt) x vm(t)]
= [kdko(xmt)]vm(t)
Therefore, vo(t) a vm(t)
(9-83)
where vm(t) is the transmitted information signal.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
57
Phase-Locked Loop Demodulator
 A sinusoidal information signal
vm(t) = Vpk coswmt
(9-84)
frequency modulated on a carrier of wc results in
sFM(t) = Acos(wct + mfsinwmt)
(9-85)
where
mf = Dfc(pk)/fm
= 2pkoVpk/wm
(9-86)
 To recover vm(t) at the demodulator output, the off-resonant slope
of the tuned circuit differentiates sFM(t) as follows:
(9-87)
 Equation (9-87) has high-frequency variations around the carrier
and amplitude (magnitude) variations of -Awc - Amfwmcoswmt.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
58
Phase-Locked Loop Demodulator
 The output of the envelope detector is the magnitude of
vd(t), with polarity depending on the diode(s) direction.
 Hence,
Vo(t) a Awc + Amfwmcoswmt
= Vdc + A2pkoVpk coswmt
(9-88)
using Equ. (9-88) and noting that the dc voltage is
proportional to carrier amplitude and frequency.
 After coupling through a series capacitor, the
information signal is
vo(t) = kdVpk coswmt
(9-89)
where kd = 2pkoA plus any circuit losses is the
demodulator sensitivity.
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Slope Detection/FM Discriminator
 In Figure 9-24a, the center frequency fo of the tuned
circuit is set such that the input carrier signal falls on
the slope of the resonance curve.
 With fc on the low side of fo and the diode in the
direction shown in Figure 9-24a, a positive S curve
transfer characteristic is realized.
 As frequency varied from fc, the RF amplitude varied.
These RF amplitude variations are peak-detected with
the diode and RF low-pass filter.
 Figure 9-24b illustrates the balanced slope detector. It
provides better linearity than the single slope detector,
but it still gas a somewhat limited frequency range, and
the two variable capacitors increase the complexity.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
60
Slope Detection/FM Discriminator
Figure 9-24. (a) Slope detector. (b) Balanced slope detector.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
61
Foster-Seeley Phase-Shift Discriminator
 In Figure 9-25, the input signal voltage across the
transformer primary winding vp is coupled by
capacitor Cc directly across the RF choke.
 The secondary induced-voltage vs is divided
equally between v1 and v2 by a secondary-winding
center-tap.
 The secondary (as well as the primary) circuit is
tuned to resonance at the carrier frequency. L2C2
from a high-Q resonant circuit which at resonance
circulates a current is with the same phase as vs.
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Foster-Seeley Phase-Shift Discriminator
Figure 9-25. Foster-Seeley phase-shift discriminator.
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Foster-Seeley Phase-Shift Discriminator
 The vector vs = v1 + v2 is in phase quadrature (90o)
to vp at resonance. The resultant vector diagram is
shown in Figure 9-26, where va and vb are the
phasor signals applied to the upper and lower peak
detectors, respectively.
 The detector output voltage Vo will be
Vo = Vao – Vbo
where
Vao = √2 |va(rms)|
and
Vbo = √2 |vb(rms)|
(9-90)
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Foster-Seeley Phase-Shift Discriminator
Figure 9-26. Vector diagram for phasors in FosterSeeley discriminator for fin at circuit resonance.
65
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Foster-Seeley Phase-Shift Discriminator
 When the FM signal frequency reaches its peak
deviation f = fc + Dfc(pk), then one calculates r(max)
and the phase shift of v1 from the argument of
1/(1+jQsr); that is,
Dq = -tan-1Qsr
(9-94)

Then the vector diagram of Figure 9-26 is modified as
illustrated in Figure 9-27 (for f > fo).

The peak output Vo(pk) and the detector sensitivity
kd = DVo/Df for this phase-shift discriminator can then
be determined graphically or by calculation using the
law of cosines.
66
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Foster-Seeley Phase-Shift Discriminator
Fig. 9-27. Identical vector diagrams for discriminator with fin
above circuit resonance. (a) corresponds to Fig. 9-26.
67
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Ratio Detector
 The ratio detector is a circuit very similar to the FosterSeeley discriminator, except for the balanced peakdetector configuration that makes it much less sensitive
to AM.
 As seen in Figure 9-29, diode Db is reversed, a large-C
(～10mF) capacitor has been connected from diode Da
to Db, and the demodulated output information is taken
from between the resistors as shown.
 Reversing the direction of Db allows for a current path,
which keeps the voltage Vc across the large-capacity
smoothing capacitor constant; that is, capacitor C
absorbs AM.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
68
Ratio Detector
Figure 9-29. Ratio detector.
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Ratio Detector
 If the FM signal frequency increases, vs remains
constant, but the vectors shift as in Figure 9-27.
 Then Da will conduct more heavily and Vao will
increase, while Db conducts less and Vbo decreases -the same as for the Foster-Seeley circuit.
 Thus, the sum of Vao and Vbo stays constant, but the
ratio varies with input frequency.
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Quadrature FM Detectors
 Quadrature FM detectors use a high-reactance
capacitor (C1 in Figure 9-30) to produce two signals
with a 90o phase difference. The phase-shifted signal is
then applied to an LC-tuned circuit resonant at the
carrier frequency.
 Frequency changes will then produce an additional
leading or lagging phase shift that is detected by
comparing zero-crossings (coincidence detector) or
by analog phase detection.
 The quadrature FM detectors of Figures 9-30 and 9-32
are the coincidence and analog phase-shift detectors,
respectively.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
71
Quadrature FM Detectors
Figure 9-30. FM quadrature (coincidence) detector.
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Quadrature FM Detectors
 The coincidence detector of Figure 9-30 provides two
equal-frequency, phase-quadrature signals to a digital
AND gate.
 As seen in Figure 9-31 the pulse width of the AND gate
output varies as a function of the resonant-circuit
phase-shifts due to frequency changes at pin B.
 The RC output circuit provides the pulse integration
for deriving the average output voltage. The AND gate
can also be thought of as a digital multiplier producing
the product AB.
73
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Quadrature FM Detectors
Fig. 9-31. Coincidence detector. Input B is shown with two different
phase shifts relative to input A. The AND gate outputs are also shown.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
74
Quadrature FM Detectors
Figure 9-32. Analog phase-shift type of
Quadrature FM detector.
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Quadrature FM Detectors
 Figure 9-32 shows the same circuit as Figure 9-30 except
for the analog product detector. The multiplication of
two periodic signals with the same frequency produces a
dc voltage that is directly proportional to the signal
phase difference.
 For small phase shifts (narrowband FM), the output will
be reasonably linear and expressed approximately by
Vo = (V1Vin/2).sinQr
a Qr
= KvcQr
(9-96)

where Q is the effective resonant-circuit factor, r = f/fo –
fo/f is the fractional deviation frequency from resonance,
and the phase-to-voltage constant for the circuit is Kvc.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
76
IF Amplifiers and Limiters
 Parasitics are common in medium-frequency RF work.
The most commonly applied preventative is a low-Q
ferrite bead (Z-bead), usually placed on the base lead.
 Another practical design consideration is a resistor
placed in series with the collector circuit. Their main
function is to maintain at least 100 ohms of impedance
across the tank when the collector is in saturation.
 Too large a value of resistance will unnecessarily
reduce the output. Too low a value will allow the
circuit Q to drop excessively.
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Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
IF Amplifiers and Limiters
 Figure 9-34 illustrates a typical bipolar BPL stage. Note
that R2 may be eliminated entirely. This would result in
class C operation.
 Class C operation is appropriate in this application
because the input signal is rectified and the remaining
input circuit will provide a self-adjusting bias for
constant amplifier gain.
 Care must be taken with the RC time constant and
the transformer/transistor selection to ensure that
the reverse bias on the base does not exceed the baseemitter breakdown voltage.
78
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
IF Amplifiers and Limiters
Figure 9-34. Bandpass limiter/amplifier.
79
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
IF Amplifiers and Limiters
Figure 6-35. Amplitude limiter.
80
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
IF Amplifiers and Limiters
Figure 6-36. Amplitude limiter transfer characteristic.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
81
IF Amplifiers and Limiters
Figure 6-37. Typical limiter response characteristic.
82
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
Preemphasis for FM
 The main difference between FM and PM is in the
relationship between frequency and phase.
f = (1/2p).dq/dt.
 A PM detector has a flat noise power (and voltage) output
versus frequency (power spectral density). This is
illustrated in Figure 9-38a.
 However, an FM detector has a parabolic noise power
spectrum, as shown in Figure 9-38b. The output noise
voltage increases linearly with frequency.
 If no compensation is used for FM, the higher audio signals
would suffer a greater S/N degradation than the lower
frequencies. For this reason compensation, called emphasis,
is used for broadcast FM.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
83
Preemphasis for FM
Figure 9-38. Detector noise output spectra
for (a). PM and (b). FM.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
84
Preemphasis for FM
 A preemphasis network at the modulator input
provides a constant increase of modulation index mf
for high-frequency audio signals.
 Such a network and its frequency response are
illustrated in Figure 9-39.
Fig. 9-39. (a)Premphasis network, and (b) Frequency response.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
85
Preemphasis for FM
 With the RC network chosen to give t = R1C = 75ms in North
America (150ms in Europe), a constant input audio signal will
result in a nearly constant rise in the VCO input voltage for
frequencies above 2.12 kHz. The larger-than-normal carrier
deviations and mf will preemphasize high-audio frequencies.
 At the receiver demodulator output, a low-pass RC network
with t = RC = 75ms will not only decrease noise at higher
audio frequencies but also deemphasize the high-frequency
information signals and return them to normal amplitudes
relative to the low frequencies.
 The overall result will be nearly constant S/N across the 15kHz audio baseband and a noise performance improvement of
about 12dB over no preemphasis. Phase modulation systems
do not require emphasis.
86
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
STEREO FM
 In Figure 9-40, audio signals from both left and right
mircrophones are combined in an linear matrixing
network to produce an L+R signal and an L-R signal.
 Both L+R and L-R are signals in the audio band and
must be separated before modulating the carrier for
transmission. This is accomplished by translating the
L-R audio signal up in the spectrum.
 As seen in Figure 9-40, the frequency translation is
achieved by amplitude-modulating a 38-kHz subsidiary
carrier in a balanced modulator to produce DSB-SC.
87
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
STEREO FM
Figure 9-40. FM stereo generation block diagram.
88
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
STEREO FM
 The stereo receiver will need a frequency-coherent 38kHz reference signal to demodulate the DSB-SC.
 To simplify the receiver, a frequency- and phasecoherent signal is derived from the subcarrier oscillator
by frequency division (÷2) to produce a pilot.
 The 19-kHz pilot fits nicely between the L+R and DSBSC L-R signals in the baseband frequency spectrum.
89
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
STEREO FM
 As indicated by its relative amplitude in the baseband
composite signal, the pilot is made small enough so
that its FM deviation of the carrier is only about 10%
of the total 75-kHz maximum deviation.
 After the FM stereo signal is received and
demodulated to baseband, the 19-kHz pilot is used to
phase-lock an oscillator, which provides the 38-kHz
subcarrier for demodulation of the L-R signal.
 A simple example using equal frequency but unequal
amplitude audio toned in the L and R microphones is
used to illustrate the formation of the composite stereo
(without pilot) in Figure 9-41.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
90
STEREO FM
Figure 9-41. Development of composite stereo signal. The 38 kHz alternately
multiplies L-R signal by +1 and –1 to produce the DSB-SC in the balanced AM
modulator (part d). The adder output (shown in e without piot) will be filtered
91
to reduce higher harmonics before FM modulation.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
STEREO FM
Fig. 9-45. Stereo FM multiplex generator
with optional SCA.
92
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
STEREO FM
Fig. 9-46. Stereo FM multiplex demodulation
with optional SCA output.
93
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
STEREO FM
Figure 9-47. Spectrum of stereo FM signal.
94
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
STEREO FM
Figure 9-48. Stereo FM decoder using the CA3090 IC
with PLL pilot synchronizing.
Prof. J.F. Huang, Fiber-Optic Communication Lab.
National Cheng Kung University, Taiwan
95