TELE3013 TELECOMMUNICATION SYSTEMS 1
Lab 3: Frequency Modulation
1. INTRODUCTION
A sinusoidal carrier signal Ac cos ωct frequency modulated by a message m(t) is defined by,
t


s( t ) = Ac cos ω c t + D f ∫ m ( τ )dτ + θ 0 


0
(1)
where Df is the frequency modulator constant (or sensitivity) in radians per second per volt.
For a sinusoidal message m(t) = Am cos ωm t, equation (1) becomes (taking θo = 0 for simplicity),
s( t ) = Ac cos( ω c t + β sin ω m t )
where β = AmDf /ωm = ω /ωm = F / fm is the frequency modulation index and F
(instantaneous) frequency deviation from the carrier frequency fc.
(2)
is the peak
When β « 1 we have a Narrow–Band Frequency Modulated (NBFM) signal,
s( t ) |NBFM = Ac cos ω c t − Ac β sin ω m t sin ω c t
(3)
2. PREPARATION
P1. Consider an FM signal with a sinusoidal message as given by equation (2). Given
that the message parameters Am = 2 volts, fm = 5 kHz and the carrier parameters
Ac = 1 volt, fc = 100 kHz and that the peak frequency deviation ∆F = 20 kHz,
determine the following:
a.
The instantaneous phase of the FM signal and the peak phase deviation
from the carrier phase. Sketch the instantaneous phase as a function of time.
b.
The instantaneous frequency of the FM signal. Sketch it as a function of
time.
c.
The FM signal modulation index.
d.
The approximate bandwidth of the FM signal.
e.
The average power of the message signal (in 1 ohm).
f.
The average power of the unmodulated carrier signal.
g.
The average power of the FM signal.
P2. Use Armstrong's method to generate Wide-Band Frequency Modulated (WBFM)
signal from NBFM. Starting from a NBFM signal with a carrier frequency of 10
kHz, a tone message signal of 2 kHz and a frequency deviation ∆F = 40 Hz give
a block diagram for the generation of a WBFM signal with a carrier frequency of
10 MHz and ∆F = 50kHz, showing the necessary frequency multipliers and
frequency converters (mixers).
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TELE3013: Lab 3
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3. EXPERIMENTS
EQUIPMENT
MODULES
QTY
TRUNK SIGNALS
Audio Oscillator
1
Signal #1: Unknown Signal
Adder
2
Signal #2: Unknown Signal
Multiplier
2
Signal #3: Speech Signal
Voltage Controlled Oscillator (VCO)
2
Tuneable Low Pass Filter
1
3.1 Generation of Narrow–Band Frequency Modulation (NBFM)
The block diagram below shows how the TIMS system can be used to generate a NBFM signal
given by equation (3) above.
multiplier
Osc.
(VCO)
adder
βsinωmt
NBFM
signal
sinωct
message
Audio
Osc.
cosωct
carrier
Use a VCO module with its input grounded and its centre frequency set to 2 kHz as the message
source. The quadrature outputs of the Audio Oscillator with its frequency set to about 10 kHz
are used as the carrier source. Recall that this method will only work for small values of the
modulation index β. For large values of β a combination AM/FM signal is obtained.
Observe and sketch the time domain waveform and amplitude spectrum of the NBFM signal.
Observe and record the change resulting when in–phase carriers are used instead of quadrature
carriers. Draw phasor diagrams for the two cases.
3.2 WBFM generation using a Voltage Controlled Oscillator (VCO)
By definition a voltage controlled oscillator (VCO) is a frequency modulator. We now use the
VCO module to investigate the properties of wideband FM.
Audio
Osc.
message
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VCO
FM signal
Centre freq. fc
TELE3013: Lab 3
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3.3 Characteristics of FM
3.3.1 Time Domain Waveform of Tone FM Signal
Set the VCO centre frequency to about 10 kHz and use the Audio Oscillator to modulate this
frequency with a 1 kHz sine wave. Observe the time domain waveform – set the oscilloscope to
display about ten carrier cycles. Note the effect of increasing the VCO sensitivity (gain) from
zero upwards.
Note:
If the oscilloscope display is triggered on this waveform then the left most carrier cycle
is ‘stationary’ more or less by definition. What is the next carrier cycle that is
approximately ‘stationary’ on the oscilloscope display? Vary the frequency of the
modulating signal and see how this result changes. A tone modulated FM signal is
described by equation (2) which is repeated below
s( t ) = Ac cos( ω c t + β sin ω m t )
A ‘stationary’ cycle corresponds to ωmt = nπ (n an integer), the maximum phase
deviation occurs when ωmt = (2n + 1)π/2 and successive peaks of the unmodulated
carrier signal correspond to ωct = 2nπ. Triggering the FM signal at a certain level, the
CRO displays a blurred signal with its maximum phase blur, Φ, corresponding to
Φ = 4 ∆φ =
∆t
2π
T
Here ∆φ =β is the peak phase deviation. ∆t is the maximum blur in time and T=2π /ωc
is the period of the carrier.
3.3.2 FM Signal Spectrum and Bandwidth
Using the time waveform set the VCO gain so that the peak (instantaneous) phase deviation ∆φ
=β ~ 1 radian. What is the value of the modulation index β? Observe and sketch the amplitude
spectrum of the of the FM signal. What is the measured bandwidth of the FM signal and how
does it compare with the bandwidth predicted by Carson’s rule? Repeat the measurements of
peak phase deviations ∆φ = 2.4, 3 and 3.8 radians. Draw up a table listing the modulation index,
the measured FM signal bandwidth and the bandwidth predicted by Carson’s rule. Comment on
the validity of Carson’s rule.
Note: The FM signal in equation (2) can be expressed as a Fourier series,
s( t ) = Ac
∞
∑ J n ( β ) cos( ω c + n ω m )t
n = −∞
where Jn(β) is an n-th order Bessel function of the first kind and it has the
relation
J–n(β
β ) = (–1)nJn(β
β ).
You may find it instructive to use MATLAB to plot the values of a few Bessel functions, eg:
b = 0:0.1:10;
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{Modulation index β: 0 to 10 in 0.1 steps}
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j0 = besselj(0, b);
{J0(β)}
j1 = besselj(1, b);
{J1(β)}
j2 = besselj(2, b);
{J2(β)}
plot ( b, [j0’ j1’ j2’] ), grid
{ J2’ means the transpose of J2}
What values of β give Jo(β) = 0?
3.4 FM Demodulation using a Phase–Locked Loop
Construct a first order Phase–Locked Loop (PLL) from a second VCO, a tuneable LPF and a
multiplier, as shown in the following plot. Use this PLL to recover the message from the FM
modulated signal generated by the first VCO.
Phase Comparator
FM
Signal
Tuneable
Lowpass
Filter
Multiplier
Recovered
Message
VCO
NOTE: You have initially to adjust the second VCO to obtain lock, i.e. you need to
approximately match the centre frequency and choose a reasonable value for the
sensitivity. You may like to compare the analog level outputs of the two VCOs on your
oscilloscope to observe when lock is obtained. Change the message frequency and
check that the receiver recovers the message accurately. Drive the PLL just out of
lock and note the transition time, Lock–No Lock–Lock.
3.5 Modulation and Demodulation using Speech Signals
Using the speech signal available from the trunk panel as your message, carry out FM modulation
and demodulation as described in sections 3.2 and 3.4.
3.6 Recovery of Unknown FM Signals
There are two FM signals with unknown messages available from the trunks panel. Demodulate
these and identify the message signals.
Sketch and explain your strategies to demodulate the unknown FM signals.
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