2. Continuous-wave modulation
1. Application goal
We study representations in time and frequency domain for two types of continuous wave
modulation: amplitude modulation (AM) and frequency modulation (FM).
2. Continuous-wave modulation: basic notions
The purpose of a communication system is to transport a signal representing a message over
a channel and deliver a reliable estimate of that signal to a user. This message may be a speech
signal, the channel maybe a cellular phone channel or a satellite channel.
The message signal (called the modulating wave) that contains information, generated by
sources of information, is a baseband signal and it is transmitted using a high frequency signal, also
called carrier signal. A parameter of the carrier wave is modified according the modulating wave.
The two most common used forms of carriers are the sine wave and the periodic pulse wave.
Correspondingly, we identify two main classes of modulation: continuous-wave modulation (CW)
and pulse width modulation.
A continuous wave has three parameters: amplitude, frequency and initial phase; the
corresponding modulation is amplitude modulation (AM), frequency modulation (FM) and phase
modulation (PM). FM and PM are also known as angle modulation.
Modulation provides the means for:
(1) shifting the range of frequencies contained in the message signal into another frequency
range suitable for transmission over the channel, centered onto the carrier frequency
(2) performing a corresponding shift back to the original frequency range after reception of
the signal
Example: A radio system operates efficiently into a frequency range that is higher than 30
KHz; on the other hand baseband signals are audio signals below 20kHz (0-20kHz). Frequency
shifting is made using modulation; in order to listen to an FM radio channel, we set the frequency to
a very high value (for instance 100MHz).
3. Amplitude modulation
Consider a carrier wave:
c ( t ) = Ac cos ( ωct )
(1)
and the modulating wave x ( t ) . The amplitude modulated signal is:
⎡ K
⎤
s AM ( t ) = ⎡⎣ Ac + K a x ( t ) ⎤⎦ cos ( ωc t ) = Ac ⎢1 + a x ( t ) ⎥ cos ( ωc t ) = Ac ⎡⎣1 + ka x ( t ) ⎤⎦ cos ( ωc t )
Ac
⎣
⎦
(2)
The constant K a , measured in ⎡⎣V −1 ⎤⎦ , is called the amplitude sensitivity of the modulator. The
amplitude modulation works by varying the strength of the transmitted signal in relation to the
1
information being sent. The amplitude of the carrier signal is varied according to the modulating
wave through the term ka x ( t ) . The amplitude of the sine wave is by definition a positive value
which means we impose the condition:
ka x(t ) ≤ 1
(3)
If this condition is not satisfied then an overmodulation is obtained, whose effect is distorting the
envelope of the AM signal and the correct recovery (demodulation) of the modulating wave is no
longer possible. The maximum value of ka x(t ) is the modulation degree, usually expressed as a
percentage:
m = ka x ( t ) max ⋅100 [ % ]
(4)
Fig. 1 shows the sine carrier wave, a modulating wave (also sine wave), and two cases of amplitude
modulation. Fig. 1 c) presents a correct modulation while in the next case we have overmodulation
(fig. 1 d), when the modulation degree is higher than 1 (m>1). For the 2nd case, we notice distortions
of the envelope as a direct result of phase inversion in the carrier signal
The condition to recover correctly the message signal from the envelope of the modulated signal is
that the carrier frequency to be high enough compared to the maximum variation speed of the
modulating signal:
fc fm
(5)
Carrier signal, c(t)
1
a)
0
-1
0
1
2
3
4
Modulating wave, x(t)
5
6
-4
x 10
1
b)
0
-1
0
1
2
3
AM signal, m=50%
4
5
6
-4
x 10
2
c)
0
-2
0
1
5
d)
2
3
4
5
AM signal, m=120% (overmodulation)
6
-4
x 10
0
-5
0
1
2
3
4
5
timp
6
-4
x 10
Fig. 1: Amplitude modulation for modulating sine wave x(t) and overmodulation.
2
If this condition is satisfied, the modulation degree can be determined using the time representation
of the signals, on the oscilloscope. The signal from fig. 1c is given below, its envelope with a dotted
line (fig 2).
Amax
1.5
Amin
1
amplitudine
0.5
0
-0.5
-1
-1.5
0
1
2
3
4
timp
5
6
-4
x 10
Fig. 2: Measuring the modulation degree using the oscilloscope.
The approximation is:
m=
Amax − Amin
⋅100 [%]
Amax + Amin
(6)
In the example above, we notice that Amax = 1.5 V and Amin = 0.5 V , so we have m=50%.
The spectrum for the AM signal results from the Fourier transform applied on the signal in relation
(2):
S AM (ω) = πAc [ δ(ω − ωc ) + δ(ω + ωc ) ] +
ka Ac
[ X (ω − ωc ) + X (ω + ωc )] (7)
2
An example is given in fig. 3.
3
X(ω)
1
SAM(ω)
πAcδ(ω-ωc)
kaAc/2
lower
upper
sideband
lower
sideband
sideband
ωc- ωm ωc ωc+ ωm
-ωc -ωc+ ωm
πAcδ(ω+ωc)
upper
sideband
-ωc- ωm
Fig. 3: The spectrum of the AM signal.
The spectrum has two Dirac impulses corresponding to the carrier; the effect of the modulation is
the frequency shifting of the spectrum from the baseband on the carrier frequency. The bandwidth
occupied by the AM signal above the carrier is the upper sideband and the bandwidth occupied by
the AM spectrum below the carrier is the lower sideband. We can see from the figure that the
bandwidth of the modulated signal, BAM is double of the bandwidth of the message (modulating)
signal, B:
BAM = 2 ⋅ BM = 2 ⋅ ωm
(8)
The main advantage of amplitude modulation is the simple implementation. As a consequence it
was used from the beginning in the field of radio transmission. Transmission of the radio signal
using AM was cheaper.
However, amplitude modulation has low energy efficiency. Although the carrier does not have
itself information, it is present in the AM spectrum, so there is a waste of power. For the
transmission of the information, only one sideband is necessary; if both the carrier and the other
sidebands are suppressed at the transmitter, no information is lost. This way, the channel needs to
provide only the same bandwidth as the message signal. The types of AM modulation that “save”
bandwidth are:
• suppressed-carrier Amplitude Modulation
• single sideband Amplitude Modulation, SSB-AM
• vestigial sideband Amplitude Modulation,VSB-AM
For the AM spectrum, consider the case of a sine modulating wave. The waveform of the AM
signal is shown in fig. 1 and the modulation degree is m = ka Am / Ac . The signal’s expression is
obtained from (2) using the particular form for the modulating wave x(t ) = Am cos(ωmt ) :
s AM (t ) = Ac (1 + m cos ( ωmt ) ) cos(ωc t )
(9)
or:
s AM (t ) = Ac cos(ωc t ) +
mAc
mAc
cos(ωc + ωm )t +
cos(ωc − ωm )t
2
2
(10)
4
SAM(ω)
πAc
πmAc/2
-ωc- ωm -ωc -ωc+ ωm
ωc- ωm ωc ωc+ ωm
ω
Fig. 4: The spectrum of the AM signal for modulating sine wave.
This spectrum corresponds in time to a sum of three sine waves with different frequencies, ωc ,
ωc − ωm and ωc + ωm :
k A
(11)
S AM (ω ) = π Ac ⎡⎣δ (ω − ωc ) + δ (ω + ωc ) ⎤⎦ + a c ⎡⎣ X (ω − ωc ) + X (ω + ωc ) ⎤⎦
2
The power of the AM signal is
A2 m 2 Ac2 m 2 Ac2 Ac2
2 + m2 )
Ps = c +
+
=
(12)
(
2
8
8
4
4. Frequency modulation
Another way to modulate a carrier is to alter its angle–phase, according to the message;
whereas the amplitude remains constant. The advantage is the signal is more robust against noise
and interference. The disadvantage is the increase in bandwidth. Again, consider a carrier wave:
c ( t ) = Ac cos ( ωct )
The modulated signal is a rotating vector with amplitude Ac and angle θi :
t
⎡
⎤
sFM (t ) = Ac cos θi ( t ) = Ac cos ⎢ωc t + 2πk f ∫ x ( τ )d τ ⎥
0
⎣
⎦
(13)
The angular velocity of this vector is the instantaneous frequency of the modulated signal
ωi ( t ) =
dθi ( t )
dt
(14)
For the FM case, the instantaneous frequency is
ωi ( t ) = ωc + 2π k f x ( t )
(15)
k F [ Hz / V ] is the frequency sensitivity. For comparison, we consider AM signals: the phase is (even
considering the no modulation case) θi (t ) = ωc t + ϕc and the amplitude isn’t constant, it depends on
time; while for FM signals, the amplitude is constant and the information is sent through the phase
or the instantaneous frequency.
5
t
If we denote y (t ) = ∫ x(τ)d τ , relation (12) becomes:
0
sFM (t ) = Ac ( cos(ωc t ) cos(2 πk F y (t )) − sin(ωc t ) sin(2πk F y (t )) )
(16)
There are two types of frequency modulation:
- if max t {k f y (t )} << π / 2 , the effect of the modulation on the carrier is small and the FM
-
bandwidth isn’t much larger than the spectrum of the carrier. This is narrow band frequency
modulation.
if max t {k f y (t )} >> π / 2 , the FM bandwidth is much larger than the spectrum of the carrier.
This is wide band frequency modulation.
For a sine modulating wave
x ( t ) = Am cos ωmt
the instantaneous frequency is:
ωi ( t ) = ωc + 2π k f Am cos ωmt
(17)
The frequency deviation is the maximum instantaneous difference between an FM modulated
carrier frequency and the nominal carrier frequency: Δω = 2π k f Am ; and the modulation index is
β=
The FM signal becomes:
or:
Δω
2π k f Am
=
ωm
ωm
sFM ( t ) = Ac cos [ωc t + β sin ωmt ]
sFM ( t ) = Ac cos(ωct ) cos(β sin(ωmt )) − Ac sin(ωc t ) sin(β sin(ωmt ))
(18)
(19)
(20)
An essential characteristic for FM signals is that the frequency deviation Δf is proportional with the
modulating signal’s amplitude; but it does not depend on its frequency. Again we have narrow band
modulation or wide band modulation for β << 1 radian or β >> 1 radian respectively.
The functions cos(β sin(ωmt )) and sin(β cos(ωmt )) are periodic with period 2π / ωm ; hence
they are decomposed into Fourier series:
∞
cos(β sin(ωmt )) = J 0 (β) + 2∑ J 2k (β) cos(2k ωmt )
(21)
k =1
and:
∞
sin(β sin(ωmt )) = 2∑ J 2k +1 (β) cos((2k + 1)ωmt )
(22)
k =0
where J P (β) are Bessel functions of the first kind, and order p. We have:
sFM (t ) = Ac ⋅
∞
∑J
n =−∞
n
(β) cos(ωc + nωm )t
(23)
6
Although the modulating wave is band limited, we can see that sFM (t ) is nonband limited,
composed of an infinite series of sine waves, separate on the frequency axis by ωm .
For β << 1 (narrow band modulation), only J 0 (β) and J 1 (β) have significant values, the
other Bessel functions can be approximated with zero. The FM signal is:
sFM (t ) = Ac J 0 (β) cos(ωc t ) + Ac J1 (β) cos(ωc + ωm )t − Ac J1 (β) cos(ωc − ωm )t
(24)
or:
t
t
⎛
⎞
sFM ( t ) = Ac cos ⎜ ωc t + 2π k f ∫ x (τ ) dτ ⎟
0
⎝
⎠
For narrow band modulation,
∫
0
=
y ( t ) = x (τ ) dτ
y(t ) ≤ A
Ac cos ωc t cos ( 2π k f y ( t ) ) − Ac sin ωc t sin ( 2π k f y ( t ) )
π
⇒ sFM ( t ) ≅ Ac cos ωc t − Ac 2π k f y ( t ) sin ωc t.
(25)
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The FM spectrum for narrow band modulation resembles the AM spectrum, obtained in the same
particular case (modulating wave is a sine wave).
⎡ X (ω − ωc ) X (ω + ωc ) ⎤
(26)
−
S FM (ω ) = π Ac ⎡⎣δ (ω − ωc ) + δ (ω + ωc ) ⎤⎦ + π Ac ⎢
⎥
ω + ωc ⎦
⎣ ω − ωc
Recall that:
k A
S AM (ω ) = π Ac ⎡⎣δ (ω − ωc ) + δ (ω + ωc ) ⎤⎦ + a c ⎡⎣ X (ω − ωc ) + X (ω + ωc ) ⎤⎦
2
2π k f A ≤
For higher values of β , the Bessel functions of superior order can no longer be neglected, and the
FM spectrum is of wide band:
A ∞
S FM (ω ) = c ∑ J n ( β ) ⎡⎣δ (ω − ωc − nωm ) + δ (ω + ωc + nωm ) ⎤⎦
(27)
2 n =−∞
The amplitude of the component with the frequency ωc depends on the factor J 0 ( β ) . Opposed to
the AM modulation, the amplitude of the corresponding FM signal spectrum isn’t constant, but
depends on β . The power of the FM signal is
1 T 2
(28)
(t )dt
Ps = ∫ xFM
T 0
Supposing the carrier frequency is a multiple of the maximum frequency of the spectrum of the
modulating wave ωc = mωm , we have :
1 2 ∞ 2
1
Ac ∑ J n ( β ) = Ac2
2 n =−∞
2
We have used the property of the Bessel functions of first kind:
Ps =
∞
(29)
∑ J (β ) = 1
k = −∞
2
K
The power of the modulated signal is the same with the power of the carrier signal.
Nearly all of the power (∼98%) of the FM signal lies within a bandwidth of:
7
⎛
1⎞
BT ≅ 2Δf + 2 f m = 2Δf ⎜ 1 + ⎟
(30)
⎝ β⎠
For wide band modulation ( β 1 ), we can approximate the transmission bandwidth as
BT ≅ 2 Δf
(31)
This means the bandwidth of the FM signal is constant and doesn’t depend on the modulating
wave’s maximum frequency.
5. Practical part
We make measurements and graphical representations in time and frequency for the AM and
FM signals. We use two signal generators, GEN1 and GEN2, an oscilloscope and a spectrum
analyzer. One of the signal generators is the source of the modulating wave whose waveform can be
sine wave, square wave or triangle wave.
The fundamental frequency of the modulating wave is set to fm=20kHz. The message signal
will modulate a carrier with a carrier frequency of fc=500kHz. The carrier is generated by GEN2 and
the modulation is made by bringing the modulating signal (output 50Ω) at the AM or FM input,
depending on the type of modulation.
Both signals (modulating wave and modulated wave) are visualized in time on the
oscilloscope using both channels. The spectral analysis is made using the spectral analyzer. The
following parameters are set:
Center frequency: 0.5MHz; Reference level 10dBm
Span width: 20kHz/div;
3kHz RBW
Tasks for the AM study:
- Represent the modulating wave and the modulated signal as seen on the oscilloscope. This
operation is made for three types of modulating wave: sine wave, triangle wave and square
wave.
- For the modulating signal of type sine wave, we measure the modulation degree using the
experimental method from relation (6). Each time, the modulating and carrier frequencies are
notated on the graphical representation.
- Graphical representation of the AM spectrum as seen on the spectrum analyzer. Attention is
paid to the magnitude and to the positioning of the spectral components on the frequency
axis. The representation is made only for a sinusoidal modulating wave.
- Compute the modulation degree from this spectrum. For this, measure the magnitude from
the carrier frequency ωc and the magnitude from the frequency ωc ± ωm :
Magnitude vs. frequency
U [ dBm ]
U [V]
Am = mAc / 2 (freq. ωc ± ωm )
Ac
(freq. ωc )
8
20 lg
U [V]
U ref
= U [ dBm ]
The reference level is U ref = 226mV .
20 lg ( mAc / 2 ) − 20 lg Ac = 20 lg m + 20 lg Ac − 20 lg 2 − 20 lg Ac = 20 lg m − 20 lg 2
-
Change the modulating wave to a square wave. What happens in the spectrum of the AM
signal, and how can you explain this.
Tasks for the FM study:
- Represent the modulating wave and the modulated signal as seen on the oscilloscope
modulating wave of type sine wave. How do you explain the spectrum on the spectral
analyzer?
- Visualize and make a graphical representation of the FM spectrum as seen on the spectrum
analyzer, for the case of narrow band modulation.
- For narrow band modulation, compute the modulation index β . Using the spectrum
analyzer, measure the magnitude from the carrier frequency ωc for the carrier signal (no
modulation). Reconnect the modulating wave generator and measure the magnitude from the
frequency ωc for narrow band FM modulation.
U [ dBm ]
Magnitude vs. signal
Carrier signal’s spectrum
U [V]
πAc
Narrow band FM spectrum πAc J 0 ( β )
20 lg
U
= U [ dBm ]
U ref
The magnitude from the carrier frequency is πAc for the carrier wave.
The magnitude from ωc is πAc J 0 ( β ) for narrow band modulation.
20 lg ( πAc J 0 ( β ) ) − 20 lg ( πAc ) = 20 lg ( πAc ) + 20 lg ( J 0 ( β ) ) − 20 lg ( πAc ) = 20 lg ( J 0 ( β ) )
-
Increase the amplitude of the modulating wave (thus increasing β, the modulation index).
Repeat the graphical representation. Compare the spectrum with the previous case and with
the AM spectrum.
9