118
Angle Modulation 1 Angle Modulation (Phase & Frequency Modulation) EE 442 Lecture 7 Spring Semester Stamp issued 1983 FM invented 1933
Angle Modulation
1
Angle Modulation (Phase & Frequency Modulation)EE 442 Lecture 7
Spring Semester
Stamp issued 1983FM invented 1933
2Review
Summary of Lecture 6 – Page 1
Modulation is the systematic alteration of a carrier wave so that it “carries” the information
of the message or data signal m(t).
Modulation allows for the designated frequency bands (with the carrier frequency at the
center of the band) to be utilized for communication and allows for signal multiplexing.
Amplitude modulation (AM) is an analog and linear modulation process as opposed to
frequency modulation (FM) and phase modulation (PM).
AM involves the variation of the carrier signal’s amplitude in direct proportion to the
modulating signal m(t).
AM is simple to implement and can be accomplished inexpensively with a small number of
components; but AM has a low power efficiency (ratio of power in the message signal
relative to the total transmitted power) and is very susceptible to noise and interference.
The landline telephone (PSTN or POTS) uses and voice signal bandwidth of 300 Hz to
3,400 Hz and a transmission voice channel of 0 to 4,000 Hz.
The Foxhole radio (from World War I) consists of an antenna, inductive coil (paired with
parasitic capacitance to form a frequency selective resonator), earphones, and rectifier
made from a razor blade and sharply pointed needle from a safety pin).
3Review
Summary of Lecture 6 – Page 2
An amplitude modulation time-varying signal (double sideband with carrier – DSB-WC) is
AM can be interpreted using phasors where the carrier of the AM signal is a phasor of
constant amplitude AC rotating CCW at frequency fC and the modulating signal m(t) made
up of a collection of slower rotating Fourier components of m(t) attached to the tip of the
carrier phasor. The vector sum of the phasors gives the AM phasor.
The corresponding amplitude modulation spectrum is
which is related to the frequency shift property of the Fourier transform.
The AM modulation index is defined as = mp/AC, where mp is the peak amplitude of
m(t). When > 100% overmodulation results in an AM waveform (i.e., envelope
distortion).
The power efficiency of AM is defined as
where Pm is the message power. The power efficiency is 11.1% when = 0.5 and is
33.3% when = 1.0 (best case).
( ) ( ) cos( )AM C Ct A m t t = +
1 1
( ) ( ) ( ) ( ) ( )2 2
AM C C C C CM M A = − + + + − + +
2
message power,
total powerm
C m
P
A P = =
+
4Review
Summary of Lecture 6 – Page 3
There are two ways to improve on the power efficiency of amplitude modulated signals: (a)
suppress the carrier power (known as DSB-SC) in the transmission, and (b) eliminate both
the carrier and one of the sidebands (SSB-SC).
Modulators: (a) Nonlinear component modulator, (b) switching modulator and (c) electronic
multipliers (such as using a Gilbert cell).
A nonlinearity generates Taylor series terms beyond the term linear in variable v, such as
v2, v3 and so on. Terms of v2 (so-called square law behavior) and higher generate new
frequencies that produce amplitude modulation.
Square-law modulators are very useful because they produce the DSB-SC AM signal that
can be demonstrated from
which is the DSB-SC AM signal as desired.
The switching modulator relies upon the generation of a square-wave pulse train p(t) to
generate new frequencies as required to perform modulation, namely
( )2
( ) cos( ) ( ) cos( )C C CA m t t m t t + →
1 2 1 1( ) cos( ) cos(3 ) cos(5 )
2 3 5C C Cp t t t t
= + − + −
5
Summary of Lecture 6 – Page 4
The switching modulator generates DSB-SC AM signals directly.
A pn-junction is a nonlinear device in its forward biased state that makes a widely used
modulator (and detector also).
An AM signal can always be demodulated using a coherent demodulator (needs a
coherent carrier that exactly matches the carrier of the transmitter to recover the message
signal m(t).
However, there are two non-coherent methods to perform AM demodulation. These are (a)
envelope detection and (b) rectifier detection.
Envelope demodulation depends upon performing half-wave rectification and letting the
peaks of the AM waveform charge a capacitor which decays at a rate allowing for the
capacitor voltage to approximately follow the envelope of the half-wave rectified waveform.
The envelope recovery is proportional to message signal m(t).
For rectifier demodulation the capacitor of the envelope detector is omitted and the
rectified AM signal is fed directly into a low-pass filter which recovers [AC + m(t)]. The DC
component AC may be removed using a series blocking capacitor.
With DSB-SC the power efficiency approaches 100% because the square of amplitude
AC is zero from the elimination of the carrier.
Review
6Review
Summary of Lecture 6 – Page 5
A mixer can be used to generate DSB-SC AM where the RF port is driven by the message
signal m(t) and the LO port is driven by the carrier signal cos(Ct). The baseband
message signal is centered about the carrier frequency in the LSB and USB even though
the carrier power is zero.
For DSB-SC AM demodulation, again a mixer can be used to receive DSB-SC AM where
the RF port is driven by the message signal m(t)cos(Ct) and the LO port is driven by the
carrier signal cos(Ct). The IF port outputs m(t) + m(t)cos(2Ct), thus allowing for m(t) to
be filtered out and recovered.
Synchronous demodulation requires detection of the carrier frequency using the DSB-SB
AM signal. One way to do this is to square the incoming AM signal, filter it with a
bandpass filter and divide the signal by two, thereby recovering a signal in step with the
transmitted carrier frequency and use this signal to drive the LO port of the demodulating
mixer.
Multipliers may be built using log and anti-log block (with op amps) to sum two inputs to
give a modulated output. Another very widely used method uses the Gilbert cell which is
integrated to produce a linear modulator.
BPF( )DSB SC t −
7Review
Summary of Lecture 6 – Page 6A commonly used method to generate DSB-SC AM is shown in the block diagram:
The ring diode modulator for DSB-SC AM is a balanced modulator that operates as a
switching modulator – shown below.
You shouldunderstand howthis mixer works.
8Review
Summary of Lecture 6 – Page 7
Mixers are used for frequency conversion and for heterodyning. Heterodyning uses an
electronic circuit to combine an input radio frequency (RF) with one that is generated (LO)
in order to produce new frequencies: one that is the sum of the two and the other the
difference of the two. Heterodyning is typically used to band-shift incoming frequencies
into intermediate frequencies (IF) for demodulation.
Heterodyne receivers can provide (a) selectivity in signal reception, (b) handle a wide
range of modulation formats, and (c) are capable of accommodating very high frequencies
(even into the millimeter frequency bands).
One problem in heterodyne receivers is image signal pickup. Mixers convert two RF
signals to the IF signal using a single LO signal. Thus, both RF signals (RF1 and RF2)
combine with the LO signal LO to give two IF outputs [(LO - RF1) and (RF2 - LO)].
The superheterodyne receiver is universally used in radio and a single conversion stage
superheterodyne receiver is shown below.
https://en.wikipedia.org/wiki/Superheterodyne_receiver
9Review
Summary of Lecture 6 – Page 8
Bandwidth efficiency can be improved is with quadrature amplitude modulation (QAM).
QAM involves two data streams: the I-channel and the Q-channel. Bandwidth efficiency is
improved because two signals to share the same bandwidth of a channel. But this can
only be done if the two modulated signals are orthogonal to each other.
Modulating one message (call it the in-phase message mI) with cos(Ct) and another
message (call it the quadrature message mQ) with sin(Ct) makes the two signals
orthogonal to each other. Thus, both messages can be independently modulated and
demodulated.
The QAM signal is of the form,
QAM transmits two DSB-SC signals in the bandwidth of one DSB-SC signal. Interference
between the two modulated signals of the same frequency is prevented by using two
carriers in phase quadrature. The In-phase (I-phase) channel modulates the cos(Ct)
signal and the Quadrature-phase (Q-phase) channel modulates the sin (Ct) signal. The
carriers used in the transmitter and receiver are synchronous with each other. In fact, they
must be almost exactly in quadrature with each other; otherwise, they experience
cochannel interference. Low-pass filters are used to extract the baseband signals mI(t)
and mQ(t) in the receiver.
QAM is used extensively as a modulation scheme for digital telecommunication systems,
such as in 802.11 Wi-Fi standards.
= + ( ) ( ) cos( ) ( ) sin( )QAM I C Q Ct m t t m t t
10Review
Summary of Lecture 6 – Page 9QAM transmitter and receiver block diagram:
Effect of error in carrier frequencies between the in-phase and the quadrature
channels.
2 cos( )Ct
2 sin( )Ct
( )Qz t
Transmitter Receiver
Channel
( )Iz t
( )QAM tcos( )Ct
sin( )Ct
( ) ( ) cos( ) ( )sin( )
( ) ( ) cos( ) ( )sin( )
= −
= +
I I Q
Q Q I
y t m t t m t t
y t m t t m t t
11Review
Summary of Lecture 6 – Page 10
Single-sideband AM (SSB AM) is the most efficient AM signal of any AM transmission
format with respect to efficient use of bandwidth (100% efficient).
The phase-sift method of generating of AM SSB.
= +( ) ( )cos( ) ( )sin( )
where minus sign applies to USB
and plus sign applies to the LSB.
( ) is ( ) phase delayed by - /2
SSB C h C
h
t m t t m t t
m t m t
HilbertTransformer
12
Summary of Lecture 6 – Page 11
Review
The phase shift function labelled Hilbert Transform performs the following phase shift
function: Given a signal, for positive frequencies, multiply it by –j (phase shift by -90 deg)
and for negative frequencies, multiply by +j (or +90 deg).
The phase shift method uses two balanced (and identical) to eliminate the carrier. Then
the phase shift is used to cancel one of the sidebands (it can be either the upper sideband
or the lower sideband).
With digital signals the closest digital modulation format is pulse amplitude modulation
(PAM).
f
( )H f
o90j or− −
o90j or+ +
13
Angle Modulation
https://semesters.in/tag/equation-for-pm-wave/
14
Some Applications for Various Modulation Techniques
We have studied AM, next is FM and PM.
Not a complete list of applications.
15
With few exceptions,Phase Modulation (PM)
is used predominantly in digital communication
Amplitude, Frequency and Phase Modulation
Remember that d
fdt
=
16
Illustrating AM, PM and FM Signals
Carrier Wave
Modulating Signal m(t)
AM Modulated Signal
PM Modulated Signal
FM Modulated Signal time
Carrier signal
m(t)
AM
PM
FM
AngleModulation
100% modulation
shown
Reference:Lathi & Ding
( )dm t
dt
( )m t
~
~
t
17
Focus Upon an FM Signal Modulated by a Single-Tone
BA
SEB
AN
D
SIG
NA
L
FM
SI
GN
AL
Single-tone modulating signal( )m t
18
carriercos(Ct)
messagem(t)
amplitudemodulation
phasemodulation
frequencymodulation
Comparing AM, PM and FM for a Ramp m(t)
https://www.princeton.edu/~mvaezi/ece3770/ECE3770_Lecture7.pdf
( )dm t
dt
( )m t
19
General Observations on FM and PM Waveforms
1. Both FM and PM waveforms are identical except for a time shift, when m(t) is a sinusoidal signal.
2. For FM, the maximum frequency deviation occurs when modulatingsignal is at its peak values (i.e., at + mp and – mp).
3. For PM, the maximum frequency deviation takes place at the zerocrossings of the modulating signal m(t).
4. It is generally difficult to know from looking at a waveform whether the modulation is FM or PM.
5. The message resides in the zero-crossings alone, provided the carrier frequency is large compared to frequency content of m(t).
6. The modulated waveform doesn’t resemble the message waveform.
Reference: Carlson & Crilly, 5th ed., Section 5.1, pages 208 to 212.
20
1. Angle modulation is resistant to propagation-induced selective fadingbecause the amplitude variations don’t contain information.
2. Angle modulation is very efficient in rejecting interference (i.e., it minimizes the effect of amplitude noise on the signal transmission).
3. Angle modulation allows for more efficient use of transmitter power.
4. Angle modulation can handle a greater dynamic range in the modulating signal without distortion (as would occur in AM).
5. Wideband FM gives significant improvement in the signal-to-noise ratioat the output and is proportional to the square of the modulationindex , where
Advantages of Angle Modulation
f
B
=
Bandwidth
Frequency deviation
B
f
21
Phase-Frequency Relationship When Frequency is Constant
0
+ 0
0 is constantCt
( )t
time t
=
= =( )
Slope: ( )i
i
Ct t
d tt
dt
(t) is generalized angle
=( ) cos( ( ))Ct A t
= + 0( ) cos( )C Ct A t
No modulation
Concept of Instantaneous Frequency
22
(t) is generalized angle
=( ) cos( ( ))Ct A tAngle
Modulationin red line
= + 0( ) cos( )C Ct A t
0
( )t
( )t
time t
=
= ( )
Slope: ( )i
i
Ct t
d tt
dt
ti
+ 0
0 is constantCt
Angle Modulation Gives PM and FM
23
−=
= = ( )
( ) and ( ) ( )t
i i
it t
d tt t d
dt
AngleModulation
PhaseModulation
FrequencyModulation
Frequency modulation and phase modulation are closely related!
24
No. Frequency Modulation (FM) Phase Modulation (PM)
1 Frequency deviation is proportional to modulating signal m(t)
Phase deviation is proportional to modulating signal m(t)
2 Noise immunity is superior to PM (and of course AM)
Noise immunity better than AM, but not FM
3 Signal-to-noise ratio (SNR) is better than PM (and of course AM)
Signal-to-noise ratio (SNR) is not quite as good as with FM
4 FM is widely used for commercialbroadcast radio (88 MHz to 108 MHz)
PM is primarily used for mobile radio services
5 Modulation index is proportional to modulating signal m(t) as well as the modulating frequency fm
Modulation index is proportional to modulating signal m(t)
Comparing Frequency Modulation to Phase Modulation
25
FM has better noise (or RFI)
rejection than AM, as shown in
this dramatic New York
publicity demonstration by
General Electric in 1940. The
radio contained both AM and FM
receivers. With a million-volt arc
as a source of interference
behind it, the AM receiver
produced only a roar of static,
while the FM receiver clearly
reproduced a music program
from Armstrong's experimental
FM transmitter W2XMN in New
Jersey.
FM has superior noise immunity compared to AM
https://en.wikipedia.org/wiki/Frequency_modulation
Note: RFI stands for radio frequency interference.
26
Phase Modulation (PM)
= + + =0 0( ) ( ) ; Usually we set 0,i C pt t k m t
= +( ) cos( ( ))PM C C pt A t k m t
The instantaneous angular frequency (in radians/second) is
= = + = +
( ) ( )( ) '( )ii C p C p
d t dm tt k k m t
dt dt
In phase modulation (PM) the instantaneous angular frequency i
varies linearly with the time derivative of the message signal m(t) [denoted here by m’(t)].
kp is the phase-deviation (sensitivity) constant. Units: radians/volt[Actually it is radians/unit of the parameter m(t).]
Agbo & SadikuSection 4.2; p. 159
27
Frequency Modulation (FM)
= +( ) ( )i C ft k m t
−
= +
( ) cos ( )t
FM C C ft A t k m d
But in frequency modulation the instantaneous angular frequency i varies linearly with the modulating signal m(t),
− −
= + = + ( ) ( ) ( )t t
i C f C ft k m d t k m d
Then
FM and PM are related to each other. In PM the angle is directly proportional to m(t).In FM the angle is directly proportional to the integral ( ) .m t dt
kf is frequency-deviation (sensitivity) constant. Units: radians/volt-sec.
Agbo & SadikuSection 4.2; p. 159
28
Summary
Angle
Frequency
Phase Modulation Frequency Modulation
= +( ) ( )i C pt t k m t −
= + ( ) ( )t
i C ft t k m d
( )i C p
dm tk
dt = + ( )i C fk m t = +
In phase modulation m(t) drives the time variation of phase i.
In frequency modulation m(t) drives the time variation of frequency fC.
=
( )( ) ii
d tt
dtDefinition: Instantaneous frequency is
Message signal is m(t)
29
A Pictorial View of FM and PM Generation
We require that H(j) be a reversible (or invertible) operationso that m(t) is recoverable.
Phase Modulator
Frequency Modulator
d
dt
( )m t
( )m t
( )dm t
dt
PM ( )t
FM ( )t
−
( )t
m d
Frequency Modulator
Phase Modulator
H(j) = 1/j
H(j) = j
Generation of PM
Generation of FM
cos( )C CA t
cos( )C CA t
Agbo & SadikuFigure 4.1
p. 160
30
Both FM and PM Generation are Nonlinear Processes
( )
( )
( )
1 2
1 1
2 2
1 2
Consider a phase modulated signal:
Let ( ) cos [ ( ) ( )]
If ( ) cos ( ) , and
( ) cos ( )
It then holds that
( ) ( ) ( ) additivity fails
The same
So PM can't be linear.
= + +
= +
= +
+
C C p
C C p
C C p
s t A t k m t m t
s t A t k m t
s t A t k m t
s t s t s t
argument holds for FM.
Note: Linearity requires both additivity and homogeneity to hold.
31
Modulation Index for Angle Modulation
Let the peak values of the message signal m(t) and its first derivative m’(t) berepresented by
Peak value of m(t) = mp = ½(mmax – mmin)
Peak value of m’(t) [= dm(t)/dt] = m’p
Frequency Deviation is the maximum deviation of the instantaneous modulated carrier frequency relative to the unmodulated carrier frequency.It is (symbolically) represented by either or f.
The ratio of the frequency deviation f to the message signal’s bandwidth BIs called the Frequency Deviation Ratio or the Modulation Index, and is denoted by (unitless).
FM: or2
PM: or2
f p
f p
p p
p p
k mk m f
k mk m f
= =
= =
2
f
B B
= =
32
=
=
Carrier signal cos( ) (volts)
Carrier frequency 2 (radians/sec)
Modulating wave ( ) cos( )
Modulating frequency 2 (radians/sec)
Deviation sensitivity (radian
Single-tone modulation
C C
C C
m m
m m
f
A t
f
m t A t
f
k
−
=
= = =
= + = +
= +
s/volt-second)
Frequency deviation (radians/sec)
Modulation Index (unitless)
Instantaneous frequency cos( ) cos( )2
Remember ( ) cos ( )
f m
f m
m m m
mi C f m C m
t
FM C C f
k A
k Af
f
Af f k t f f t
t A t k m d
( )
= +
= +
, generally
Tone modulated wave ( ) cos sin( )
or ( ) cos sin( )
f m
FM C C m
m
FM C C m
k At A t t
t A t t
Equations for FM Wave with Single-Tone Modulation
33
Summary of Mathematical Equations for FM and PM
Type of Modulation
Modulating Signal Angle Modulated Wave
Phase modulation
m(t)
Frequencymodulation
m(t)
Phase modulation
Tone:
m(t) = Am cos(mt)
Frequencymodulation
Tone:
m(t) = Am cos(mt)
( )cos ( )C C pA t k m t +
( )cos ( )t
C C fA t k m d −
+
( )cos cos( )C C p m CA t k A t +
cos sin( )f m
C C C
m
k AA t t
+
=f m
m
k A
34
( )6 3( ) 10 cos 2 (10 ) 8sin(2 (10 ) )FM t t t = +
( )FM t
Example
f
-
Cf
35
Solution to Example
( )( ) cos 2 sin(2 )FM C C mt A f t f t = +
Start with the basic FM equation:
Compare this to
(a) We see that fC = 1,000,000 Hz & fm = 1000 Hz.(b) The modulation index is = 8.(c) The peak deviation frequency f is
Note: f /fC is 0.008 or 0.8 % deviation frequency to carrierfrequency.
( )6 3( ) 10 cos 2 (10 ) 8sin(2 (10 ) )FM t t t = +
8 1000 8,000 Hzmf f = = =
36
Average Power of a FM or PM Wave
The amplitude AC is constant in a phase modulated or a frequency modulated signal. RF power does not depend upon the frequency or the phase of the waveform.
= +( ) cos ( , ( ))FM or PM C C kt A t g k m t
=2
Average Power (always)2CA
This is a result of FM and PM signals being constant amplitude.
Note: kk becomes kf for FM and kp for PM.
37
Problem:
Consider an angle modulated signal given by
What is the average power of this signal?
Solution:
( )6( ) 6 cos 2 10 2 sin(8000 voltst t t = +
2
2
Average power where 6 volts2
6 36Therefore, 18 watts (assumes 1 ohm resistance)
2 2
Note that the result does not depend upon it being FM or PM.
CC C
C
AP A
P
= = =
= = =
Average Power of a FM or PM Wave
38
# Frequency Modulation (FM) Amplitude Modulation (AM)
1 FM receivers have better noise immunity
AM receivers are very susceptible to noise
2 Noise immunity can be improved by increasing the frequency deviation
The only option in AM is to increase the transmission power
3 Bandwidth requirement is greater and depends upon modulation index
AM bandwidth is less than FM or PM and doesn’t depend upon a modulation index
4 FM (or PM) transmitters and receivers are more complex than for AM
AM transmitters and receivers are less complex than for FM (or PM)
5 All transmitted power is useful so FM is very efficient
Power is wasted in transmitting the carrier and double sidebands in DSB (but DSB-SC & SSB addresses this)
Comparison of FM (or PM) to AM
39
AM, FM and PM Waveforms for Single-Tone m(t)
Carrier Wave
Modulating Signal m(t)
AM Modulated Signal
PM Modulated Signal
FM Modulated Signal time
Carrier signal
m(t)
AM
PM
FM
AngleModulation
100% modulation
shown
Reference:Lathi & Ding
( )dm t
dt
( )m t
~
~
Focus uponfrequency
40
FM and PM Examples
Sketch FM and PM waveforms for the modulating signal m(t). The constantskf and kp are 2 105 and 10, respectively. Carrier frequency fc = 100 MHz.
( )
( )
8 5
min max
8 5
min
8 5
max
( ) 1 10 1 10 ( );2
1 and 1
10 10 ( 1) 99.9 MHz,
10 10 ( 1) 100.1 MHz
f
i C
i
i
kf f m t m t
m m
f
f
= + = +
= − =
= + − =
= + + =
( )
( )
= + = +
= − =
= + − =
= + + =
8
min max
8
min
8
max
'( ) 1 10 5 '( );2
' 20,000 and ' 20,000
10 5( 20,000) 99.9 MHz,
10 5( 20,000) 100.1 MHz
p
i C
i
i
kf f m t m t
m m
f
f
FM PM
m’(t)m(t)m(t)( )d m t
dtseconds
Fig. 5.4; p. 256 of 4th ed., Lathi & Ding
41
Digital Frequency Shift Keying is Related to FM
Sketch the FM waveform for the modulating signal m(t). The constant kf is2 105. Carrier frequency fc = 100 MHz.
Since m(t) switches from +1 to -1 and vice versa, the FM wavefrequency switches between 99.9 MHz and 100.1 MHz. This is calledFrequency Shift Keying (FSK) and is a digital communication format.
8 5( ) 1 10 1 10 ( )2
f
i C
kf f m t m t
= + = +
FM
m(t)
Fig. 5.5; p. 258 of 4th ed., Lathi & Ding
42
Example – continued
Fig. 5.5; p. 258 of 4th ed., Lathi & Ding
This is carrier PM by a digital signal – it is Phase Shift Keying (PSK)because the digital data is represented by phase of the carrier wave.
= + = +8( ) ( )1
1 102 4
p
i C
k dm t dm tf f
dt dt
PM
Sketch the PM waveform for the modulating signal m(t) from prior slide.The constant kp equals /2. Carrier frequency fc = 100 MHz.
= + = +
= = −
= − =
= − − = − − = −
( ) cos ( ) cos ( )2
( ) sin( ) when ( ) 1
( ) sin( ) when ( ) 1
where jump in ( ) (1) ( 1) 2 or ( 1) (1) 2
PM C C p d C C d
PM C C
PM C C
d
t A t k m t A t m t
t A t m t
t A t m t
m t
m’(t)( )dm t
dt
Evaluate the instantaneous jumps by considering:
( )0
dm t
dt=
Instantaneous jumpsin phase by radians.( ) ~ ( , )p dk m t −
43
Generalized Angle Modulation
Start with equation (4.8) on page 159, which is
( ) cos[ ( )] where ( ) ( ) ( ) ( ) ( )
with ( ) ( ) for PM; ( ) ( ) for FM
Suppose we use the exponential carrier e instead of
−
= + = = −
= =
C
t
A C C
j t
C C
t A t k t t m t h t m h t d
h t t h t u t
A A
( ( )) ( ) ( )
( )
cos( ),
then the form for generalized angle modulation becomes
( )
where for PM; for FM
( ) Re ( ) Re ; where ( ) ( ) ( ) ( )
and
(
+= =
→ →
= = = =
C C
pC
C
j t k t j t jk t
A C C
p f
jk tj t
PM A C
FM
t
t A e A e e
k k k k
t t A e e t m t t m t
t ( )
) Re ( ) Re ; where ( ) ( )
−
= = =
fC
tjk tj t
A Ct A e e t m d
Agbo & Sadiku; Section 4.2 & 4.3 on pages 158 to 168
44
Generalized Angle Modulation (continued)
0
2 2 3 3
( )( ) Re ( ) Re
!
( ) ( )( ) Re 1 ( )
2! 3!
Now take the real part of the expression above,
( ) cos( ) ( )sin( )
C
C
n n n
fj t
FM A C
n
f fj t
FM C f
FM C C f C
j k tt t A e
n
k t jk tt A e jk t
kt A t k t t
=
= =
= + − + −
= − −
2 2 ( )cos( )
2!
Note: ( ) has a bandwidth = Hz and (t) has a bandwidth = Hz,
but ( ) has a bandwidth = Hz; as , bandwidth
f
C
n
tt
m t B B
t nB n
+
→ →
Consider first Frequency Modulation (FM),
Conclusion: The instantaneous frequency deviations are symmetricalabout carrier frequency C, thus, FM is double side-banded. The effective FM bandwidth = 2nB Hz.
Agbo & Sadiku; Section 4.2 & 4.3 on pages 158 to 168
45
Generalized Angle Modulation (continued)
( ) cos( ) ( )sin( )
( ) cos( ) ( ) sin( )
FM C C f C
t
FM C C C f C
t A t k t t
t A t A k m d t
−
−
−
Consider the case where kf is small. meaning that |kf (t)| << 1.It is commonly referred to as narrowband FM (NBFM). We takeonly the first two terms in the expansion for FM(t).
Equation (4.15)
By analogy, we can apply same analysis for Phase Modulation (PM).For PM, if kp is small, then |kp (t)| << 1. This is known asnarrowband PM (NBPM).
( ) cos( ) ( ) sin( )PM C C C p Ct A t A k m t t − Equation (4.16)
FM:
PM:
Using these results allows us to generate narrowband FM and PMwith the block diagrams on the next slide (slide #46):
Agbo & Sadiku; Section 4.2 & 4.3 on pages 158 to 168
46
Generation of Narrowband FM and PM
/2
NBFM
ACcos(ct)
m (t)
+
kf
-ACsin(ct)
+
/2
NBPM
ACcos(ct)
m (t)
+
kp
-ACsin(ct)
+
Agbo & Sadiku; Figure 4.5 on page 168
47
Modulation Index Parameter in Angle Modulation
Parameter is the modulation index for angle modulation.
is used to differentiate between narrowband angle modulation andwideband angle modulation.
Narrowband angle modulation requires << 1 (Typically < 0.3)Wideband angle modulation requires >> 1 (Typically > 5.0)
Equivalently,
Narrowband angle modulation requires f << BWideband angle modulation requires f >> B
Comments:1. Narrowband FM has about the same bandwidth as that of AM.2. Commercial (broadcast) FM is wideband FM (required due to
its superior noise performance).3. Why even consider narrowband FM? Two reasons:
a. NBFM is easier to generate that WBFM.b. It is commonly used as the first step in generating WBFM.
48
Narrowband FM with Tone Modulation
Let ( ) cos( ), then ; 2 ; and
Then ( ) cos( ) sin( )
The time-domain NBFM signal is
( ) cos( ) sin( ) sin( )
1( ) cos( ) cos (
2
f p f m
m m p m m
m m
t tf m
f f m m m
m
FM C C C m C
FM C C C
k m k Am t A t m A B
k Ak m d k A d t
t A t A t t
t A t A
− −
= = = = =
= =
−
+
( ) ( )
nd rd
1) cos ( )
2
The 2 term is the upper sideband and the 3 term is the lower sideband.
In comparing to AM:
C m C C mt A t + − −
Equation (4.18)
Equation (4.19)
49
( ) ( )1 1
( ) cos( ) cos ( ) cos ( )2 2
FM C C C C m C C mt A t A t A t + + − −
(C + m)C
AC = 0.2
Narrowband FM (NBFM)
(C - m)
Tone modulation cos(mt)
NBPM requires << 1 radian(generally less than 0.3 radian)
2
CA
2
CA−
( )FM t
Sidebands arein quadrature.
50
Narrowband FM (NBFM)
Cm -m
C rotates faster than m
Phasor lengths adjust to keep constant AC.
51
Review: Phasor Interpretation of AM DSB with Carrier
C
us
ls
cos(Ct)
cos(mt)
DSB AM
C
C + mC - m
m = |us| = |ls|
C rotates faster than m
Spectrum:
uppersideband
lowersideband
52
Narrowband FM Example (Example 4.4)
Exercise: The message signal input to a modulator is m(t) = 4cos(2104t)and the carrier is 10cos(108t). If frequency modulation is performed with kf = 1000, verify that the modulated signal meets the criteria of being narrowband FM. Also, obtain an expression for its spectrum and sketch this spectrum.
Solution: First we calculate the modulation index
Using the equation from slide #49:
4
41000 0.2; 0.3 NBFM
2 10
1 110 thus, (0.2)(10) 1
2 2
mf
m
C C
Ak
A A
= = =
= = =
( ) ( )
( ) ( )
1 1( ) cos( ) cos ( ) cos ( )
2 2
( ) 10 cos( ) cos ( ) cos ( )
FM C C C C m C C m
FM C C m C m
t A t A t A t
t t t t
+ + − −
+ + − −
From: Agbo & Sadiku; page 170
53
Narrowband FM Example (Example 4.4 continued)
( ) ( )
8
( ) 10 cos( ) cos ( ) cos ( )
The corresponding expression for the specturm becomes
( ) 10 ( ) ( ) ( ) ( )
( ) ( )
where 10 radians/sec and 2 10
FM C C m C m
FM C C C m C m
C m C m
C m
t t t t
+ + − −
= + + − + + + + − −
− + − + − +
= = 4 radians/sec
C
10
-C
(C + m)(C - m)
-(-C + m)(-C - m)
( )FM
Bandwidth = 2m
0
54
Wideband FM (WBFM)
WBFM requires >> 1 radian (much more complicated)
( )
= +
For wideband FM we have a nonlinear process, with single-tone
modulation:
( ) Re exp sin( )
We need to in order to
analyze ( ). The solut
expand the exponential in a Fourier series
WBFM C C m
WBFM
t A j t j t
t
( )
=−
= +
ion has an expansion in Bessel functions:
( ) ( ) cos 2 ( )
where the coefficients ( ) are Bessel functions.
WBFM C C m
nn
n
t A J f nf t
J
Spectral analysis for tone modulation of WBFM: Agbo & Sadiku, pp. 171-180.We will not cover this section in EE 442 but rather focus upon the physicalinterpretation of FM spectrum spread.
m
f f
f B
= =
Modulation Index
55
Digression: Bessel Functions (of the 1st kind)
Bessel functions have many applications: cylindrical waveguides, vibrationalmodes on circular membrane, FM modulation synthesis, acoustic vibrations, etc.
https://www.cefns.nau.edu/~schulz/Bessel/J02.html
http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html
WBFM (or WBPM) Requires More bandwidth Than AM
56
AM
WBFM
Carrier Signal (frequency fc )
Message Signal (frequency fm)
Amplitude Modulated Signal
Frequency Modulated (FM) Signal
f
f
f
f
fC
fm
A
A
A
A
AC
Am
A
t
t
t
t
A Amplitude Modulated Signal
Frequency Modulated (FM) Signal
Message Signal (frequency fm)
Carrier Signal (frequency fC)
57
= 0.2
= 1.0
= 5
= 10
Number of Sidebands¶
Bandwidth
0.1 2 2 fm
0.3 4 4 fm
0.5 4 4 fm
1.0 6 6 fm
2.0 8 8 fm
5.0 16 16 fm
10.0 28 28 fm
Single-Tone FM Spectra as Function of Modulation Index
= =
m m
f
f
Single-tone Modulation Index
TB or BW
NBFM
WBFM
fm constant
¶Both upper and lowersidebands about fC.
58
Spectra of FM Signals
From A. Bruce Carlson, Communication Systems, An Introduction to Signals and Noise inElectrical Communication, 2nd edition, 1975; Chapter 6, Figure 6.5, Page 229.
= 0.2
= 1.0
= 5
= 10
A A
Single-toneModulation Index
= =
m m
f
f
is constant &
is decreasingm
f
f
increasing &
is constantm
f
f
ff
WBFM
59
Broadcast FM Radio covers from 88 MHz to 108 MHz100 stations – 200 kHz spacing between FM stations
Selecting an FM Station
Service Type Frequency Band
Channel Bandwidth
MaximumDeviation
Highest Audio
Commercial FM Radio Broadcast
88.0 to 108.0 MHz
200 kHz 75 kHz 15 kHz
Note: 0 dBu = 0.775 volt into 600 ohms(which is equivalent to 1 mW powerdelivered into the 600 ohm resistor)
60
Measured Spectrum of an FM Radio Signal
noise
Voice modulation
200 kHz
61
Service Type Frequency Band
Channel Bandwidth
MaximumDeviation
Highest Audio
Commercial FM Radio Broadcast
88.0 to 108.0 MHz
200 kHz 75 kHz 15 kHz
Television Sound (analog)
Digital TV has replaced
4.5 MHz above the
picture carrier frequency
100 kHz 25 kHz monaural &50 kHz stereo
15 kHz
Public safety – Police, Fire, Ambulance, Taxi, Forestry, Utilities, & Transportation
50 MHz and 122 MHz to
174 MHz
20 kHz 5 kHz 3 kHz
Amateur, CE class A & Business band Radio
216 MHz to 470 MHz
15 kHz 3 kHz 3 kHz
Specifications for Some Commercial FM Transmissions
62
The Three Important Parameters in FM and PM
The three important frequencies in FM and PM are
1. Carrier frequency fC (or C)2. Maximum modulation frequency fm (or m), and3. Peak frequency deviation f (or )
Two Definitions of importance:
1. Modulation index
2. Deviation ratio D
Remember: For FM = kf mp & for PM = kp m’p
(can be a very large number)2m m m m
f f
B f B
= = = =
(always much less than unity)C C
fD
f
= =
63
FM Bandwidth and the Modulation Index
A. Narrowband FM (NBFM) – << 1 radian
2 where is the bandwidth of ( )NBFM m mB B B m t
+ = = =
=
+ = +
+
2( 1) , where
is the peak frequency deviation max ( )
2( ) 2( 1)
For PM we have analogous equation,
2( 1
Carson's Ru
)
le
WBFM m
m m m
f
WBFM m m
WBPM m
f fB B
B f
f f k m t
B f B B
B B
B. Wideband FM (WBFM) – >> 1 radian
64
Example: Bandwidth of FM Signal
The message signal input to a modulator is 10cos(2 104t). If frequencymodulation with frequency deviation constant kf = 104 is performed, findthe bandwidth of the resulting FM signal.
Solution:
( )
( ) ( )
4
4
10 1015
2 2 10
2 1 2 5 1 10 kHz 120 kHz,
using to calcuC laarson's rule te bandwidth
f m
m
FM m
FM
k A
f
B f
B
= = =
= + = + =
65
Example: Equal Bandwidth for FM & PM Signals
If phase modulation is performed using the message signal 10cos(2 104t)used in the previous slide, find the phase deviation constant kp giving thePM signal the same bandwidth, namely, 120 kHz.
Solution:
For both the FM and PM signals to have the same bandwidth, and f mustbe the same. For FM, = kf Am ; but for PM, = kp m’p .Expressing the message signal m(t) = Am cos(mt) gives
( )
4
4
'( ) cos( ) sin( ) '
,
10 1
2 10 2
' 1Check: = (10) 5
2
m m m m m p m m
f
f m p m m p
m
p p p m m
p m
m m
dm t A t A t m A
dt
Thus
kk A k A k
k m k Ak A
= = − = −
= − → = = =
= = = =
66
Example: Commercial FM Radio Stations
For commercial FM radio, the audio message signal has a spectral range of30 Hz to 15 kHz, and the FCC allows a frequency deviation of 75 kHz. Estimate the transmission bandwidth for commercial FM using Carson’sRule.
Solution:
( ) ( )
We start by calculating
75 kHz 5
15 kHz
Using Carson's rule gives
2 1 2 5 1 15 kHz 180 kHz
The allowed bandwidth for commercial FM is 200 kHz.
Note that Carson's rul
m
FM m
f
B
B B
= = =
= + = + =
e slightly underestimates the
bandwidth.
67
Why Does FM and PM Take Much More Bandwidth?
Observation: The bandwidth required for AM and NBFM are the same.
However, WBFM (wideband FM) requires much more bandwidth. Why?
A Fourier spectrum of an FM signal shows that to keep the amplitude constant of an FM signal that many components are required to represent the FM waveform. The frequency spectrum of an actual FM signal has components extending infinitely, although their amplitude decreases forsufficiently higher frequencies. Sufficiently higher frequencies appliesto frequencies above the Carson bandwidth rule.
Next we examine the Fourier components this using phasors.
+ = + Carson's Rule2( ) 2( 1)WBFM m mB f B B
Review
Note: Only magnitudes of spectral lines shown.:
FM Tone-Modulated Signal Spectrum
68
For = 2.0
Remember:
69
NBFM Constructed From Phasors in FM Modulation
fCAC
2
CA
2
CA
-fmfm
(t)
AC
2
CA
fC fC + fm
fC - fm
2
CA−
0
NBFM with tone modulation
70
WBFM Phasor Diagram for Arbitrary
AC
carrier
even-order sideband
odd-order sideband
After A. Bruce Carlson and Paul B. Crilly, CommunicationSystems, 5th ed., McGraw-Hill Book Company, New York, NY.
0
71
Sidebands Constructed From Phasors in FM Modulation
Animation showing how phase modulation works in the phasor picture -- phase modulation with a sinusoidal modulation waveform and a modulation depth of π/4 radians. The blue line segments represent the phasors at the carrier and the harmonics of the modulation frequency.
72
Generating FM Signals
There are two basic methods to generate FM:
1. Direct Method (uses voltage-controlled oscillator to vary thethe frequency linearly with the message signal m(t))
Advantage: Can generate large frequency deviation.Disadvantage: Carrier frequency tends to drift and
must be stabilized.
2. Armstrong’s Indirect Method (first generate NBFM with the message signal with a small frequency deviation and thenfrequency multiplication is used to increase the frequencyand frequency deviation to desired levels (generates WBFM)
Advantage: More stable carrier frequency.Disadvantage: More complex hardware and cost.
73
Direct Generation of FM Signal Using a VCO
Ceq is capacitance CD plus capacitance of other capacitors.
VCO is “voltage-controlled oscillator”
1
~osc
eqLC
Q
CD
RFC
m(t)
VCO
Varactor diode
( )FM t
+VCC
LC Resonator
74
Direct Generation of FM Signal Using a VCO and PLL
WBFM Output Signal
FrequencyDivider N
Input: m(t)
VCO
LPF
Crystal Oscillatorfosc = fC / N
Mixer used as phase detector
12
1 12 2
cos( ) cos( ) cos( ) cos( )
cos( ) cos(2 ) cos( ) cos(2 )
+ = − − + + +
= − + + = + +
X X X X X X
X X
t t t t t t
t t
Mixer used as phase detector:
75
Varactor diode
Narrowband FM Generated by Pulling a Crystal Oscillator
Q1
CD
m(t)
( )FM t
+VCC
Xtal
R4R2
R3
A crystal filter is placed in the feedback loop to stabilize the oscillator.The frequency of oscillation can be pulled slightly from the high-Q crystal resonator’s frequency. The frequency deviates only slightly andis typically only up to about 100 ppm. However, the oscillator is very stable for m(t) = 0.
A crystal is anelectro-mechanical
resonator.
76
Digression: Q-Values for Quartz Crystals in Electronics
A crystal oscillator is an electronic oscillator circuit that uses the mechanical
resonance of a vibrating crystal of piezoelectric material to create an electrical signal with a precise frequency.
A major reason for the wide use of crystal oscillators is their high Q factor. A
typical Q value for a quartz oscillator ranges from 104 to 107, compared to
perhaps 102 for an LC oscillator.
The maximum Q for a high stability quartz oscillator can be estimated
as Q = 1.6 × 107/f, where f is the resonant frequency in megahertz.
resfQ
f=
Xtal
https://en.wikipedia.org/wiki/Crystal_oscillator
https://txccrystal.com/term.html
77
Generation of Narrowband Frequency Modulation (NBFM)
NBFM is limited to << 1 radian
Agbo & SadikuFigure 4.5; page 168
/2
NBFM
ACcos(ct)
m(t)
kf
-ACsin(ct)
DSB-SCmodulator
Carrier
+
+
Oscillator
( )( ) cos ( ) sin( )t
FM C C C f Ct A t A k m d t −
= −
78
Generation of Narrowband Phase Modulation (NBPM)
( )( ) cos ( ) sin( )FM C C C p Ct A t A k m t t = −
Agbo & SadikuFigure 4.5; page 168
/2
NBPM
ACcos(ct)
m(t)
kf
-ACsin(ct)
DSB-SCmodulator
Carrier
+
+
Oscillator
79
Indirect Generation of FM Using Frequency Multiplication
( )FMNB t ( )FM
WB t
NBFMFrequencyMultiplier
( )m t
In this method, a narrowband frequency-modulatedsignal is first generated and then a frequency multiplier is used to increase the modulation index.The concept is shown below:
A frequency multiplier is used to increase both thecarrier frequency and the modulation index by integer N.
80
Frequency Multipliers
A frequency multiplier is a nonlinear component followed by a bandpassfilter at the multiplied frequency desired.
0
0
( ) cos ( ) , and
( ) cos ( )
t
in C C f
t
out C C f
t A t k m d
t A n t nk m d
= +
= +
We select the nth order nonlinear component of y(t) and pass it throughthe bandpass filter.
Nonlinear Device
Bandpass Filter @ nC
( )out t( )in t ( )y t
Conclusion: Carrier frequency is now nfC and frequency deviation is now nf.Commercial frequency multipliers are generally 2 and 3.
Note: m(t) isnot distortedby multiplier.
Section 4.4; Page 181 ofAgbo & Sadiku
81
Armstrong Indirect FM Transmitter Example
These numbers correspondto an FM broadcast radio station.
( )FMNB t
( )FMWB t
( )m tNBFM
generation
64Multiplier
BPF
Crystal Oscillator
1
1
200 kHz
25HzCf
f
=
=2
2
12.8 MHz
1.6 kHzCf
f
=
=
3
3
1.9 MHz
1.6 kHzCf
f
=
=
48Multiplier
4
4
91.2MHz
76.8 kHzCf
f
=
=
PA
A mixerdoes not
change f
Crystal stabilizedvoltage-controlled
oscillator
82
Why are Two Multiplication Chains Used?
NBFM
generatorMultiplierChain A
MultiplierChain B
Mixer ( )FMWB t
( )FMNB t
Oscillator
83
Many Ways to Perform Frequency Multiplication
In electronics, a frequency multiplier is an electronic circuit that
generates an output signal whose output frequency is a harmonic
(multiple) of its input frequency. Frequency multipliers consist of a
nonlinear circuit that distorts the input signal and consequently
generates harmonics of the input signal.
Most multipliers are doublers or triplers
XOR
84
Frequency Multiplication Using Comb Generation
From our discussion on Fourier series and pulse trains:
Tp
T
Amplitude
Comb frequencies shown
1
pT
2
pT1
T
85
Simple Comb Generator
A step recovery diode (SRD) is a p-n junction diode having the ability to generate
extremely short pulses. It is also called snap-off diode or charge-storage diode,
and has a variety of uses in microwave electronics (e.g., pulse generator or
parametric amplifier).
Comb Generator
Circuit
https://www.edn.com/electronics-blogs/the-emc-blog/4402169/DIY-6-GHz-comb-generator
86
Step Recovery Diode Based Comb Generation
http://www.mwrf.com/analog-semiconductors/designing-step-recovery-diode-based-comb-generator
The key to generating a wide comb of frequencies is togenerate very narrow pulses which step recovery diodesare designed to do.
Time (nanoseconds)
Vo
lts
87
Generation of Narrowband Phase Modulation (NBPM)
= +( ) cos( ( ))PM C C pt A t k m t
/2
NBPM
ACcos(ct)
m (t)kp
-ACsin(ct)
+
+
Agbo & SadikuFigure 4.5; page 168
88
Generation of Narrow Band Phase Modulation
CD
m(t)
+VD
Varactor diode
kp
Carrierfrequency fC
( )PM t
https://www.slideshare.net/sghunio/chapter06-fm-circuits
Limitation 1: Only a small amount of phase shift is generated (low-deviation)Limitation 2: All phase-shift circuits produce amplitude variations.
Advantages of frequency modulation
1. Resilient to noise: The main advantage of frequency modulation is a reduction in noise. As most noise is amplitude based, this can be removed by running the received signal through a limiter so that only frequency variations remain.
2. Resilient to signal strength variations: In the same way that amplitude noise can be removed, so too can signal variations due to channel degradation because it does not suffer from amplitude variations as the signal level varies. This makes FM ideal for use in mobile applications where signal levels constantly vary.
3. Does not require linear amplifiers in the transmitter: As only frequency changes contain the information carried, amplifiers in the transmitter need not be linear.
4. Enables greater efficiency : The use of non-linear amplifiers (e.g., class C and class D/E amplifiers) means that transmitter efficiency levels can be higher. This results from linear amplifiers being inherently inefficient.
89
Advantages of FM
Disadvantages of frequency modulation
1. Requires a more complicated demodulator: One of the disadvantages is that the demodulator is a more complicated, and hence more expensive than the very simple diode detectors used in AM.
2. Sidebands extend to infinity: The sidebands for an FM transmission theoretically extend out to infinity. To limit the bandwidth of the transmission, filters are used, and these introduce some distortion of the signal.
90
Disadvantages of FM
91
Ideal FM Differentiator Demodulator
The ideal FM detector converts the FM signal‘s instantaneous frequency i
to an amplitude that is proportional to i.
( )
( )
−
−
−
= + = +
= +
= − + +
Input: ( ) cos ( ) cos ( )
( )Output: cos ( )
( )( ) sin ( )
t
FM C C C C f
t
FMC C f
t
FMC C f C f
t A t t A t k m d
d t dA t k m d
dt dt
d tA k m t t k m d
dt
Differentiation performs FM to AM conversion
EnvelopeDetector
d
dt
Limiter Differentiator
( )FM t
( )FMd t
dt ( )( )C C C fA A k m t +
After DC removal
Both AM and FM included
AM allows theenvelope detector
to be used
92
Bandpass Limiter at the Receiver
For an envelope detector to work well the FM signal’s amplitudeshould be constant or flat. We can accomplish with a “hard limiter.”Factors such as channel noise, interference and channel fading resultin amplitude variations in an FM signal’s amplitude at the receiver.
Band-passFilter@ C
Limiter
( )FM t
t →
Output ofLimiter
( )FM tInput
t→
VL
ConstantAmplitude Output
( )4
( ) ( ) cos ( )t
FM C fx t t t k t m d −
= = +
From Leon W. Couch, II, Digital and Analog Communication Systems, 8th edition, 2013; Figure 4-7 (page 265).
93
Practical FM Differentiator Demodulator
( )( )
= = =+ +
=
3
3
3
3
Multiplica
s
tion by in
/ 1( ) ; where
1 1 /
1F
s l
or << ; t
hen ( )
The high-pass filter
the frequency domain i equiva ent
to differentiation in t
a
ime domai acts n!
dB
dB
dB
dB
jj RCH j
j RC j RC
H j j C
j
RRC
+
a differentiator for an FM signal. Therefore,
y(t) = ( )C C C C fA RC A RCk m t
RC+
_x(t) Envelope
Detector
+
_y(t)Differentiator
at low frequencies
Envelope detector extracts m(t)
94
Bode Plot of CR High-Pass Filter
Log
amp
litu
de
Normalized frequency
45
3dBA
mp
litu
de
(dB
)P
has
e (d
egre
es)
95
Practical Frequency Demodulators
Frequency discriminators can be built in various ways:
• Time-delay demodulator (uses differentiation)
• FM slope detector (FM to AM conversion)
• Balanced discriminator
• Quadrature demodulators
• Phase locked loops (a superior technique)
• Zero crossing detector
96
Time Delay Demodulator
This is an implementation of discrete time approximation to differentiation.
( )
( )0 0
1( ) ( ) ( )
( ) 1lim ( ) lim ( ) ( )
FM FM
FMFM FM
y t t t
d ty t t t
dt
→ →
= − −
= = − −
It can be shown that an adequate value for is less than T/4, where T is theperiod of the unmodulated carrier for the FM signal. Again, this relies uponFM to AM conversion after which the envelope detector recovers m(t).
Amplifier(Gain = 1/)
EnvelopeDetector
yd(t)( )FM t
Time Delay Time-Delay Differentiator
+ y(t)
97
An FM Slope Detector Performs FM to AM Conversion
( )FM t
Slope sets frequency toamplitude conversion scale
EnvelopeDetector Comment: The
differentiationoperation is performed by anycircuit acting as a frequency-to-amplitude converter.
Slopeapproximation
x(t) y(t)
Operates on theskirt of the LCresonance curve
98
Balanced Discriminator (Foster-Seeley Discriminator) – 1936
( )FM t
Centered around fc
•••
Transfer Characteristic
f
Two tuned circuits
Envelopedetectors
Another example of the use of symmetry in design.
km(t)
0U Cf f
0L Cf f
0Uf
0Lf
99
Quadrature Demodulator – Block Diagram
FM signal is converted into PM signal
PM signal is used to recover the message signal m(t)
Phase Shifting Circuit
Low-Pass Filter
Phase Comparator
Circuit
FM ( )t
( )m t
Phase DetectorSignal delay 0 timescarrier frequency fC
= 90 degrees (or /2).
100
Using a XOR Gate for Phase-Frequency Detector
XOR
A B Output
0 0 0
0 1 1
1 0 1
1 1 0
Exclusive OR gate
A
B
Logic Table
101
Quadrature Demodulator – Implementation
The signal is split into two components. One passes through a network
providing a basic 90° phase shift in addition to the phase shift from the
signal’s frequency deviation. The mixer output is dependent upon the phase difference between the two signals; that is, it acts as a phase detector producing a voltage output proportional to the phase difference and thus the frequency deviation on the FM signal.
http://www.radio-electronics.com/info/rf-technology-design/fm-reception/fm-quadrature-detector-demodulator.php
C1
C2
R L
Low-passFilter
IncomingFM signal Demodulated
output
Phase Detector
102
Phase-Locked Loops (Using Feedback)
A PLL consists of three basic components:❑ Phase detector❑ Loop filter❑ Voltage-controlled oscillator (VCO)
PLL Diagram:
cos( ( ))C C iA t t +
2 cos( ( ))VCO C oA t t +( )oe t
( )H s
Phase Detector
Output signal isphase difference
Low-PassFilter
Oscillator(VCO)
Bias Generator
e
103
Zero-Crossing Detectors
An example is shown on the next slide.
104
Zero-Crossing Detector Illustration
https://www.slideshare.net/avocado1111/angle-modulation-35636989
Averagingcircuit
m(t)( )FM tMulti-
vibrator
Zero-crossingcircuit
Hard Limiter
More frequentZC’s gives higherinstantaneousfrequency whichcauses greateraverage signal.
m(t)
105
Noise in Frequency Modulation
In FM systems noise has a greater effect on the higher modulating frequencies. It is common practice to boost the signal level of the higher modulating frequencies to improve the signal-to-noise ratio of the overall transmitted FM signal.
This artificial boosting at the transmitter is called “pre-emphasis” and the removal of the boost at the receiver is called “de-emphasis.”
The result is an improvement in the discernible quality of received FM signals.
Power Spectral Density (PSD) of output noise in an FM receiver.(Increases because noise is differentiated in FM receiver)
( )2
2( ) 2 for
o
T
N O
BfS f N f
fB-B
0
( )oNS f
-f - 2B f +2B
Fig. 10.9; p. 578 of 4th ed., Lathi & Ding
Differentiator Demodulator (Slides 91 & 93)
106
Pre-Emphasis and De-Emphasis in FMIn
terf
eren
ce
Frequency f
PM
FM
FM with Pre- and De-emphasis filters
Channel noise acts as interference inFM and is uniform over the entire BW.Voice and music have more energy atlower frequencies, so we need to “emphasize “their upper frequencies by filtering. However, the HF emphasis must be removed at the receiver usinga “de-emphasis” filter.
Pre-emphasis Filter
De-emphasis Filter
FM Transmitter
FM Receiver
AWG Noise
Channel
m(t)
R1
R2C
R1 C
Filtering improves SNR in FM transmission.
(Widely used commercially in the recording industry)
107
Typical Pre-Emphasis and De-Emphasis Filters
Transmitter Receiver
Pre-emphasis Filter De-emphasis Filter
1
1( )
1out
in
VH
V j R C
= =
+( )1
1 2
1( )
1out
in
j R CVH
V j R R C
+= =
+
1
1
R C
( ) ( )H dB
log( )
-6 dB/octave
1
1
R C ( )1 2
1
R R C
( ) ( )H dB
log( )
+6 dB/octave
2.1 kHz 2.1 kHz33 kHz
R1
R2C
R1 C
( )1 2
1
R R C
108
Analog and Digital FM Cellular Telephones
1G analog cellular telephone (1983) – AMPS (Advanced Mobile Phone Service)First use of cellular concept . . . Used 30 kHz channel spacing (but voice BW was B = 3 kHz)
Peak frequency deviation f = 12 kHz, and BT = 2(f + B) = 2(12 kHz + 3 kHz) = 30 kHz
Two channels (30 kHz each); one for uplink and one for downlinkUsed FM for voice and FSK (next slide) for data communicationNo protection from eavesdroppers!
Successor to AMPS was GSM (Global System for Mobile) in early 1990sGSM is 2G cellular telephone Still used by nearly 50% of world’s population (as of 2017)GSM was a digital communication system
Modulating signal is a bit stream representing voice signalUses Gaussian Minimum Shift Keying (GMSK)Channel bandwidth is 200 kHz (simultaneously shared by 32 users
This is 4.8 times improvement over AMPS
More to come on cellular . . .
Digital Carrier Modulation – ASK, FSK and PSK
109
AmplitudeShift Keying
FrequencyShift Keying
PhaseShift Keying
Dig
ital
Sig
nal
s
https://slideplayer.com/slide/12711804/
( ) ~ ( , )p dk m t −
m(t)
110
Digital Phase Shift Modulation
Binary Phase Shift Keying (BPSK)
The wave shape is ‘symmetrical’ at each phase transition because the bit rate is a sub-multiple of the carrier frequency ωC/(2π). In addition, the message transitions are timed to occur at the zero-crossings of the carrier.
( ) ~ ( , )p dk m t −
111
Questions?
https://www.tutorialspoint.com/principles_of_communication/principles_of_communication_modulation.htm
112
Additional slides
113-1 -1
Switch
Integrator
m(t)
Schmitt Trigger
Inverter
FM(t)
time
Triangular-Wave FM Generation
vin
vout
vinvout
Able to generateFM up to 30 MHz
After A. Bruce Carlson and Paul B. Crilly, CommunicationSystems, 5th ed., McGraw-Hill Book Company, New York, NY.
Schmitt Trigger characteristic
A
114
Switching-Circuit Phase Modulator
m(t) Sawtooth wave
Comparator Output
Flip-Flop Output
BPF+ Flip-
Flop
Comparatorm(t)
Sawtooth Generator
Carrier Oscillator
PM(t)
time
And filtering removes harmonics. After A. Bruce Carlson and Paul B. Crilly, Communication
Systems, 5th ed., McGraw-Hill Book Company, New York, NY.
115
FM System Improvement in SNR
The signal-to-noise ratio (SNR) improvement in an FM system is afunction of modulation index ,
Example: For FM transmission bandwidth BT of 200 kHz and a messagebandwidth Bm of 15 kHz ( = 5.67), the improvement in the SNR at theoutput of an FM receiver to have an FM gain of 27 dB above the CNR.
This is essentially a tradeoff between message signal quality (SNR) andFM transmission bandwidth. Thus, greater transmission bandwidth is the key to FM’s superior performance.
33 ( 1) , where is carrier-to-noise ratio
32
FM
TFM
m
SNR CNR CNR
BSNR CNR
B
= +
=
116
PSD of AWG Noise Through Differentiator Network
PSD of input noise (Uniform White Noise):
The transfer function of a differentiator is given by
The PSD of the output noise is calculated by
( )iNS f K=
( ) 2H f j f=
2 2
22 3
( ) ( ) ( ) 2 ,
Therefore,
82 watts
3
o iN N
B
o
B
S f H f S f j f K
KN j f K df B
−
= =
= =
d
dt
( )iN t ( )oN t
( )NoS f( )Ni
S f
K
BB− f BB− f
Therefore,
117
FM System As Special Case of PM System
( )cos ( )C C fA t k m d +
( )n t
O O,S N
Phase
Modulator
PM Receiver
d
dt
FM Modulator FM Demodulator
( )m d
( )m t
After: Lathi & Ding, 4th ed.; page 577
( )
22 2 2
O O
2
O
23 where22
Cf C
Ak mS A N
N B BB
= =
NN
118
Phase Modulator Circuit
Limitation 1: Only a small amount of phase shift is generated (low-deviation)Limitation 2: All phase-shift circuits produce amplitude variations.
CDm(t)
Varactor diode
Carrierfrequency fC
( )PM t
Co L
+
_
