Modulation - Demodulation - Electrical Engineering (EE)

 Page 1


Modulation and Demodulation of Analog Signals
Martin Kumm
March 5, 2009
Contents
1 Introduction 2
2 Mathematical Representation of Modulated Signals 2
3 Modulation 3
3.1 Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 Single Sideband Modulation . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 Phase and Frequency Modulation . . . . . . . . . . . . . . . . . . . . 5
4 Demodulation 5
4.1 Amplitude Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.2 Single Sideband Demodulation . . . . . . . . . . . . . . . . . . . . . . 8
4.3 Phase and Frequency Demodulation . . . . . . . . . . . . . . . . . . . 11
References 11
1
Page 2


Modulation and Demodulation of Analog Signals
Martin Kumm
March 5, 2009
Contents
1 Introduction 2
2 Mathematical Representation of Modulated Signals 2
3 Modulation 3
3.1 Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 Single Sideband Modulation . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 Phase and Frequency Modulation . . . . . . . . . . . . . . . . . . . . 5
4 Demodulation 5
4.1 Amplitude Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.2 Single Sideband Demodulation . . . . . . . . . . . . . . . . . . . . . . 8
4.3 Phase and Frequency Demodulation . . . . . . . . . . . . . . . . . . . 11
References 11
1
1 Introduction
The motivation for writing this article was that I started implementing blocks for
demodulating RF signals on Field Programmable Gate Arrays (FPGAs). During that
I was asking me what are the best methods and structures for doing that. Most of the
text books dealing with the topic of receivers either focussing on practical concerns
of analog receivers or have a mathematical point of view that is hard to realize in
hardware. So I started some calculations that are summarized in this document.
2 Mathematical Representation of Modulated Signals
An unmodulated radio frequency (RF) signal can be described as cosine waveform
oscillation at an RF frequency f
RF
= !
RF
=(2) with a peak amplitude ^ a and an
initial phase 
0
1
:
s
RF
(t) = ^ a cos(!
RF
t +
0
) (1)
This RF waveform can be independently modulated in amplitude and phase (the
argument of the cosine), leading to the representation
s
RF
(t) = ^ a(t) cos ['(t)] : (2)
A further representation that is very usefull can be found by extending the real
signal with an imaginary part which is its own Hilbert Transform (e.g. see
[Kam04],[KK06],[OS99]) leading to the so-called analytic signal
s
RF
(t) =s
RF
(t) +jHfs
RF
(t)g : (3)
In the case of a cosine function, the Hilbert Transform is a sine function
s
RF
(t) = ^ a(t)

cos ['(t)] +j sin ['(t)]
	
= ^ a(t)e
j'(t)
: (4)
The analytic signal is marked with an underline in the following. The real RF wave-
form can be obtained by simply taking the real part of the analytic signal
s
RF
(t) =Refs
RF
(t)g : (5)
The amplitude and phase components for common analog modulation modes are
summarized in Table 1. The low frequency modulation signal is called s
LF
and is
normalized to an absolute peak value of one. The signals s
+
LF
(t) and s

LF
(t) are the
upper and lower sidebands of the analytic representation of the modulation signal.
The upper one is the natural representation of an analytic signal
s
+
LF
(t) =s
LF
(t) =s
LF
(t) +jHfs
LF
(t)g (6)
1
A representation using the sine function can be made by simply changing the denition of 
0
.
Using the cosine function eases the converting between real an complex signals.
2
Page 3


Modulation and Demodulation of Analog Signals
Martin Kumm
March 5, 2009
Contents
1 Introduction 2
2 Mathematical Representation of Modulated Signals 2
3 Modulation 3
3.1 Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 Single Sideband Modulation . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 Phase and Frequency Modulation . . . . . . . . . . . . . . . . . . . . 5
4 Demodulation 5
4.1 Amplitude Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.2 Single Sideband Demodulation . . . . . . . . . . . . . . . . . . . . . . 8
4.3 Phase and Frequency Demodulation . . . . . . . . . . . . . . . . . . . 11
References 11
1
1 Introduction
The motivation for writing this article was that I started implementing blocks for
demodulating RF signals on Field Programmable Gate Arrays (FPGAs). During that
I was asking me what are the best methods and structures for doing that. Most of the
text books dealing with the topic of receivers either focussing on practical concerns
of analog receivers or have a mathematical point of view that is hard to realize in
hardware. So I started some calculations that are summarized in this document.
2 Mathematical Representation of Modulated Signals
An unmodulated radio frequency (RF) signal can be described as cosine waveform
oscillation at an RF frequency f
RF
= !
RF
=(2) with a peak amplitude ^ a and an
initial phase 
0
1
:
s
RF
(t) = ^ a cos(!
RF
t +
0
) (1)
This RF waveform can be independently modulated in amplitude and phase (the
argument of the cosine), leading to the representation
s
RF
(t) = ^ a(t) cos ['(t)] : (2)
A further representation that is very usefull can be found by extending the real
signal with an imaginary part which is its own Hilbert Transform (e.g. see
[Kam04],[KK06],[OS99]) leading to the so-called analytic signal
s
RF
(t) =s
RF
(t) +jHfs
RF
(t)g : (3)
In the case of a cosine function, the Hilbert Transform is a sine function
s
RF
(t) = ^ a(t)

cos ['(t)] +j sin ['(t)]
	
= ^ a(t)e
j'(t)
: (4)
The analytic signal is marked with an underline in the following. The real RF wave-
form can be obtained by simply taking the real part of the analytic signal
s
RF
(t) =Refs
RF
(t)g : (5)
The amplitude and phase components for common analog modulation modes are
summarized in Table 1. The low frequency modulation signal is called s
LF
and is
normalized to an absolute peak value of one. The signals s
+
LF
(t) and s

LF
(t) are the
upper and lower sidebands of the analytic representation of the modulation signal.
The upper one is the natural representation of an analytic signal
s
+
LF
(t) =s
LF
(t) =s
LF
(t) +jHfs
LF
(t)g (6)
1
A representation using the sine function can be made by simply changing the denition of 
0
.
Using the cosine function eases the converting between real an complex signals.
2
as it has only signal components for positive frequencies. The lower sideband is the
complex conjugate of the analytic signal:
s

LF
(t) =s

LF
(t) =s
LF
(t)jHfs
LF
(t)g (7)
Furthermore, the modulation modes are characterized by the parameters a
0
, which
is the peak RF amplitude, the modulation index m (m =f0:::1g), the peak phase
variation 
 and the peak frequency variation 
!. It should be noticed that the AM
mode only in
uences the amplitude and PM and FM only in
uences the phase while
the single sideband modulation modes USB and LSB in
uences both magnitudes
[Kam04].
3 Modulation
In this section, the modulation and demodulation schemes are shown for continuous
signals and ideal components (e.g. lters, oscillators, etc.).
3.1 Amplitude Modulation
The structure of an amplitude modulator is shown in Figure 1. It is the direct
realization of the formula for amplitude modulation in Table 1. The corresponding
spectra of intermediate signals are shown in Figure 2.
For a better understanding, the intermediate signals of Figure 1 are calculated for
a modulating sinwave signal
s
LF
(t) = cos(!
LF
t +
LF
) : (8)
s
RF
(t) = cos(!
RF
t)

1
2
+
m
2
cos(!
LF
t +
LF
)

=
1
2
cos(!
RF
t) +
m
4
cos((!
RF
+!
LF
)t +
LF
) +
m
4
cos((!
RF
!
LF
)t
LF
) (9)
Modulation Mode ^ a(t) '(t)
Amplitude Modulation (AM) a
0
1
2
[1 +ms
LF
(t)] !
RF
t +
0
Upper Sideband Modulation (USB) a
0
m s
+
LF
(t) !
RF
t +
0
Lower Sideband Modulation (LSB) a
0
m s

LF
(t) !
RF
t +
0
Phase Modulation (PM) a
0
!
RF
t + 
s
LF
(t) +
0
Frequency Modulation (FM) a
0
[!
RF

!s
LF
(t)]t +
0
Table 1: Modulation Modes
3
Page 4


Modulation and Demodulation of Analog Signals
Martin Kumm
March 5, 2009
Contents
1 Introduction 2
2 Mathematical Representation of Modulated Signals 2
3 Modulation 3
3.1 Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 Single Sideband Modulation . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 Phase and Frequency Modulation . . . . . . . . . . . . . . . . . . . . 5
4 Demodulation 5
4.1 Amplitude Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.2 Single Sideband Demodulation . . . . . . . . . . . . . . . . . . . . . . 8
4.3 Phase and Frequency Demodulation . . . . . . . . . . . . . . . . . . . 11
References 11
1
1 Introduction
The motivation for writing this article was that I started implementing blocks for
demodulating RF signals on Field Programmable Gate Arrays (FPGAs). During that
I was asking me what are the best methods and structures for doing that. Most of the
text books dealing with the topic of receivers either focussing on practical concerns
of analog receivers or have a mathematical point of view that is hard to realize in
hardware. So I started some calculations that are summarized in this document.
2 Mathematical Representation of Modulated Signals
An unmodulated radio frequency (RF) signal can be described as cosine waveform
oscillation at an RF frequency f
RF
= !
RF
=(2) with a peak amplitude ^ a and an
initial phase 
0
1
:
s
RF
(t) = ^ a cos(!
RF
t +
0
) (1)
This RF waveform can be independently modulated in amplitude and phase (the
argument of the cosine), leading to the representation
s
RF
(t) = ^ a(t) cos ['(t)] : (2)
A further representation that is very usefull can be found by extending the real
signal with an imaginary part which is its own Hilbert Transform (e.g. see
[Kam04],[KK06],[OS99]) leading to the so-called analytic signal
s
RF
(t) =s
RF
(t) +jHfs
RF
(t)g : (3)
In the case of a cosine function, the Hilbert Transform is a sine function
s
RF
(t) = ^ a(t)

cos ['(t)] +j sin ['(t)]
	
= ^ a(t)e
j'(t)
: (4)
The analytic signal is marked with an underline in the following. The real RF wave-
form can be obtained by simply taking the real part of the analytic signal
s
RF
(t) =Refs
RF
(t)g : (5)
The amplitude and phase components for common analog modulation modes are
summarized in Table 1. The low frequency modulation signal is called s
LF
and is
normalized to an absolute peak value of one. The signals s
+
LF
(t) and s

LF
(t) are the
upper and lower sidebands of the analytic representation of the modulation signal.
The upper one is the natural representation of an analytic signal
s
+
LF
(t) =s
LF
(t) =s
LF
(t) +jHfs
LF
(t)g (6)
1
A representation using the sine function can be made by simply changing the denition of 
0
.
Using the cosine function eases the converting between real an complex signals.
2
as it has only signal components for positive frequencies. The lower sideband is the
complex conjugate of the analytic signal:
s

LF
(t) =s

LF
(t) =s
LF
(t)jHfs
LF
(t)g (7)
Furthermore, the modulation modes are characterized by the parameters a
0
, which
is the peak RF amplitude, the modulation index m (m =f0:::1g), the peak phase
variation 
 and the peak frequency variation 
!. It should be noticed that the AM
mode only in
uences the amplitude and PM and FM only in
uences the phase while
the single sideband modulation modes USB and LSB in
uences both magnitudes
[Kam04].
3 Modulation
In this section, the modulation and demodulation schemes are shown for continuous
signals and ideal components (e.g. lters, oscillators, etc.).
3.1 Amplitude Modulation
The structure of an amplitude modulator is shown in Figure 1. It is the direct
realization of the formula for amplitude modulation in Table 1. The corresponding
spectra of intermediate signals are shown in Figure 2.
For a better understanding, the intermediate signals of Figure 1 are calculated for
a modulating sinwave signal
s
LF
(t) = cos(!
LF
t +
LF
) : (8)
s
RF
(t) = cos(!
RF
t)

1
2
+
m
2
cos(!
LF
t +
LF
)

=
1
2
cos(!
RF
t) +
m
4
cos((!
RF
+!
LF
)t +
LF
) +
m
4
cos((!
RF
!
LF
)t
LF
) (9)
Modulation Mode ^ a(t) '(t)
Amplitude Modulation (AM) a
0
1
2
[1 +ms
LF
(t)] !
RF
t +
0
Upper Sideband Modulation (USB) a
0
m s
+
LF
(t) !
RF
t +
0
Lower Sideband Modulation (LSB) a
0
m s

LF
(t) !
RF
t +
0
Phase Modulation (PM) a
0
!
RF
t + 
s
LF
(t) +
0
Frequency Modulation (FM) a
0
[!
RF

!s
LF
(t)]t +
0
Table 1: Modulation Modes
3
Figure 1: Amplitude modulation structure
Figure 2: Spectra of intermediate signals for amplitude modulation
3.2 Single Sideband Modulation
As Table 1 shows, the single sideband modulation can be realized by taking the
Hilbert Transform of the modulation signal followed by a complex multiplication
with the complex RF generator signal. This structure is shown in Figure 3. The
upper sideband (USB) can be selected by a plus sign and the lower sideband (LSB)
by the minus sign.
The spectra of the intermediate signals of Figure 3 are shown in Figure 4. The
output is a complex signal { but we are only interested in the real part of that signal.
This simplies the complex multiplication as the real part of the multiplication result
Figure 3: SSB modulation structure
4
Page 5


Modulation and Demodulation of Analog Signals
Martin Kumm
March 5, 2009
Contents
1 Introduction 2
2 Mathematical Representation of Modulated Signals 2
3 Modulation 3
3.1 Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 Single Sideband Modulation . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 Phase and Frequency Modulation . . . . . . . . . . . . . . . . . . . . 5
4 Demodulation 5
4.1 Amplitude Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.2 Single Sideband Demodulation . . . . . . . . . . . . . . . . . . . . . . 8
4.3 Phase and Frequency Demodulation . . . . . . . . . . . . . . . . . . . 11
References 11
1
1 Introduction
The motivation for writing this article was that I started implementing blocks for
demodulating RF signals on Field Programmable Gate Arrays (FPGAs). During that
I was asking me what are the best methods and structures for doing that. Most of the
text books dealing with the topic of receivers either focussing on practical concerns
of analog receivers or have a mathematical point of view that is hard to realize in
hardware. So I started some calculations that are summarized in this document.
2 Mathematical Representation of Modulated Signals
An unmodulated radio frequency (RF) signal can be described as cosine waveform
oscillation at an RF frequency f
RF
= !
RF
=(2) with a peak amplitude ^ a and an
initial phase 
0
1
:
s
RF
(t) = ^ a cos(!
RF
t +
0
) (1)
This RF waveform can be independently modulated in amplitude and phase (the
argument of the cosine), leading to the representation
s
RF
(t) = ^ a(t) cos ['(t)] : (2)
A further representation that is very usefull can be found by extending the real
signal with an imaginary part which is its own Hilbert Transform (e.g. see
[Kam04],[KK06],[OS99]) leading to the so-called analytic signal
s
RF
(t) =s
RF
(t) +jHfs
RF
(t)g : (3)
In the case of a cosine function, the Hilbert Transform is a sine function
s
RF
(t) = ^ a(t)

cos ['(t)] +j sin ['(t)]
	
= ^ a(t)e
j'(t)
: (4)
The analytic signal is marked with an underline in the following. The real RF wave-
form can be obtained by simply taking the real part of the analytic signal
s
RF
(t) =Refs
RF
(t)g : (5)
The amplitude and phase components for common analog modulation modes are
summarized in Table 1. The low frequency modulation signal is called s
LF
and is
normalized to an absolute peak value of one. The signals s
+
LF
(t) and s

LF
(t) are the
upper and lower sidebands of the analytic representation of the modulation signal.
The upper one is the natural representation of an analytic signal
s
+
LF
(t) =s
LF
(t) =s
LF
(t) +jHfs
LF
(t)g (6)
1
A representation using the sine function can be made by simply changing the denition of 
0
.
Using the cosine function eases the converting between real an complex signals.
2
as it has only signal components for positive frequencies. The lower sideband is the
complex conjugate of the analytic signal:
s

LF
(t) =s

LF
(t) =s
LF
(t)jHfs
LF
(t)g (7)
Furthermore, the modulation modes are characterized by the parameters a
0
, which
is the peak RF amplitude, the modulation index m (m =f0:::1g), the peak phase
variation 
 and the peak frequency variation 
!. It should be noticed that the AM
mode only in
uences the amplitude and PM and FM only in
uences the phase while
the single sideband modulation modes USB and LSB in
uences both magnitudes
[Kam04].
3 Modulation
In this section, the modulation and demodulation schemes are shown for continuous
signals and ideal components (e.g. lters, oscillators, etc.).
3.1 Amplitude Modulation
The structure of an amplitude modulator is shown in Figure 1. It is the direct
realization of the formula for amplitude modulation in Table 1. The corresponding
spectra of intermediate signals are shown in Figure 2.
For a better understanding, the intermediate signals of Figure 1 are calculated for
a modulating sinwave signal
s
LF
(t) = cos(!
LF
t +
LF
) : (8)
s
RF
(t) = cos(!
RF
t)

1
2
+
m
2
cos(!
LF
t +
LF
)

=
1
2
cos(!
RF
t) +
m
4
cos((!
RF
+!
LF
)t +
LF
) +
m
4
cos((!
RF
!
LF
)t
LF
) (9)
Modulation Mode ^ a(t) '(t)
Amplitude Modulation (AM) a
0
1
2
[1 +ms
LF
(t)] !
RF
t +
0
Upper Sideband Modulation (USB) a
0
m s
+
LF
(t) !
RF
t +
0
Lower Sideband Modulation (LSB) a
0
m s

LF
(t) !
RF
t +
0
Phase Modulation (PM) a
0
!
RF
t + 
s
LF
(t) +
0
Frequency Modulation (FM) a
0
[!
RF

!s
LF
(t)]t +
0
Table 1: Modulation Modes
3
Figure 1: Amplitude modulation structure
Figure 2: Spectra of intermediate signals for amplitude modulation
3.2 Single Sideband Modulation
As Table 1 shows, the single sideband modulation can be realized by taking the
Hilbert Transform of the modulation signal followed by a complex multiplication
with the complex RF generator signal. This structure is shown in Figure 3. The
upper sideband (USB) can be selected by a plus sign and the lower sideband (LSB)
by the minus sign.
The spectra of the intermediate signals of Figure 3 are shown in Figure 4. The
output is a complex signal { but we are only interested in the real part of that signal.
This simplies the complex multiplication as the real part of the multiplication result
Figure 3: SSB modulation structure
4
can be realized by two real multiplications and one real subtraction:
Ref(a
r
+ja
i
)
 (b
r
+jb
i
)g =Refa
r
b
r
a
i
b
i
+j(a
i
b
r
+a
r
b
i
)g =a
r
b
r
a
i
b
i
(10)
This is shown in Figure 5. The upper sideband (USB) is now selected by a minus
sign and the lower sideband (LSB) by the plus sign.
For a better understanding, the intermediate signals of Figure 5 are calculated for
a modulating sinwave signal
s
LF
(t) = cos(!
LF
t +
LF
) : (11)
The Hilbert Transform of the input signal is
Hfs
LF
(t)g = sin(!
LF
t +
LF
) : (12)
Multiplying the input signal with the cosine RF waveform results in
s
LF
(t)
 cos(!
RF
t) =
1
2
cos((!
RF
+!
LF
)t +
LF
) +
1
2
cos((!
RF
!
LF
)t
LF
) : (13)
Multiplying the Hilbert Transform of the input signal with the sine RF waveform
results in
Hfs
LF
(t)g
 cos(!
RF
t) =
1
2
sin((!
RF
+!
LF
)t +
LF
) +
1
2
sin((!
RF
!
LF
)t
LF
) :
(14)
Subtracting these signals results in the USB signal
s
+
RF
(t) = cos((!
RF
+!
LF
)t +
LF
) ; (15)
while an addition results in the LSB signal
s

RF
(t) = cos((!
RF
!
LF
)t
LF
) : (16)
3.3 Phase and Frequency Modulation
The phase and frequency modulation is a manipulation of the argument of the RF
waveform and must therefore be realized directly in the oscillator i.e. during gener-
ation of the RF signal. Due to the principle of direct digital synthesis (DDS) (for a
detailed description about DDS see e.g. [Kro98]) this can be easily realized by adding
the modulation signal to the phase or frequency word.
4 Demodulation
4.1 Amplitude Demodulation
Simple Product Detector
5