Omar Siddiqui
Department of Electrical Engineering College of Engineering
Taibah University Madinah
Email:[email protected]
EE 372 – Communication Theory and Systems I Lecture 7: Single Side Band (SSB) Modulation
ةبيط ةعماج
SSB Block diagram and equation
Hilbert Transform
Pre-envelope of a signal
Representation of the SSB signal
SSB Generation
SSB Demodulation
SSB+C
Double-Sideband Suppressed Carrier Modulation (DSB-SC)
The DSB-SC is obtained by a multiplication of the message signal with the carrier
) 2
cos(
) ( )
(t m t A f t
s c c
Block Diagram X
Accos(2fct)
m(t) s(t)
Mathematically
DSB-SC Frequency and Time domains
Time Domain Frequency Domain
m(t) M(f)
0 f
fc
fc
2 Ac
c(t)=Accos(2fct) ( ) ( )
2 c c
c f f f f
A
Message
Carrier
DSB-SC
2 ) 0 ( M Ac
Phase reversals when m(t) crosses zero
DSB-SC Bandwidth Issue
( ) ( )
) 2
( Ac M f fc M f fc f
S
-fc fc-W fc fc+W
-fc-W -fc+W
0 +W
-W
) 0 ( M )
( f M
BW=W
Frequency spectrum of Message Frequency spectrum of DSB-SC
LSB
BW=2W
Bandwidth of message signal = W
Bandwidth of the DSB-SC signal = 2W
USB
0
2 ) 0 ( M Ac
Double the message bandwidth is required Q. How to reduce the bandwidth?
A. Only transmit one of the side bands and filter the other i.e. the Single Side Band Modulation
BPF BPF
Problem: Highly Selective filters are required which increase complexity
Single Side Band (SSB)
The SSB signal can be generated by first generating the DSB and then using a BPF to filter out the USB or LSB
The filters have to be very highly selective
Since all filters have a stop band, there will always be a portion of the other side band
Sharp cut off
Single Side Band (SSB) with an energy Gap
natural cut off
Baseband signal with energy gap
USB - To avoid the sharp filters, SSB
is used with the baseband
signals that have an energy gap at the origin
- if a signal does not naturally have an energy gap, it has to be passed through a BPF in order to create one
Single Side Band Representation
Block Diagram X
Accos(2fct)
m(t) BPF s(t)
Mathematically, the SSB signal can be written as:
) 2
sin(
) ˆ( ) 2
2 cos(
) 2 (
)
( A m t f t
t f t
A m t
sSSB c c c c
Where ‘–’ is for USB and ‘+’ for LSB
) ˆ t(
m is the Hilbert transform of m(t)
Hilbert Transform of a time domain function g(t) is also a time domain function defined by:
Hilbert’s Transform
t d t g
g
1 ( ) )
ˆ( 1 ( )
)
ˆ( g t
t t
g
The convolution in time domain corresponds to multiplication in frequency domain We have gˆ(t) Gˆ( f ) and 1 sgn( )
f t j
) ( ) sgn(
)
ˆ( f j f G f
G
Fourier Transform of the Hilbert Transform
The sgn function is defined as:
1 0
0 0
0 ,
1 )
sgn( f
f f f
) sgn( f
f
1
1
Pre-envelopes of a signal defined in terms of Hilbert’s Transform:
Pre Envelope of a Signal
) ˆ( )
( )
(t g t jg t
g G( f ) G( f ) jGˆ( f )
Properties of the Pre Envelope of a Signal
0 ) 0 (
) ( 2 )
(
G
f G f
G f>0
f=0 f<0
) ˆ( )
( )
(t g t jg t
g G( f ) G( f ) jGˆ( f )
0 ) 0 (
) ( 2 )
(
G
f G f
G f<0
f=0 f>0
) ( f G
2 ) ( f G
Writing a SSB signal from pre-envelopes
) ( f M W W
fc fc W
fc
W fc
2 ) (f fc M 2
) (f fc M
) 2 (
) 1 2 (
) 1
( c c
USB f M f f M f f
S
t f j t
f j USB
c
c m t e
e t m t
s 2 ( ) 2
2 ) 1
2 ( ) 1
(
Taking inverse FT
t f j t
f j USB
c
c m t jm t e
e t m j t m t
s 2 [ ( ) ˆ( )] 2
2 )] 1
ˆ( ) ( 2[ ) 1
(
2 ] )[
ˆ( 2 ]
)[
( ) (
2 2
2 2
j e t e
e m t e
m t s
t f j t f j t
f j t f j USB
c c
c
c
t f t
m t f t
m t
s ( ) ( )cos2 ˆ( )sin2 Message signal
SUSB(f)
0
USB Representation LSB Representation
2 ) (f fc M 2
) (f fc M
) 2 (
) 1 2 (
) 1
( c c
LSB f M f f M f f
S
Taking inverse FT
t f j t
f j LSB
c
c m t e
e t m t
s 2 ( ) 2
2 ) 1
2 ( ) 1
(
t f j t
f j LSB
c
c m t jm t e
e t m j t m t
s 2 [ ( ) ˆ( )] 2
2 )] 1
ˆ( ) ( 2[ ) 1
(
2 ] )[
ˆ( 2 ]
)[
( ) (
2 2
2 2
j e t e
e m t e
m t s
t f j t f j t
f j t f j LSB
c c
c
c
t f t
m t f t
m t
sLSB( ) ( )cos2 c ˆ( )sin2 c
Generation of SSB Signals
Two methods are generally used:
A. Selective Filtering Method B. Phase shift Method
A. Selective Filtering Method: A sharp cut-off filter is applied after
generating the DSB signal. It is suitable for voice signals because voice has negligible frequency contents close to zero as shown in the spectral density plot of voice signal
Generation of SSB Signals
B. Phase Shift Method: This method uses the definition of the Hilbert Transform. Consider the equation of the SSB waveform:
The Hilbert Transform in frequency domain is given by:
Therefore, every spectral component of m(t) has to be delayed by -90o which is a difficult task and can be done only for a finite range of frequencies
t f t
m t f t
m t
sLSB( ) ( )cos2 c ˆ( )sin2 c
) ( ) sgn(
) ( ) sgn(
)
ˆ ( f j f M f e 2 f M f
M j
Detection of SSB Signals
Synchronous Detection:
A DSB detector can also be used to detect the SSB signal.
It can be shown that the output of the following circuit is proportional to the message signal
However, the frequency and phase synchronization is needed
Sometimes a carrier is transmitted with the SSB signal so that the signal can be detected using envelope detection. This type of SSB is called SSB+C
)
1(t
v vo(t)
t f t
m t f t
m t
sSSB( ) ( )cos2 c ˆ( )sin2 c LPF
SSB Example Tone Modulation
) 2 sin(
) ˆ( ) 2
2 cos(
) 2 (
)
( A m t f t
t f t
A m t
sSSB c c c c
Example 4.6: Find the SSB signal in time domain if m(t) cos(2fmt)
) 2 1 cos(
)
ˆ( f t
t t
m m
) ˆ (
)
ˆ(t M f
m
) 1 sgn(
f t j
2 ( )
) 1 2 (
) 1 2
cos( fmt f fm f fm
1 cos(2 ) t
t fm
jsgn( f ) 2 ( ) ) 1
2 ( 1
m
m f f
f
f
)
1 sgn(
j f
( )
2 ) 1 2 (
1
m
m f f
f
f
What is sgn(f) multiplied by a function?
Solution
We need to calculate mˆ t( ) which is the Hibert Transform of m(t)
Let us do the calculations in Frequency Domain
1 cos(2 ) t
t fm
SSB Example Tone Modulation
) 2 sin(
) ˆ( ) 2
2 cos(
) 2 (
)
( A m t f t
t f t
A m t
sSSB c c c c
Example 4.6: Find the SSB signal in time domain if m(t) cos(2fmt)
The positive side will be multiplied by +1 and negative by -1 Solution
) sgn( f
f
1
1
)
sgn( f
( )
2 ) 1 2 (
1
m
m f f
f j
j f
1 cos(2 ) t
t fm
j 2
1 j
2 1
fm
fm
x
j 2
1
j 2
1 fm
fm
=
( )
2 ) 1 2 (
1
m
m f f
f j
j f
1 cos(2 ) t
t fm
SSB Example Tone Modulation
) 2 sin(
)
ˆ(t f t
m m sin(2 )sin(2 )
) 2 2
cos(
) 2
2 sin(
)
( A f t f t
t f t
A f t
sSSB c m c c m c
) 2 sin(
) 2
2 sin(
) 2 cos(
) 2
2 cos(
)
( A f t f t
t f t
A f t
sUSB c m c c m c
USB Signal
] ) (
2 cos[
)
(t A f f t
sUSB c c m
( )
) 2 2 (
)
( c m c m
USB A f f f
f f A f
f
S
) 2 sin(
) 2
2 sin(
) 2 cos(
) 2
2 cos(
)
( A f t f t
t f t
A f t
sLSB c m c c m c
LSB Signal
] ) (
2 cos[
)
(t A f f t
sLSB c c m
SLSB( f )
2 Ac
m
c f
f )
( fc fm
fc
fc
2
Ac
) ( f SUSB
0
( )
) 2
2 ( m
c m
c A f f f
f f
A f
2 Ac
m
c f
f )
( fc fm
fc
2
Ac SLSB( f ) 0
SSB+C Signals
v v2 DC
Filter
) (t m A
-The carrier is added to DSB and SSB modulations to facilitate the envelope detection - The envelope detector is a simple RC circuit with a ripple filter and a DC block
) (t s
DSB Modulator
m(t) m(t)cos(2fct)
) 2 cos( f t
A c
Am(t)cos(2fct)
DSB+C (AM) SSB+C (AM)
SSB Modulator
m(t)
) 2 sin(
) ˆ( ) 2 cos(
)
(t f t mt f t
m c c
) 2 cos( f t
A c
Am(t)cos(2fct)mˆ(t)sin(2fct)
DSB+C (AM) SSB+C
The Envelope Detector
Consider the SSB+C signal:
The envelope of this signal is given by:
It will be shown below that this envelope can be approximated to A+m(t) if the amplitude of the carrier A is much larger than the message signal
Envelope Detection of SSB+C Signals
t f t
m t f t
m t f A
t
s( ) cos2 c ( )cos2 c ˆ( )sin2 c
A m t f t m t f t
t
s( ) ( ) cos2 c ˆ( )sin2 c
( ) ˆ ( )
)
(t A m t 2 m2 t
E
) ˆ( ), (t m t m
A
( ) ˆ ( )
)
(t A m t 2 m2 t
E
Expanding and taking A2 as a common factor ) ˆ ( ) ( 2 ) ( )
(t A2 m2 t Am t m2 t
E
2 2(2 ) 2 2( ) ˆ2(2 ) 1
)
( A
t m A
t Am A
t A m
t
E 2
2 2
2( ) 2 ( ) ˆ ( )
1 A
t m A
t m A
t
A m
) ˆ( ), (t m t m
A
Applying
A t A m
t
E 2 ( )
1 )
( 2
1
) (
1 2
A
t A m
Applying the Binomial expansion (1x)n 1nx for small x
A
t A m
t
E 2 ( )
2 1 1 )
( E(t) Am(t)
Hence envelope detection can be used to detect SSB+C signals
Reading from the book
Chapter 4: Amplitude Modulations and Demodulations, Pages 198 - 204 B.P. Lathi, “Modern Digital and analog Communication Systems”, 4th Edition
