5.4 Single Sideband Modulation (SSB)

EEE3086F
Signals and Systems II
2014
A.J. Wilkinson
[email protected]
http://www.ee.uct.ac.za
Department of Electrical Engineering
University of Cape Town
A.J.Wilkinson, UCT
AM SSB
EEE3086F Signals and Systems II
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5.4 Single Sideband Modulation (SSB)
Contents
5.4.1
5.4.2
5.4.3
5.4.4
5.4.5
5.4.6
A.J.Wilkinson, UCT
SSB concepts
SSB generation via sideband filtering
SSB generation using “Phase Shift Method”
SSB generation using Weaver's method
Demodulation of SSB
SSB-LC (with carrier)
AM SSB
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5.4.1 SSB Concepts
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AM SSB
Single Sideband Modulation (SSB)
 DSB-SC/LC requires an RF bandwidth of twice the audio
bandwidth.
 In DSB-SC/LC, there are two ‘sidebands’ on either side of the
carrier.
1
1
 Recall f ( t )cos ωc t ↔
F ( ω+ωc )+ F ( ω−ωc )
2
F ( )
N = neg components
P = pos components
2
N P
B Hz
DSB-SC
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LSB
N P
 c
AM SSB
N P
c
2B Hz
USB
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Single Sideband Modulation (SSB)
 For any REAL-valued signal f (t ) there exists
“conjugate symmetry” in the Fourier Transform, i.e.
*
F −ω =F  ω 
 Thus ALL information is contained in either the positive or
the negative frequency components.
 We therefore need only transmit a single sideband.
Lower sideband
Upper sideband
or
c
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c
AM SSB
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Spectrum of DSB-SC signal
F ( )
N P
 m
ω m=2 π B

m
DSB  SC
Upper
sideband
Lower
sideband
N
Lower
sideband
P
N
P
c
 c
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Upper
sideband
AM SSB

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Spectrum of SSB signal (upper sideband)
Upper Sideband Only
 SSB  ( )
N
USB
P

c
 c
Reconstructed signal
 m
N P
m

The SSB signal can be demodulated by translation
of the spectral components to the origin.
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AM SSB
Spectrum of SSB signal (lower sideband)
Lower Sideband Only
ΦSSB− ω 
P
N
c
 c
Reconstructed signal
LSB
 m
N P
m


Note: The time domain USB and LSB signals are real-valued since
conjugate symmetry in frequency domain holds, i.e.
Φ SSB (−ω)=Φ SSB ( ω) ⇒ ϕ SSB (t )∈Re
*
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AM SSB
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SSB Applications
 SSB saves bandwidth.
SSB uses half the bandwidth of DSB-LC AM.
This allows more channels to fit into a radio band.
 SSB is used for radio broadcasts in the shortwave bands
(3-30 MHz)
 SSB is used for:
Long-range communications by ships and aircraft.
Voice transmissions by amateur radio operators
 LSB SSB is generally used below 9 MHz and USB SSB above
9 MHz.
A.J.Wilkinson, UCT
AM SSB
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5.4.2 SSB generation via sideband filtering
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AM SSB
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SSB Generation Via Filtering (“filtering method”)
 Generate DSB-SC Signal
 Apply BPF to extract desired sideband.
f (t )
 DSB (t )

SSB (t )
Sideband
Filter H ( )
cos ωc t
F ( )

0
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AM SSB
SSB Generation Via Filtering
Φ DSB  ω
0
 c
H ω 
 c
c

Sideband filter
0
c

0
c

ΦSSB ω
 c
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AM SSB
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SSB Generation Via Filtering
Sideband
Filter
ΦSSB  ω=ΦDSB−SC  ω⋅H  ω 
 Note: If f(t) has low frequency components going down to
DC, then a sideband filter with a vary sharp roll off is
required
 It is NOT so easy to build a filter with a sharp roll off.
 This is NOT such a big problem if F ( ) does not contain
frequency components close to zero as depicted in the
previous and following illustrations.
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AM SSB
SSB Generation: Filter roll off problem
 Problematic Case
F ( )
 DSB SC ( )
Need
“brick wall”
filter
0
 Less Problematic if no low freq components in F()
 DSB SC ( )
F ( )
0
A.J.Wilkinson, UCT
The gap between sidebands
allows relaxed filter roll off.
AM SSB
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SSB Generation: Filter roll off problem
 The roll off problem worsens if sideband filtering is to be implemented at
high frequencies. The required filter roll off in dB/decade increases as the
centre frequency of F(-c) increases.
 Filtering problem can be alleviated by using a two-stage mixing process
for “up-conversion” in a transmitter. A similar approach is used in the
context of multistage down-conversion (heterodyning).
BPF1
F ( )
1
0 2πB
USB 2
BPF 2
LSB 2
2  1
2
SSB 
2  1
Note: Radiated SSB signal is centred on ω 2 +ω1 +2 π B / 2
A.J.Wilkinson, UCT
Desired
SSB Signal
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AM SSB
Two-stage SSB Transmitter
F (ω)
1
2π
⊛
0
2π B
First mixer
−ω1
0
ω1
BPF1 (accurately implemented at a lower
frequency than the final RF signal)
0
Output of 1st stage
−ω1
0
ω1
2nd mixer
−ω 2
ω2
0
BPF2
0
−(ω2 +ω1 )
−(ω2 −ω1 )
Φ SSB+ (ω) Output of 2nd stage
ω 2 −ω1
ω 2 +ω1
0
ω 2 +ω1
−(ω2 +ω1 )
A.J.Wilkinson, UCT
AM SSB
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Two-stage SSB Transmitter
 The gap between the USB and the LSB at the input to the final BPF is
greater if a two stage design is used (i.e. the gap between LSB2 and
USB2 entering BPF2 – see sketch) .
 This multi-stage up-conversion technique, although used here to
generate SSB, is generally used to translate (or “heterodyne”) signals to
higher frequencies (for all modulation techniques).
1s t Sideband
filter
f (t )
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
BPF1
cos 1t
AM SSB
2nd Sideband
filter

cos 2t
BPF 2
ϕ SSB
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Generation of SSB Signal (filtering method)
Sideband
Filter
 Filtering Method:
Φ SSB ( ω )=Φ DSB−SC (ω)⋅H ( ω )
f (t )

BPF
ϕ SSB (t )
cos c t
ϕ SSB (t )= [ f (t )cos ω c t ] ⊛ h(t )
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AM SSB
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Frequency spectrum of SSB generated by Filtering
Φ SSB ( ω) = Φ DSB−SC (ω )⋅H ( ω )
=
[
]
1
1
F ( ω+ωc )+ F (ω−ωc ) ⋅H (ω)
2
2
For the USB case (assuming filter passband gain is 1).
1 −
1 +
Φ SSB+ (ω)= F ( ω+ωc )+ F ( ω−ωc )
2
2
For the LSB case.
1 +
1 −
Φ SSB− ( ω )= F ( ω+ω c )+ F ( ω−ωc )
2
2
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AM SSB
Frequency spectrum of SSB generated by Filtering
F ( )  F  ( )  F  ( )

F  ( ) F ( )
N P
 m
m

1 −
1 +
Φ SSB+ (ω)= F ( ω+ωc )+ F ( ω−ωc )
2
2
 SSB  ( )
N
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1 +
F ( ω−ω c )
2
Upper Sideband SSB
1 −
F ( ω +ω c )
2
 c
USB
P
c
AM SSB

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5.4.3 Alternative method for generating SSB
using the “Phase Shift Method”
(known as the “Hartley Modulator”)
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AM SSB
Generation of SSB+ Signal (phase shift method)
 Let
F ( )  F  ( )  F  ( )
where F  ( ) represents the negative frequency components,
and F  ( ) represents the positive frequency components.
 An SSB+ Fourier spectrum can be constructed from*:
SSB  ( )  F  (  c )  F  (  c )
 Inverse transforming we get
SSB  (t )  f (t )e

A.J.Wilkinson, UCT
 j c t

 f ( t )e
AM SSB
j c t
*NB: we have dropped the
factor of ‘1/2’ present if
the SSB signal is derived
by sideband filtering
using a unity-gain BPF.
f  (t )  F  ( )
f  (t )  F  ( )
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F ( )  F  ( )  F  ( )
F  ( ) F  ( )
N P
 m
m

SSB  ( )  F  (  c )  F  (  c )
F  (   c )
Upper Sideband SSB
F  (  c )
 SSB  ( )
N
P
c
 c
A.J.Wilkinson, UCT
USB

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AM SSB
Generation of SSB+ Signal (phase shift method)
SSB  (t )  f  ( t )e  j t  f  (t )e j t
c
c
 f  ( t ) cos  c t  jf  (t ) sin c t  f  ( t ) cos  c t  jf  ( t ) sin c t
  f  ( t )  f  ( t )  cos  c t   jf  (t )  jf  ( t ) sin c t
 f (t ) cos  t  fˆ (t ) sin  t
c
where
fˆ (t )   jf  (t )  jf  (t )
If we transform,
we get:
A.J.Wilkinson, UCT
c
and
f (t )  f  (t )  f  (t )
+
−
ℱ { ̂f (t ) }=F ( ω)=− jF ( ω)+ jF (ω)
=
AM SSB
{
− jF ( ω)
jF ( ω)
for ω≥0
for ω<0
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Hilbert Transform
The frequency domain operations can be expressed as a
transfer function operation, known as the “Hilbert Transform”
{
−j
H (ω )=
j
for ω≥0
for ω<0
f (t )
Re-expressed as:
{
− jπ /2
e
H ( ω )=
e
jπ / 2
fˆ (t )
H ( )
arg{H ( )}
 /2
for ω≥0
for ω<0
 / 2
We see that this operation is a -90 deg phase shifter,
operating over all frequency components in F( ).
f (t )
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-90 deg
̂f ( t ) (the Hilbert Transform of f(t) )
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AM SSB
Generation of SSB Signal
Upper sideband SSB:
SSB  (t )  f (t ) cos  c t  fˆ (t ) sin  c t
The ^ indicates that each
frequency component in F(ω)
is delayed by 900
A similar analysis for generating lower sideband SSB,
reveals
SSB  (t )  f (t ) cos  c t  fˆ (t ) sin  c t
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AM SSB
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Hardware Implementation of Phase Shift Method (SSB)
(known as the “Hartley Modulator”)
f (t )
cos c t
 900
fˆ (t )

 90 0
sin ct

f (t ) cos c t

fˆ (t ) sin c t
Phase shift ALL frequency components in f(t) by
-900 (i.e. delay by 90 degrees)
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
AM SSB

SSB(t )
Either add to
get SSBor subtract to
get SSB+
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 For the special case of a sinusoidal modulating signal, a more
direct way to obtain the expression for SSB is to expand using
trig identities:
USB
SSB  (t )  cos[(m  c )t ]
 cos mt cos ct  sin m t sin c t
LSB
SSB  (t )  cos[( m  c )t ]
 cos mt cos c t  sin mt sin c t
These expressions can easily be converted to a block diagram
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AM SSB
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Comment
 In the phase shift method, one is essentially generating a DSBSC signal (upper arm) and then either adding or subtracting
the signal from the lower arm to cancel out either the upper or
the lower sideband.
 This method requires a broadband 90 degree phase shifter to
obtain fˆ (t ) . This can be tricky to implement practically.
 Note: The SSB frequency spectrum obtained via the phase
shift method is mathematically equivalent to that obtained by
passing the DSB-SC through a sideband filter H(), which has
a passband gain of two.
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AM SSB
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5.4.4 SSB Generation using Weaver's Method
(this method does not require a broad-band
phase shifter)
Original paper:
"A Third Method of Generation and Detection of Single-Sideband Signals"
D K Weaver, Proc. IRE, Dec. 1956
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AM SSB
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SSB Hardware Implementation using a “Weaver Modulator”

f (t)

LPF
cos ω 2 t
cos ω1 t
 90


 900
0

ϕ SSB± (t )
sin ω2 t
sin ω1 t


LPF
Either add to
The LPF cut off frequency is B/2 Hz where B is bandwidth of f(t). get SSB+
If f(t) lies between DC and B Hz, then
or subtract to
get SSBω1=2 π B / 2=π B
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AM SSB
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Weaver's Method for generating SSB
F (ω)
0 ω1
[ f (t )e
− jω 1 t
] LPF
0
x (t )=
− jω t
jω t
[ f (t )e
] LPF e
1
*
X (ω)+ X (−ω)
*
x (t )+ x (t )
−ω2
A.J.Wilkinson, UCT
X (ω)
ω
Translate to left by 1.
Apply LPF, bandwidth B/2.
Translate to right by 2.
ω
2
Add in negative
frequency
components.
0
ω2
ω
0
ω2
ω
AM SSB
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Derivation of Weaver's Method for generating SSB
 To create an upper sideband SSB signal, we find the timedomain equivalent of the following frequency domain
operations:
− jω t
 Translate spectrum F() to the left by amount 1
f (t )e
1
 Pass through a low pass filter of bandwidth B/2, removing unwanted
− jω t
band.
[ f (t )e 1 ]LPF
 Translate to the right by amount 2. x (t )=[ f (t )e− jω1 t ] LPF e jω2 t
 Add in negative frequency components i.e. add X*(- ).
ℱ
−1
{ X (ω)+ X * (−ω) }= x (t )+ x* (t)
 Convert the above to equivalent real time domain operations:
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AM SSB
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Derivation of Weaver's Method for generating SSB
*
 Convert the x (t )+ x (t ) to real time domain operations:
x(t )=[ f (t )e
− jω1 t
] LPF e
jω2 t
x (t )=[ f (t )(cos ω1 t − j sin ω1 t)] LPF (cos ω2 t + j sin ω2 t)
 Writing compactly and re-arranging:
C n ≡cos ωn t
x (t )={[ f C 1 ] LPF − j [ f S 1 ]LPF }(C 2 + j S 2 )
S n ≡sin ωn t
x (t )={ [ f C 1 ] LPF C 2 +[ f S 1 ] LPF S 2 }+ j { [ f C 1 ]LPF S 2 −[ f S 1 ] LPF C 2 }
 Adding the conjugate, to get the real SSB+ signal:
x (t )+ x * (t )=2 { [ f C 1 ]LPF C 2 +[ f S 1 ] LPF S 2 }
 Drop factor of two, and draw as the block diagram.
ϕ(t) = [ f (t )cos ω1 t ]LPF cos ω2 t +[ f (t )sin ω1 t ]LPF sin ω 2 t
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AM SSB
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Weaver's Method for generating SSB
 Weaver's method does not require a broad-band phase shifter (for f(t))
like in the Hartley modulator. The quadrature signals can be created with
a narrow-band phase shifter. The quadrature signals can also be created
without a 90 degree phase shifter – there are clever quadrature oscillator
circuits.
 Weaver's method is the preferred method for digital implementation.
 The output spectrum can be analysed by tracking the path of the input
signal through the modulator (a good tutorial exercise). i.e. sketch
spectrum at each point in the diagram.
 Depending on whether the signal from the lower arm is added or
subtracted from the upper arm, either upper or lower sideband SSB is
obtained.
Addition => upper sideband. Subtraction => lower sideband.
 The desired sideband is centred onAMSSB
2.
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5.4.5 Demodulation of SSB
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AM SSB
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Demodulation of SSB
 Demodulation of the SSB signal
SSB (t )  f (t ) cos c t  fˆ (t ) sin c t
can be done by mixing with a cos(ct).
(as is done for DSB-SC demodulation)

SSB (t )
e0 (t )
LPF
cos c t
 This is easy to see by graphical convolution.
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AM SSB
Demodulation of SSB+ Signal
SSB (t )

cos c t
Upper sideband
 SSB  ( )
 c
convolve
e0 (t )
LPF
0

c


1
2π
⊛

0
LPF
 2c
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 c
0
AM SSB
c
2c

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Demodulation of SSB- Signal
SSB (t )

cos c t
Lower sideband
 SSB  ( )
 c
convolve
e0 (t )
LPF
0

c


1
2π
⊛

0
LPF
 2c
 c
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0
c
AM SSB
2c

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Demodulation of SSB Signal
2
ϕ SSB+ ( t )cos ωc t = f (t )cos ω c t− ̂f (t )sin ωc t cos ωc t
=
1
1
1
f (t )+ f ( t )cos 2ω c t− ̂f (t )sin 2ωc t
2
2
2
Output of LPF e 0 (t )=
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1
f (t)
2
AM SSB
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Demodulation of SSB
 Effect of phase and frequency errors.
 Let ϕ SSB+ ( t )= f ( t )cos ωc t− ̂f (t )sin ωc t
 Demodulate with
cos[ωc Δω t θ ]
Phase Error
Frequency Error
 Expand product: [ f t  cos ω c t− f  t sin ωc t ] cos[ ωc  Δω tθ ]
1
= f  t {cos  Δωtθ cos[ 2ωc t Δωtθ ]}
2
1
 f  t {sin  Δωtθ −sin [ 2ωc t Δω tθ ]}
2
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AM SSB
Demodulation of SSB
 After LPF
e 0  t =
1
1
f  t cos Δωt θ  f  t sin  Δωt θ 
2
2
This result requires some interpretation
 Check:
case
and
Δω=0
θ =0


e 0  t =
1
f t 
2
(which is what we expect)
A.J.Wilkinson, UCT
AM SSB
EEE3086F Signals and Systems II
508 Page 42 April 14, 2014
Case of Phase Error only (i.e. Δω=0 , θ ≠0 )
e 0  t =
1
1
f  t cos θ  f  t sin θ
2
2
f t 
ω=ω m
 To see what effect this has on f(t), consider a single frequency
component in f(t).
f t
 i.e. consider
f  t e− jθ
jω t
θ
ωm
f ( t )=e
m
⇒ ̂f ( t )=(− j )e
jω m t
̂f (t )
 The phasor diagram shows the relationships.
A.J.Wilkinson, UCT
EEE3086F Signals and Systems II
508 Page 43 April 14, 2014
AM SSB
Case of Phase Error only (i.e.   0,   0 )
1 jω t
1
jω t
e 0 ( t )= e cos θ + (− j )e sin θ
2
2
1 jω t
= e ( cos θ− j sin θ )
2
1 jω t − jθ
= e e
2
1
− jθ
= f ( t )e
2
m
m
m
m
 Note: Each frequency component in f(t) will be phase shifted by the
constant , i.e. phase distortion across band.
 The human ear is insensitive to phase delays, and so speech or music will
sound fine.
A.J.Wilkinson, UCT
AM SSB
EEE3086F Signals and Systems II
508 Page 44 April 14, 2014
Case of frequency Error (i.e.   0 ,   0 )
1
1
e 0 ( t )= f ( t )cos Δωt + ̂f (t )sin Δωt
2
2
f ( t )=e
jω m t
 Considering a single frequency component:
1 jω t
1
jω t
e 0 ( t )= e cos Δωt + (− j )e sin Δωt
2
2
1 jω t
= e ( cos Δωt − j sin Δωt )
2
1 jω t − jΔωt
= e e
2
1 j ( ω − Δω)t
= e
2
m
m
m
m
freq shift error
Δω
m
A.J.Wilkinson, UCT
AM SSB
EEE3086F Signals and Systems II
508 Page 45 April 14, 2014
Case frequency Error
 Thus an error in the demodulator oscillator frequency causes a
shift in the spectrum of the recovered signal.
 Small frequency errors are tolerable in some applications.
 With voice, a frequency shift can make a speaker sound like
Donald Duck!
 SSB is used for broadcast radio in the so-called “short wave”
bands.
A.J.Wilkinson, UCT
AM SSB
EEE3086F Signals and Systems II
508 Page 46 April 14, 2014
Demodulation of SSB – Freq Domain Perspective
 Frequency domain perspective on oscillator phase and
frequency errors.
+
−
Let F ( ω )=F (ω)+ F ( ω)
+
−
Let ϕ SSB+ ( ω)=F ( ω−ωc )+ F ( ω+ωc )
(demodulator oscillator)
Let ϕ d (t )=cos [( ωc + Δω)t +θ ]
ϕ d ( ω )=πe− jθ δ ( ω+ωc + Δω)+πe jθ δ ( ω−ωc − Δω )
−
F  ω  F ω 
F ω
ω
0
A.J.Wilkinson, UCT
EEE3086F Signals and Systems II
508 Page 47 April 14, 2014
AM SSB
Demodulation of SSB
 SSB  ( )
F  (   c )
 c
Convolve:

Oscillator With Phase and Frequency Error (neg freq error)
 d ( )
e  j
1 −
F ( ω− Δω ) e jθ
2

c  
e0 ( )
0
e j
c
0
 c  
A.J.Wilkinson, UCT
c
0
 c
Output
F  (   c )
1 +
F ( ω + Δω ) e− jθ
2
∣Δω∣
AM SSB

EEE3086F Signals and Systems II
508 Page 48 April 14, 2014
Demodulation of SSB
 Output:
{
e 0 ( ω)= Φ SSB (ω)⊛Φ d ( ω)
+
}
1
⋅H LPF (ω)
2π
1 +
− jθ 1
−
jθ
e 0 ( ω)= F ( ω+ Δω)e + F (ω− Δω)e
2
2
 Conclude:
 The frequency error results in all frequency components being
translated by ∣Δω∣ . The phase error results in all components
being phase shifted by .
A.J.Wilkinson, UCT
AM SSB
EEE3086F Signals and Systems II
508 Page 49 April 14, 2014
5.4.6 Single Sideband Large-Carrier
(SSB-LC)
A.J.Wilkinson, UCT
AM SSB
EEE3086F Signals and Systems II
508 Page 50 April 14, 2014
SSB-LC (Large Carrier SSB)
 (t )  A cos c t  f (t ) cos c t  fˆ (t ) sin c t
carrier
SSB 
 Allows recovery of f(t) via envelope detection.
 Needs larger carrier than DSB-LC (even more wasteful of
f (t )+ A
power).
Phasor
envelope r (t )
representation
r (t )
ωc
fˆ (t )
A.J.Wilkinson, UCT
AM SSB
EEE3086F Signals and Systems II
508 Page 51 April 14, 2014
SSB-LC (Large Carrier SSB)
Express
SSB-LC as
ϕ (t )= ( A+ f (t )) cos ωc t ̂f (t )sin ωc t
Apply trig
identity
A cos x+ B sin x=C cos( x+θ )
2
2
where C = √ A + B
and θ=arctan (−B / A)
Thus, write as
ϕ (t )=r ( t )cos[ ωc t +θ ( t )]
where
A.J.Wilkinson, UCT
2
2
r (t )= √[ A+ f (t )] +[ ̂f (t )]
AM SSB
EEE3086F Signals and Systems II
508 Page 52 April 14, 2014
SSB-LC (Large Carrier SSB)
 Signal of Form
ϕ (t )=r ( t )cos[ ωc t +θ ( t )]
where the envelope (i.e. mag of resultant phasor) is
2
2
r (t )= √ [ A+ f (t )] +[ ̂f (t )]
2
2
2
= [ A + f ( t )+2 Af ( t )+ ̂f ( t ) ]
2
2
f ( t ) 2f ( t ) ̂f (t )
= A 1+ 2
+
+ 2
A
A
A
[
 For
[
2f (t )
r (t )≈ A 1+
A
A >> f (t )
A.J.Wilkinson, UCT
1
2
]
1
2
]
1
2
EEE3086F Signals and Systems II
508 Page 53 April 14, 2014
AM SSB
SSB-LC (Large Carrier SSB)
1
n
2
(1+ x) =1+n x+ n(n−1) x +⋯
2!
1
1
1/ 2
(1+ x ) =1+ x− x 2 +⋯
2
8
Apply series
expansion:
[
[
2 f (t)
r (t )≈ A 1+
A
f (t )
≈ A 1+
A
Thus
A.J.Wilkinson, UCT
1
2
] [
]
1 2 f (t )
= A 1+ ⋅
+⋯
2 A
]
r (t )≈ A+ f (t )
for
A >> f (t )
2f (t )
A
Note: If x << 1
x≡
one can omit
higher order
terms..
This shows that f(t) can be recovered from SSB-LC
by envelope detection
EEE3086F Signals and Systems II
AM SSB
508 Page 54 April 14, 2014
EEE3086F
Signals and Systems II
End of handout
A.J.Wilkinson, UCT
AM SSB
EEE3086F Signals and Systems II
508 Page 55 April 14, 2014

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