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AM SSBA.J.Wilkinson, UCT EEE3086F Signals and Systems II508 Page 1 April 14, 2014
EEE3086FSignals and Systems II
2014
A.J. [email protected]
http://www.ee.uct.ac.zaDepartment of Electrical Engineering
University of Cape Town
AM SSBA.J.Wilkinson, UCT EEE3086F Signals and Systems II508 Page 2 April 14, 2014
5.4 Single Sideband Modulation (SSB)
5.4.1 SSB concepts5.4.2 SSB generation via sideband filtering5.4.3 SSB generation using “Phase Shift Method”5.4.4 SSB generation using Weaver's method 5.4.5 Demodulation of SSB5.4.6 SSB-LC (with carrier)
Contents
AM SSBA.J.Wilkinson, UCT EEE3086F Signals and Systems II508 Page 3 April 14, 2014
5.4.1 SSB Concepts
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 4 April 14, 2014
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.
Recall
Hz2B
f ( t )cos ωc t ↔12
F (ω+ωc )+12
F (ω−ωc )
N P
N PN = neg componentsP = pos components
DSB-SCcc
B Hz
USBLSB N P
)(F
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 5 April 14, 2014
Single Sideband Modulation (SSB)
For any REAL-valued signal there exists“conjugate symmetry” in the Fourier Transform, i.e.
Thus ALL information is contained in either the positive or the negative frequency components.
We therefore need only transmit a single sideband.
)(tf
F −ω =F* ω
sidebandUpper
cor
sidebandLower
c
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 6 April 14, 2014
Spectrum of DSB-SC signal
sidebandLower
sidebandUpper
N
)(F
m
SCDSB
sidebandLower
sidebandUpper
m
c c
P
N P N P
ωm=2π B
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 7 April 14, 2014
Spectrum of SSB signal (upper sideband)
OnlySidebandUpper
c c N P
USB)(SSB
m mN P
Reconstructed signal
The SSB signal can be demodulated by translationof the spectral components to the origin.
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 8 April 14, 2014
Spectrum of SSB signal (lower sideband)
Note: The time domain USB and LSB signals are real-valued since conjugate symmetry in frequency domain holds, i.e.
N
OnlySidebandLower
c cP
m m
LSB
ΦSSB(−ω)=ΦSSB* (ω) ⇒ ϕSSB ( t )∈Re
N P
ΦSSB− ω
Reconstructed signal
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 9 April 14, 2014
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.
AM SSBA.J.Wilkinson, UCT EEE3086F Signals and Systems II508 Page 10 April 14, 2014
5.4.2 SSB generation via sideband filtering
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 11 April 14, 2014
SSB Generation Via Filtering (“filtering method”)
Generate DSB-SC Signal Apply BPF to extract desired sideband.
0
)(tDSB
cosωc t
)(FilterSideband
H)(tf )(tSSB
)(F
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 12 April 14, 2014
SSB Generation Via Filtering
c
ΦDSBω
0
0
H ω c
c c
c
ΦSSB ω
0 c
Sideband filter
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 13 April 14, 2014
SSB Generation Via Filtering
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 does not contain
frequency components close to zero as depicted in the previous and following illustrations.
)(F
ΦSSB ω=ΦDSB−SC ω ⋅H ω FilterSideband
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 14 April 14, 2014
SSB Generation: Filter roll off problem
Problematic Case
Less Problematic if no low freq components in F()
)(F
0
)(SCDSB
0
)(F)(SCDSB
The gap between sidebandsallows relaxed filter roll off.
Need “brick wall” filter
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 15 April 14, 2014
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).
1BPF2LSB
112 2
12
2BPFSSB
Desired SSB Signal
2USB)(F
Note: Radiated SSB signal is centred on ω2+ω1+2π B /2
2π B0
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 16 April 14, 2014
Two-stage SSB Transmitter
F (ω)
2π B0
0
0
0
ω2+ω1ΦSSB+(ω)
0
0
−ω1 ω1
−ω1 ω1
ω2
First mixer
Output of 1st stage
−ω2
ω2−ω1−(ω2+ω1) −(ω2−ω1)
ω2+ω10
−(ω2+ω1)
Output of 2nd stage
2nd mixer
BPF1 (accurately implemented at a lowerfrequency than the final RF signal)
BPF2
12π ⊛
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 17 April 14, 2014
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).
ϕ SSB
t1cos
1BPF)(tf
t2cos
2BPF
1s t Sideband filter
2nd Sideband filter
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 18 April 14, 2014
Generation of SSB Signal (filtering method)
Filtering Method:
ΦSSB(ω )=ΦDSB−SC (ω)⋅H (ω )FilterSideband
tccos
ϕ SSB ( t )=[ f ( t )cosωc t ]⊛ h( t )
BPF)(tf ϕ SSB ( t )
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 19 April 14, 2014
Frequency spectrum of SSB generated by Filtering
ΦSSB(ω) = ΦDSB−SC (ω )⋅H (ω)
For the USB case (assuming filter passband gain is 1).
ΦSSB+ (ω)=12
F−(ω+ωc )+12
F +(ω−ωc )
For the LSB case.
= [ 12 F (ω+ωc )+ 12 F (ω−ωc )]⋅H (ω)
ΦSSB−(ω )=12
F + (ω+ωc )+12
F−(ω−ωc )
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 20 April 14, 2014
Frequency spectrum of SSB generated by Filtering
N
m mP
SSBSidebandUpper
c c P
USB)(SSBN
ΦSSB+ (ω)=12
F−(ω+ωc )+12
F +(ω−ωc )
12
F− (ω+ωc )12
F+(ω−ωc)
)(F)()()( FFF
)(F
AM SSBA.J.Wilkinson, UCT EEE3086F Signals and Systems II508 Page 21 April 14, 2014
5.4.3 Alternative method for generating SSB using the “Phase Shift Method”
(known as the “Hartley Modulator”)
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 22 April 14, 2014
Generation of SSB+ Signal (phase shift method)
Let
where represents the negative frequency components, and represents the positive frequency components.
An SSB+ Fourier spectrum can be constructed from*:
Inverse transforming we get
)()()( FFF
)()()( ccSSB FF
)(F)(F
tjtjSSB
cc etfetft )()()( )()()()(
FtfFtf
*NB: we have dropped the factor of ‘1/2’ present if the SSB signal is derived by sideband filteringusing a unity-gain BPF.
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 23 April 14, 2014
N
m mP
SSBSidebandUpper
c c P
USB)(SSBN
)()()( ccSSB FF
)( cF )( cF
)(F)()()( FFF
)(F
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 24 April 14, 2014
Generation of SSB+ Signal (phase shift method)
ttfttf
ttjftjfttftfttjfttfttjfttf
etfetft
cc
cc
cccc
tjtjSSB
cc
sin)(ˆcos)(
sin)()(cos)()(
sin)(cos)(sin)(cos)()()()(
and )()()( tftftf )()()(ˆ tjftjftf where
ℱ { f̂ ( t )}=F (ω )=− jF +(ω)+ jF−(ω)
={− jF (ω) for ω≥0jF (ω) for ω
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 25 April 14, 2014
Hilbert Transform
H (ω )={− j for ω≥0j for ω
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 27 April 14, 2014
Hardware Implementation of Phase Shift Method (SSB)(known as the “Hartley Modulator”)
tccos090
)(tf090
ttf csin)(ˆ
tcsin
ttf ccos)(
)(tSSB
Phase shift ALL frequency components in f(t) by -900 (i.e. delay by 90 degrees)
)(ˆ tf
Either add toget SSB-or subtract toget SSB+
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 28 April 14, 2014
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:
tttttt
cmcm
cmSSB
sinsincoscos])cos[()(
tttttt
cmcm
cmSSB
sinsincoscos])cos[()(
USB
LSB
These expressions can easily be converted to a block diagram
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 29 April 14, 2014
Comment
In the phase shift method, one is essentially generating a DSB-SC 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 . 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.
)(ˆ tf
AM SSBA.J.Wilkinson, UCT EEE3086F Signals and Systems II508 Page 30 April 14, 2014
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
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 31 April 14, 2014
SSB Hardware Implementation using a “Weaver Modulator”
090
ϕSSB±(t )
Either add toget SSB+or subtract toget SSB-
090
sin ω1 t
LPF
LPF
The LPF cut off frequency is B/2 Hz where B is bandwidth of f(t).If f(t) lies between DC and B Hz, then
ω1=2π B /2=π B
f (t)
sin ω2 t
cos ω1 t cos ω2 t
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 32 April 14, 2014
Weaver's Method for generating SSB
F (ω)
0
ωω1
0
0
ω2
ω2−ω2
0Translate to left by 1.Apply LPF, bandwidth B/2.
Translate to right by 2.
Add in negative frequency components.
[ f (t )e− jω1t ]LPF
x (t )=
x (t )+ x*(t )
ω
ω
ω
X (ω)+ X *(−ω)
X (ω)[ f (t )e− jω1 t ]LPF e
jω2 t
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 33 April 14, 2014
Derivation of Weaver's Method for generating SSB
To create an upper sideband SSB signal, we find the time-domain equivalent of the following frequency domain operations:Translate spectrum F() to the left by amount 1 Pass through a low pass filter of bandwidth B/2, removing unwanted
band.Translate to the right by amount 2.Add in negative frequency components i.e. add X*(-).
Convert the above to equivalent real time domain operations:
f (t )e− jω1 t
[ f (t )e− jω1 t ]LPFx (t)=[ f (t )e− jω1 t ]LPF e
jω2 t
ℱ −1 {X (ω)+ X *(−ω) }=x (t )+ x*(t)
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 34 April 14, 2014
Derivation of Weaver's Method for generating SSB
Convert the to real time domain operations:
Writing compactly and re-arranging:
Adding the conjugate, to get the real SSB+ signal:
Drop factor of two, and draw as the block diagram.
x (t )=[ f (t)(cos ω1 t− j sin ω1 t)]LPF (cos ω2 t+ j sin ω2 t)
x (t )={[ f C1]LPF− j [ f S 1]LPF }(C 2+ j S 2)
x (t )={ [ f C1]LPF C 2+[ f S 1]LPF S 2 }+ j { [ f C 1]LPF S 2−[ f S 1]LPF C 2 }
x (t )+ x*(t )=2 { [ f C1]LPF C 2+[ f S1]LPF S 2 }
x(t )=[ f (t )e− jω1 t ]LPF ejω2 t
ϕ(t) = [ f (t )cos ω1 t ]LPF cos ω2 t+[ f (t )sin ω1 t ]LPF sin ω2 t
C n≡cos ωn tS n≡sin ωn t
x (t )+ x*(t )
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 35 April 14, 2014
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 on 2.
AM SSBA.J.Wilkinson, UCT EEE3086F Signals and Systems II508 Page 36 April 14, 2014
5.4.5 Demodulation of SSB
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 37 April 14, 2014
Demodulation of SSB
Demodulation of the SSB signal
can be done by mixing with a cos(ct). (as is done for DSB-SC demodulation)
This is easy to see by graphical convolution.
ttfttft ccSSB sin)(ˆcos)()(
)(tSSBtccos
LPF )(0 te
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 38 April 14, 2014
Demodulation of SSB+ Signal
)(SSB
)(tSSBtccos LPF )(0 te
c c
c2 c2
LPF
0
0
cc
0 convolve
Upper sideband
12π⊛
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 39 April 14, 2014
Demodulation of SSB- Signal
)(tSSBtccos LPF )(0 te
c2
LPF
0 c
)(SSB
c 0
0 convolve
Lower sideband
c
c2c
1
2π⊛
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 40 April 14, 2014
Demodulation of SSB Signal
= 12
f ( t )+ 12
f ( t )cos 2ωc t−12
f̂ ( t )sin 2ωc t
Output of LPF e0( t )=12
f ( t )
ϕSSB+ ( t )cos ωc t = f ( t )cosωc2 t− f̂ ( t )sin ωc t cos ωc t
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 41 April 14, 2014
Demodulation of SSB
Effect of phase and frequency errors. Let
Demodulate with
Expand product:
Frequency Error
cos [ ωcΔω tθ ]
ϕSSB+( t )= f ( t )cos ωc t− f̂ ( t )sin ωc t
Phase Error
[ f t cosωc t− f t sin ωc t ]cos[ ωcΔω tθ ]
=12
f t {cos Δωtθ cos[ 2ωc tΔω tθ ]}
12
f t {sin Δωtθ −sin [ 2ωc tΔω tθ ]}
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 42 April 14, 2014
Demodulation of SSB
After LPF
Check: Δω=0case
e0 t =12
f t cos Δωtθ 12
f t sin Δωtθ
and θ =0 e0 t =12 f t
(which is what we expect)
This result requires some interpretation
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 43 April 14, 2014
Case of Phase Error only (i.e. , )
To see what effect this has on f(t), consider a single frequency component in f(t).
i.e. consider
The phasor diagram shows the relationships.
f t
Δω=0
e0 t =12
f t cos θ 12
f t sin θ
θ ≠0
ω=ωm
f ( t )=e jωm tf t e− jθ
f t
f̂ (t )
ωm θ
⇒ f̂ ( t )=(− j )e jωm t
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 44 April 14, 2014
Case of Phase Error only (i.e. , )
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.
0 0
e0( t )=12
e jωm t cos θ+12
(− j )e jωm t sin θ
=12
e jωm t(cos θ− j sin θ )
=12
e jωm t e− jθ
=12
f ( t )e− jθ
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 45 April 14, 2014
Case of frequency Error (i.e. , )
Considering a single frequency component:
e0( t )=12
f ( t )cos Δωt+ 12
f̂ ( t )sin Δωt f ( t )=e jωm t
e0( t )=12
e jωm t cos Δωt+12
(− j )e jωm t sin Δωt
=12
e jωm t (cos Δωt− j sin Δωt )
=12
e jωm t e− jΔωt
=12
e j (ωm−Δω)t
freq shift errorΔω
0 0
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 46 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 II508 Page 47 April 14, 2014
Demodulation of SSB – Freq Domain Perspective
Frequency domain perspective on oscillator phase and frequency errors.
Let Let Let ϕd ( t )=cos [(ωc+Δω) t+θ ]
ϕd (ω )=πe− jθ δ (ω+ωc+Δω)+πe
jθ δ (ω−ωc−Δω )
F ω
F (ω )=F + (ω)+F−(ω)ϕSSB+ (ω)=F
+ (ω−ωc )+F−(ω+ωc )
F−ω
0 ω
Fω
(demodulator oscillator)
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 48 April 14, 2014
Demodulation of SSB
)( cF
c c
c c
)(SSB
0
0
cc
)( cF
Oscillator With Phase and Frequency Error (neg freq error))(d
12
F−(ω−Δω)e jθ
∣Δω∣0
)(0 e12
F+ (ω+Δω )e− jθ
je jeConvolve:
Output
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 49 April 14, 2014
Demodulation of SSB
Output:
Conclude: The frequency error results in all frequency components being
translated by . The phase error results in all components being phase shifted by .
∣Δω∣
e0(ω)={ΦSSB+(ω)⊛Φd (ω) 12π }⋅H LPF (ω )e0(ω)=
12
F + (ω+ Δω)e− jθ+ 12
F−(ω−Δω)e jθ
AM SSBA.J.Wilkinson, UCT EEE3086F Signals and Systems II508 Page 50 April 14, 2014
5.4.6 Single Sideband Large-Carrier (SSB-LC)
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 51 April 14, 2014
SSB-LC (Large Carrier SSB)
Allows recovery of f(t) via envelope detection. Needs larger carrier than DSB-LC (even more wasteful of
power).
carrier SSB
ttfttftAt ccc sin)(ˆcos)(cos)(
envelope )(tr )(tr
f (t )+ A
)(ˆ tf
ωc
Phasorrepresentation
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 52 April 14, 2014
SSB-LC (Large Carrier SSB)
r ( t )=√[ A+ f ( t )]2+[ f̂ ( t )]2
ϕ ( t )=r ( t )cos [ωc t+θ ( t ) ]
ϕ ( t )=( A+ f ( t )) cos ωc t f̂ ( t )sin ωc t
Acos x+Bsin x=C cos( x+θ )where C=√ A2+B2and θ=arctan (−B / A)
ExpressSSB-LC as
Apply trigidentity
Thus, write as
where
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 53 April 14, 2014
SSB-LC (Large Carrier SSB)
Signal of Form
where the envelope (i.e. mag of resultant phasor) is
For A>> f ( t )
r ( t )=√[ A+ f ( t )]2+[ f̂ ( t )]2
=[ A2+ f 2( t )+2 Af ( t )+ f̂ 2( t ) ]12
=A[1+ f 2( t )A2 +2f ( t )A + f̂2( t )
A2 ]12
ϕ ( t )=r ( t )cos[ωc t+θ ( t )]
r ( t )≈A [1+ 2f ( t )A ]12
A.J.Wilkinson, UCT AM SSB EEE3086F Signals and Systems II508 Page 54 April 14, 2014
SSB-LC (Large Carrier SSB)
r ( t )≈A+ f ( t )Thus
x > f ( t )
x≡2f ( t )A
This shows that f(t) can be recovered from SSB-LC by envelope detection
Apply series expansion:
(1+ x )1/2=1+ 12
x−18
x2+⋯
Note: If one can omithigher order terms..
AM SSBA.J.Wilkinson, UCT EEE3086F Signals and Systems II508 Page 55 April 14, 2014
EEE3086FSignals and Systems II
End of handout