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Single Side Band (SSB) Modulation
• In DSB-SC it is observed that there is symmetry in the
bandstructure. So, even if one half is transmitted, the other
half can be recovered at the received. By doing so, the
bandwidth and power of transmission is reduced by half.
Depending on which half of DSB-SC signal is transmitted,
there are two types of SSB modulation
1. Lower Side Band (LSB) Modulation
2. Upper Side Band (USB) Modulation
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M( ω )
Baseband signal
−2 π B
2 πB
DSBSC
−ω c
ωc
USB
−ω c
ωc
LSB
−ω c
ωc
Figure 1: SSB signals from orignal signal
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• Mathematical Analysis of SSB modulation
M(ω )
−2 B
M+ (ω )
0
2
B
M (ω )
−
0
0
M ( ω + ωc )
−
M ( ω − ωc )
+
−ω
ωc
c
M (ω − ω )
−
M ( ω + ωc )
+
−ω c
&
c
c
ωc
Figure 2: Frequency analysis of SSB signals
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• From Figure 2 and the concept of the Hilbert Transform,
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ΦU SB (ω) = M+ (ω − ωc ) + M− (ω + ωc )
φU SB (t) = m+ (t)ejωc t + m− (t)e−jωc t
But, from complex representation of signals,
m+ (t) = m(t) + j m̂(t)
m− (t) = m(t) − j m̂(t)
So,
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φU SB (t) = m(t) cos(ωc t) − m̂(t) sin(ωc t)
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Similarly,
φLSB (t) = m(t) cos(ωc t) + m̂(t) sin(ωc t)
• Generation of SSB signals A SSB signal is represented by:
φSSB (t) = m(t) cos(ωc t) ± m̂(t) sin(ωc t)
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m(t)
DSBSC
+
cos(ω c t)
Σ
π/2
sin( ω c t)
π /2
SSB signal
+
−
DSBSC
Figure 3: Generation of SSB signals
As shown in Figure 3, a DSB-SC modulator is used for SSB
signal generation.
• Coherent Demodulation of SSB signals ΦSSB (t) is multiplied
with cos(ωc t) and passed through low pass filter to get back the
orignal signal.
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1
1
m(t) [1 + cos(2ωc t)] ± m̂(t) sin(2ωc t)
2
2
1
1
1
= m(t) + cos(2ωc t) ± m̂(t) sin(2ωc t)
2
2
2
φSSB (t) cos(ωc t) =
M(ω)
M −(ω + ω c )
−2ωc
0
M ( ω −ω c )
+
2 ωc
Figure 4: Demodulated SSB signal
The demodulated signal is passed through an LPF to remove
unwanted SSB terms.
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Vestigial Side Band (VSB) Modulation
• The following are the drawbacks of SSB signal generation:
1. Generation of an SSB signal is difficult.
2. Selective filtering is to be done to get the original signal
back.
3. Phase shifter should be exactly tuned to 900 .
• To overcome these drawbacks, VSB modulation is used. It can
viewed as a compromise between SSB and DSB-SC. Figure 5
shows all the three modulation schemes.
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Φ
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(ω)
VSB
−ω
B
c
0
ωc
B
ν
Figure 5: VSB Modulation
• In VSB
1. One sideband is not rejected fully.
2. One sideband is transmitted fully and a small part (vestige)
of the other sideband is transmitted.
• The transmission BW is BWv = B + v. where, v is the
vestigial frequency band. The generation of VSB signal is
shown in Figure 6
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m(t)
H (ω)
i
Φ
( t)
VSB
cos( ω t)
c
Figure 6: Block Diagram - Generation of VSB signal
Here, Hi (ω) is a filter which shapes the other sideband.
ΦV SB (ω) = [M (ω − ωc ) + M (ω + ωc )] .Hi (ω)
To recover the original signal from the VSB signal, the VSB
signal is multiplied with cos(ωc t) and passed through an LPF
such that original signal is recovered.
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Φ
( t)
VSB
LPF
H (ω)
0
m(t)
cos( ω c t)
Figure 7: Block Diagram - Demodulation of VSB signal
From Figure 6 and Figure 7, the criterion to choose LPF is:
M (ω) = [ΦV SB (ω + ωc ) + ΦV SB (ω − ωc )] .H0 (ω)
= [Hi (ω + ωc ) + Hi (ω − ωc )] .M (ω).H0 (ω)
=⇒ H0 (ω) =
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Hi (ω + ωc ) + Hi (ω − ωc )
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Appendix: The Hilbert Transform
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The Hilbert Transform on a signal changes its phase by ±900 . The
Hilbert transform of a signal g(t) is represented as ĝ(t).
ĝ(t) =
1
π
Z
1
g(t) = −
π
+∞
−∞
Z
g(τ )
dτ
t−τ
+∞
−∞
ĝ(τ )
t−τ
dτ
We, say g(t) and ĝ(t) constitute a Hilbert Transform pair. If we
observe the above equations, it is evident that Hilbert transform is
1
.
nothing but the convolution of g(t) with πt
The Fourier Transform of ĝ(t) is computed from signum
function sgn(t).
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2
sgn(t) ←→
jω
1
←→ −jsgn(ω)
=⇒
πt
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Where,


ω>0

1,
sgn(ω) = 0,
ω=0



−1, ω < 0
Since, ĝ(t) = g(t) ∗
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1
πt ,
Ĝ(ω) = G(ω) × −jsgn(ω)
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Properties of Hilbert Transform
1. g(t) and ĝ(t) have the same magnitude spectrum.
2. If ĝ(t) is HT of g(t) then HT of ĝ(t) is −g(t).
3. g(t) and ĝ(t) are orthogonal over the entire interval −∞ to
+∞.
Z
+∞
g(t)ĝ(t) dt = 0
−∞
Complex representation of signals
If g(t) is a real valued signal, then its complex representation g+ (t)
is given by
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g+ (t)
G+ (ω)
= g(t) + jĝ(t)
= G(ω) + sgn(ω)G(ω)
Therefore,
G+ (ω) =



2G(ω), ω > 0



G(0),
ω=0
0,
ω<0
g+ (t) is called pre-evelope and exists only for positive frequencies.
For negative frequencies g− (t) is defined as follows:
g− (t)
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G− (ω)
= g(t) − jĝ(t)
= G(ω) − sgn(ω)G(ω)
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Therefore,



2G(ω),
G− (ω) = G(0),



0,
ω<0
ω=0
ω>0
• Essentially the pre-envelope of a signal enables the suppression
of one of the sidebands in signal transmission.
• The pre-envelope is used in the generation of the SSB-signal.
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