ECE 453 – Jose Schutt‐Aine 1

ECE 453Single Sideband Modulation

Jose E. Schutt-AineElectrical & Computer Engineering

University of Illinoisjesa@illinois.edu

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( ) 0, 0h t t

( ) ( ) ( )e oh t h t h t

1( ) ( ) ( )2e

h t h t h t

1( ) ( ) ( )2o

h t h t h t

( ), 0( )

( ), 0e

oe

h t th t

h t t

( ) sgn( ) ( )o eh t t h t

Consider a function h(t)

Even function

Odd function

Every function can be considered as the sum of an even function and an odd function

Causality Principle

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( ) ( ) sgn( ) ( )e eh t h t t h t

1( ) ( ) * ( )e eH f H f H fj f

ˆ( ) ( ) ( )e eH f H f jH f

ˆ ( ) is the Hilbert transform of ( )e eH f H fImaginary part of transfer function is related to the real part through the Hilbert transform

In frequency domain this becomes

Hilbert Transform

1 1 ( )ˆ ( ) ( )* HH f H f df f

1sgn( )tj f

Make use of

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( ) ( ) ( ) 2 ( )f t F F t f

ˆsgn( ) ( )t m t jM

( )m t M

ˆ ( ) sgnm t jM

Hilbert Transform

Using symmetry property of Fourier transforms

ˆ

( )

Mis Hilbert transform ofM

ˆ ( )

( )

m tis Hilbert transform ofm t

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Hermitian Property

( ) *f t F F F Real

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Modulation Theorem

1 1( )cos( )2 2

j jc c cA t t e A e A

The spectrum of the modulated signal can be obtained by superimposing two copies of the spectrum of A(t) that have been displaced by +c and –c on the frequency axis.

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Motivation for SSB

• Observations on SSBUSB and LSB are uniquely related to each other, as they are symmetric with respect to fc.Therefore, it is enough to transmit only one side band. For demodulation SSB can be coherently demodulated by multiplying with cos(ct) and followed by LPF.

Amplitude modulation and DSB-SC techniques require transmission bandwidth of twice the bandwidth of the modulating signal m(t).

In both cases the transmission bandwidth is occupied by the upper sideband (USB) and lower sideband (LSB)

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SSB - Frequency Domain Representation

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( ) ( )m t M f USB signal

( ) ( )m t M f LSB signal

SSB - Time Domain RepresentationBaseband modulating signal (real)( ) ( )m t M f

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SSB - Time Domain Representation

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Using the spectrum relationship

1 1( ) ( ) ( ) ( ) 1 sgn( ) ( ) ( )2 2

M f M f u f M f f M f jM f

1 1( ) ( ) ( ) ( ) 1 sgn( ) ( ) ( )2 2

M f M f u f M f f M f jM f

1 1( ) ( )sgn( )2 2

jM f M f f

( ) ( ) sgn( )M f M f j f

SSB – Hilbert Transform

where

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1sgn( )tj f

( ) ( ) sgn( )M f M f j f We have

The Fourier series pair

1 1 ( )ˆ ( ) ( )* mm t m t dt t

ˆ ( )m t is the Hilbert transform of m(t)Thus,

SSB – Hilbert Transform

ˆ ( ) ( )m t M f

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0( ) sgn( )

0j f

H f j fj f

H(f): wideband phase shifter (Hilbert Transform)

SSB – Hilbert Transform

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1 ˆ( ) ( ) ( )2

m t m t jm t

1 ˆ( ) ( ) ( )2

m t m t jm t

ˆ ( )m t is the Hilbert transform of m(t)

SSB – Hilbert Transform

By delaying the phase of every component of m(t) by /2 we get the Hilbert transform of m(t)

Hilbert transformer is ideal phase shifter

ˆ ( )m t

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( ) ( ) ( )USB c cS f M f f M f f

1 1( ) ( ) ( ) ( ) ( )2 2USB c c c c

S f M f f M f f M f f M f fj

The inverse Fourier transform is then

ˆ( ) ( )cos ( )sinUSB c cs t m t t m t t Similarly, we can show that

ˆ( ) ( )cos ( )sinLSB c cs t m t t m t t So, in general, we have

ˆ( ) ( )cos ( )sinSSB c cs t m t t m t t (USB and LSB)

SSB – Time-Domain Representation

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Generation of SSB Signals

• Selective Filtering MethodUse m(t) to generate DSB‐SC (m(t)cosct)Feed DSB‐SC through a band‐pass filter (BPF)We must have B 

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Generation of SSB Signals

Gap between sidebands must exist

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Phase-shift Method

ˆ( ) ( )cos ( )sinSSB c cs t m t t m t t

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Demodulation of SSB Signals

2 ˆ( )cos ( )cos ( )sin cos1 1 1 ˆ( ) ( )cos 2 ( )sin 22 2 2

SSB c c c c

c c

s t t m t t m t t t

m t m t t m t t

If we filter signal with a LPF, we can eliminate components centered at 2fc and filter output will be ~ m(t)

Any DSB-SC coherent demodulation technique can be used

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Demodulation of SSB Signals

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Demodulation of SSB Signals