Tele3113 wk4tue

TELE3113 Analogue and DigitalCommunications

SSB Modulation

Wei Zhang

[email protected]

School of Electrical Engineering and Telecommunications

The University of New South Wales

Last week ...

We have studied:

Amplitude Modulation:

s(t) = [1 + kam(t)]c(t).

Simple envelope detection, but low power efficiency.

DSB-SC Modulation:

s(t) = m(t)c(t).

High power efficiency, but requiring a perfect phase

recovery for coherent detection.

Both AM and DSB-SC have two symmetric sidebands in the

modulated wave, thereby causing the wastage of bandwidth.

TELE3113 - SSB Modulation. August 11, 2009. – p.1/13

From DSB-SC to SSB (1)

First, let us review DSB-SC modulation of a single-tonemessage signal m(t) = Am cos(2πfmt). The modulated signal is

sDSB(t) = m(t)c(t) = AmAc cos(2πfmt) cos(2πfct)

=1

2AmAc cos[2π(fc + fm)t] +

1

2AmAc cos[2π(fc − fm)t].

The FT of the DSB-SC modulated signal is given by

SDSB(f) =AmAc

4δ(f − fc − fm) +

AmAc

4δ(f + fc + fm)

+AmAc

4δ(f − fc + fm) +

AmAc

4δ(f + fc − fm).

TELE3113 - SSB Modulation. August 11, 2009. – p.2/13

From DSB-SC to SSB (2)

Suppose that we want to generate a sinusoidal SSB modulated

wave that retains the upper side-frequency at fc + fm. By

suppressing the second term in the equation of sDSB(t), we get

sUSSB(t) =1

2AmAc cos[2π(fc + fm)t].

It can be further expressed as (using the trigonometric identity

cos(x + y) = cos x cos y − sinx sin y)

sUSSB(t) =1

2AmAc cos(2πfct) cos(2πfmt)

−1

2AmAc sin(2πfct) sin(2πfmt).

TELE3113 - SSB Modulation. August 11, 2009. – p.3/13

From DSB-SC to SSB (3)

Suppose that we want to generate a sinusoidal SSB modulated

wave that retains the lower side-frequency at fc − fm. By

suppressing the first term in the equation of sDSB(t), we get

sLSSB(t) =1

2AmAc cos[2π(fc − fm)t].

We further express it as

sLSSB(t) =1

2AmAc cos(2πfct) cos(2πfmt)

+1

2AmAc sin(2πfct) sin(2πfmt).

TELE3113 - SSB Modulation. August 11, 2009. – p.4/13

From DSB-SC to SSB (4)

Combining the equations of sUSSB(t) and sLSSB(t), we get the

SSB modulated wave of a single-tone message signal

m(t) = Am cos(2πfmt) as follows:

sSSB(t) =1

2AmAc cos(2πfct) cos(2πfmt)

∓1

2AmAc sin(2πfct) sin(2πfmt),

where the minus and plus signs apply to the upper SSB and

lower SSB, respectively.

TELE3113 - SSB Modulation. August 11, 2009. – p.5/13

SSB

For a periodic message signal m(t) =∑

mam cos(2πfmt),

the SSB modulated wave is

sSSB(t) =1

2Ac cos(2πfct)

∑

m

am cos(2πfmt)

∓1

2Ac sin(2πfct)

∑

m

am sin(2πfmt).

Generally, for a Fourier transformable message signal m(t),

the SSB modulated wave is

sSSB(t) =1

2Acm(t) cos(2πfct) ∓

1

2Acm̂(t) sin(2πfct),

where m̂(t) is Hilbert transform of m(t). (See next page)TELE3113 - SSB Modulation. August 11, 2009. – p.6/13

Hilbert Transform (1)

The signal m̂(t) is the Hilbert transform of the signal m(t),

defined as

m̂(t) =1

π

∫

∞

−∞

m(τ)

t − τdτ

= m(t) ?1

πt. (convolution)

If m(t) ⇔ M(f), then

m̂(t) ⇔ M̂(f) = −jsgn(f)M(f),

where the sign function is sgn(f) =

1, f > 0

0, f = 0

−1, f < 0TELE3113 - SSB Modulation. August 11, 2009. – p.7/13

Hilbert Transform (2)

tth

π1

)( = )(tm )(ˆ tm

Il lustration of Hilbert transform in time domain

)sgn()( fjfH −= )( fM )(ˆ fM

Illustration of Hilbert transform in frequency domain

TELE3113 - SSB Modulation. August 11, 2009. – p.8/13

Hilbert Transform (3)

Note that the frequency response of Hilbert transformer

h(t) = 1

πtis

H(f) = −jsgn(f).

The magnitude of H(f) is given by

|H(f)| =

1, f > 0

1, f < 0

and the phase is given by

∠H(f) =

−90◦, f > 0

90◦, f < 0

TELE3113 - SSB Modulation. August 11, 2009. – p.9/13

Spectra of SSB

For positive frequencies, the spectra of the two kinds of SSB

modulated waves are as follows:

For the upper SSB,

S(f) =

Ac

2M(f − fc), f ≥ fc

0, 0 < f < fc

For the lower SSB,

S(f) =

0, f ≥ fc

Ac

2M(f − fc), 0 < f < fc

TELE3113 - SSB Modulation. August 11, 2009. – p.10/13

Modulation of SSB (1)

Frequency Discrimination Method

Product modulator

Carrier wave

Band-pass filter

Message signal )(tm SSB-Modulated

signal )(ts

)2cos( tfA cc π

TELE3113 - SSB Modulation. August 11, 2009. – p.11/13

Modulation of SSB (2)

Phase Discrimination Method

Product modulator

)2cos( tfc

π

Product modulator

∑

090− Phase-shi fter

)2sin( tfc

π

Oscillator

Message signal )(tm

Wideband Phase-shifter

)(ˆ tm

m

+

SSB-Modulated signal )(ts

TELE3113 - SSB Modulation. August 11, 2009. – p.12/13

Demodulation of SSB

Product modulator

Local oscillator

Low-pass filter

Modulated wave )(ts )(tv

Demodulated signal )(tvo

)2cos(' φπ +tfAcc

Suppose in the receiver the local oscillator can provide thesame frequency, but arbitrary phase difference φ,

measured with respect to the carrier wave c(t).

It applies equally well to the demodulation of both DSB-SC

and SSB.TELE3113 - SSB Modulation. August 11, 2009. – p.13/13