ssb generation

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AM SSB A.J.Wilkinson, UCT EEE3086F  Signals and Systems II 508 Page 1 June 2, 2012 EEE3086F Signals and Systems II 2012 Andrew Wilkinson [email protected] http://www.ee.uct.ac.za Department of Electrical Engineering University of Cape Town AM SSB A.J.Wilkinson, UCT EEE3086F  Signals and Systems II 508 Page 2 June 2, 2012 5.4 Single Sideband Modulation (SSB) 5.4.1 SSB concepts 5.4.2 SSB generation via sideband filtering 5.4.3 SSB generation using “Phase Shift Method” 5.4.4 Demodulation of SSB 5.4.5 SSB-LC (with carrier) Contents

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Single Side Band Generation AM

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AM SSBA.J.Wilkinson, UCT EEE3086F  Signals and Systems II508 Page 1 June 2, 2012

EEE3086FSignals and Systems II

2012

Andrew Wilkinson

[email protected]

http://www.ee.uct.ac.za

Department of Electrical Engineering

University of Cape Town

AM SSBA.J.Wilkinson, UCT EEE3086F  Signals and Systems II508 Page 2 June 2, 2012

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 Demodulation of SSB5.4.5 SSB-LC (with carrier)

Contents

AM SSBA.J.Wilkinson, UCT EEE3086F  Signals and Systems II508 Page 3 June 2, 2012

5.4.1 SSB Concepts

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 4 June 2, 2012

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 ωωc 12

F ω−ωc

N P

N PN = neg componentsP = pos components

DSB-SC

cc

HzB

USBLSB N P

)(F

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 5 June 2, 2012

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 June 2, 2012

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 June 2, 2012

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 June 2, 2012

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 c

P

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 June 2, 2012

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 June 2, 2012

5.4.2 SSB generation via sideband filtering

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 11 June 2, 2012

SSB Generation Via Filtering (“filtering method”)

Generate DSB-SC Signal

Apply BPF to extract desired sideband.

0

)(tDSB

cosωc t

)(Filter

Sideband

H

)(tf )(tSSB

)(F

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 12 June 2, 2012

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 June 2, 2012

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 June 2, 2012

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 June 2, 2012

SSB Generation: Filter roll off problem

The roll off problem is 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

2BPF

SSB

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 June 2, 2012

Two-stage SSB Transmitter

)(F

2π B0

0

0

0

ω2+ω1

ΦSSB+(ω)

0

0

−ω1 ω1

−ω1 ω1

ω2

First mixer

Output of 1st stage

−ω2

ω2−ω1−(ω2+ω1) −(ω2−ω1)

ω2+ω1

0

−(ω2+ω1)

Output of 2nd stage

2nd mixer

BPF1 (accurately implemented at a lowerfrequency than the final RF signal)

BPF2

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 17 June 2, 2012

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 heterodyne signals to higher frequencies (for all modulation techniques).

ϕSSB

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 June 2, 2012

Generation of SSB Signal (filtering method)

Filtering Method:

ΦSSB(ω )=ΦDSB−SC (ω)⋅H (ω )

FilterSideband

tccos

ϕSSB( t )=[ f ( t )cos ωc t ]⊛ h( t )

BPF)(tf ϕSSB ( t )

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 19 June 2, 2012

Frequency spectrum of SSB generated by Filtering

)()(2

1)(

2

1

)()()(

HFF

H

cc

SCDSBSSB

For the USB case (assuming filter passband gain is 1).

)(2

1)(

2

1)( ccSSB FF

For the LSB case.

)(2

1)(

2

1)( ccSSB FF

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 20 June 2, 2012

Frequency spectrum of SSB generated by Filtering

N

m mP

SSBSidebandUpper

c c P

USB)(SSB

N

ΦSSB+ (ω)=12

F−(ω+ωc )+

12

F +(ω−ωc )

)(2

1cF )(

2

1cF

)(F)()()( FFF

)(F

AM SSBA.J.Wilkinson, UCT EEE3086F  Signals and Systems II508 Page 21 June 2, 2012

5.4.3 Alternative method for generating SSB using the “Phase Shift Method”

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 22 June 2, 2012

Generation of SSB+ Signal (phase shift method)

Let

where represents the negative components, and represents the positive components.

An SSB+ Fourier spectrum can be constructed from*:

Inverse transforming we get

)()()( FFF

)()()( ccSSB FF

)(F)(F

tjtjSSB

cc etfetft )()()(

)()(

)()(

Ftf

Ftf

*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 June 2, 2012

N

m mP

SSBSidebandUpper

c c P

USB)(SSB

N

)()()( ccSSB FF

)( cF )( cF

)(F)()()( FFF

)(F

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 24 June 2, 2012

Generation of SSB+ Signal (phase shift method)

ttfttf

ttjftjfttftf

ttjfttfttjfttf

etfetft

cc

cc

cccc

tjtjSSB

cc

sin)(ˆcos)(

sin)()(cos)()(

sin)(cos)(sin)(cos)(

)()()(

and )()()( tftftf )()()(ˆ tjftjftf where

F { f t }=F ω=− jFω jF−

ω

={− jF ω for ω≥0jF ω for ω0

If we transform,we get:

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 25 June 2, 2012

Hilbert Transform

H (ω )={− j for ω≥0j for ω<0

H (ω )={e− jπ /2

for ω≥0e

jπ /2

for ω<0

Re-expressed as:

We see that this operation is a -90 deg phase shifter, operating over all frequency components in F().

-90 deg)(tf

)(tf)(H

)(ˆ tf

)(ˆ tf

The frequency domain operations can be expressed as a transfer function operation, known as the “Hilbert Transform”

(the Hilbert Transform of f(t) )

2/

2/)}(arg{ H

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 26 June 2, 2012

Generation of SSB Signal

ttfttft ccSSB sin)(ˆcos)()(

The ^ indicates that each frequency component in F(ω) is delayed by 900

ttfttft ccSSB sin)(ˆcos)()(

A similar analysis for generating lower sideband SSB, reveals

Upper sideband SSB:

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 27 June 2, 2012

Hardware Implementation of Phase Shift Method (SSB)(known as the “Hartley Modulator”)

tccos090

)(tf

090

ttf csin)(ˆtcsin

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 June 2, 2012

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:

tttt

tt

cmcm

cmSSB

sinsincoscos

])cos[()(

tttt

tt

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 June 2, 2012

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 lower or the upper sideband.

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.

AM SSBA.J.Wilkinson, UCT EEE3086F  Signals and Systems II508 Page 30 June 2, 2012

5.4.4 Demodulation of SSB

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 31 June 2, 2012

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 32 June 2, 2012

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 33 June 2, 2012

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 34 June 2, 2012

Demodulation of SSB Signal

ttfttftf cc 2sin)(ˆ2

12cos)(

2

1)(

2

1

)(2

1)( LPFofOutput 0 tfte

tttfttftt ccccSSB cossin)(ˆcos)(cos)( 2

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 35 June 2, 2012

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− f 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 36 June 2, 2012

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 37 June 2, 2012

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 t

⇒ f t =− j ejωm t

f t e− jθ

f t

f t

ωm θ

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 38 June 2, 2012

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 39 June 2, 2012

Case of frequency Error (i.e. , )

Considering a single frequency component:

e0 t =12

f t cos Δωt12

f t sin Δωt f t =e jωm t

e0 t =12

e jωm t cos Δωt12

− j e jωm t sin Δωt

=12

ejωm t

cos Δωt− j sin Δωt

=12

ejωm t

e− jΔωt

=12

e j ωm−Δω t

freq shift error

Δω

0 0

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 40 June 2, 2012

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 41 June 2, 2012

Demodulation of SSB – Freq Domain Perspective

Frequency domain perspective on oscillator phase and frequency errors.

Let

Let

Let φd t =cos [ ωcΔω tθ ]

φd ω =πe− jθ δ ωωcΔω πe jθ δ ω−ωc−Δω

F ω

F ω =Fω F−ω

φSSBω =Fω−ωc F−ωωc

F−ω

0 ω

(demodulator oscillator)

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 42 June 2, 2012

Demodulation of SSB

)( cF

c c

c c

)(SSB

0

0

cc

)( cF

Oscillator With Phase and Frequency Error (neg freq error))(d

jeF )(21

∣Δω∣0

)(0 e jeF )(

2

1

je jeConvolve:

Output

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 43 June 2, 2012

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)

e0ω =12

FωΔω e− jθ 12

F−ω−Δω e jθ

AM SSBA.J.Wilkinson, UCT EEE3086F  Signals and Systems II508 Page 44 June 2, 2012

5.4.5 Single Sideband Large-Carrier (SSB-LC)

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 45 June 2, 2012

SSB-LC (Large Carrier SSB)

Allows recovery via envelope detection.

Needs larger carrier than DSB-LC (even more wasteful of power).

carrier SSB

ttfttftAt ccc sin)(ˆcos)(cos)(

envelope )(tr )(tr

Atf )(

)(ˆ tf

ωc

Phasorrepresentation

A.J.Wilkinson, UCT AM SSB EEE3086F  Signals and Systems II508 Page 46 June 2, 2012

SSB-LC (Large Carrier SSB)

22 )](ˆ[)]([)( tftfAtr

φ( t )=r ( t )cos [ωc t+ θ ( t )]

ttfttfAt cc sin)(ˆcos)()(

A cos x+ B sin x=C cos( x+ θ )

where C=√ A2+ B2

and 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 47 June 2, 2012

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

+f̂ 2( 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 48 June 2, 2012

SSB-LC (Large Carrier SSB)

r t ≈A f t Thus

x << 1

(1+ x)n=1+ n x+

12!

n(n−1) x2+ ⋯

r (t )≈ A [1+ 2 f (t )A ]

12 =A [1+

12⋅2 f (t )A

+ ⋯] ≈ A [1+

f (t )A ]

for A>> f t

x≡2f ( t )

A

Thus 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 49 June 2, 2012

EEE3086FSignals and Systems II

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