5.4 single sideband modulation (ssb)castor.am.gdynia.pl/~pik/prokom/am_ssb.pdf · 2019. 2. 10. ·...

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


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


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


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


Spectrum of DSB-SC signal

sidebandLower

sidebandUpper

N

)(F

m

SCDSB

sidebandLower

sidebandUpper

m

c c

P

N P N P

ωm=2π B


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.


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


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.


5.4.2 SSB generation via sideband filtering


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


SSB Generation Via Filtering

c

ΦDSBω

0

0

H ω c

c c

c

ΦSSB ω

0 c

Sideband filter


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


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


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


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 ⊛


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


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 )


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 )


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


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

(known as the “Hartley Modulator”)


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.


N

m mP

SSBSidebandUpper

c c P

USB)(SSBN

)()()( ccSSB FF

)( cF )( cF

)(F)()()( FFF

)(F


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


Hilbert Transform

H (ω )={− j for ω≥0j for ω


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+


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


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


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


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


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


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)


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 )


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.


5.4.5 Demodulation of SSB


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


Demodulation of SSB+ Signal

)(SSB

)(tSSBtccos LPF )(0 te

c c

c2 c2

LPF

0

0

cc

0 convolve

Upper sideband

12π⊛


Demodulation of SSB- Signal

)(tSSBtccos LPF )(0 te

c2

LPF

0 c

)(SSB

c 0

0 convolve

Lower sideband

c

c2c

1

2π⊛


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


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 ]}


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


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


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


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


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.


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)


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


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


5.4.6 Single Sideband Large-Carrier (SSB-LC)


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



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



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



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..


EEE3086FSignals and Systems II

End of handout

5.4 single sideband modulation (ssb)castor.am.gdynia.pl/~pik/prokom/am_ssb.pdf · 2019. 2. 10. ·...

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