Lecture 6: Signals Transmission

Aristotle University of ThessalonikiAristotle University of Thessaloniki – – Department of Geodesy and SurveyingDepartment of Geodesy and Surveying

A. DermanisA. Dermanis Signals and Spectral Methods in GeoinformaticsSignals and Spectral Methods in Geoinformatics

Lecture 6:Signals Transmission

Signals and Spectral Methodsin Geoinformatics



Signal transmission

1 MODULATION : Placing the signal on a monochromatic signal (carrier frequency)

2 TRANSMISSION

3 RECEPTION

4 DEMODULATION : Signal recovery (removal of carrier frequency)



modulation



modulation

Modulation = placement of signal m(t) on a monochromatic signal xC(t) = aC cos(φ0C+ωCt) with carrier frequency ωC



modulation


Α. Amplitude modulation (general form) : )cos()()( 0 ttatx CC



modulation


Α. Amplitude modulation (general form) : )cos()()( 0 ttatx CC

m(t)



modulation


Α. Amplitude modulation (general form) :

Β. Angle modulation (general form) :

)cos()()( 0 ttatx CC

)](cos[)( 0 ttatx CCC

m(t)



modulation






m(t)



modulation






Α. AM = Amplitude Modulation :

m(t)



modulation







)()( tmkAta a

m(t)



modulation







)()( tmkAta a )cos()]([)( 0 ttmkAtx CCa

m(t)



modulation








Β. Angle modulation

m(t)



modulation









Β1. PM = Phase Modulation :

Β2. FM = Frequency Modulation :

m(t)



modulation











)()( tmkt p

m(t)



modulation











)()( tmkt p )](cos[)( 0 tmktatx pCCCC

m(t)



modulation












)()()( tmktdt

dt f

m(t)



modulation












)()()( tmktdt

dt f

])()(cos[)(

0

0 t

tfC dttmktttx

m(t)



modulation












)()()( tmktdt

dt f

])()(cos[)(

0

0 t

tfC dttmktttx

)(t

m(t)



Example: Modulation of a sinusoidal signal m(t) = cosωt



signal to be modulated


)(tm




carrier frequency


)(tm

)cos( tC




carrier frequency

amplitude modulation


)cos()]([)( ttmkAtx Ca

)(tm

)cos( tC

AM




carrier frequency


phase modulation



])(cos[)( tmkttx fC

)(tm

)cos( tC

AM

PM




carrier frequency


phase modulation

frequency modulaion



])(cos[)( tmkttx fC

])(cos[)(0

t

tpC dttmkttx

)(tm

)cos( tC

AM

PM

FM




carrier frequency


phase modulation

frequency modulaion



])(cos[)( tmkttx fC

dt

d

])(cos[)(0

t

tpC dttmkttx

dt

)(tm

)cos( tC

AM

PM

FM



demodulation



demodulation

Demodulation = separation of main signal m(t) from the received modulated signal x(t)



demodulation


Spectrum of signal m(t) = Fourier transform : dtetmM ti

)()(



demodulation



)()(

)(tm



demodulation



)()(

)(tm

)(tmA A



demodulation



)()(

)(tm

)(tmA A)(M0M

A2

mm 0



demodulation



)()(

)cos()()cos()cos()]([)( ttmtAttmAtx CCC

)(tm

)(tmA

)(tx

)(tmA A)(M0M

A2

mm 0



demodulation



)()(


)()()(2

1)(

2

1)( CCCC AAMMX

)(tm

)(tmA

)(tx

)(X

A02

1 M

CC 0

)(tmA A)(M0M

A2

mm 0



demodulation



)()(


)()()(2

1)(

2

1)( CCCC AAMMX

Properties used :

)(2)(21 AA

)(tm

)(tmA

)(tx

)(X

A02

1 M

CC 0

)(tmA A)(M0M

A2

mm 0



demodulation



)()(


)()()(2

1)(

2

1)( CCCC AAMMX

Properties used :

)(2)(21 AA

Modulation theorem

)]()([2

1cos)(

)()(

000

ZZtz

Ztz

)(tm

)(tmA

)(tx

)(X

A02

1 M

CC 0

)(tmA A)(M0M

A2

mm 0



demodulation



)()(


)()()(2

1)(

2

1)( CCCC AAMMX

Properties used :

)(2)(21 AA

Modulation theorem

)]()([2

1cos)(

)()(

000

ZZtz

Ztz

from which follows

)()()cos( CCC AAtA

)(2

1)(

2

1)cos()( CCC MMttm

)(tm

)(tmA

)(tx

)(X

A02

1 M

CC 0

)(tmA A)(M0M

A2

mm 0



Double Band demodulation




)(2

1)(

2

1)()cos()()( CCC MMXttmtx

Modulation = multiplication of the signal m(t) with the carrier frequency cosωC t




)(2

1)(

2


Demodulation = multiplication again with the carrier frequency cosωC t + low pass filter


)(cos)()()cos()()( 2 ttxtmttxtd CC )(2

1)(

2

1)( CC XXD




)(2

1)(

2





1)(

2

1)( CC XXD

)(2

1)(

2

1)( CC MMX




)(2

1)(

2





1)(

2

1)( CC XXD

)(2

1)(

2

1)( CC MMX

)(2

1)2(

2

1)(

2

1)(

2

1)( MMMMX CCCCCCω ωωC




)(2

1)(

2





1)(

2

1)( CC XXD

)(2

1)(

2

1)( CC MMX

)(2

1)2(

2

1)(

2

1)(

2

1)( MMMMX CCCCCC

)2(2

1)(

2

1)(

2

1)(

2

1)( CCCCCC MMMMX ω ω+ωC

ω ωωC




)(2

1)(

2





1)(

2

1)( CC XXD

)(2

1)(

2

1)( CC MMX

)(2

1)2(

2

1)(

2

1)(

2

1)( MMMMX CCCCCC

)2(2

1)(

2

1)(

2

1)(

2

1)( CCCCCC MMMMX

)2(

2

1)(

2

1

2

1)(

2

1)2(

2

1

2

1)( CC MMMMD

ω ω+ωC

ω ωωC




)(2

1)(

2





1)(

2

1)( CC XXD

)(2

1)(

2

1)( CC MMX

)(2

1)2(

2

1)(

2

1)(

2

1)( MMMMX CCCCCC

)2(2

1)(

2

1)(

2

1)(

2

1)( CCCCCC MMMMX

)2(

2

1)(

2

1

2

1)(

2

1)2(

2

1

2

1)( CC MMMMD

)2(4

1)(

2

1)2(

4

1)( CC MMMD

ω ω+ωC

ω ωωC




)(2

1)(

2





1)(

2

1)( CC XXD

)(2

1)(

2

1)( CC MMX

)(2

1)2(

2

1)(

2

1)(

2

1)( MMMMX CCCCCC

)2(2

1)(

2

1)(

2

1)(

2

1)( CCCCCC MMMMX

)2(

2

1)(

2

1

2

1)(

2

1)2(

2

1

2

1)( CC MMMMD

)2(4

1)(

2

1)2(

4

1)( CC MMMD

After the low pass filter remains : )(2

1)(

2

1tmM

ω ω+ωC

ω ωωC






original signal

| M(ω) | Double Band demodulation



original signal

modulated signal

MODULATION

| M(ω) |

| Χ(ω) |

ωCωC




original signal

modulated signalTRANSMISSION - RECEPTION

MODULATION

| M(ω) |

| Χ(ω) |

ωCωC




original signal


MODULATION

DEMODULATION

Multiplication with carrier frequency

| M(ω) |

| Χ(ω) |

ωCωC

| D(ω) |

2ωC2ωC




original signal


MODULATION

DEMODULATION


Application of low pass filter

| H(ω) |

| M(ω) |

| Χ(ω) |

ωCωC

| D(ω) |

2ωC2ωC




original signal


MODULATION

DEMODULATION


Application of low pass filter

| H(ω) |

½ | M(ω) |

| M(ω) |

| Χ(ω) |

ωCωC

| D(ω) |

2ωC2ωC




Double Band demodulation- preservation of outer parts



| M(ω) | Double Band demodulation- preservation of outer parts



Modulation = multiplication with cosωCt+ high pass filter

| M(ω) |

| Χ(ω) |

ωCωC





| M(ω) |

| H(ω) || Χ(ω) |

ωCωC





modulated signal

| M(ω) |

| H(ω) |

| Χ(ω) |

ωCωC

| Χ(ω) |

ωCωC





Demodulation = multiplication with cosωCt+ low pass filter

modulated signal

| M(ω) |

| H(ω) |

| Χ(ω) |

ωCωC

| Χ(ω) |

ωCωC

| D(ω) |

2ωC2ωC






modulated signal

| M(ω) |

| H(ω) |

| H(ω) |

| Χ(ω) |

ωCωC

| Χ(ω) |

ωCωC

| D(ω) |

2ωC2ωC






modulated signal

demodulated signal

| M(ω) |

¼| M(ω) |

| H(ω) |

| H(ω) |

| Χ(ω) |

ωCωC

| Χ(ω) |

ωCωC

| D(ω) |

2ωC2ωC




Double Band demodulation- preservation of inner parts



| M(ω) | Double Band demodulation- preservation of inner parts



| M(ω) |

| Χ(ω) |

ωCωC


Modulation = multiplication with cosωCt+ low pass filter



| H(ω) |

| M(ω) |

| Χ(ω) |

ωCωC





| H(ω) |

| M(ω) |

| Χ(ω) |

ωCωC

| Χ(ω) |

ωCωC



modulated signal



| H(ω) |

| M(ω) |

| Χ(ω) |

ωCωC

| D(ω) |

2ωC2ωC

| Χ(ω) |

ωCωC



Demodulation = multiplication with cosωCt+ high pass filter

modulated signal



| H(ω) |

| M(ω) |

| H(ω) |

| Χ(ω) |

ωCωC

| D(ω) |

2ωC2ωC

| Χ(ω) |

ωCωC




modulated signal



| H(ω) |

| M(ω) |

| H(ω) |

| Χ(ω) |

ωCωC

| D(ω) |

2ωC2ωC

| Χ(ω) |

ωCωC

¼| M(ω) |




modulated signal

demodulated signal



multiplexing



multiplexing

band limited signals :

m

mmMtm||0

0)()(



multiplexing


m

mmMtm||0

0)()(

(spectrum concentrated in a band 2ωm wide centered at zero)

| M(ω) |

ωmωm



multiplexing


m

mmMtm||0

0)()(


Modulation : 0)()cos()()( Xttmtx C mCmC

mCmC

για

και

(spectrum concentrated in two bands 2ωm wide centered at –ωC and ωC )

| M(ω) |

ωmωm



multiplexing


m

mmMtm||0

0)()(



mCmC

για

και


ωCωC

| Χ(ω) || M(ω) |

ωmωm



multiplexing


m

mmMtm||0

0)()(



mCmC

για

και


ωCωC

| Χ(ω) || M(ω) |

ωmωm

ωC ωmωC ωm ωC ωmωC ωm



multiplexing


m

mmMtm||0

0)()(



mCmC

για

και




multiplexing


m

mmMtm||0

0)()(



mCmC

για

και


THEREFORE: Possibility of simultaneous moduletion of more signalsas long as there spectra do not overlap !

Separation with band pass filters + (usual) demodulation



multiplexing


m

mmMtm||0

0)()(



mCmC

για

και




Signals to be transmitted : )(,),(),( 21 tmtmtm n



multiplexing


m

mmMtm||0

0)()(



mCmC

για

και




Signals to be transmitted :

Corresponding carrier frequencyes :

)(,),(),( 21 tmtmtm n

n ,,, 21



multiplexing


m

mmMtm||0

0)()(



mCmC

για

και






Modulated signals :

)(,),(),( 21 tmtmtm n

n ,,, 21

)cos()(,),cos()(),cos()( 2211 ttmttmttm nn



multiplexing


m

mmMtm||0

0)()(



mCmC

για

και






Modulated signals :

Multiplexing = sum of modulated signals with non overlapping spectra

)cos()()cos()()cos()()( 2211 ttmttmttmtx nn

)(,),(),( 21 tmtmtm n

n ,,, 21

)cos()(,),cos()(),cos()( 2211 ttmttmttm nn



multiplexing



)(1 M

)(2 M

)(3 M

)(1 m

)(2 m

)(3 m

multiplexing



)(1 M

)(2 M

)(3 M

)(1 m

)(2 m

)(3 m

~1cos

~2cos

~3cos

multiplexing



)(1 M

)(2 M

)(3 M

)(1 m

)(2 m

)(3 m

~1cos

~2cos

~3cos

multiplexing



)(1 M

)(2 M

)(3 M

)(1 m

)(2 m

)(3 m

~1cos

~2cos

~3cos

)(tx

multiplexing



)(1 M

)(2 M

)(3 M

)(1 m

)(2 m

)(3 m

~1cos

~2cos

~3cos

)(tx

)(X

1 1 22 33

multiplexing



)(1 M

)(2 M

)(3 M

)(1 m

)(2 m

)(3 m

~1cos

~2cos

~3cos

)(tx

)(X

1 1 22 33

BPF = Band Pass Filter (inside band)

multiplexing



)(1 M

)(2 M

)(3 M

)(1 m

)(2 m

)(3 m

~1cos

~2cos

~3cos

)(tx

)(X

1 1 22 33

BPF = Band Pass Filter (inside band)

multiplexing



)(1 M

)(2 M

)(3 M

)(1 m

)(2 m

)(3 m

~1cos

~2cos

~3cos

)(tx

)(X

1 1 22 33

multiplexing



multiplexing – mathematical description




)(2

1)(

2

1)(

2

1)(

2

1)(

2

1)(

2

1)( 22221111 nnnn MMMMMMX

)(tx





)(2

1)(

2

1)(

2

1)(

2

1)(

2

1)(

2


BPF)(tx

BPF

BPF

Application of band pass filter (BPF, inside band) == preservation of a single term :

)cos()()()(2

1)(

2

1)( ttmtxMMX kkkkkkkk




~2cos


)(2

1)(

2

1)(

2

1)(

2

1)(

2

1)(

2


BPF

~3cos

~1cos

LPF

)(tx

BPF

BPF

LPF

LPF


)cos()()()(2

1)(

2


Usual demodulation =

= [ cosωi ] + [ LPF ] =

= retrieval of signal mk(t)




~2cos


)(2

1)(

2

1)(

2

1)(

2

1)(

2

1)(

2


BPF

~3cos

~1cos

LPF

)(tx

BPF

BPF

LPF

LPF

BPF = Band Pass Filter, inside band

LPF = Low Pass Filter


)cos()()()(2

1)(

2



= [ cosωi ] + [ LPF ] =





~2cos


)(2

1)(

2

1)(

2

1)(

2

1)(

2

1)(

2


BPF

)(1 m

)(2 m

)(3 m

~3cos

~1cos

LPF

)(tx

BPF

BPF

LPF

LPF

BPF = Band Pass Filter, inside band

LPF = Low Pass Filter


)cos()()()(2

1)(

2



= [ cosωi ] + [ LPF ] =





heterodyning



heterodyning

Heterodyning = Mixing (sum) of received signal having frequency f, with a signal of slightly different frequency fR produced in the receiver ( fR f )



heterodyning

Rfff RRfff 222




heterodyning

Rfff RRfff 222

)sin()sin()2sin()2sin()( 0000 RRRRRR tatatfatfatx




heterodyning

Rfff RRfff 222


:RΜε


2

sin2

sinsin2

sin)( 00 ttattaatx RR



signal of frequency f (angular ω) with amplitude which varies periodicallywith frequency Δf angular Δω)

heterodyning

Rfff RRfff 222


:RΜε


2

sin2

sinsin2




heterodyning

Rfff RRfff 222


:RΜε


2

sin2

sinsin2


signal of frequency f (angular ω) with amplitude which varies periodically with frequency Δf angular Δω)



heterodyning

Rfff RRfff 222


:RΜε

2

sin2

sinsin2






heterodyning

Rfff RRfff 222


:RΜε

Δf = f fR (angular Δω = ω ωR ) = beat frequency

2

sin2

sinsin2






heterodyning

Rfff RRfff 222


:RΜε


Δf = f fR (angular Δω = ω ωR ) = beat frequency

Application: observations in space geodesy utilizing the Doppler phaenomenon (variation of frequency caused by the variation of the receiver-transmitter relative position)

2

sin2

sinsin2





Beat frequency :

Δf = f – fR = 1 – 5/6 = 1/6

(TΔf = 6)

T = 1 f = 1

TR = 6/5 fR = 5/6

Δf = f fR = 1/6

TΔf = 6

Example :

Received frequency : f = 1 (T = 1)

Frequency at receiver : fR = 5/6 (T = 6/5)

8

6

4

2

-2

-4

-6

-8

2 4 86 10 12 14 16

2 4 86 10 12 14 16

2 4 86 10 12 14 16

4

2

-2

-4

4

2

-2

-4

0

0

0



END

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Lecture 6: Signals Transmission