Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What...

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Modulation/demodulation (1) Why modulate?

To match the signal to the available transmission channel What is modulation/demodulation?

Linear/non-linear up-conversion/down-conversion of the information-bearing signal

What characterises a linear modulation?

The spectral shape of the modulating signal is preserved, it is only shifted to the carrier frequency The bandwidth of the modulate signal is typically B = 2Wx The modulated signal envelope varies as a function of time

�� ( ) Re ( )exp 2 ( )cos 2 ( )sin 2c p s q cs t z t j f t z t f t z t f t� � ��

( ) ( ) ( )p qz t z t jz t� �� is the complex modulating signal

The later expression gives the quadrature representation

Modulation in radio communication systems Important considerations bandwidth efficiency, spectrum efficiency noise interference tolerance nonlinear amplification tolerance implementation factors Reference receiver error performance in AWGN-channel, assumptions: additive white zero-mean Gaussian noise the receiver filter is matched to the noiseless received digital pulse shape or

correlation is performed with the noiseless received digital pulse shape single symbol or ISI-free transmission Problems when applying reference receiver performance in radio systems multipath propagation �� ISI present dynamic channel �� channel estimation needed for matched filtering AWGN-model not valid for radio interference carrier and symbol timing recovery methods impact performance received power variations �� average BER, BER-statistics important

Modulation/demodulation (2) What characterises a non-linear modulation?

The spectral shape of the modulating signal is changed The bandwidth of the modulate signal is typically B > 2Wx The modulated signal envelope is mostly constant

�� ( ) Re 2 exp 2 exp ( ) 2 cos 2 ( )tx c tx cs t P j f t jf x t P f t f x t� ��

where �� f mostly is a linear function (integration, convolution) Generalized modulation �� ( ) Re ( )exp 2 exp ( )cs t z t j f t jf x t�� contains both linear and non-linear modulation Every modulation can be generated by a quadrature modulator, where the modulating signal is an unprocessed/linearly/non-linearly processed information signal

Modulation/demodulation (3)

The basic digital quadrature modulator

pulse shapingfilter

��/2CG

�� kk

a t kT ��

�� kk

b t kT ��

�� cos 2 cf t��

�� sin 2 cf t��

�� kk

a x t kT��

�� kk

b x t kT��

��

a x t kT f t

b x t kT f t

��

� ��

Modulation/demodulation (4)

The offset digital quadrature modulator

pulse shapingfilter

��/2CG

�� kk

a t kT ��

�� kk

b t kT ��

�� kk

a x t kT��

�� 0.5kk

b x t T kT� ��

��

0.5 sin 2

a x t kT f t

b x t T kT f t

��

� � ��

delayT/2

Modulation/demodulation (5) The ��/4-shifted digital quadrature modulator

The staircase function �� 1

0stc( , ) u

k kt T t kT

��

pulse shapingfilter

��/2CG

�� kk

a t kT ��

�� kk

b t kT ��

�� kk

a x t kT��

�� kk

b x t kT��

��

cos 2 stc( , )4

sin 2 stc( , )4

a x t kT f t t T

b x t kT f t t T

��

� � � � � ��

� ��

� � �� stc( , )

��

QAM-methods (Quadrature Amplitude Modulation) QAM is a linear modulation where the expression for the modulated signal is:

2( ) Re ( ) ( ) cj f tk k

ks t a jb x t kT e ��

��

� ��

�� ( )cos(2 ) ( )sin(2 )k c k ck

a x t kT f t b x t kT f t� ��

�� (2)

where - , 1, 3, ... ( 1), ( ) ( ) 1/k k k ka b M where P a P b M� � � � � � �� 0,k k k k k l k l klE a b E a E b E a a E b b � � � �� - x(t) is the base-band pulse.

5 4 3 2 1 0 1 2 3 4 55

5Phasor diagram of 16QAM with raised cosine filtering, �� = 0.25

5 4 3 2 1 0 1 2 3 4 55

Constellation diagram of 16QAM with ideal decision sampling, �� = 0.25

Basic QAM-constellations having about same average power4QAM 16QAM

64QAM 256QAM

Examples of non-quadratic QAM-constellations

8QAM 32QAM

128QAM 512QAM

QPRS-constellations

9QPRS 49QPRS 225QPRS(4QAM ) (16QAM ) (64QAM )

(Quadrature Partial Response Signaling)

MQAM_MPSK_MQPRS.dsf

MPSK. (M-ary Phase Shift Keying)

Phase modulation representation:

s t P j f t a x t kTc c kk

( ) Re exp ( ( ))��

��

��2 2��

0.5( ) ,t Tx t rectT M

��

� ��

Quadrature modulation representation t kT k T!!!! ��, ( )1 :

�� 0.5( ) 2 cos cos 2c k ck

t T kTs t P a rect f tM T��

��

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�� 0.5sin sin 2k ct T kTa rect f t

M T��

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MPSK-CONSTELLATIONS2PSK 4PSK

8PSK 16PSK

An alternative MQAM modulator

5 3 1 1 3 5

7 9��7��9��9

��7

4QAMmodu-lator

A12 dB

(((( 64QAM

Carrier demodulator Tasks: To demodulate the modulating signal To produce signal quadrature components for coherent demodulation/data

detection Carrier recovery for coherent demodulation Analog to digital conversion (ADC) of the signal components Characteristics: Coherent detection is a multiplication procedure Low-pass filtering rejects the mixing result at twice the IF-frequency Sampling for ADC at symbol rate Problems: Phase quadrature over the whole signal bandwidth Quadrature amplitude balance over the whole signal bandwidth Recovered carrier phase bias and noise

The quadrature demodulator

pulse shapingfilter

��/2LO

�� kk

a t kT ��

�� kk

b t kT ��

�� kk

a x t kT��

�� kk

b x t kT��

powerdivider

dec.circuit

ADC dec.circuit

controlsignals

Optimum receiver bit error probability BEP-expression are derived under certain given conditions: single symbol reception or intersymbol-interference free sequence reception, The pulse waveforms used for each possible symbol are represented by

minimum size orthonormal function set spanning a vector space, receiver filter(s) matched to received pulse waveforms, Maximum a posteriori signal (MAP) processing )))) the vector space is

divided into non-overlapping decision sub-spaces for each symbol )))) the decision will be the symbol corresponding to the subspace where the noisy received signal vector is )))) minimum symbol error probability,

white Gaussian noise, optimum bit combination mapping to the symbols (Gray coding) fully known received carrier phase/four-fold phase ambiguity, in case of average SEP in the Rayleigh-fading AWGN channel, the channel

amplitude will not change during one symbol, but all channel states are visited during the period over which SEP is determined

��

� � � � ��

��

1 Q 1 Q

P C m z z

P E m z z

� � ��

� �� MQAM_MPSK_MQPRS.dsf

4 ( M - 2) ��

� � � � ��

MQAM_MPSK_MQPRS.dsf

��

�� 2

1 Q( ) 1 2Q( )

3Q( ) 2Q ( )

P C m z z

P E m z z

� � ��

� ��

� � � � ��

��

�� 3

1 2Q( ) 1 2Q( )

4Q( ) 4Q ( )

P C m z z

P E m z z

� � ��

� �� MQAM_MPSK_MQPRS.dsf

�� 22M ��

� �� !��"�!�!��

�� 1 1

1M Ms k k k

k kP E P m P E m P E m

M� ��

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m m mN P E m N P E m N P E mM

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1 4 2Q( ) ( ) 4 2 3Q( ) 2 ( )

2 4Q( ) 4 ( )

z Q z M z Q zM

M z Q z

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%%%%� � �� &&&&''''

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1 8 12 24 4 16 16 Q( )

4 8 16 4 16 16 Q ( )

M M M zM

M M M z

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%%%%� � � � � � �� ''''

� � �� 21 4 4 Q( ) 4 8 4 Q ( )M M z M M zM

" %" %" %" %� � � � �� $ '$ '$ '$ '

� � �� 2 24 Q 1 Qn n

d dM M MM * ** ** ** *

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22 21 14 1 Q 4 1 Qo os

E EP EN NM M

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�� %&� � �&�� '(&� � �

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MQAM_MPSK_MQPRS.dsf

QAM-error probability in the LTI-AWGN-channel With matched filtering and a fully known carrier phase and ideal symbol timing the symbol error probability of general quadratic QAM is

221 3 1 34 1 4 1

1 1sP Q QM MM M

+ ++ ++ ++ +� � � � � � � � � � � � � � � �

� � � � � ��

++++ ��

With Gray-coding (giving a minimum of different bits in adjacent symbols) the bit error probability (BEP) in 4QAM and 16QAM is:

�� ,4b QAMP Q ++++��

�� ,16 0.75 0.2 0.5 1.8 0.25 12.5b QAMP Q Q Q+ + ++ + ++ + ++ + +� � ��

. �//��

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log 4 1 3 (0)1 1 Q

log 12 1

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+�� ,�-� ��

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sinrxE � � � � � ��

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/ � ��

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� �� 2

1sin 2

Q Q Q sin2

E M EdP e m

N N M��

� � � � � � � � � � � � � � � � � ��

/ � �� ' ��

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/ � �� ' ��

' ��

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( ) ( )

22Q sin

P e m P e P r A B

P r A P r B P r AB

EP r A P r B

N M��

� � ! �� ! �� ! �� ! �

� ! � ! � !� ! � ! � !� ! � ! � !� ! � ! � !

� � � � � � � � . ! � ! �. ! � ! �. ! � ! �. ! � ! � � ��

/ � ��

�2 2

( ) 22( ) Q sin

log log

2 log ( )2Q sin

P e EP e

M M N M

E MM N M

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24( ) Q sin

2 log ( )4Q sin

rxb de

E MM N M

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/ � ��

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y(t)r(t)

/ � ��

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� / � ��

r t x t kT f t a w t

x t kT f t a

x t kT f t a w t

x t kT f t a isi t w t

c k ok

c k oll k

c k o k

( ) ( ) cos ( )

( ) cos

( ) cos ( )

( ) cos ( ) ( )

��

��////

��

� �

3��

� ak ��$��$��&�� 1, ��3,...,��(M – 1) � o �� &�

� w t n t f t n t f tc c s c( ) ( ) cos ( ) sin�� 2 2�� 4 �� ' ��

�� fc �� isik �� T = 1/Rs ��

/ � ��

r t T x t kT f t T a w t T

x t kT f t a

x t kT f t a w t T

x t kT f t a isi t w t T

c k ok

c k oll k

c k o k

( ) ( ) cos ( ) ( )

( ) cos

( ) cos ( )

( ) cos ( ) ( )

��

��////

��

� �

��

r t r t T x t kT f t a isi t w t

x t kT f t a isi t w t Tc k o k

c k o k

( ) ( ) ( ) cos ( ) ( )

( ) cos ( ) ( )

��

� �

5 �� #0 #�� ' ��

�� 21( ) cos 2 ( )rx c k k

t kTy t P f T a a xT

� ��

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#� 2�� f Tc ��

�� 21( ) cos ( )rx k k

t kTy t P a a x

T��

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ak ak-1 y –1 –1 –2Prx –1 +1 0 +1 –1 0 +1 +1 +Prx

• #�� , y = 0 / � �� , y //// 0��

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/ � �� &� � ��% �0 1�

-.�0 1&��

�� 0.5expbP ++++� ��

-$�0 1&��

11 Q 1 sin , 1 sin

Q 1 sin , 1 sin

bPM M M

� �� + ++ ++ ++ +

"""" � � � � � � � � � � � � - � � �- � � �- � � �- � � �#### � �� #### � �� $$$$

%%%%� � � � � � � � � � � � � � �� &&&&� �� &&&&� �� ''''

% !�&�"��$��6��% ��' ��

�� 2 2

0Q( , ) exp I

x y xx y xy

��

� � � � � � � � ��

�� !�� %&� � � �

Transmitted constellation

Possible received constellations withphase unambiguuity

$ ��

Adding a pilot signal Changing one constellation point

Adding a learning sequence Differential encoding and decoding

symbol phase change

. �44��1 � �� 1 5� . �. �� 0�

� -��

diff.encoder

diff.decoder

k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20ak 2 3 1 2 0 2 1 3 3 3 3 2 0 0 1 2 1 0 0 1 bk-1 0 2 1 2 0 0 2 3 2 1 0 3 1 1 1 2 0 1 1 1 bk 0 2 1 2 0 0 2 3 2 1 0 3 1 1 1 2 0 1 1 1 2 phase state 1 ck 0 2 1 2 0 0 2 3 2 1 0 3 1 1 1 2 0 1 1 1 2 dk 0 2 3 1 2 0 2 1 3 3 3 3 2 0 0 1 2 1 0 0 1

phase state 2 ck 0 3 2 3 1 1 3 0 3 2 1 0 2 2 2 3 1 2 2 2 3 dk 0 3 3 1 2 0 2 1 3 3 3 3 2 0 0 1 2 1 0 0 1 phase state 3 ck 0 0 3 0 2 2 0 1 0 3 2 1 3 3 3 0 2 3 3 3 0 dk 0 0 3 1 2 0 2 1 3 3 3 3 2 0 0 1 2 1 0 0 1 phase state 4 ck 0 1 0 1 3 3 1 2 1 0 3 2 0 0 0 1 3 0 0 0 1 dk 0 1 3 1 2 0 2 1 3 3 3 3 2 0 0 1 2 1 0 0 1 phase state 4 with symbol decision errors ck 0 1 0 1 2 3 1 2 1 0 1 2 0 2 1 1 3 0 1 0 1 dk 0 1 3 1 1 2 2 1 3 3 1 1 2 2 3 0 2 1 1 3 1

�� )��

�� )�� Ps,MDPSK(E) ----2Ps,MPSK(E) Pb,MDPSK(E) ----2Pb,MPSK(E)

��

� �� $��

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pulse shapingfilter

��/2CG

�� kk

a t kT ��

�� kk

a t kT ��

�� cos kk

a x t kT� � � �

��

�� sin kk

a x t kT� � � �

��

cos cos 2

sin sin 2

a x t kT f t

f t a x t kT

��

� � � � ��

� ��

� � � � � � ��

� ��

cos( )

sin( )

sineoscillatorf1

sineoscillatorf2

binarydata

FSK-signal

FSK_tx.dsf

VCObinarysignal

CPFSK-signal

FSK_tx.dsf

G FM-mod.

1 0 0 01 1

FSK-transmitter

MSK-transmitter

FSK_MSK_principle.dsf

1 � 1 $5� 6 ��1 � �4� 0$��5��7 ��

.�� 20 1��

#��

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envelopedemodu-lator

sampler

samplerenvelopedemodu-lator

( ) jh t

s T t e 0000��

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( ) jh t

s T t e 0000��

��FSK_MSK_principle.dsf

��"��/�� /�4� 0$��

�7�� 20 1�� Q rx

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� 0.5exp2

N� � � �

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7�� $0 1��

� � � � ��

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Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What...

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