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Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

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Page 1: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

1

�������� ��� ���� � ���

Page 2: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

2

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

Page 3: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

3

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

Page 4: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

4

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

Page 5: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

5

Modulation/demodulation (3)

The basic digital quadrature modulator

S

P

pulse shapingfilter

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��������

���� ���� ���� ����

���� ���� ���� ����

cos 2

sin 2

k ck

k ck

a x t kT f t

b x t kT f t

����

����

��������

� �� �� �� �����

Page 6: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

6

Modulation/demodulation (4)

The offset digital quadrature modulator

S

P

pulse shapingfilter

pulse shapingfilter

����/2CG

���� ����kk

a t kT ��������

���� ����kk

b t kT ��������

���� ����cos 2 cf t����

���� ����sin 2 cf t����

���� ����kk

a x t kT��������

���� ����0.5kk

b x t T kT� �� �� �� �����

���� ���� ���� ����

���� ���� ���� ����

cos 2

0.5 sin 2

k ck

k ck

a x t kT f t

b x t T kT f t

����

����

��������

� � �� � �� � �� � �����

delayT/2

Page 7: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

7

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

The staircase function ���� ����1

0stc( , ) u

k

k kt T t kT

����� �� �� �� �����

S

P

pulse shapingfilter

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

k ck

k ck

a x t kT f t t T

b x t kT f t t T

��������

��������

� � � � � �� �� �� ����� � �� �� �� �

� �� �� �� �� � � �

� � �� � �� � �� � ����� � �� �� �� �� �� �� �� �stc( , )

4t T

��������

Page 8: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

8

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.

Page 9: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

9

5 4 3 2 1 0 1 2 3 4 55

4

3

2

1

0

1

2

3

4

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

Page 10: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

10

5 4 3 2 1 0 1 2 3 4 55

4

3

2

1

0

1

2

3

4

5

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

Page 11: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

11

Basic QAM-constellations having about same average power4QAM 16QAM

64QAM 256QAM

Page 12: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

12

Examples of non-quadratic QAM-constellations

8QAM 32QAM

128QAM 512QAM

Page 13: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

13

QPRS-constellations

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

(Quadrature Partial Response Signaling)

MQAM_MPSK_MQPRS.dsf

Page 14: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

14

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����

��������

������������

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

���� ����0.5sin sin 2k ct T kTa rect f t

M T���� ����

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

Page 15: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

15

MPSK-CONSTELLATIONS2PSK 4PSK

8PSK 16PSK

Page 16: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

16

An alternative MQAM modulator

5 3 1 1 3 5

5

3

1

1

3

5

7 9����7����9����9

����7

9

7

G

4QAMmodu-lator

4QAMmodu-lator

4QAMmodu-lator

A6 dB

A12 dB

(((( 64QAM

Page 17: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

17

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

Page 18: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

18

The quadrature demodulator

S

P

pulse shapingfilter

pulse shapingfilter

����/2LO

���� ����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��������

powerdivider

ADC

DAC

dec.circuit

ADC dec.circuit

controlsignals

r(t)

Page 19: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

19

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

Page 20: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

20

��������������������

� � � � �� �� �� ������ ����� �� ����� �

���� ���� ���� �������� ���� ���� �������� ����

���� ���� ���� ���� ���� ����

12

1

1 Q 1 Q

2Q Q

P C m z z

P E m z z

� � �� � �� � �� � �

� �� �� �� � MQAM_MPSK_MQPRS.dsf

Page 21: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

21

4 ( M - 2) ����������������

� � � � �� �� �� ������ � ���� �� ����� �

MQAM_MPSK_MQPRS.dsf

���� ���� ���� ���� ���� ����

���� ����2

22

1 Q( ) 1 2Q( )

3Q( ) 2Q ( )

P C m z z

P E m z z

� � �� � �� � �� � �

� �� �� �� �

Page 22: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

22

� � � � �� �� �� ������ ����� �� ����� �

������������������

���� ���� ���� ���� ���� ����

���� ����3

23

1 2Q( ) 1 2Q( )

4Q( ) 4Q ( )

P C m z z

P E m z z

� � �� � �� � �� � �

� �� �� �� � MQAM_MPSK_MQPRS.dsf

���� ����22M ����

Page 23: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

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� �������� � !�������"�!�!���

���� ���� ���� ���� ���� ���� ���� ����1 1

1M Ms k k k

k kP E P m P E m P E m

M� �� �� �� �� �� �� �� �� �� �� �� � �

� � ���� ���� ���� ���� ���� ����1 2 31 2 31

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

" %" %" %" %� � �� � �� � �� � �$ '$ '$ '$ ' �

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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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%%%%� � � � � � �� � � � � � �� � � � � � �� � � � � � �''''

Page 24: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

24

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

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� � ���� ���� ���� ����2 24 Q 1 Qn n

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

" %" %" %" %� � � � � � � � � � � �� � � �� � � �� � � �# &# &# &# &� � � �� � � �� � � �� � � �

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22 21 14 1 Q 4 1 Qo os

o o

E EP EN NM M

� � � � � � � � � � � � � � � � � � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �

oE ������ � � ���������������� �� ���� ����������� ���������� ���� �

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Page 25: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

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

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MQAM_MPSK_MQPRS.dsf

Page 26: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

26

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

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

� � � � � �� � � � � �� � � � � �� � � � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� �� �� �� �� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �

where

rxo s

PN R

++++ ����

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

Page 27: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

27

. �//���� ���� ������� � � � �

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Page 28: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

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���� ����

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2,

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log1

2 1

log 4 1 3 (0)1 1 Q

log 12 1

b de b

n

MP P

M

M xM MMM ****

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Page 29: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

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Page 39: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

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Page 41: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

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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

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Page 44: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

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Page 47: Lecture Linear NonlinearModulations v2 · 2016-01-28 · 4 Modulation/demodulation (2) What characterises a non-linear modulation? The spectral shape of the modulating signal is changed

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