ENSC327

Communication Systems

27: Digital Bandpass Modulation

(Ch. 7)

1

Jie Liang

School of Engineering Science

Simon Fraser University

Outline

� 7.1 Preliminaries

� 7.2 Binary Amplitude-Shift Keying (BASK)

� 7.3 Phase-Shift Keying (PSK)

� 7.4 Frequency-shifting Keying (FSK)

2

� 7.4 Frequency-shifting Keying (FSK)

� 7.7 M-ary Digital Modulation

� 7.8 Mapping of digitally modulated waveforms onto

constellations of signal points

7.1 Preliminaries

� If the channel is low-pass (e.g., coaxial cable), we can transmit

the pulses corresponding to digital data directly.

� If the channel is band-pass (e.g. wireless, satellite), we need to

use the digital data to modulate a high-freq sinusoidal carrier:

)2cos()( tfAtc φπ +=

3

)2cos()(ccc

tfAtc φπ +=

1. and 0represent to and 0 use : πφc

� Amplitude-Shift Keying (ASK):

� Use two Ac’s to represent 0 and 1.

� Phase-Shift Keying (PSK):

� Frequency-Shift Keying (ASK):

� use two fc’s to represent 0 and 1.

7.1 Preliminaries

� The amplitude of the carrier is usually chosen as

such that the carrier has unit energy measured over

.2

b

c

TA =

4

such that the carrier has unit energy measured over

one bit duration.

7.2 Binary Amplitude-Shift Keying

(BASK)

� In BASK, the modulated wave is

==

0. symbolfor ,0

1, symbolfor ),2cos(2

)2cos(2

)()(tf

T

E

tfT

tbts c

b

b

c

b

ππ

5

� This is a special case of Amplitude Modulation (AM):

( ) ,)2cos()(1)( tftmkAtscacπ+=

� Therefore the BASK spectrum has a carrier component.

� Envelope detector can be used to demodulate the digital signal.

� b(t) is the on-off signalling coding of the input binary data.

7.2 Binary Amplitude-Shift Keying

(BASK)

� The average transmitted signal energy is

6

7.3 Phase-Shift Keying (PSK)

� We first consider binary PSK (BPSK):

−=+

=

0. symbolfor ),2cos(2

)2cos(2

1, symbolfor ),2cos(2

)(

tfT

Etf

T

E

tfT

E

ts

c

b

c

b

c

b

b

πππ

π

7

−=+ 0. symbolfor ),2cos()2cos( tfT

tfT

c

b

c

b

πππ

� The two possible values are called antipodal signals.

� A special case of DSB-SC:

� No carrier component in the freq domain.

� BPSK has constant envelope � constant transmitted power. Desired in

many systems.

� But cannot use envelope detector in the receiver, need coherent

detection.

7.3 Phase-Shift Keying (PSK)

� Detection of BPSK signals:

� Coherent DSB-SC receiver

� Sample & decision-making: new to digital communication

� Can reduce error rate. Advantage over analog comm.

8

Quadriphase-Shift Keying (QPSK)

� Recall Chap 3.5: Quadrature-amplitude modulation (QAM):

� Transmit two DSB-SC signals in the same spectrum region.

� Use two modulators with orthogonal carriers.

9

)2sin()()2cos()()( :signal dTransmitte21

tftmAtftmAtsccccππ +=

� The two signals do not affect each other.

Quadriphase-Shift Keying (QPSK)

� QAM can be generalized to digital modulation

� In QPSK, the transmitted signal has four possible phases:

π/4, 3π/4, 5π/4, 7π/4.

≤≤−+= ,0 ),

4)12(2cos(

2)(

TtitfT

Ets c

i

π

π

i=1i=2

10

=

elsewhere. ,04)( Tts

i

� Index i: 1, 2, 3, 4.

� Each signal can represent two bits of binary data, called dibits.

� Tb: bit duration.

� T: Symbol duration.

� It’s easy to see that the energy of si(t) is E. This is the Symbol Energy.

� Since each symbol represents 2 bits, the average transmitted energy per bit

is

i=3 i=4

Quadriphase-Shift Keying (QPSK)

�To see the link with QAM:

≤≤−+=

elsewhere. ,0

,0 ),4

)12(2cos(2

)(Ttitf

T

Ets c

i

π

π

22 EEπ

π

π

π

−−−=

11

)2sin(2

)()2cos(2

)(

)2sin()4

)12sin((2

)2cos()4

)12cos((2

)(

21tf

Ttatf

Tta

tfiT

Etfi

T

Ets

cc

cci

ππ

π

π

π

π

+=

−−−=

Quadriphase-Shift Keying (QPSK)

� Detection of QPSK:

� Similar to QAM

� Two coherent BPSK detectors.

)2sin(2

)()2cos(2

)()(21

tfT

tatfT

tatscciππ +=

12

� Two coherent BPSK detectors.

7.4 Frequency-Shift Keying (FSK)

=

=

=

2. i toscorrespond 0 symbolfor ),2cos(2

1, i toscorrespond 1 symbolfor ),2cos(2

)(

1

tfE

tfT

E

tsb

b

b

i

π

π

� Binary FSK (BFSK): symbol 0 and 1 are represented

by two sinusoidal waves with different frequencies

13

� f1 and f2 can be chosen such that neighboring signals

have continuous phases. This can reduce the bandwidth of

the transmitted waveforms.

� This is called the Sunde BFSK.

= 2. i toscorrespond 0 symbolfor ),2cos(

22tf

T

E

b

bπ

Frequency-Shift Keying (FSK)

� Example:

14

Continuous phase

can reduce bandwidth

7.7 M-ary Digital Modulation

� M-ary PSK

� M-ary QAM

� M-ary FSK

� Mapping waveforms to signal points

15

� Mapping waveforms to signal points

7.7 M-ary Digital Modulation

� During each symbol interval of duration T, the

transmitter sends one of M possible signal s1(t), …,

sM(t). M is usually a power of 2: M = 2^m.

� M-ary modulation is necessary if we want to conserve

16

M-ary modulation is necessary if we want to conserve

the bandwidth.

� But M-ary system needs more power and more

complicated implementation to achieve the same

error rate as binary system.

M-ary Phase-Shift Keying

� Generalization of the QPSK

.0 1,-M0,...,i ),2

2cos(2

)( TtiM

tfT

Ets

ci≤≤=+=

π

π

� This can be expressed as

17

Signal Space Diagram

� As the increase of M, the receiver of the M-ary

modulation can become more complicated, because

for each input symbol, a naive receiver needs to

compare with M references.

18

� It is thus necessary to simplify the signal

representation and therefore reduce the complexity of

the receiver.

� The concept of signal space is useful here.

Signal Space Diagram

� The signals si(t) can be written as

19

� We can visualize the transmitted signals as points in a

K-dimensional space, with axes { })(tjφ

M-ary Phase-Shift Keying

� In M-ary PSK: ( ) ( ).2sin22

sin2cos22

cos)( tfT

iM

EtfT

iM

Etscciπ

π

π

π

−

=

� We can define two orthonormal basis functions:

20

� {si(t)} can be represented by

points on a signal space diagram.

� The coordinate of each point:

8-PSK

� In MPSK, the distance from the origin to

each point is equal to the signal energy E.

M-ary QAM

� Recall Chap 3.5: QAM

)2sin()()2cos()()(21

tftmAtftmAtsccccππ +=

� If m1(t) and m2(t) are discrete, we get digital QAM:

22 EEππ −=

21

).2sin(2

)2cos(2

)( 00 tfbT

Etfa

T

Ets

ciciππ −=

� Example of signal space

diagram: 16-QAM

Possible values for ai, bi:

-3, -1, 1, 3.

Envelope is not constant.

Mapping of Modulated Waveforms

to Constellations of Signal Points

� The correlator method is used in receiver in many systems:

� Calculate the correlation of input with a pulse template,

� Sample the output of the correlator,

� Compare the sample with some thresholds to decode the bits.

� For example, in BPSK, the template is simply the basis function:

22

function:

).2cos(2

)(1

tfT

tc

b

πφ =

� If the transmitter sends s1(t):

its correlation with the basis function is:

Mapping of Modulated Waveforms

to Constellations of Signal Points

� If the transmitter sends s2(t):

its correlation with the basis function is:

� This can be represented by a one-dimensional diagram:

23

� This can be represented by a one-dimensional diagram:

Mapping of Modulated Waveforms

to Constellations of Signal Points

� This diagram is useful in studying the effect of the noise.

� When noise is considered, the received signal will be

noise. is )( ),()()( tntntstriiii

+=

� The output of the correlator will be

24

( ) .)()()()()('0

10

1 ii

T

ii

T

iiivsdtttnsdtttntss

bb

+=+=+= ∫∫ φφ

� The output of the correlator will be

The noise ni(t) introduces some disturbs to the position of the desired point on the signal space diagram.

Decoding could be wrong if the noise is too large.

BPSK:

Mapping of Modulated Waveforms

to Constellations of Signal Points

� BFSK:

).2cos(/2)(

),2cos(/2)(11

tfTEts

tfTEtsbb

π

π

=

=

� The transmitted signals can be written as:

25

).2cos(/2)(

),2cos(/2)(

22

11

tfTt

tfTt

b

b

πφ

πφ

=

=

).2cos(/2)(22tfTEts

bbπ=

� The receiver takes correlation of the received signal with two basis functions:

Mapping of Modulated Waveforms

to Constellations of Signal Points

� If s1(t) is sent, the outputs of the two correlators are:

26

� If s2(t) is sent, the outputs are:

Mapping of Modulated Waveforms

to Constellations of Signal Points

� So each signal can be represented by a point on a 2-D diagram:

φ

27

1φ

2φ

The noise introduces some disturbs to the position of the desired point on the signal space diagram.

Decoding could be wrong if the noise is too large.

Mapping of Modulated Waveforms

to Constellations of Signal Points

� Compare the diagrams of BPSK and BFSK, we can see that the distance of the two points are

� Since noise changes the position of the signal in the signal

28

� Since noise changes the position of the signal in the signal space diagram at the receiver, we can see from these figures that BPSK is more robust to noise than the BFSK.

� This will be studied in details in Chapter 10.

2bE

2bE

BPSK:BFSK: