digital communications systems - topic 4.pdf4 digital systems for bandpass communication amplitude...

1

DIGITAL

COMMUNICATIONS

SYSTEMS

MSc in Electronic Technologies and

Communications

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DIGITAL SYSTEMS FOR

BANDPASS COMMUNICATION Bandpass binary signalling

The common techniques of bandpass binary signalling are:

- On-off keying (OOK), also known as amplitude shift keying (ASK), which consists in activating and deactivating a sinusoidal carrier wave by using a unipolar binary signal. It is equivalent to a DSB-SC signal, where the modulating signal is a unipolar binary signal.

- Binary phase shift keying (BPSK), which consists in shifting the phase of a sinusoidal carrier wave 0º or 180º by using a unipolar binary signal. It is equivalent to PM with a unipolar digital signal or modulating a DSB-SC signal by means of a polar digital waveform.

- Frequency shift keying (FSK), which consists in shifting the frequency of a sinusoidal wave from a mark frequency (corresponding to sending a binary digit ‘1’) until a space frequency (corresponding to sending a binary digit ‘0’), according to a baseband digital signal. It is equivalent to modulating an FM carrier by using a binary digital signal.

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DIGITAL SYSTEMS FOR

BANDPASS COMMUNICATION Bandpass binary signalling

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DIGITAL SYSTEMS FOR

BANDPASS COMMUNICATION

Amplitude shift keying (ASK)

- An OOK signal is given by:

- The complex envelope of the OOK signal is:

- The power spectral density of this complex envelope will be given by:

when m(t) is a unipolar signal with peak amplitude of √2 , in such a way that s(t)

has an average normalized power equal to Ac2/2. The power spectral density of

the bandpass signal is obtained by means of a simple shift of the spectrum of the

complex envelope towards frequencies fc and –fc, additionally to multiplying them

by a scale factor of ¼.

ttmAts cc cos)()(

OOKfor )()( tmAtg c

OOKfor sin

)(2

)(

22

b

bb

cg

fT

fTTf

Af

P

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DIGITAL SYSTEMS FOR



- When R = 1/Tb is the data rate, the null-to-null bandwidth for an OOK signal is 2R,

that is, exactly the double of that of the baseband signal (BT = 2B).

- If we use a raised-cosine filter, for the case of binary signalling (D = R) we will

have

RrBRrDrB T )1()1()1(21

21

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DIGITAL SYSTEMS FOR



- An OOK signal can be demodulated by using an envelope detector (non-coherent

detection) or by using a product detector (coherent detection), since it is basically

an AM signal. However, for optimal detection of an OOK signal corrupted by

additive white Gaussian noise (AWGN), it is required from product detection and

processing with a correlation filter.

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DIGITAL SYSTEMS FOR


Binary phase shift keying (BPSK)

- A BPSK signal is given by

where m(t) is a baseband polar data signal. If we expand the previous expression,

we will be able to see that it consists in an AM signal

- If we suppose that m(t) only takes values of 1, and that cos(x) and sin(x) are

even and odd functions of x, the representation of the BPSK signal is reduced to

- The peak phase deviation Dp = Dq sets the value for the pilot carrier term. The

digital modulation index, h, is defined as

where 2Dq is the maximum peak-to-peak phase deviation (radians) along a

symbol period (in the case of binary signals, the bit period).

)(cos)( tmDtAts pcc

ttmDAttmDAts cpccpc sin )(sin cos)(cos)(

termdatarmcarrier tepilot

sin sin )(coscos)( tDtmAtDAts cpccpc

qD

2h

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DIGITAL SYSTEMS FOR


Binary phase shift keying (BPSK)

- It is clear that the smaller is Dp the larger the power is wasted in the pilot carrier.

The efficiency of the modulation can be increased to its maximum value by

making Dp = Dq = /2 (h = 1). In that case, the BPSK signal is transformed in

- Hence, BPSK in this optimal case is equivalent to a DSB-SC signal with a

baseband polar waveform. The complex envelope of this signal is

and its power spectral density

where m(t) takes values of 1, in such a way that s(t) has an average normalized

power of Ac2/2. The spectrum of the BPSK signal can be easily obtained from the

complex envelope in the same way as it was previously described. The BPSK

signal has a null-to-null bandwidth of 2R, just like the OOK signals.

ttmAts cc sin )()(

BPSKfor )()( tmjAtg c

BPSKfor sin

)(

2

2

b

bbcg

fT

fTTAf

P

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DIGITAL SYSTEMS FOR


Differential phase shift keying (DPSK)

- The demodulation of BPSK signals requires from synchronous detection.

However, despite a BPSK signal cannot be incoherently demodulated, if this one

is differentially encoded previous to transmission, it will be possible to recover the

data signal by using the differential decoder shown in the figure. The differentially

encoded BPSK signal is known as DPSK. For optimal detection, we have to

substitute the low-pass filter by a correlation filter with integrating and dumping,

and the DPSK input signal has to be pre-filtered with a bandpass filter with

impulse response function h(t)=P[(t-0.5Tb)/Tb]cos(ct). In practice, it is usual to

work with DPSK instead of BPSK, because it does not require from a carrier

recovering circuit.

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DIGITAL SYSTEMS FOR


Frequency shift keying (FSK)

- We can distinguish between two kinds of FSK signals, discontinuous-phase FSK

and continuous-phase FSK. In the first case, we only need a switching device,

which commutes, according to the value of the baseband binary signal, between

two oscillators working at different frequencies. For this reason, the phase of the

signal is usually discontinuous. The discontinuous-phase FSK is given by

where f1 is the mark frequency and f2 the space frequency. The continuous-phase

FSK signals are generated by feeding a frequency modulator with the data signal.

This continuous-phase FSK signal is given by

where m(t) is the baseband signal. Although m(t) is discontinuous in the

commutation instant, q(t) is continuous, because it is proportional to the

integral of m(t). If the modulating signal is binary, it is called binary FSK.

sent is '0'digit binary a when ,cos

sent is '1'digit binary a when ,cos)(

22

11

q

q

tA

tAts

c

c

t

f

tj

c

tj

t

fcc

dmDteAtgetgts

dmDtAts

c q

q)()(,)(,)(Re)(

FSKfor )(cos)(

)(

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DIGITAL SYSTEMS FOR



- The spectra of FSK signals, in the same way as those of FM, are difficult to

evaluate, since g(t) is a non-linear function of m(t). However, we are going to

suppose that we use as binary modulating signal a square waveform whose

period is T0 = 2Tb, being the data rate R = 1/Tb. The peak frequency deviation is

DF = max[(1/2)dq(t)/dt] = Df /2 when m(t) takes values of 1. From the input

square waveform, we will obtain a triangular phase function, and the digital

modulation index is

where the equality with the modulation index for FM is only fulfilled if we take as

bandwidth B = 1/T0. The Fourier series of the complex envelope is

where f0 = 1/T0 = R/2 and D = 2DF = 2h/T0.

Frequency modulation

index (FM)

n

tjn

nectg 0)(

nh

nh

nh

nhAc

ncn

2/

2/sin 1

2/

2/sin

2

fB

F

T

F

R

FFTh

q

D

D

DD

D

0

0/1

22

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DIGITAL SYSTEMS FOR



- The spectrum of the complex envelope is

- From the previous expression is easy to deduce the spectrum of the FSK signal.

Notice that it will be constituted by a series of delta functions which are R/2 apart

and grouped in the vicinity of the mark and space frequencies. The bandwidth of a

FSK signal is given by the Carson’s rule: BT = 2( + 1)B, where = DF/B.

Therefore, if we consider B = R (first null bandwidth)

- FSK signals can be demodulated with a frequency detector (non coherent) or by

using a product detector (coherent detection). In this second case, we need two

product detectors tuned to the mark and space frequencies, each followed by a

low-pass filter. The output signals form the filters are subtracted by using a linear

adder so as to obtain the demodulated binary signal. For optimal detection in front

of AWGN, we need coherent detection and processing with a correlation filter in

conjunction with a threshold device (comparator).

n

n

n

n

nRfcnffcfG

2)()( 0

RFBFBT DD 222

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Estimated spectrum for an ASK signal (R = 1, fc = 10)

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Estimated spectrum for a BPSK signal (R = 1, fc = 10)

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Estimated spectrum for an FSK signal (R = 1, f1 = 10, f2 = 20)

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DIGITAL SYSTEMS FOR


Estimated spectrum for an FSK signal (R = 1, f1 = 9, f2 = 11)

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Estimated spectrum for an FSK signal (R = 1, f1 = 9.5, f2 = 10.5)

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DIGITAL SYSTEMS FOR


Estimated spectrum for an FSK signal (R = 1, f1 = 9.9, f2 = 10.1)

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DIGITAL SYSTEMS FOR


Quadrature phase shift keying (QPSK) and M-ary phase shift

keying (M-PSK)

- If by means of a binary signal, making use of a DAC, we generate a

multilevel signal of M levels and apply this one to a PM transmitter, we

will have the so-called M-ary phase shift keying (M-PSK). M-PSK for

M = 4 is a special case known as quadrature phase shift keying (QPSK).

In the figure are shown the two possible symbol constellations for a

QPSK signal, that is, the allowed values for the complex envelope.

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DIGITAL SYSTEMS FOR


Quadrature phase shift keying (QPSK) and M-ary phase shift

keying (M-PSK)

- M-PSK transmission can also be generated by means of two carrier waves in

quadrature, which modulate the x and y components of the complex envelope

(instead of using a phase modulator)

where:

being i = 1, 2, ... , M, and qi are the phase angles which are allowed for the

M-PSK signal.

)()()( )( tjytxeAtg tj

c q

ici

ici

Ay

Ax

q

q

sen

cos

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DIGITAL SYSTEMS FOR


Quadrature amplitude modulation (QAM)

- The signal generated using two quadrature carriers (see previous figure) is called

quadrature amplitude modulation (QAM). The generalized QAM signal is

- In the figure is shown a rectangular QAM constellation of 16 symbols (M = 16

levels), where the relationship between (Ri, qi) and (xi, yi) can easily be evaluated

from this figure. Components xi and yi are allowed to have four levels by

dimension.

ttyttxts cc sen )(cos)()(

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DIGITAL SYSTEMS FOR



- The x and y waveforms are represented by

where D = R/l and (xn, yn) denote one of the allowed values (xi, yi) during the time

which is necessary to transmit a symbol centred in t = nTs = n/D (we need Ts s for

sending each symbol), h1(t) is the pulse waveform used for each symbol.

Sometimes the synchronization between components x(t) and y(t) is

compensated by Ts/2 = 1/(2D) s, then y(t) would be given by

n

n

n

n

D

nthyty

D

nthxtx

1

1

)(

)(

n

nDD

nthyty

2

1)( 1

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DIGITAL SYSTEMS FOR



- A popular type of staggered modulation for the case of QPSK (QAM where

M = 4) is the so-called offset QPSK (OQPSK), where the data stream is divided in

even and odd-numbered bits, each being transmitted via the cosine or the sine of

the carrier, respectively. Moreover, an offset of Ts/2 s (Tb s) is introduced between

both components. With this technique the maximum phase jumps are equal to

±90º, instead of the phase jumps of ±180º which can be obtained with

conventional QPSK schemes. The phase jumps of 180º during symbol transitions

can yield problems during the signal reception under highly dispersive channels or

due to non-linear amplification, etc. A special case of OQPSK when h1(t) is a sine

pulse is minimum shift keying (MSK).

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DIGITAL SYSTEMS FOR



- Example. Comparison between QPSK and OQPSK signals

0 1 2 3 4 5 6 7 8 9 10

-1

0

1

m

0 1 2 3 4 5 6 7 8 9 10

-1

0

1

x1

0 1 2 3 4 5 6 7 8 9 10

-1

0

1

y1

0 1 2 3 4 5 6 7 8 9 10-2

0

2

OQ

PS

K

0 1 2 3 4 5 6 7 8 9 10-2

0

2

QP

SK

x y x y x y x y x y

00

10

11

01

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DIGITAL SYSTEMS FOR


Power spectral density of MPSK and QAM

- The complex envelope of an MPSK or QAM is given by

- where cn is a complex random variable that represents the multilevel value of the

pulse during the nth symbol, f(t) = P(t/Ts) is equivalent to a rectangular pulse with

duration Ts, D = 1/Ts is the symbol rate (or baud rate). The Fourier transform of

the square pulse is

where Ts = lTb (there exist l bits corresponding to each allowed multilevel value).

In the case of symmetrical modulation, the power spectral density of the complex

envelope for MPSK or QAM with data modulation with rectangular bit pattern is:

n

sn nTtfctg )(

b

bb

s

ss

fTl

fTllT

fT

fTTfF

sin sin )(

QAM andMPSK for sin

)(

2

b

bg

fTl

fTlKf

P

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DIGITAL SYSTEMS FOR


Power spectral density of MPSK and QAM

- This power spectral density coincides with the power spectral density of QPSK

when l = 1. The null-to-null bandwidth for MPSK or QAM is:

- The spectral efficiency is:

DlRBT 2/2

Hz

bits/s

2

l

B

R

T

h

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DIGITAL SYSTEMS FOR


Minimum shift keying (MSK)

- Minimum shift keying (MSK) is another bandwidth-conservative technique, which

is equivalent to OQPSK using sine pulses h1(t). The MSK signal is a continuous-

phase FSK (CPM, continuous phase modulation) with the minimum modulation

index (h = 0.5) which ensures orthogonality of the modulated signals.

- The peak frequency deviation is

- The complex envelope of the MSK signal is

- where m(t) = 1, 0 < t < Tb. Then

where the signs denote the possible data values during the time interval (0, Tb).

MSKfor 4

1

4

1

2R

TT

hF

bb

D

Dt

dmFj

c

tj

c eAeAtg 0)(2

)()(q

b

Ttj

c

tj

c TttjytxeAeAtg b

0),()()()2/()( q

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DIGITAL SYSTEMS FOR



- Therefore:

and the MSK signal is

- It can be observe as the change in the sign of m(t) during the interval (0, Tb) only

affects to y(t), but not to x(t), in the signalling interval (0, 2Tb). Moreover, the pulse

sin[t/(2Tb)] of y(t) is 2Tb-second width. In addition, we can see as the sign of m(t)

during the interval (Tb , 2Tb) only affects to x(t) but not to y(t) in the signalling

interval (Tb, 3Tb). That is, the data modulate alternatively the components x(t) and

y(t), hence MSK is equivalent to OQPSK in which the pulse shape is one-half

cycle of a sinusoid.

b

b

c

b

b

c

TtT

tAty

TtT

tAtx

0,2

sin )(

0,2

cos)(

ttyttxts cc sin )(cos)()(

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DIGITAL SYSTEMS FOR



- Example. MSK signalling

y x y x y x y x y x

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DIGITAL SYSTEMS FOR



- Example. Comparison of the spectra of the MSK, QPSK and OQPSK signals

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DIGITAL SYSTEMS FOR


Error performance in bandpass communication systems

- In the same way as in baseband communication systems, the optimal detector for

bandpass communication systems is the correlation filter. In the figure is shown

the receiver structure based on the correlation filtering for the binary case, where

two parallel correlation branches are required.

- The output signals from both correlators can be subtracted, such as is shown in

the figure:

)()()( 21 TzTzTz

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DIGITAL SYSTEMS FOR



- The output signal z(T) is given by a signal component ai(T) which is related with

the transmitted symbol in that moment (for the binary case, a1 or a2) plus a noise

component n0(T):

- The detector has to decide if s1 or s2 has been sent according to if z(T) is bigger

or smaller than g0:

)()()( 0 TnTaTz i

021

2)(

1

2

g

aa

Tz

s

s

34

DIGITAL SYSTEMS FOR



- Any detector which uses the previous structure to determine which symbol has

been sent will present an optimal error performance under AWGN. Suppose, for

example, the case of M-ary PSK (MPSK). In this case, we have that si(t) can be

expressed as:

- The factor is added to normalize the expected output of the detector,

then making the energy per symbol of the MPSK signal equal to E. On the other

hand, we have that any of the previous signals si(t) can be written in terms of the

next set of orthonormal basis functions:

Mi

Tt

M

it

T

Etsi

,,1

02cos

2)( 0

)cos(2

)( 01 tT

t

)(sin2

)( 02 tT

t

TE /2

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DIGITAL SYSTEMS FOR



- We will have that:

being fi = 2i/M the phase shift of the transmitted symbol, with respect to zero

degrees.

Mi

TttEtE

tM

iEt

M

iE

tatats

ii

iii

,,1

0)(sen)(cos

)(2

sen)(2

cos

)()()(

21

21

2211

ff

36

DIGITAL SYSTEMS FOR



- Observe as, after the coherent demodulation by using the two correlation

branches, we obtain the in-phase and quadrature baseband components X and Y

of the complex envelope for the received symbol. Therefore, it is logical to think of

obtaining an error performance of this bandpass system equivalent to that of its

baseband counterpart. Thus, for binary schemes we did have that:

0

)1(

N

EQP b

B

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DIGITAL SYSTEMS FOR



- In the previous expression, is the cross-correlation coefficient between the

transmitted symbols, which for antipodal-signalling such as BPSK is equal to -1.

Thus, the bit error probability of a BPSK communication system is:

0

2

N

EQP b

B

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DIGITAL SYSTEMS FOR



- For binary ASK or OOK, we have that = 0, hence:

- For binary FSK, due to modulate the possible transmitting symbols by using

orthogonal basis functions, we will have that = 0 again and, therefore, the bit

error rate coincides with that obtained for OOK when coherent detection is used.

Both ASK and FSK signals can be non-coherently demodulated by using

envelope detectors and oscillators tuned to the frequencies corresponding to that

of the transmitted symbols. These detectors, since they do not require from

phase-synchronization with the received carrier waves, are less complex but, on

the other hand, present a worse error performance than their optimal coherent

counterparts. At best, the bit error rate of a non-coherent detector of FSK signals

(and for OOK signals too) will be given by:

0N

EQP b

B

02exp

2

1

N

EP b

B

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DIGITAL SYSTEMS FOR



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DIGITAL SYSTEMS FOR



- When we are working with M-ary schemes, the symbol error performance PE(M)

for coherently detected MPSK signals, for large energy-to-noise ratios, can be

expressed as

where Es = Eb·log2M is the energy per symbol and M = 2k is the size of the symbol

set, being k the number of bits per symbol. For the case of MPSK signalling, the

relationship between the probability of symbol error PE and the probability of bit

error PB, when Gray code assignment is used, is approximately

- Hence, for QPSK (OQPSK and MSK), considering the previously said, we have

that:

2,sin2

2)(0

M

MN

EQMP s

E

)1(for log2

EEE

B Pk

P

M

PP

BPSKB

BPSKBQPSKEQPSKE

QPSKB PPP

M

PP ,

,,

2

,

,2

2

2log

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DIGITAL SYSTEMS FOR



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DIGITAL SYSTEMS FOR



- Taking into account that the symbol rate Rs is decreased by increasing the

number of bits k represented per symbol, for a given data rate R:

and , in the case of using Nyquist filtering, the minimum-required bandwidth for

transmitting at a symbol rate Rs is:

- The spectral efficiency for MPSK systems, and MQAM in general, taking into

account the previous relationships, is:

kRRs /

s

s

T RT

B 1

bit/s/HzMkkR

R

R

R

B

R

sT

2log/

h

43

DIGITAL SYSTEMS FOR



- For rectangular constellation, Gaussian channel and matched filter reception, the

bit-error probability of M-QAM, where M = 2k and k is even, is:

0

2

2

2

2

1

log3

log

)/11(2

N

E

L

LQ

L

LP b

B

44

DIGITAL SYSTEMS FOR



- For coherently detected MFSK signals, the symbol-error probability satisfies that:

- For non-coherently detected MFSK, an upper bound for the error probability is the

following (this upper bound of PE is, logically, also valid for coherent detection):

- For MFSK signalling, the relationship between the probability of symbol error PE

and the bit-error probability PB is:

0

1)(N

EQMMP s

E

02exp

2

1)(

N

EMMP s

E

2

1

1

2/

12

2

12

2

12

2

bits ofnumber

bits erroneous ofnumber

1

11

kk

k

E

B

k

k

Ek

k

EEB

M

M

P

P

Pk

k

PPP

45

DIGITAL SYSTEMS FOR



46

DIGITAL SYSTEMS FOR



- The MFSK schemes have the inconvenient, in contrast to MPSK and MQAM

schemes, that by increasing M the required bandwidth is larger and larger. For

non-coherently detected MFSK, the minimum tone spacing (separation between

the frequencies of the different tones or carriers) is Rs = 1/Ts, being Rs the symbol

rate and Ts the symbol duration. Hence, if we use M carriers, the minimum

required bandwidth will be:

- Therefore, the spectral efficiency for MFSK schemes is:

M

MR

k

RMMR

T

MB s

s

T

2log

bit/s/HzM

M

B

R

T

2logh

47

DIGITAL SYSTEMS FOR



- Comparison between the spectral efficiency for the different schemes of

bandpass digital signalling

digital communications systems - topic 4.pdf4 digital systems for bandpass communication amplitude...

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