Ece414 chapter3 w12

ECE414 Wireless Communications, University of Waterloo, Winter 2012 1

Bandpass Transmission Techniques for Wireless Communication

Chapter 3


Outline

Introduction to Digital Communications

Signal (Vector) Space Representations

Digital Modulation Schemes (M-ASK, M-PSK, M-FSK)

Performance Measures for Modulation Schemes

- Bandwidth (spectral) efficiency

- Power efficiency

- Temporal characteristics (e.g., dynamic power range, peak/average ratio)

Power Spectral Density of Digital Modulation Schemes

Error Rate Performance of Digital Modulation Schemes

Comparison of Digital Modulation Schemes in terms of Spectral Efficiency and Power Efficiency

Temporally Efficient Digital Modulation Schemes


Original message signal (analog)

Recovered message signal (analog)

A/DSource

EncoderModulator

Channel

De-modulator

Channel Decoder

SourceDecoder

D/A

Block Diagram for a Digital Communication System

ChannelEncoder

Analog-to-Digital (A/D) Conversion: Analog (i.e., continuous-time continuous-amplitude) message signal is converted into a discrete-time discrete-amplitude digital signals by time-sampling and amplitude-quantization. The resulting signals are then mapped to binary sequences.


Source Encoding: Removes the redundant information embedded in the message signal, therefore represents the message with as few binary digits as possible, i.e., data compression

Channel Encoding: Introduces redundancy in a “controlled” manner which can be used at the receiver to overcome the effects of noise, interference and fading. Provides “noise immunity” to transmitted information.

Source coding and channel coding will not be studied in this course…

Modulation: Converts (maps) codewords to high-frequency analog waveforms. A certain parameter of the carrier signal (i.e., modulated signal) is varied in accordance with message signal (i.e. modulating signal) e.g. amplitude shift keying (ASK), phase shift keying (PSK), frequency shift keying (FSK)

Receiver Blocks: Perform the inverse of the transmitter operations in order to recover the original analog message (continuous-time continuous-amplitude) signal.

In a practical digital communication receiver, there are also additional sub-blocks such as channel estimation, synchronization (frame/frequency/phase), authentications, crypto, multiplexing, etc.

Block Diagram for a Digital Communication System (cont’d)


Why is Modulation Required?

To achieve easy radiation: Dimensions of the transmit/receive antennas are limited by the corresponding wavelength. The frequency conversion allows the use of practical antenna lengths.

To accommodate for simultaneous transmission of several baseband signals: Simultaneous transmission of different baseband signals which are possibly overlapping can be facilitated by assigning slightly different frequency carriers for each one.

Modulatio shifts the baseband signal to a higher frequency band, centered at the so-called “carrier frequency”.

Large bandwidths require high carrier frequencies: Practical requirements in front-end filter design dictates the bandwidth-to-frequency carrier ratio (i.e., fractional bandwidth) be kept within a certain range.

1.001.0 cf

B

cf

B: Fractional bandwidth


To (possibly) expand the bandwidth of the transmitted signal for better transmission quality: When the bandwidth increases, the required SNR (for fixed noise level, corresponding signal power) to achieve a specific transmission rate decreases

Why is Modulation Required? (cont’d)

SNRBC 1log2 12 B

C

SNR

SNRBC 1log2

Channel capacity Bandwidth Signal-to-noise ratio

High-rate transmission requires larger bandwidths (therefore, higher carrier frequencies): According to Shannon Theorem, channel capacity is defined as the maximum achievable information rate that can be transmitted over the channel. For the additive white Gaussian noise (AWGN) channel,


Signal-Space Representations

Consider a modulation format where the transmitted signal waveforms belong to the modulation set .

Each of the waveform can be represented as a point (vector) in an N-dimensional signal space (sometimes called as vector space) defined by the orthonormal basis functions

Mmm ts 1

Nnn t 1 MN

sTt 0

jidttt jT

is *

0

tstsdtttss n

N

nnmn

T

mnm

s

1

,*

0,

Nmmmmm sssts ,2,1, ,...,, s

The Gram-Schmidt procedure (See Appendix A of the textbook) provides a systematic approach to construct the set of orthonormal functions, which span the signal space.


2

1

2,

0

2m

N

nnm

T

ms sdttsEs

s

Mmlk ,...2,1,

Energy

Correlation

Euclidean Distance

Signal-Space Representations (cont’d)

Nmmmm sssts ,2,1, ,...,, ms

Nkkkkk sssts ,2,1, ,...,, s

Nlllll sssts ,2,1, ,...,, s


M-ary Amplitude Shift Keying (M-ASK)

tfAts cmm 2cos sTt 0 MmMmAm ...2,1,12

otherwise ,0

0 ,2cos2 scs TttfTt

Basis Function(s) (Obtained through Gram-Schmidt procedure)

Signal-Space (Vector Space) Representation (Obtained through the use of basis functions)

2smmm TAts s

mlpmtfj

mcmm AtseAtfAts c ,2Re2cos

Baseband (Equivalent Lowpass) Representation

1-dimensional

22

0

2sm

T

ms TAdttsEs

m Signal Energy


M-ASK (cont’d)

Examples of M-ASK Signal Constellations

M=4

Bandpass Modulation Signal

Equivalent Lowpass Signal

11 10 00 01


M-ary Phase Shift Keying (M-PSK)

mcm tfAts 2cos sTt 0 MmMmm ...2,1,12

otherwise ,0

0 ,2cos21

scs TttfTt

otherwise ,0

0 ,2sin22

scs TttfTt

Basis Functions

Signal-Space Representation

msmsmm TATAts sin2,cos2 s

mcm jlpm

tfjjmcm AetseAetfAts ,

2Re2cos


2-dimensional

22

0

2s

T

mss TAdttsEEs

m Signal Energy


tfAts

tfAts

c

c

2cos0

2cos1

2

1

Example: Binary Phase Shift Keying (BPSK)

sEsE t1

t2 1 0 1 1 0

1 0 1 1 0

A

-A

A

-A

0,2 sEs


0,1 sEs

t

t

Bandpass Modulation Signal

Equivalent Lowpass Signal


Example: Quadrature Phase Shift Keying (QPSK)

s

s

s

s

Es

Es

Es

Es

,0 232cos 11

0, 2cos10

,0 22cos01

0, 2cos00

44

33

22

11

tfAts

tfAts

tfAts

tfAts

c

c

c

c

sEsE t1

t2Signal-Space Representation

sE

sE


Quadrature Amplitude Modulation (QAM)

tfAtfAts cimcrmm 2sin2cos ,,

Am,r, Am,i: Information-bearing signal amplitudes of the quadrature carriers

sTt 0Mm ,...2,1

Alternatively, QAM can be considered as a combination of ASK and PSK.

2,

2, imrmm AAA rmimm AAarctg ,,

mcmm tfAts 2cos where sTt 0Mm ,...2,1

Examples of QAM Signal Constellations


QAM (cont’d)

tfjimrmcimcrmm

cejAAtfAtfAts 2,,,, Re 2sin2cos

mjarctgmimrmlpm eAjAAts ,,,

otherwise ,0

0 ,2cos21

scs TttfTt

otherwise ,0

0 ,2sin22

scs TttfTt

Basis Functions


2,2 ,, simsrmmm TATAts s 2-dimensional


22,

2,

0

2simrm

T

ms TAAdttsEs

m

Signal Energy


M-ary Frequency Shift Keying (M-FSK)

tffAts mcm 2cos sTt 0 Mmfmfm ...2,1,

tfjlpm

tfjtfjmcm

mcm AetseAetffAts 2,

22Re2cos

Cross Correlation

flkTjT

ftlkj

slk e

flkT

flkTdte

T

s

sin1

0

2,

flkT

flkTlk

2

2sin,


0, lkFor andTf 21 lk

Tf 21Therefore, the minimum frequency separation between adjacent signals for orthogonality of the M signals is


M-FSK (cont’d)

0......0011 sEts s

Tf 21 Assuming frequency separation , the signal-space representation for the M-FSK signals are given as N-dimensional vectors, where N=M.

otherwise ,0

0 ,2cos2 smcsm

TttffTt

0......0022 sEts s

sMM Ets ......000 s

.

.

.

22

0

2s

T

mss TAdttsEEs

mwhere


Performance Measures for Modulation Schemes

Bandwidth (spectral) efficiency: How much bandwidth is needed for a given data rate?

zBits/sec/Hlog2

W

TM

W

R sss

: Bandwidth efficiency

: Data rate W : Bandwidth

The bandwidth depends on the modulation scheme and pulse shaping. Power spectral density (PSD) is typically used to determine the bandwidth of the transmitted signal. There are various definitions for bandwidth:

• Main lobe (null-to-null) bandwidth: The width of the main spectral lobe.

• Fractional power-containment bandwidth: The frequency interval that contains (1-ε) of the total signal power, e.g. 99.9% of the total power.

• Bounded PSD bandwidth: The frequency interval where the PSD stays above a prescribed certain threshold, e.g. sidelobes peaks 40 dB below its maximum value

Roughly speaking, bandwidth of the modulation scheme is proportional to the dimension number.

s

sR


Power efficiency: How much power is needed for reliable transmission with a specified fidelity?

The fidelity for a digital communication system is usually measured in terms of symbol- or bit-error probability. For a given SNR, we aim to achieve a low error probability (how low? it depends on the application). Symbol error probability (SEP) is in general easier to evaluate. Bit error probability (BEP) depends on the mapping of source bits onto modulation signals. A bound on BEP is given as

Performance Measures for Modulation Schemes (cont’d)

ePePM

ePb

2log

Two common mapping forms are “natural mapping” and “Gray mapping”. In Gray mapping, the neighbour points differ in only one digit. It should be noted that Gray mapping is not possible for every signal constellation.


Temporal efficiency: How wide are the time variations of the transmitted signal?

Temporal efficiency=Peak power/Average power

The choice of amplifier depends on the temporal characteristics of the signal.

Other considerations:

• Hardware/software implementation complexity & cost of implementation

• Sensitivity to interference

• Robustness to impairments encountered in a wireless channel

Performance Measures for Modulation Schemes (cont’d)

In most practical scenarios, these performance measures conflict with each other. The communication system designer should be able to find the best “trade-off” for a given application under specific constraints.


Comparison of Spectral Efficiency of Modulation Schemes

M-PSK and QAM

bits/seclog

rate Data 2

T

M

Hz2

nulltonullBWT

M-FSK

bits/seclog

rate Data 2

T

M

Hz2

roughlyBWT

M

M: “Modulation order”, “Constellation size”


Power Spectral Density (PSD) In practical, pulse shaping should be considered for a precise bandwidth measurement and considered in the spectral efficiency calculations.

Power spectral density (PSD) describes the distribution of signal power in the frequency domain. If the baseband equivalent of the transmitted signal sequence is given as

ksk kTtpatg ka : Baseband modulation symbol

sT : Signal interval tp : Pulse shape

ffPT

f as

g 21

then the PSD of g(t) is given as

tpFfP

sfnTj

naa enRf 2

where

See Ch.4 of Digital Communications by Proakis for the proof


Example: PSD of BPSK with Rectangle Pulse Shaping

ksk kTtpatg Aak

0 ,0

0,

0,

0 , 2

*

2*

n

nA

naEaE

naEaaEnR

nkk

knkka

22 AenRnRFf sfnTj

naaa

p(t)

T/2 T

Autocorrelation of data sequence

Pulse shaping

t

Baseband equivalent of BPSK sequence

Independent data symbols are assumed


Example: PSD of BPSK with Rectangle Pulse Shaping (cont’d)

PSD of baseband BPSK sequence


PSD of bandpass BPSK sequence

ccFT ffGffGfS *

2

1 tfj cetgts 2Re

scsscs

cgcgs

TffTATffTA

fffff

2222

*

sinc4

1sinc

4

1

4

1

Example: PSD of BPSK with Rectangle Pulse Shaping (cont’d)

Bandpass BPSK sequence and its Fourier transform (spectral density)

Null-to-null bandwidth

See Tutorial 1

See Ch.4 of Digital Communications by Proakis for the proof


Example: PSD of QAM with Rectangle Pulse Shaping

kskk kTtpjbatg AAba kk 3,,

Baseband equivalent of QAM sequence

0 ,0

0,10

0,

0 ,

22

n

nA

njbajbaE

njbaE

jbajbaEnR

nknkkk

kk

nknkkka

Autocorrelation of data sequence

PSD of baseband QAM sequence

fTTAfg22 sinc10

PSD of bandpass QAM sequence

scsscss TffTATffTAf 2222 sinc4

10sinc

4

10

Note that PSD of QAM has the same general form as BPSK.


Some Practical Pulse Shapes

Below are some pulse shapes commonly used in communication systems:

TtT

tAtp

0 ,sin

Half-Sinusoid Pulse

2

1sinc

2

1-sinc

2fTfTe

ATfP fTj

Full-Cosine Pulse

TtT

tAtp

0 cos1

2

1sinc1sinc2sinc4

fTfTfTeAT

fP fTj


Some Practical Pulse Shapes (cont’d)

Gaussian Pulse

22

22ln

2exp

TtBAtp

2

2

2lnexp

2

2ln

B

fe

T

AfP fTj

where B is defined as the “3dB bandwidth of pulse”

Raised Cosine Pulse

22241

cossin

Tt

Tt

Tt

Tttp

Tf

Tf

TTf

TTT

fT

fP

2

1 ,0

2

1

2

1,

2

1cos1

2

2

10 ,

10 α: Roll-off factor


Comparison of Pulse Shapes

Time-Domain

Gaussian

Half-sinusoid

Full-cosine

Square


Comparison of Pulse Shapes (cont’d)

Frequency-Domain

Square

Gaussian

Half-sinusoid

Full-cosine

2/T

3/T

4/T

• Square

BW=2/T

• Half-sinusoid

BW=3/T

• Full-cosine

BW=4/T


Comparison of Pulse Shapes (cont’d)

Raised Cosine

10

α: Roll-off factor

TBW

T

21

1/T

2/T

TBW

1


For a given SNR (i.e. a given signal power for fixed noise power), we aim to achieve a low error probability. To calculate error probability, first we need to identify the receiver structure.

The receiver consists of a demodulator and a detector:

• The demodulator converts the received waveform r(t) into a N dimensional vector where N is the dimension of the signal-space for the given modulation type.

• The detector decides which of the possible M signal waveforms was transmitted based on r, where M is the constellation size.

Optimum Receiver for AWGN

Nrrr ,..., 21r

Demodulator Detector tr r ms tsm

tn nsr m tntstr m


Optimum Receiver for AWGN (cont’d)

Correlation-type demodulator Matched-filter demodulator

For details, see Proakis’ Digital Communications Chapter 5


Optimum Receiver for AWGN (cont’d) We want to design a signal detector that makes a decision based on the observation of the vector r such that the probability of a correct decision is maximized. The optimal decision rule is based on the maximization of so-called “a posteriori probabilities”

rsmp : The probability of choosing sm m=1,2…M based on the observation of r

This decision criterion is called the Maximum A Posteriori Probability (MAP) rule.

mMm

mmMm

mm

Mm

mMm

p

pp

p

pp

p

sr

ssr

r

ssr

rs

...2,1

...2,1

...2,1

...2,1

max

max

max

max

Bayes Theorem

rp : Common for all

Mp m 1s , i.e. Equally probable messages

The conditional pdf is called the likelihood function and the decision criterion based on the maximization of over the M signals is called the maximum likelihood (ML) criterion.

mp sr mp sr


nsr m

0

2

022

2

2exp

1

2exp

2

1

02 N

n

N

nnf k

N

kk

For an AWGN channel, the components of the noise vector n are zero-mean Gaussian random variables with variance N0/2

N

kkmkN

N

kkmk

N

kkmk

N

kkmkm

srNN

srNN

srfsrpp

1

2,

02

0

1

2,

00

1,

1,

1exp

1

1exp

1

sr

The received signal will have a Gaussian conditional distribution


Nk ...2,1


2

1

2, minminmax m

m

N

kkmk

mm

msrp srsr

The ML rule is then given as

The ML receiver decides in favor of the signal which is closest in Euclidean distance to the received vector, r.


222 2minmin mmm

mm

ssrrsr

Expanding the decision rule,

where is the signal energy. Neglecting terms which do not affect the decision and under the assumption that constant-energy modulation set (e.g. PSK) is used

2mmE s

mm

mm

srsr maxmin 2

“Distance” metrics

“Correlation” metrics


Example: Error Probability for BPSK

11 2cos0 tfAtsb c

22 2cos1 tfAtsb c

01 where

2where

tfTEtfAts cc 2cos22cos1

tfTEtfAts cc 2cos22cos2

tsts 12

Unlike other M-PSK for M>2, we can represent this special form of BPSK signal as 1-dimensional signal. The basis function is given as

otherwise ,0

0 ,2cos21

TttfTt c

trdt

T

0

.Euclidean Distance Decoder

t1

Therefore, the optimal receiver has the following form of

r

i.e. antipodal signaling


Example: Error Probability for BPSK (cont’d)

Assume s1(t) is sent. Under the assumption of AWGN, the received signal

twtstr 1

The output of demodulator

nEdtttwtsdtttrTT

0

110

1

where

2,0~ 00

1 NNdtttwnTdef

Assume s2(t) is sent. The output of demodulator is now

nEdtttrT

0

1


Example: Error Probability for BPSK

Eb

Eb

1

00

0

1r

Here we have two possible alternatives, therefore we can use a “zero threshold detector” as an optimal detector.

001ˆ110ˆ

0,1ˆ1,0ˆ

bPbbPbPbbP

bbPbbPeP

2/110 bPbP

01ˆ10ˆ bbPbbP Due to symmetry

Equally probable messages 10ˆ bbPeP

Under the assumption that b=1 is sent zEr

drbrfbrPbbP

011010ˆ

Let P(e) denote the error probability

EE

Decision regions

0ˆb1ˆb

Bayes Theorem


Example: Error Probability for BPSK (cont’d)

0

2

N

EQ

drbrfbbP

0110ˆ

dr

N

Er

N

0 0

2

0

exp1

20N

Ery

dyy

NE

02

2

2exp

2

1

where Q-function is defined as dyexQx

y 22

2

1

E E


Example: Error Probability for QPSK

22cos 11

232cos10

22cos01

2cos00

4

3

2

1

tfAts

tfAts

tfAts

tfAts

c

c

c

c

tr

dtT

0

. Detector

t1

dtT

0

.

t2

otherwise ,0

0 ,2cos21

scs TttfTt

otherwise ,0

0 ,2sin22

scs TttfTt

2

4,3,2,1minˆ msrs m


Example: Error Probability for QPSK (cont’d)

Under the assumption of AWGN (which exhibits symmetry), rotating and moving the signal constellation does not change the error probability. Therefore, we can rotate/move our signal constellation in such a way that the resulting constellation allows easy mathematical derivation.

Here, we move our constellation as the “target” signal is located on the origin. If there is no symmetry in the signal constellation, this should be repeated for each signal.

Decision regions

First, we calculate P(c), i.e. the probability of making a correct decision. Then, probability of error is simply found as P(e)=1-P(c).


Assume that the signal located at the origin has been transmitted. If the received signal is in the shaded area, this means we will make a correct decision.

2

0

2

0

1

1

22

2,2

N

EQ

N

EQ

dnPdnP

dndnPscP

s

s

QI

QI


sEd 2d

2d

2d 2,0~ 00

1 NNdtttwnTdef

I

2,0~ 00

2 NNdtttwnTdef

Q

QnP

2,~ Nn

QQ 1


Due to symmetry,

4321 scPscPscPscPcP

2

00

2

2

00

222

21

N

EQ

N

EQ

N

EQ

N

EQcPeP

bb

EE

ss

bs



Example: Error Probability for BFSK

t

TfAts c 2

12cos0 1

tr

dtT

0

. Detector

t1

dtT

0

.

t2

2

4,3,2,1minˆ msrs m

t

TfAts c

12cos1 2

otherwise ,0

0 ,212cos21

scs TttTfTt

otherwise ,0

0 ,12cos22

scs TttTfTt


sE t1

t2

sE

Example: Error Probability for BFSK (cont’d)

sEd 2

0N

EQeP s

By rotation, it can be easily shown that

Now, we will study the same problem without rotation:

Assume was sent. The received signal is 0,11 sEts s QIs nnE ,r

Decision is based on mm

mm

srsr maxmin 2

0121 N

EQEnPEnnPPeP s

ssIQsrsrs

2,0~, 0NNnn IQ 0,0~ NNnnn IQ

def

Due to symmetry,

01 N

EQePeP ss


A Union Bound on Error Probability

2mmd sr

In most cases, probability of error can not be obtained in closed form. Therefore, one needs to find some bounds or approximations which can work for any signal constellation.

We have already shown that the optimal decoder for any signal constellation over AWGN is given by the Euclidean distance decoder, i.e.

M

mmm

M

mm eP

MPePeP

11

1sss meP s : Probability of making a

decision error when sm was sent

M

lmlm

M

lmml

M

lmmlm

ll

ll

ml

P

ddP

ddPeP

1

1

1

sss

s

ss

ii

ii APAP

Union Bound (U-B)

lmP ss : The probability of choosing sl

instead of the originally transmitted sm


A Union Bound on Error Probability (cont’d)

M

l

N

dBUB

M

l

mlBU

N

M

l

mlM

lmlmm

ml

ml

ml

mlml

e

N

dQ

dQPeP

1

4

1 0

,

2

1

,

1

0

2,

02

2

2

ssss

U-B: Union Bound

M

m

M

l

mlM

mm

mlN

dQ

MeP

MeP

1 1 0

,

1 2

11s

Assuming equal-probable message signals, the probability of error is

UB-B: Union-Bhattacharyya Bound

2, mlmld ss where

2/2xexQ


The U-B requires the computation of all distances dl,m among signals in the constellation. A looser bound can be obtained as follows

0

min

1 0

,

21

2 N

dQM

N

dQeP

M

l

mlBU

m

ml

s

0

min

21

N

dQM

P(e) is dominated by the minimum Euclidean distance of the signal constellation.

A Union Bound on Error Probability (cont’d)

M

m

M

l

mlM

mm

mlN

dQ

MeP

MeP

1 1 0

,

1 2

11s

Then the probability of error is found as

mlml

dd ,,

min minwhere is the minimum Euclidean distance of the constellation.

“Minimum Euclidean distance” bound


An Approximation for Error Probability

As an alternative, we can also the following approximate upper bound

0

min,

~

1 0

,

2

2 min N

dQN

N

dQeP md

M

l

mlBU

m

ml

s

Approximate upper bound

mdNmin,

: Number of signals at distance dmin from sm

0

min

2min N

dQNd

M

mmdd N

MN

1,minmin

1

M

mmeP

MeP

1

1s

M

mmd N

dQN

M 1 0

min, 2

1~

min


Error Probability for M-PSK

sE

MME

MEd bs

22

22min sinlog4sin4

2 ,2

2 ,1min M

MNd

2 ,sinlog2

2

2 ,2

22

0

0

MM

MN

EQ

MN

EQ

ePb

b

Replacing and into the formula on p.50, we obtain2mind

mindN


Error Probability for M-PSK (cont’d)

Error rate degrades as M increases.

Recall that spectral efficiency increases as M increases.


Error Probability for QAM

tfAtfAts cimcrmm 2sin2cos ,, 2,2 ,, simsrmm TATA s

AAAA imrm 3,, ,,

sssss TAEEEE 29151230

sssss TAEEEE 210965

sssss

ssss

TAEEEE

EEEE

25

1413117

8421

s

M

mss TAE

ME

mavg

2

15

1

avg

bs

avg bEE

ss EETAd5

8

5

222

4min

0s 1s 2s 3s

4s 5s 6s 7s

8s 9s 10s 11s

12s 13s 14s 15s

mind


Error Probability for QAM (cont’d)

0s 1s 2s 3s

4s 5s 6s 7s

8s 9s 10s 11s

12s 13s 14s 15s

2 neighbours

3 neighbours

4 neighbours

151230 ,,, ssss

10965 ,,, ssss

11714138421 ,,,,,,, ssssssss

• 2 neighbours

• 4 neighbours

• 3 neighbours

00

min

5

43

2~

min N

EQ

N

dQNeP avgb

d

Using the result from p.50, we obtain an approximate upper bound

31

1min,min

M

mdd m

NM

N

3 neighbours


Error Probability for QAM (cont’d)

Power efficiency decreases with increasing M, but not early as fast as M-PSK.

Recall that spectral efficiency increases as M increases.


tffAts mcm 2cos sTt 0 MmTmfm ...2,1,2

Error Probability for M-FSK

Each signal occupies its own dimension. Therefore, each signal has M-1neighbours, separated from each other by

sEd 2min

0

2

0

log11

N

MEQM

N

EQMeP bs


Error Probability for M-FSK (cont’d)

As M increases, power efficiency improves (i.e. less Eb is required).

Recall that spectral efficiency decreases as M increases.

For M=2, BFSK requires 3dB more energy/bit to achieve the same P(e) as BPSK. In other words, BPSK is 3dB more power efficient that BFSK.

0

2

N

EQeP b

• BPSK

0N

EQeP b

• BFSK


Comparison of Power Efficiency of Modulation Schemes

We will use BPSK/QPSK as a benchmark with which to compare the power efficiency of other modulation schemes.

BPSK/QPSK has . Now, define the power efficiency of a modulation scheme (relative to BPSK/QPSK) as

bEd 42min

bP E

d

4log10

2min

10


Differential Phase Shift Keying (DPSK) So far, we assumed that coherent demodulation is performed, i.e. that the

carrier phase is perfectly known at the receiver. This normally requires carrier phase estimation.

An alternative is “differentially encoding”, where the data is encoded in phase difference from one symbol to the next. Assuming binary signalling,

kkkkk baabd 11,11,0

dk 0 1 1 1 0 1

bk +1 -1 -1 -1 +1 -1

ak +1 +1 -1 +1 -1 -1 +1

This diagram might correspond to either PSK or DPSK!


Transmitter and Receiver for DPSK

Mapper Differential Encoder

1,0kd 1kb 1ka tsDPSK

tA ccos

tccos2

tcsin2

dt

dt 1 Symbol Delay *

tr kTt

*1ky

kzky .Resgn

θ represents any mismatch between transmitter/receiver oscillators or phase introduced by the channel. In our system model, (independent of where it comes from) we included in the transmitter block.

In coherent systems, we need to estimate and compensate this phase error at the receiver. Here, we simply ignore it!

tyI

tyQ


Error Probability of DPSK

tAats ck cos kTtTk 1

kkj

Tk

T

j

Tk NaATedttNdtaAedttyy TNNNk 02,0~

tNAea jk

211 kk

def

yy 212 kk

def

yy Defining the decision variable can be written as

22

21

*1Resgn kkk yyz

110Re11ˆ21

*1 kkkkkk bPbyyPbbPeP

tjytyty QI


We need to find statistical properties of and : 1 2

22/ 111 kkkkj NNaaATe

22/ 112 kkkkj NNaaATe

2/11 kkj aaATeE

2/12 kkj aaATeE

TNNNNNEEEVar kkkk 0*

112

111 4

1

TNNNNNEEEVar kkkk 0*

112

222 4

1

Error Probability of DPSK (cont’d)

211 kk

def

yy 212 kk

def

yy

First, we recall the definitions


kkk baa 1 Encoding scheme:

+1 +1 +1 +1 0

-1 +1 -1 0 -1

-1 -1 +1 -1 0

+1 -1 -1 0 1

ka 1ka kb 21 kk aa 21 kk aa

01 E

jATeE 2

TNVar 01

TNVar 02

211 ,0~, NIR

22 ,cos~ ATNR 2

2 ,sin~ ATNI

Rician: Rayleigh,: 21

Under the assumption that is sent1kb

202 TNwhere

IR j 111

IR j 222

Complex Gaussian

Complex Gaussian



Rayleigh:1

Rician:2

2

21

21

12

exp

f

22

022

2222

22

1exp

If

ATTA 22222 sincoswhere the non-zero mean is found as

dfPPeP2

02121

Now, we return to P(e) computation

22

2

121

21

21

2exp

exp

2exp

22

duu

dP

21

2def

u


Rayleigh: 1Rician: 2


dxmx

Imxxm

dxx

Ixx

dIeP

202

22

022

2

202

22

02

202

22

022

2

2exp

2exp

2

1

22exp

2

1

2exp

2exp

2def

x

Variable change

2def

m

=1

0

0

2

exp2

1

2exp

2

1

N

E

N

TA

222 TAdttsET


202 TN

AT


0 2 4 6 8 10 1210

-6

10-5

10-4

10-3

10-2

10-1

100

Coherent vs. Differential PSK

SNR [dB]

BE

R

CoherentDifferential


There is some performance degradation due to differential detection, but now a less complex receiver can be used (i.e. no need for phase tracking).


Temporal Characteristics of Modulation Schemes

So far, we have considered pulse shapes which are strictly limited in the symbol interval. By using a pulse shape to “spill over” into adjacent symbol intervals, better spectral efficiency can be achieved, however this also results in intersymbol interference (ISI).

The following block diagram is commonly used for studying ISI. Assuming matched filter type implementation for the demodulator,

tw

thT thC thR

slTt

Detector

ksk kTtpatg

“Actual” Channel

“Equivalent” Channel

sseqk

kl lTnTklhaz where thththh RCTeq thtwtn R


Temporal Characteristics of Modulation Schemes (cont’d)

Here, we use in pulse shapes which spill over adjacent symbols. This will bring ISI terms:

otherwise ,0

0 ,1 nnTh seq

constant

sleq T

lfH

This condition is known as “Nyquist pulse-shaping criterion” or “Nyquist condition for zero ISI”.

sseqlk

keqlsseqk

kl lTnTklhahalTnTklhaz

0

ISI terms The condition for no ISI is

In frequency domain, this requires

See proof Proakis “Digital Communications” Chapter 9



WTs 2

1

WTs 2

1

For this case, there is no choice for Heq to satisfy Nyquist criterion.

sleq T

lfH

sleq T

lfH

otherwise,0

, WfTfH s

eq

For this case, there is only one solution:

W: Bandwidth of equivalent ch.

In the following, we consider three distinct cases:


WTs 2

1

sleq T

lfH


For this case, there exists many solutions as to satisfy cons.

sleq T

lfH

A particular pulse shape which satisfies the above property and has been widely used in practical applications is “raised cosine”. (See page 28) The “Nyquist” pulse takes zero at the sampling points for adjacent signalling intervals.

cons. sl

TlfX



Under the matched-filter assumption (i.e. which maximizes the output signal-to-noise ratio), the transmit and receive filters satisfy

fHfH RT

Under the ideal channel assumption , i.e.

fHfHfH eqRT

1fHC

For “raised-cosine” equivalent channel response, we can divide it into two “root-raised-cosine” (RRC) filters.

s

sss

ss

ss

RCRRC

Tf

Tf

TTf

TT

TfT

fHfH

2

1 ,0

2

1

2

1 ,

2

1cos

2

2

1 ,


Temporal Characteristics of BPSK

sk

k kTtpatg

Consider the baseband BPSK modulated signal with RRC pulse shape

1ka

“Eye pattern” is a sketch of g(t) for all possible combinations of ,...,, 321 aaa

• Minimum instantaneous power=0

• Maximum instantaneous power=(1.6)2=4.1 [dB]

• Dynamic range=4.1 [dB]

• Average power=1=0 [dB]

• dB1.4power Avg.powerPeak

For this example, we observe large “dynamic range of instantaneous power” and large “peak/average ratio”. These make the design of TX power amplifier difficult.


The QPSK signal with pulse shaping can be written as

tkTtpbtkTtpatg ck

skck

sk sincos

1, kk ba

The instantaneous power of the QPSK signal is

22

ksk

ksk kTtpbkTtpa

Hence, a QPSK signal suffers similar time-domain problems as a BPSK signal. Now assume, different pulses are used for I&Q channels. If Q channel pulse is delayed by 1/2 symbol relative to I channel pulse, i.e. the instantaneous power is

2sTtptq

22

2

kssk

ksk TkTtpbkTtpa

Both terms can not pass through zero simultaneously, hence significantly increasing the minimum instantaneous power and reducing dynamic range of the signal. PSD and BER remain unchanged. This is known as “Offset QPSK (OQPSK)”.

Temporal Characteristics of QPSK


+ =

jj

kk

ee

jba

,2 434 , jj

kk

ee

jba

kee

kee

jba

jj

jj

kk

oddfor ,,

even for ,,2

434

Another variant of QPSK is “π/4-QPSK”. This modulation scheme is a superposition of two QPSK signal constellations offset by π/4 relative to each other.

PSD and BER of π/4-QPSK are the same as QPSK.

Temporal Characteristics of QPSK (cont’d)


In QPSK, transitions between opposite points in the signal constellation cause the instantaneous power to zero, leading to a large dynamic range.

The special structure of π/4-QPSK avoids transitions which pass the origin, reducing dynamic range and peak-to-average power ratio.

Temporal Characteristics of QPSK (cont’d)


Continuous FSK

We can get perfect temporal properties by using continuous FSK (CFSK)

sk

k kTtpatg 1ka

k

skc

t

c kTtqahtfdghtfts 22cos 22cos

where

dkTptqt

s

def

Instantaneous power= constant

Dynamic range=0dB

Peak-to-average power ratio=0dB

There is no abrupt switching from one phase to another, avoiding phase discontinuities.

h: Modulation index


1/2Ts

t

1/2

t

p(t): “Frequency pulse”

Ts

Ts

Here, we assume a rectangle pulse shape for p(t).

q(t): “Phase pulse”

ns

sn

kk

ksk

aT

nTthah

kTtqaht

22

2;

1

0

a

-3πh

-2πh

-πh

πh

2πh

3πh

0

………

………

Ts 2Ts 3Ts

+1

-1

+1

-1

+1

-1

The shaded path illustrates the phases for the input sequence {+1,+1,-1}

ss TntnT 1

“Phase Tree”

Continuous FSK (cont’d)

n=0,1,..


nns

nn

s

sn

kk hnat

T

haa

T

nTthaht

22

222 ;

1

0a ss TntnT 1

n

nn

s

ncc hnat

T

hafttfts

22cos ;2cos a

1na

hnt

T

hfts n

sc

22cos

1na

hnt

T

hfts n

sc

22cos

sThf

For orthogonality, the minimum value for h should be chosen as h=1/2. This special case is known as “Minimum Shift Keying” (MSK).

The separation between two carriers is



We have already introduced MSK as a special case of modulation family of CFSK.

An MSK signal can be also considered as a special form of OQPSK where the rectangular pulses are replaced with half-sinusoidal pulses.

tfTkTtpatfkTtpats cssk

kcsk

k 2sin22cos2even odd

otherwise ,0

20,2

cos ss

TtT

ttp

The transmission rate on the two orthogonal carriers is 1/2Ts bits/sec so that the combined transmission rate is 1/Ts bits/sec.



Comparison of MSK, QPSK and OQPSK

Continuous phase is assured in MSK while 90 and 180 phase changes are observable for OQPSK and QPSK respectively.


In terms of temporal efficiency, MSK obviously outperforms QPSK and OQPSK.

The main lobe of MSK is wider than that of QPSK and OQPSK and, in terms of null-to-null bandwidth MSK is less spectral efficient.

MSK has lower sidelobes than QPSK and OQPSK Less adjacent channel interference

MSK, QPSK and OQPSK have the same power efficiency.

Comparison of MSK, QPSK and OQPSK (cont’d)


Gaussian MSK

The spectral efficiency of MSK can be further improved by prefiltering.

sk

k kTtpatg Gaussian LPF

MSK Modulator

The frequency response function of Gaussian LPF filter is given as

2ln

2exp

2ln

2

2

2lnexp

2222 tBBth

B

ffH

where B is “3dB-bandwidth of the filter”.

We are interested in how a rectangle pulse passed through a Gaussian LPF will look like.


T

Tt

T

TtQtf

2/2/

Frequency pulse

2ln2 BT

2

12

exp2

111

2x

x

xxQx

tq

Phase pulse

T

Ttx

2/1

T

Ttx

2/2

Phase pulse corresponding to rectangular pulse shaping (i.e. no filtering) is also included in the figure.

Gaussian MSK (cont’d)

BT: Normalized 3dB-Bandwidth


For BT ∞, the pulse shape takes its original “unfiltered” form , i.e. rectangle pulse. GMSKMSK

The frequency pulse has a duration of 2Ts although signaling rate is 1/Ts. Such a LPF will result in intersymbol interference which requires sequence estimation for optimal detection.


BT: Normalized 3dB-Bandwidth of Gaussian filter


BT should be chosen as to find a good compromise between spectral efficiency and ISI.

As BT decreases, the spectral efficiency improves (i.e. less bandwith). Also sidelobes fall off very rapidly (i.e. less adjacent channel interference).

However, reducing BT results in ISI and error rate performance degrades (i.e. observation of an “irreducible error floor” due to ISI)

In practical application, BT is typically chosen as (0.2, 0.5). GSM systems use GMSK with BT=0.35.


Engineering

Ece414 chapter3 w12