ECE414 Wireless Communications, University of Waterloo, Winter 2012 1 Bandpass Transmission Techniques for Wireless Communication Chapter 3
May 24, 2015
ECE414 Wireless Communications, University of Waterloo, Winter 2012 1
Bandpass Transmission Techniques for Wireless Communication
Chapter 3
ECE414 Wireless Communications, University of Waterloo, Winter 2012 2
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
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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.
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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)
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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
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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,
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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.
ECE414 Wireless Communications, University of Waterloo, Winter 2012 8
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
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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
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M-ASK (cont’d)
Examples of M-ASK Signal Constellations
M=4
Bandpass Modulation Signal
Equivalent Lowpass Signal
11 10 00 01
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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
Baseband (Equivalent Lowpass) Representation
2-dimensional
22
0
2s
T
mss TAdttsEEs
m Signal Energy
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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
Signal-Space Representation
0,1 sEs
t
t
Bandpass Modulation Signal
Equivalent Lowpass Signal
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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
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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
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QAM (cont’d)
tfjimrmcimcrmm
cejAAtfAtfAts 2,,,, Re 2sin2cos
mjarctgmimrmlpm eAjAAts ,,,
otherwise ,0
0 ,2cos21
scs TttfTt
otherwise ,0
0 ,2sin22
scs TttfTt
Basis Functions
Signal-Space Representation
2,2 ,, simsrmmm TATAts s 2-dimensional
Baseband (Equivalent Lowpass) Representation
22,
2,
0
2simrm
T
ms TAAdttsEs
m
Signal Energy
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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,
Baseband (Equivalent Lowpass) Representation
0, lkFor andTf 21 lk
Tf 21Therefore, the minimum frequency separation between adjacent signals for orthogonality of the M signals is
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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
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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
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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.
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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.
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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”
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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
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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
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Example: PSD of BPSK with Rectangle Pulse Shaping (cont’d)
PSD of baseband BPSK sequence
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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
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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.
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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
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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
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Comparison of Pulse Shapes
Time-Domain
Gaussian
Half-sinusoid
Full-cosine
Square
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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
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Comparison of Pulse Shapes (cont’d)
Raised Cosine
10
α: Roll-off factor
TBW
T
21
1/T
2/T
TBW
1
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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
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Optimum Receiver for AWGN (cont’d)
Correlation-type demodulator Matched-filter demodulator
For details, see Proakis’ Digital Communications Chapter 5
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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
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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
Optimum Receiver for AWGN (cont’d)
Nk ...2,1
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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.
Optimum Receiver for AWGN (cont’d)
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
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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
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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
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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
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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
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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
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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).
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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
Example: Error Probability for QPSK (cont’d)
sEd 2d
2d
2d 2,0~ 00
1 NNdtttwnTdef
I
2,0~ 00
2 NNdtttwnTdef
Q
QnP
2,~ Nn
QQ 1
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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 QPSK (cont’d)
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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
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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
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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
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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
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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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 50
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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 51
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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 52
Error Probability for M-PSK (cont’d)
Error rate degrades as M increases.
Recall that spectral efficiency increases as M increases.
ECE414 Wireless Communications, University of Waterloo, Winter 2012 53
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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 54
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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 55
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.
ECE414 Wireless Communications, University of Waterloo, Winter 2012 56
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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 57
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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 58
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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 59
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!
ECE414 Wireless Communications, University of Waterloo, Winter 2012 60
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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 61
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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 62
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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 63
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
Error Probability of DPSK (cont’d)
ECE414 Wireless Communications, University of Waterloo, Winter 2012 64
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
Error Probability of DPSK (cont’d)
Rayleigh: 1Rician: 2
ECE414 Wireless Communications, University of Waterloo, Winter 2012 65
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
Error Probability of DPSK (cont’d)
202 TN
AT
ECE414 Wireless Communications, University of Waterloo, Winter 2012 66
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
Error Probability of DPSK (cont’d)
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).
ECE414 Wireless Communications, University of Waterloo, Winter 2012 67
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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 68
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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 69
Temporal Characteristics of Modulation Schemes (cont’d)
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:
ECE414 Wireless Communications, University of Waterloo, Winter 2012 70
WTs 2
1
sleq T
lfH
Temporal Characteristics of Modulation Schemes (cont’d)
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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 71
Temporal Characteristics of Modulation Schemes (cont’d)
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 ,
ECE414 Wireless Communications, University of Waterloo, Winter 2012 72
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.
ECE414 Wireless Communications, University of Waterloo, Winter 2012 73
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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 74
+ =
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)
ECE414 Wireless Communications, University of Waterloo, Winter 2012 75
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)
ECE414 Wireless Communications, University of Waterloo, Winter 2012 76
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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 77
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,..
ECE414 Wireless Communications, University of Waterloo, Winter 2012 78
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
Continuous FSK (cont’d)
ECE414 Wireless Communications, University of Waterloo, Winter 2012 79
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.
Continuous FSK (cont’d)
ECE414 Wireless Communications, University of Waterloo, Winter 2012 80
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.
ECE414 Wireless Communications, University of Waterloo, Winter 2012 81
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)
ECE414 Wireless Communications, University of Waterloo, Winter 2012 82
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.
ECE414 Wireless Communications, University of Waterloo, Winter 2012 83
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
ECE414 Wireless Communications, University of Waterloo, Winter 2012 84
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.
Gaussian MSK (cont’d)
BT: Normalized 3dB-Bandwidth of Gaussian filter
ECE414 Wireless Communications, University of Waterloo, Winter 2012 85
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.
Gaussian MSK (cont’d)