CHAPTER 4 updated.ppt. Wireless communication

15/05/2008 18:10:18 Academic excellence in ICT Education
1
GTUC
MASTERS IN TELECOM ENGINEERING (MTE)
COURSE: WIRELESS COMMUNICATIONS
DATE: 21st OCT 2016
COURSE CODE: MTE 507
CREDIT HOURS: 3 LECTURER:Dr. D. M. O. Adjin
OFFICE HOURS: 08:00 – 16:00 GMT
ROOM: HOD / Telecom Eng. PHONE: 020 – 269 -8175
TIME: 08:00 – 16:00 GMT E-MAIL: dadjin@gtuc.edu.gh

CHAPTER 4 – PERFORMANCE OF
DIGITAL MODULATION OVER
WIRELESS CHANNELS
 AWGN Channels
 Fading
 Doppler Spread
 Inter-Symbol-Interference
 Diversity

AWGN CHANNELS
 This section discusses the SNR and its relation to
energy per bit (Eb) & energy per symbol (Es).
 Error probability on AWGN channels is also examined for
different modulation techniques as parameterized by these
energy metrics.
3

 Signal-to-Noise Power Ratio & Bit/Symbol Energy
 In an AWGN channel the modulated signal s(t) =
Re{u(t)ej2πfc t } has noise n(t) added to it prior to reception.
 The noise n(t) is a white Gaussian random process with
mean zero & power spectral density (PSD) N0 / B.
 The received signal is thus r(t) = s(t) + n(t ).
4

 We define the received SNR as the ratio of the
Rx’d signal power Pr to the power of the noise
within the bandwidth of the Tx’d signal s(t).
 The Rx’d power Pr is determined by the Tx’d power & the
path loss, shadowing, & multipath fading.
 The noise power is determined by the bandwidth of the
Tx’d signal & the spectral properties of n(t ).
 If the bandwidth of the complex envelope u(t) of s(t) is B,
 Then the bandwidth of the transmitted signal s(t) is 2B.
5

 Since the noise n(t) has uniform PSD N0/B,
 The total noise power within the bandwidth 2B is N =
N0/B · 2B = N0B. Hence the received SNR is given by:
SNR = Pr / N0B
 In systems with interference, we often use the Rx’d signal-
to-interference-plus-noise power ratio (SINR) in place of
SNR for calculating error probability.
6

 This is a reasonable approximation if the
interference statistics approximate those of
Gaussian noise.
 The received SINR is given by: SINR = Pr / (N0B
+ Pi ),
 Where Pi is the average power of the interference.
7

 The SNR is often expressed in terms of the signal energy
per bit Eb (or per symbol, Es) as:
SNR = Pr /N0B = Es/N0BTs = Eb / N0BTb
 Where:
 Ts is the symbol time
 Tb is the bit time (for binary modulation Ts = Tb and Es
= Eb).
8

 For pulse shaping with Ts =1/B (e.g., raised cosine pulses
with β =1),
 We have SNR = Es/N0 for multilevel signaling and SNR
= Eb/N0 for binary signaling.
 For general pulses, Ts = k/B for some constant k, in which
case k · SNR = Es/N0.
 The quantities γs = Es/N0 and γb = Eb/N0 are sometimes
called the SNR per symbol & the SNR per bit,
respectively.
9

 For performance specification, we are interested in the bit
error probability Pb as a function of γb.
 With M-ary signalling (e.g., MPAM & MPSK) the Pb
depends on both the symbol error probability & the
mapping of bits to symbols.
 Thus, we typically compute the symbol error
probability Ps as a function of γs based on the signal
space concepts
 Then obtain Pb as a function of γb using an exact or
approximate conversion.
10

 The approximate conversion typically assumes that,
 The symbol energy is divided equally among all bits
 Gray encoding is used, so that (at reasonable SNRs) one symbol
error corresponds to exactly one bit error.
 These assumptions for M-ary signaling lead to the
approximations
11

Error Probability for BPSK & QPSK
 We first consider BPSK modulation with coherent
detection & perfect recovery of the carrier freq & phase.
 With binary modulation each symbol corresponds to
one bit, so the symbol & bit error rates are the same.
 The Tx’d signal is s1(t) = Ag(t ) cos(2πfc t) to send a 0-bit
and s2(t) = −Ag(t ) cos(2πfc t) to send a 1-bit for A > 0.
 Thus, Pb = Q(√dmin / 2N0)
12

 dmin = s1 − s0 = A − (−A) = 2A.
 Let us now relate A to the energy per bit.
 We have:
 Thus, the signal constellation for BPSK in terms of energy
per bit is given by s0 = √Eb and s1 = − √Eb.
 This yields the minimum distance dmin = 2A = 2√Eb.
 By Substitution yields:
13

 QPSK modulation consists of BPSK modulation on both
the in-phase & quadrature components of the signal.
 With perfect phase & carrier recovery, the received signal
components corresponding to each of these branches are
orthogonal.
 Therefore, the bit error probability on each branch is
the same as for BPSK: Pb = Q(√2γb)
 The symbol error probability equals the probability that
either branch has a bit error:
14

FADING
 In AWGN the probability of symbol error depends
on the received SNR or, equivalently, on γs .
 In a fading environment,
 The Prx varies randomly over distance or time as a result of
shadowing and/or multipath fading.
 Thus, in fading, γs is a random variable with distribution
pγs(γ ) , thus, Ps(γs) is also random.
 The performance metric when γs is random depends on the
rate of change of the fading.
15

 There are three different performance criteria that
can be used to characterize the random variable Ps :
 The outage probability, Pout, defined as the probability that γs falls
below a given value corresponding to the maximum allowable
Ps ;
 The average error probability, Ps
- , averaged over the distribution of
γs;
 Combined average error probability & outage,
 Defined as the average error probability that can be achieved
some percentage of time or spatial locations.
16

 The Average Probability Of Symbol Error Applies When
The Fading Coherence Time Is On The Order Of A
Symbol Time (Ts ≈ Tc),
 Thus, The Signal Fade Level Is Roughly Constant Over
A Symbol Period.
 The Average Error Probability Is A Reasonably Good
Figure Of Merit For The Channel Quality Under These
Conditions.
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 If The Signal Fading Is Changing Slowly (Ts << Tc) Then
A Deep Fade Will Affect Many Simultaneous Symbols.
 Thus, Fading May Lead To Large Error Bursts,
 Which Cannot Be Corrected For With Coding Of Reasonable
Complexity.
 Hence, These Error Bursts Can Seriously Degrade End-to-
end Performance.
18

 Outage & Average Error Probability Are Often
Combined,
 When The Channel Is Modeled As A Combination Of Fast & Slow
Fading
E.g., Log-normal Shadowing With Fast Rayleigh Fading.
 Note That,
 If Tc << Ts, Then The Fading Will Be Averaged Out By
The Matched Filter In The Demodulator.
 Thus, For Very Fast Fading, Performance Is The Same As
In AWGN.
19

Outage Probability
 The outage probability relative to γ0 is defined
as:
Where γ0 typically specifies the minimum
SNR required for acceptable performance.
20

 For example, if we consider digitized voice, Pb = 10−3
is an acceptable error rate
 It Can’t Be Detected By The Human Ear.
 Thus, for a BPSK signal in Rayleigh fading,
γb < 7 dB would be declared an outage;
hence we obtain γ0 = 7 dB.
21

 In Rayleigh fading the outage probability
becomes:
 Inverting this formula shows that, for a given
outage probability, the required average SNR
 γ-
s is:
22

 In decibels this means that,
10 log γs must exceed the target 10 log γ0 by Fd = −10
log[−ln(1− Pout )],
In order to maintain acceptable performance more
than 100(1− Pout ) percent of the time.
 The quantity Fd is known as the dB Fade Margin.
23

Average Probability of Error
 The average probability of error is used as a
performance metric when Ts ≈ Tc.
 Thus, assume that γs is constant over a symbol time.
 Then the average probability of error is computed by
integrating the error probability in AWGN over the
fading distribution:
24

 Where Ps(γ ) is the probability of symbol error
inAWGN with SNR γ,
 For a given distribution of the fading amplitude r
E.g., In Rayleigh, Rician & log-normal, we compute pγs(γ )
by making the change of variable:
25

 Combined Outage and Average Error Probability
 When The Fading Environment Is A Superposition Of
Both Fast And Slow Fading (E.g., Log-normal Shadowing
& Rayleigh Fading),
A Common Performance Metric Is, “Combined Outage
& Average Error Probability”,
Where Outage Occurs When The Slow Fading Falls Below
Some Target Value And The Average Performance In Non-
outage Is Obtained By Averaging Over The Fast Fading.
26

 We Use The Following Notation:
27

 We can specify an average error probability P-
s with some
probability 1 − Pout.
 An outage is declared,
 When the received SNR per symbol due to shadowing &
path loss alone, γ-
s , falls below a given target value γ-
s0 .
28

 When not in outage (γ-
s ≥ γ-
s0 ),
The average probability of error is obtained by
averaging over the distribution of the fast fading
conditioned on the mean SNR:
29

 The criterion used to determine the outage target γ-
s0 is
typically based on a given maximum acceptable average
probability of error P
-s0 .
 The threshold γ-
s0 must then satisfy the average probability
model below:
30

DOPPLER SPREAD (DS)
 DS Is The Range Of Freqs Over Which The Rx’ved
Spectrum Is Essentially Non-zero.
 It Is The Measure Of Spectral Broadening Caused By The Time Rate
Of Change Of Mobile Radio Channel.
 One Consequence Of DS Is An Irreducible Error Floor For
Modulation Techniques,
 Using Differential Detection.
 Since In Differential Modulation The Signal Phase Associated
With One Symbol Is Used As A Phase Reference For The
Next Symbol.
31

 If The Channel Phase Decorrelates Over A Symbol,
 Then The Phase Reference Becomes Extremely Noisy,
 Leading To A High Symbol Error Rate That Is Independent Of Prx.
 The Phase Correlation B/n Symbols & Consequent
Degradation In Performance Are Functions Of The
Doppler Frequency fD = v/λ & the symbol time Ts .
32

 The channel correlation AC(τ ) over time τ equals the
inverse Fourier Transform of the Doppler power spectrum
SC(f )
As a function of Doppler frequency f.
 The correlation coefficient is thus ρC = AC(T )/AC(0)
evaluated at T = Ts for DQPSK
or at T = Tb for DPSK.
33

INTER-SYMBOL-INTERFERENCE (ISI)
 Inter-symbol Interference Frequency-selective
fading gives rise to ISI,
 Where the received symbol over a given symbol period
experiences interference from other symbols that have
been delayed by multipath.
 Since increasing signal power also increases the
power of the ISI,
 This interference gives rise to an irreducible error floor
that is independent of signal power.
34

 An approximation to symbol error probability with
ISI can be obtained by,
 Treating the ISI as uncorrelated white Gaussian noise.
 Then the SNR becomes:
 Where;
Pr is the received power associated with the LOS signal
component,
I is the received power associated with the ISI.
35

 In a static channel,
 The resulting probability of symbol error will be
Ps(γˆs), where Ps is the probability of symbol error in
AWGN.
 If both the LOS signal component & the ISI experience flat
fading,
 Then γˆs will be a random variable with distribution
p(γˆs),
The average symbol error probability is then;
 P-s = Ps(γˆs)p(γˆs) dγs .
36

 Note that,
 γˆs is the ratio of two random variables – the
LOS received power Pr & the ISI received
power I,
The resulting distribution p(γˆs) may be hard to
obtain
 Irreducible error floors due to ISI are often obtained by
simulation, which can easily
 incorporate different channel models, modulation formats,
and symbol sequence characteristics
37

 Irreducible error floors due to ISI are often obtained by
simulation,
 Which Can Easily Incorporate Different Channel
Models, Modulation Formats & Symbol Sequence
Characteristics
 BPSK, DPSK, QPSK, OQPSK & MSK modulations are
simulated for:
 Different Pulse Shapes
 Channels With Different Power Delay Profiles, Including A
Gaussian, Exponential, Equal-amplitude Two-ray,
 Empirical Power Delay Profile.
38

 The Simulation results indicate that;
The irreducible error floor is more sensitive to the rms
delay spread of the channel than to the shape of its
power delay profile.
Pulse shaping can significantly impact the error floor:
39

DIVERSITY
 We Observed From The Fading Section That,
 Both Rayleigh Fading & Log-normal Shadowing Exact
A Large Power Penalty On The Performance Of
Modulation Over Wireless Channels.
 One Of The Best Techniques To Mitigate The Effects Of
Fading Is Diversity Combination Of Independently Fading
Signal Paths.
Diversity Combination Exploits The Fact That:
Independent Signal Paths Have A Low Probability
Of Experiencing Deep Fades Simultaneously.
40

 Thus,
The Idea Behind Diversity Is To Send The Same
Data Over Independent Fading Paths.
These Independent Paths Are Combined In Such A
Way That The Fading Of The Resultant Signal Is
Reduced.
 Hence,
 The Main Purpose Of Diversity Is To:
 Coherently Combine Independent Fading Paths To
Alleviate The Effects Of Fading In Wireless Channels.
41

 E.g., Assume A System With Two Antennas At
Either The Tx’r Or Rx’r That Experience
Independent Fading.
If The Antennas Are Spaced Sufficiently Far Apart, It
Is Unlikely That They Both Experience Deep Fades At
The Same Time.
 By Selecting The Antenna With The Strongest Signal, A
Technique Known As Selection Combining,
 We Obtain A Much Better Signal Than If We Had Just
One Antenna.
42

 Diversity techniques that mitigate the effect of multipath
fading are called Micro-diversity,
 Diversity to mitigate the effects of shadowing from
buildings and objects is called Macro-diversity.
 Macro-diversity is generally implemented by combining
signals received by several BTSs or RAPs,
 Which requires coordination among these different
stations or points.
43

 Such coordination is implemented.
 As Part Of The Networking Protocols In
Infrastructure-based Wireless Networks.
44

Realization of Independent Fading Paths
 There Are Many Ways Of Achieving Independent
Fading Paths In A Wireless System.
One Method Is To Use Multiple Transmit Or Receive
Antennas, Also Called An Antenna Array,
Where The Elements Of The Array Are Separated In
Distance.
This Type Of Diversity Is Referred To As Space Diversity.
45

 The maximum diversity gain for either Tx’r
or Rx’r space diversity typically requires that,
 The separation b/n antennas be such that the fading
amplitudes corresponding to each antenna are
approximately independent.
 A second method of achieving diversity is by using either two
Tx’t antennas or two Rx’e antennas with different
polarization
E.g., vertically and horizontally polarized waves
The two transmitted waves follow the same path.
46

 A second method of achieving diversity is by using either
two Tx’t antennas or two Rx’e antennas with different
polarization
E.g., vertically and horizontally polarized waves
The two transmitted waves follow the same path.
47

 Freq Diversity
 It is achieved by Tx’g the same narrowband signal at
different carrier freqs, where the carriers are separated by
the coherence bandwidth of the channel.
 This technique requires additional transmit power to send
the signal over multiple freq bands.
48

Typical Freq Diversity Diagram
49

 Time Diversity
 This Is Achieved By:
 Tx’g The Same Signal At Different Times,
 Where The Time Difference Is Greater Than The
Channel Coherence Time (The Inverse Of The Channel
Doppler Spread).
 It Does Not Require Increased Ptx,
 But It Lowers Data Rates, Since Data Is Repeated In The Diversity
Time Slots Rather Than Sending New Data In Those Time Slots.
50

Typical Time Diversity Diagram
51

Receiver Diversity
 In receiver diversity,
 The independent fading paths associated with multiple
receive antennas are combined to obtain a signal that is
then passed thro’ a standard demodulator.
 The combination vary in complexity & overall
performance.
 Most combining techniques are linear:
 The O/p of the combiner is just a weighted sum of the
different fading paths as shown below for M-branch
diversity
52

 Combining more than one branch signal
requires co-phasing,
 where the phase θi of the ith branch is removed
through multiplication by αi = ai e−jθi for some
real-valued ai.
 This phase removal requires coherent detection of
each branch to determine its phase θi.
53

 Linear combiner
54

 Without Co-phasing,
The Branch Signals Would Not Add Up
Coherently In The Combiner,
Hence,
The Resulting Output Could Still Exhibit Significant
Fading Due To Constructive And Destructive
Addition Of The Signals In All The Branches.
55

 Transmitter Diversity
 Here, There Are Multiple Transmit Antennas, & The Pt Is
Divided Among These Antennas.
 Transmit Diversity Is Desirable In Systems Where,
More Space, Power, & Processing Capability Are
Available On The Transmit Side Than On The Receive
Side, As In Cellular Systems.
56

 Transmit diversity design depends on whether or not the
complex channel gain is known to the transmitter.
When this gain is known, the system is quite similar to
receiver diversity.
 Without this channel knowledge,
 Transmit diversity gain requires a combination of space
& time diversity via a novel technique called the
Alamouti scheme and its extensions.
57

 Typical Transmit diversity Diagram
58

END OF CHAPTER FOUR
59

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