Short notes about "Multi-user Radio Communications" part 3

Short Notes about
“Multiuser Radio Communications”
Part Three
Presented by:
Eng. Mohamed Mohy-El Din Shaheen
E-Mail; mohamedmohy24@gmail.com
Teaching Assistant, Dept. of Electrical and
Computer Engineering,
Higher Technological Institute,
Egypt

CONTENTS
8.7- Binary Signaling Over a Rayleigh Fading Channel.
8.7.1- Diversity Techniques.
8.8- TDMA and CDMA Wireless Communication Systems.
8.8.1- Rake Receiver.
8.9- Source Coding of Speech for Wireless
Communications.
8.9.1- Multi-Pulse Excited LPC.
8.9.2- Code Excited LPC.
8.10- Adaptive Antenna Arrays For Wireless
Communications.
8.10.1- Adaptive Antenna Array.
8.11- Summary and Discussion.

8.7- Binary Signaling Over a Rayleigh Fading
Channel.

8.7- BINARY SIGNALING OVER A RAYLEIGH
FADING CHANNEL
 The transmitted signal from a base
station radiates in all directions.
 Some of the transmitted waves are
reflected , diffracted or scattered as
shown in Fig 8.47.
 The phase of the received signals
varies.
 Depending on the value of the phase,
they might add constructively or
destructively.
 Consider the transmission of binary
data,
 Over a Rayleigh Fading Channel.
 The received signal is modeled as
follows;
Fig 8.47 Rayleigh Fading
Channel Concept [31].
𝒙~ 𝒕 = 𝜶 𝒆𝒙𝒑 −𝒋𝝓 𝒔∼ 𝒕 + 𝒘∼ 𝒕 (8.63)
 Where;
 𝑥∼
𝑡 is The Complex Envelope of the
Received signal.
 𝛼 is Rayleigh Distributed Random
Variable, describing Attenuation in
Transmission.
 𝑠~
𝑡 is The Complex Envelope
of the Transmitted Signal.
 𝑤~
𝑡 is a Complex Valued
White Gaussian Noise Process.
 𝜙 is Uniformly Distributed
Random Variable, describing the
Phase Shift in Transmission.

FADING CHANNEL
 It is assumed the channel is Flat in
Time and Frequency,
 So we can estimate the phase shift
𝜙 from the received signal without
error.
 Suppose the Coherent Binary Phase
Shift Keying,
 Is used to do Data Transmission.
 Under the condition that 𝛼 is
constant over a bit interval,
 We may express “The Average
Probability of symbol error”,
 Due to “White Gaussian Noise”
acting alone as follows;
𝑷 𝒆 𝜸 =
𝟏
𝟐
𝒆𝒓𝒇𝒄 𝜸 (8.64)
 𝛾 is An Attenuated Version of
The Transmitted Signal Energy
Per Bit to Noise Spectral Density
Ratio .
 𝑒𝑟𝑓𝑐 is the complementary error
function.
𝜸 =
𝜶 𝟐 𝑬 𝒃
𝑵 𝟎
(8.65)
 Where:
Per Bit to Noise Spectral Density
Ratio .
 𝛼 is Rayleigh Distributed
Random Variable, describing
Attenuation in Transmission.
 𝐸 𝑏 is The Transmitted Signal
Energy Per Bit.
 𝑁0 is Noise Spectral Density.
 Where;
 𝑃𝑒 𝛾 is The Average Probability
of Symbol Error.

FADING CHANNEL
 To Evaluate The Average
Probability,
 Of Symbol Error in The Presence
of,
 Fading and Noise as shown by,
𝑷 𝒆 =
𝟎
∞
𝑷 𝒆 𝜸 𝒇 𝜸 𝒅𝜸 (8.66)
 Where;
 𝑃𝑒 is The Average Probability
of Symbol Error.
 𝑓 𝛾 is The Probability Density
Function of 𝛾 .
 We may express 𝑓 𝛾 as
follows;
𝒇 𝜸 =
𝟏
𝜸 𝟎
𝒆𝒙𝒑 −
𝜸
𝜸 𝟎
, 𝛄 ≥ 𝟎 (8.67)
 Where;
 𝑓 𝛾 is The Probability
Density Function of 𝛾 .
 𝛾0 is The Mean Value of The
Received Signal Energy Per Bit
to Noise Spectral Density Ratio.
Per Bit to Noise Spectral
Density Ratio .

FADING CHANNEL
 𝛾0 is defined by the
following equation;
𝜸 𝟎 = 𝑬 𝜸 =
𝑬 𝒃
𝑵 𝟎
𝑬 𝜶 𝟐
(8.68)
 Where;
 𝛾0 is The Mean Value of The
Received Signal Energy Per Bit
to Noise Spectral Density Ratio.
 𝐸 is Statistical Expectation
Operator.
Density Ratio.
 𝐸 𝑏 is The Transmitted Signal
Energy Per Bit.
 𝑁0 is Noise Spectral Density.
 𝛼 is Rayleigh Distributed
Random Variable, describing
Attenuation in Transmission.
 𝐸 𝛼2
is The Mean Square
Value of 𝛼 .

FADING CHANNEL
 Using Equations (8.64), (8.66)
and (8.67),
 And carrying out integration,
 We get the final result;
𝑷 𝒆 =
𝟏
𝟐
𝟏 −
𝜸 𝟎
𝟏 + 𝜸 𝟎
(8.69)
 Where;
 𝑃𝑒 is The Average
Probability of Symbol Error
for Coherent Binary PSK.
 𝛾0 is The Mean Value of
The Received Signal Energy
Density Ratio.
 We may derive the 𝑃𝑒 for;
 Coherent Binary FSK.
 Binary DPSK.
 Non Coherent Binary FSK.
 As shown in Table 8.2

FADING CHANNEL
 In Fig 8.48, we use the
exact formulas of Table 8.2,
 To plot the Bit Error Rate
versus 𝛾0 in dB.
 We have included also the
Bit Error Rate of,
 Coherent Binary PSK,
 And Non Coherent Binary
FSK,
 For a Non Fading Channel.
 We see at Fig 8.48 that;
 The Degradation being
measured in tens of
decibels,
 Of additional 𝛾0
compared to a Non Fading
Channel for,
 The same Bit Error Rate.
 For Large 𝛾0 as in last
column of Table 8.2 ;
 Inverse relation between 𝑃𝑒 𝑎𝑛𝑑 𝛾0
 But at the case of “a Non Fading Channel”,
 Exponential relation between 𝑃𝑒 𝑎𝑛𝑑 𝛾0
Fig 8.48 Performance of Binary Signaling Scheme over a
Rayleigh Fading Channel shown as Continuous Curves; The
Dashed Curves to a Non Fading Channel.

FADING CHANNEL
 The Practical of this
Difference,
 In Mobile Radio Environment
relative to a Non Fading
Environment,
 We have to provide a Large
Increase in 𝛾0 ,
 To ensure Low 𝑃𝑒 for
Practical use.
 Which requires;
 Increase the Transmitted
Power
 And Increase Antenna Size.

FADING CHANNEL
 Diversity Techniques can be used
to,
 Improve system performance in
Fading Channels.
 Instead of Transmitting and
Receiving the desired signal
through one channel,
 We obtain 𝑳 copies of the
desired signal through 𝑳
different channels.
 The idea is that while some
copies may undergo deep Fades,
others may not.
 The Types of Diversity
Techniques are;
A. Frequency Diversity.
B. Time Diversity.
C. Space Diversity.
Fig 8.49 Causes of Fading Channels
[32].
 Fading Channels caused by ,
 The Reflection, Diffraction and Scattering
of the transmitted waves from Base
Station to Mobile Station as shown in Fig
8.49.

FADING CHANNEL
A. Frequency Diversity.
 Is implemented by transmitting same
information 𝑆 𝑡 on more than one carrier
Frequency as shown in Fig 8.50.
 The separation between the carriers at least
the coherence Bandwidth ∆𝑓 .
B. Time Diversity.
 Repeatedly transmits the same information 𝑆 𝑡
at the time spacing at least ∆𝑡 as shown in Fig
8.51
C. Space Diversity.
 Multiple Transmitting or Receiving Antennas
are used,
 With the spacing between adjacent Antennas
being chosen,
 So as to assure the independence of Fading
Events.
Fig 8.50 Frequency Diversity
Technique [33].
Fig 8.51 Time Diversity
Technique [34].

FADING CHANNEL
 Fig 8.52 shows,
 𝑳 Separate Receivers.
 It is assumed that;
 𝜶ℓ “ The Channel Attenuation
Factors” and
 𝝓ℓ “The Channel Phase
Shifts”
 Are available.
 The Linear Combiner Results in
two output complex envelopes
defined by;
𝒗 𝒌
~ 𝒕 =
ℓ=𝟏
𝑳
𝜶ℓ 𝒆𝒙𝒑 𝒋𝝓ℓ 𝒗ℓ𝒌~ 𝒕 , 𝒌 = 𝟎, 𝟏 (8.70)
 Where;
 𝒗 𝒌~ 𝒕 is Output of Linear Combiner.
 𝜶ℓ is The Channel Attenuation Factors.
 𝝓ℓ is The Channel Phase Shifts.
 𝜶ℓ 𝒆𝒙𝒑 𝒋𝝓ℓ is The Complex Conjugate of
the ℓ𝑡ℎ Channel Gain.
 𝒗ℓ𝒌~ 𝒕 is The Output of the
𝑘𝑡ℎ Matched Filter in the ℓ𝑡ℎ
Receiver.
 ℓ = 1,2, … . 𝐿
Fig 8.52 The Space Diversity Technique Block
Diagram

FADING CHANNEL
 𝒗 𝟎~ 𝒕 is The One Output Complex
Envelope Corresponds to the
Transmission of Symbol 0.
 𝒗 𝟏~ 𝒕 is The Other Output
Complex Envelope Corresponds to
the Transmission of Symbol 1.
 The Real Parts of 𝒗 𝟎~ 𝒕 and
𝒗 𝟏~ 𝒕 are then used in The
Decision Making Process.
 The situation described here applies
to Binary FSK.
 This system is designed to
Compensate only for Short Term
Effects of a Fading Channel.
 Fig 8.53 shows that at 𝛾0 = 15 𝑑𝐵 ;
 When 𝐿 = 1 , means “No Diversity”,
 Then 𝑃𝑒 =
1
100
“The worst Result”
 When 𝐿 = 4 . Means “The Number
of Fading Channels are Four”,
 Then 𝑃𝑒 =
1.3
105 ”The Best Result”.
Fig 8.53 Performance of Binary Signaling Schemes
with Diversity
 Thus we show the effectiveness of Diversity as a
means of mitigating the short term effects of
Rayleigh Fading.

8.8- TDMA and CDMA Wireless Communication
Systems

8.8- TDMA AND CDMA WIRELESS
COMMUNICATION SYSTEMS
 In wireless communications,
 A user would like to talk and
listen simultaneously.
 So some form of Duplexing
is required as FDD
“Frequency Division
Duplexing” .
 At FDD, it has two
Frequency Bands as shown
in Fig 8.54;
 One for the Forward Link
from the Base Station to the
Mobile Station (869- 894)
MHz.
 The other for the Reverse
link from the Mobile to the
Base Station (824-849)
MHz.
Fig 8.54 Frequency Division Duplexing
Bands [35].

 FDD is an integral
part of The Two
Wireless
Communication
Systems;
 (GSM and IS-95) as
shown in Table 8.3.

 GSM uses TDMA.
 in a TDMA system;
 Each subscriber is
permitted to access the
Radio Channel, During a
set of predetermined
Time Slots,
 During which the
subscriber will have full
use of the Channel as
shown in Fig 8.55.
 Note;
 At Fig 8.55; only the
Down Link Direction is
shown,
 There is also a
corresponding Frame in
the Up Link Direction.
Fig 8.55 TDMA Concept [36].

 Data are transmitted over the channel
in “bursts”,
 As shown in the Frame structure of
Fig 8.56
 The basic Frame of GSM is
composed of;
 Eight 577 𝜇𝑠 Slots.
 Each Time Slot has;
 “Guard Time” occupying 8.25 Bits
and is used to prevent Data Bursts
received at the Base Station from
Mobiles from Overlapping with each
other is achieved by transmitting no
signal during the Guard Time.
 “Tree Tail Bits” are all logical zeroes,
are used in conventional Decoding of
the channel encoded data Bits.
 “User Data” are 57 Bits.
 “1 Bit Flag” is used to identify
whether the Data Bits are Digitized
Speech or other Information Bearing
Signal.
Fig 8.56 Frame Structure of the GSM Wireless
Communication System [37].
 “26 Bits Training” sequence is used for
Channel Equalization.
 Each Time Slot contains
(8.25+3+57+1+26+1+57+3=156.25 Bits).
 Each Time Slot contains (57+57=114 User Data
Bits”.
 Each Time slot contains (156.25 – 114 = 42.25
Bit which are called “overhead”, by ignoring the
2 flag bits thus 40.25 Bits.
 Frame Efficiency of GSM = 1 −
40.25
156.25
× 100
= 74.24%

 IS-95 uses CDMA.
 In CDMA;
 Each subscriber is
assigned a distinct
Spreading Code,
 There by permitting the
subscriber full access to
the channel all of the
time as shown in Fig
8.57.
 In CDMA system we
have MAI (Multiple
Access Interference),
 Which arises because of
deviation of the
Spreading Codes from
Perfect Orthogonality.
Fig 8.57 CDMA Concept [38].

 A Related Phenomenon that
needs attention is “The
Near Far Problem”.
 Which occurs if the
Received Signals from the
Mobile units,
 Do not have equal power at
the Base Station as shown
in Fig 8.58 .
 Thus, the strongest
Received Signal from a
Mobile user,
 Captures the Demodulation
Process at the Base Station,
 To the Detriment of the other
users.
 To overcome this problem,
 The Base Station maintains
Control over The Power
Level,
Fig 8.58 Near Far Problem
[39].

8.8.1- Rake
Receiver The Rake Receiver is
designed to,
 Equalize the Effect of
Multipath.
 As shown in Fig 8.59, the
Receiver consists of;
a) A Number of
“Correlators”;
 Which are connected in
parallel and operating in a
synchronous fashion.
 Each correlator has two
inputs;
i. A Delay version of the
Received Signal.
ii. A Replica of PN
Sequence acts as a
Reference Signal.
Fig 8.59 Block Diagram of the Rake
Receiver.

8.8.1- Rake Receiver
b) Phase and Gain Adjustors;
 These Functional Blocks are used to,
 To be sure that the correlator outputs all
add Constructively,
 Through the following steps;
i. An appropriate delay is introduced
into each correlator output,
ii. So that the Phase angles
𝜙1, 𝜙2, … 𝜙 𝑀 of the correlator outputs
are in agreement with each other.
iii. The Weighting Coefficients into each
correlator output 𝛼1, 𝛼2, … 𝛼 𝑀 , are
computed according to “Maximal Ratio
Combining” as shown in Fig 8.60 so
that,
iv. The correlators responding to strong
paths are accenuated.
v. While, the correlators not
synchronizing to any strong path are
Fig 8.60 Maximal Ratio
Combining Principle [40].

8.8.1- Rake Receiver
c) Linear Combiner Output.;
 The linear Combiner output is;
𝒚 𝒕 =
𝒌=𝟏
𝑴
𝜶 𝒌 𝒛 𝒌 𝒕 (8.71)
Where;
 𝑦 𝑡 is the Combiner output,
which behaves as a single
Propagation Path between
transmitter and receiver,
 Rather than a series of Multiple
Paths spread in Time.
 𝛼 𝑘 are The Weighting
Coefficients.
 𝑧 𝑘 𝑡 is the Phase Compensated
Output of the 𝑘𝑡ℎ Correlator.
d) Integrator Block;
 Which integrates the y t over
the Bit interval Tb .
e) Decision Device Block;
 That determines whether Binary
symbol one or zero was
transmitted in that Bit
interval 𝑇𝑏 .

8.9- Source Coding of Speech for Wireless
Communications

8.9- SOURCE CODING OF SPEECH FOR
WIRELESS COMMUNICATIONS
 Speech Coding;
 Means finding a representation of
Speech,
 Which can be Transmitted
Efficiently,
 Through a Digital Channel.
 It is usually lossy coding,
 Meaning that the waveform can’t
be completely reproduced by the
decoder,
 Instead, only the information which
is useful to a human listener is
retained as shown in Fig 8.61.
 In this section we describe two
different techniques for Speech
Coding;
 Multi-Pulse Excited LPC.
 Code-Excited LPC.
 Linear Predictive Coding (LPC);
 Is a method for Encoding good
quality Speech at a Low Bit
Rate.
Fig 8.61 Speech Coding
Concept [41].

8.9.1- Multi-Pulse Excited LPC
 This form of Speech Coding
exploits the Principle of Analysis
by “Synthesis” that means,
 The Encoder includes a
Replica of,
 The Decoder in its Design.
 The Encoder Consists of three
parts as shown in Fig 8.62;
I. Synthesis Filter.
II. Excitation Generator.
III. Error Minimization.
Fig 8.62 Encoder of Multi-Pulse
Excited LPC.

I. Synthesis Filter;
 Its function is to produce,
 A Synthetic Version of the
Original Speech that is of high
Quality as shown in Fig 8.63.
II. Excitation Generator;
 For Producing the Excitation
applied to the Synthesis Filter.
 The Excitation consists of a
definite number of Pulses.
III. Error Minimization;
 For Optimizing the Error
between the Original Input
Speech and the Synthesized
Speech.
 Hence the aim of previous step
is; to Optimize the Amplitudes
and Positions of the Pulses used
 The three Parts of the Encoder;
 Form a “Closed Loop Optimization
Procedure”,
 Which permits the Encoder to
operate at a Bit Rate below 16 kb/s,
 While maintaining high quality
speech.
Fig 8.63 Synthesis Filter Concept [42].

 The Encoding Procedure has two
main Steps;
1) Computing the Free Parameters of
the Synthesis Filter Outside the
Optimization Loop over a period of
10 to 30 ms.
2) Computing the Optimum Excitation
for the Synthesis Filter by minimizing
the Error with the Closed Loop.
 Thus;
 The Free Filter Parameters of the
Synthesis Filter,
 And The Optimum Excitation for the
Synthesis Filter,
 Constitute The Transmitted Signal.

 The Decoder, located in the
Receiver, consists of Two Parts, as
shown in Fig 8.62;
1) Excitation Generator.
2) Synthesis Filter.
 These two parts are identical to the
corresponding ones in the
Encoder.
 By passing the Decoded Excitation
through,
 The Synthesis Filter whose
Parameters are set equal to those
in Encoder,
 Thus, the Function of the Decoder
is to use the Received Signal to
produce a Synthesis version of the
Original Speech Signal.
Fig 8.62 Decoder of Multi-Pulse
Excited LPC.

8.9.2- Code Excited LPC
 Fig 8.63 shows the Block
Diagram of the Code Excited
LPC
 Commonly referred to as
CELP.
 The Encoder of CELP consists
from;
I. “Excitation Code Book”
 The distinguishing Feature of
CELP,
 Is the use of Predetermined
“Code Book”,
 As the source of Excitation for
the Synthesis Filter.
Fig 8.63 Encoder of the CELP.

II. Synthesis Filter;
 It consists of Two Pole Filters connected
in cascade,
 One of which performs Short Term
Prediction,
 And the other performs Long Term
Prediction.
 The Free Parameters of Synthesis Filter
are computed first using the actual
Speech Samples as input.
III. Minimization of Perceptually
Weighted Error;
 Used for minimizing the Average Power
of The Error,
 Between the Original Speech and the
Synthesized Speech.
 This Minimized Error used to,
 The Choice of a Particular Code stored in
the Excitation Code Book.
 And The Optimization of The Gain Factor
Fig 8.64 Resonance of Vocal Tract [43].
 Note;
 “Short Term Prediction”; means The
Resonance of Vocal Tract as shown
in Fig 8.64.
 “Long Term Prediction”; means
Periodicity of Voiced Speech.

 Then, The Transmitted Signal
Consists from;
1) The Particular Code stored in the
Excitation Code Book.
2) The Optimized Gain Factor G.
3) The Optimized Filter Parameters.
 The Decoder of CELP consists from;
I. An Identical Copy of The Code
Book.
II. An Identical Copy of The
Synthesis Filter.
 Hence given the Received Signal,
 The Decoder is Enabled to,
 Parameterize its own Synthesis
Filter,
 And Determine the appropriate
Excitation for the Synthesis Filter.
 CELP is capable of producing

8.10- Adaptive Antenna Arrays For Wireless
Communications.

8.10- ADAPTIVE ANTENNA ARRAYS FOR
 The goal of Wireless Communications is;
 To Allow as many users as possible,
 To Communicate Reliably without,
 Regard to Location and Mobility.
 This Goal is Impeded by Three major
Channel Impairments as follows;
1) Multipath:
 Results from that, The signal takes many
paths to the destination.
 Multipath causes Fading due to Phase
Cancellation between Different Propagation
Paths.
 Fading Leads to a Reduction in available
Signal Power.
 Multipath Propagation shown in Fig 8.65
where;
 Reflection; occurs when a Wave Impinges
upon an Object with Large Size (relative to
Fig 8.65 Multipath Propagation Concept
[44]
 Diffraction; occurs when a
Wave is Blocked by Sharp
Edges.
 Scattering; occurs when a
Wave Impinges upon an Object
with small Size (relative to 𝜆 ).

2) Delay Spread;
 Results from Differences in
Propagation;
 Delays among the Multiple
Propagation Paths as shown in Fig
8.66.
 Delay Spread causes a Reduction in
the Attainable Data Rate.
3) Co-Channel Interference;
 Arises in Cellular Systems where,
 The available Frequency Channels
are divided into Different Sets,
 With each set being assigned to a
specific Cell as shown in Fig 8.67,
 And with several Cells using the
same set of Frequencies.
 Co-Channel Interference Limits the
System Capacity,
 (i.e. The Largest Possible Number of
users that can be served by the
System).
Fig 8.66 Delay Spread Concept [45]
Fig 8.67 Co-Channel Interference

 We may Combat The Effects of,
 Multipath Fading and Co-Channel
Interference,
 At The Base Station By using,
 Three Identical but Separate “Antenna
Arrays” as shown in Fig 8.68,
 One for Each Section of the Base Station.
 Only one user accesses a Sector of a
Base Station at a Given Frequency.
 Cellular Systems use 120 degree,
 Sectorization at each Base Station.
Fig 8.68 Cell-tower Antenna Array
[47].

 Fig 8.69 shows The Block
Diagram of,
 “An Array Signal
Processor”.
 It is assumed that there
are,
 𝑵 is the number of users
whose signals are received
at a Particular Sector of the
Base Station.
 𝑴 is the number of
Identical Antenna Elements
of the Array for a Sector.
 𝑼𝒔𝒆𝒓 𝒐𝒇 𝑰𝒏𝒕𝒆𝒓𝒆𝒔𝒕 is a
particular user is treated as
the one of Interest.
 𝑰𝒏𝒕𝒆𝒓𝒇𝒆𝒓𝒊𝒏𝒈 𝑼𝒔𝒆𝒓𝒔 are
the remaining 𝑁 − 1
users cause Co-Channel
Fig 8.69 The Block Diagram of Array Signal
Processor

 𝑨𝑾𝑮𝑵 is Additive White Gaussian
Noise,
 Which is the Cause of Corruption for,
 Each Array Signal Processor’s Input.
 This Structure is drawn for one user of
Interest.
 𝑴𝒖𝒍𝒕𝒊𝒑𝒂𝒕𝒉 𝑪𝒉𝒂𝒏𝒏𝒆𝒍 is characterized
by the Channel Matrix 𝑪 as shown by;
𝑪 = 𝒄 𝟏, 𝒄 𝟐, … , 𝒄 𝑵 (8.72)
 Where;
 𝑪 is The Channel Matrix has
dimensions 𝑀 − 𝑏𝑦 − 𝑁 .
 The Channel Matrix Represents The
Multipath Channel.
 𝑵 is the number of users whose
signals are received at a Particular
Sector of the Base Station.
 𝑴 is the number of Identical
Antenna Elements of the Array for
a Sector.
 The Goal of “The Block Diagram
of Array Signal Processor” are;
1) The Co-Channel Interference
produced by the 𝑁 − 1
Interfering users is Cancelled.
2) The Output Signal to Noise Ratio
for the user of Interest is
Maximized.
 These Two steps are called,
 Design Requirements 1 and 2.

 To Satisfy The Design
Requirement 1,
 (i.e., Cancellation of Co-Channel
Interference).
 We choose The Vector 𝒘 to be
Orthogonal to The Vectors
𝒄 𝟐, . , 𝒄 𝑵 .
 Where;
 𝒘 is the “weight vector”.
 The “weight vector” characterizes
the Array Signal Processor.
 The “weight vector” has 𝑴
Dimensions.
 𝒄 𝟐, … … , 𝒄 𝑵 are Associated with
The Interfering Users.
 To Satisfy The Design Requirement 2,
 (i.e., Maximization of SNR).
a) We first Construct a Subspace denoted
by 𝓦 .
 This Subspace has The Dimension is
equal to,
 The Difference between “The Number of
Antenna Elements 𝑴 ”,
 And “The Number of Interfering
Users 𝑵 − 𝟏 ” as follows;
 𝑀 − 𝑁 − 1 = 𝑀 − 𝑁 + 1 .
b) Next we project 𝒄 𝟏∗ on to The
Subspace 𝓦 .
 Where;
 𝒄 𝟏∗ is The Complex Conjugate of the
Channel Vector 𝒄 𝟏 Pertaining to User
1.
 𝓦 is The Subspace (a subset of
space):
 If we add two vectors in 𝓦 , their sum

 Example 8.3;
 To illustrate the method for
Determining “The weight vector
𝒘 ”.
 Consider a System involving;
 Two Users characterized by “The
Channel Vectors 𝒄 𝟏 𝒂𝒏𝒅 𝒄 𝟐 thus
𝑵 = 𝟐 .
 And an Antenna Array consists of
Three Elements thus 𝑴 = 𝟑 .
 Then The Subspace 𝓦 is Two
Dimensional as shown by, 𝑀 − 𝑁 Fig 8.70 The Signal Space Diagram
 𝒄 𝟐 is The Channel Vector Pertaining to
User 2 of Interferer.
 𝓦 is The Subspace shown Shaded in
this Fig 8.70, is Orthogonal to 𝒄 𝟐 .
 𝒘 is The “weight vector” characterizes
the Array Signal Processor, is determined
by the Projection of 𝒄 𝟏∗ on to the
Subspace.

Example 8.3;
 The Important Conclusion is;
 A Linear Receiver using,
 Optimum 𝑴 Antenna Elements.
 And involving 𝑵 − 𝟏 Interfering
Users.
 This Receiver has the same
Performance,
 As a Linear Receiver with 𝑴 − 𝑵

 If The Delay Spread
can’t be ignored,
 So we need The Array
Signal Processor shown
in Fig 8.71,
 Which combines;
A. “Spatial Processing”,
 Which is provided by
Antenna Array.
B. “Temporal
Processing”,
 Which is provided by a
bank of Finite Duration
Impulse Response (FIR)
Filters.
 𝒛−𝟏
is Unit Delay
Element Block being
equal to the Symbol
Duration.
 𝑾∗
are The Filter
Coefficients (Complex
Valued) Fig 8.71 Baseband Space Time Processor

 We need to make the Receiving
Array Signal Processor in Fig 8.69
Adaptive for the Following reasons;
 In Reality, Multipath Fading, Delay
Spread and Co-Channel Interference
are all Non-stationary.
 Also, the Channel Characterization
may be unknown.
 Fig 8.72 shows the Structure of “An
Adaptive Antenna Array”,
 Where;
 The Output of each Antenna Element
,
 Is Multiplied by a “Controllable
Weight”.
 Then, The Outputs of the “Elemental
Weights” 𝑊1∗ 𝑛 , 𝑊2∗ 𝑛 … . 𝑊 𝑀∗ 𝑛
of the Array,
 Are Summed to Produce The Array
Fig 8.72 Block Diagram of Adaptive Antenna
Array

 We have also at “Adaptive Antenna
Array”,
 “Reference Signal”,
 Is Correlated with the Desired
Signal,
 So the “Adaptive Antenna Array”
doesn’t require knowledge of,
 The Direction of Arrival of the
Desired Signal originating from a
user of Interest.
 “Error Signal”,
 Which is the Difference between
“The Output Signal of the Array”
and “The Reference Signal”.
 This “Error Signal” is used to apply
the appropriate adjustments to,
 Control the “Elemental Weight”

 To Optimize the Performance of “The
Adaptive Antenna Array”,
 We need to minimize the “Cost
Function” defined as Follows;
𝑱 = 𝑬 𝒆 𝒏 𝟐
(8.73)
 Where;
 𝑱 is the Cost Function.
 𝒆 𝒏 is the Error Signal at time 𝒕
 The Output Signal of The Array is as
follows;
𝒚 𝒏 =
𝒌=𝟏
𝑴
𝒘 𝒌∗ 𝒏 𝒙 𝒌 𝒏 (8.74)
 Where;
 𝒚 𝒏 isThe Output Signal of The Array.
 𝒌 is a specific of Element in the Array.
 𝑴 is the Total Number of Elements in the
Array.
 𝒙 𝒌 𝒏 is the Output of the specific 𝒌
Element in the Array at a Discrete time
𝒏 .
 𝒘 𝒌 𝒏 is the Corresponding value of the
Controllable Weight connected to this 𝒌
Element.
 𝒘 𝒌∗ 𝒏 𝒙 𝒌 𝒏 is the Inner Product

 We may Evaluate the Error Signal
as follows;
𝒆 𝒏 = 𝒅 𝒏 − 𝒚 𝒏 (8.75)
 Where;
 𝒆 𝒏 is the Error Signal.
 𝒅 𝒏 is the Reference Signal.
 𝒚 𝒏 is the Output Signal of The Array.
 The Controllable Weight applied to specific
kth Element in the Array is given by;
∆𝒘 𝒌 𝒏 = 𝝁 𝒆∗ 𝒏 𝒙 𝒌 𝒏 , 𝒌 = 𝟏, 𝟐, … , 𝑴 (8.76)
 Where;
 ∆𝒘 𝒌 𝒏 isThe Controllable Weight
applied to specific kth Element in the
Array.
 𝝁 𝒆∗ 𝒏 is The Step Size
Parameter.
 𝒙 𝒌 𝒏 is the Output of
the specific (𝒌) Element in
the Array at a Discrete
time (𝒏).
 𝒌 is a specific of
Element in the Array.
 𝒏 is an Integer serving
as Discrete Time.

 The updated value of specific kth
Element in the Array is given by;
𝒘 𝒌 𝒏 + 𝟏 = 𝒘 𝒌 𝒏 + ∆𝒘 𝒌 𝒏 , 𝒌 = 𝟏, 𝟐, . . , 𝑴 (8.77)
 Where;
 𝒘 𝒌 𝒏 + 𝟏 is The updated value
of specific kth Element in the
Array.
 𝒘 𝒌 𝒏 is the Corresponding
value of the Controllable Weight
connected to this (𝒌) Element.
 ∆𝒘 𝒌 𝒏 is The Controllable
Weight applied to specific kth
Element in the Array.
 𝒌 is a specific of Element in the
Array.
 𝑴 is the Total Number of
Elements in the Array.
 Equations (8.74) – (8.77), in
that order,
 Constitute the Least Mean
Square (LMS) algorithm.

 The Advantages of an Adaptive
Antenna Array using LMS algorithm
are;
 Simplicity of Implementation.
 Linear Growth in Complexity with the
Number of Antenna Elements.
 Robust Performance with respect to
Disturbances.
 The Disadvantages of an Adaptive
Antenna Array using LMS algorithm
are;
 Slow rate of Convergence.
 Sensitivity of the Convergence to
variation in Reference Signal Power.
 The Limitations of the LMS algorithm,
 Can be overcome by using the Direct
Matrix Inversion (DMI) algorithm.
 The DMI algorithm,
 Operates in the Batch Mode,
 In that the Computation of the
Elemental Weights,
 Is based on a Batch of (K)
Snapshots.
 The Size (K) should be small
enough in the Computation.
 The Size (K) should be Large
enough to approach the
optimum Solution.
 DMI is the optimum Technique
for Array Antennas,
 Currently deployed in many
Base Stations Today.

8.11- SUMMARY AND DISCUSSION
 We discussed in This Chapter the
following Items;
 Two Types of Multiuser
Communications;
1) Satellite Communications;
 Offer Global Coverage.
2) Wireless Communications;
 Offer Mobility.
 The Major Sources of Degradation
discussed in Wireless
Communications;
1) Co-Channel Interference.
2) Fading.
3) Delay Spread.
 Both Interference and Multipath
require the use of the following
Techniques;
1) Diversity.
2) Adaptive Array Antennas.
3) Rake Receiver.

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