Fundamentals of MIMO Wireless Fundamentals of MIMO Wireless Fundamentals of MIMO Wireless Fundamentals of MIMO Wireless CommunicationsCommunicationsCommunicationsCommunications
Part IPart IPart IPart I
Prof. Rakhesh Singh
Fundamentals of MIMO Wireless Fundamentals of MIMO Wireless Fundamentals of MIMO Wireless Fundamentals of MIMO Wireless CommunicationsCommunicationsCommunicationsCommunications
Part IPart IPart IPart I
Singh Kshetrimayum
Fundamentals of MIMO Wireless CommunicationsPart I
It covers
Chapter 1: Introduction to MIMO systems
Chapter 2: Classical and generalized fading distributions
Chapter 3: Analytical MIMO channel modelsChapter 3: Analytical MIMO channel models
Chapter 4: Power allocation in MIMO systems
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Fundamentals of MIMO Wireless Communications
Chapter 1: Introduction to MIMO systems
Chapter 2: Classical and generalized fading distributions
Chapter 3: Analytical MIMO channel modelsChapter 3: Analytical MIMO channel models
Chapter 4: Power allocation in MIMO systems
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
SISO Systems
Fig. 1 Single-input, single-output (SISO) system (N
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
output (SISO) system (NT =NR =1)
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
SIMO systems
Fig. 2 Single-input, multiple-output (SIMO) system Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
output (SIMO) system (NT =1 and NR≥2)Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Receiver diversity techniques
• Receiver diversity techniques (
• Equal gain combining (EGC)
• co-phases signals on each branch
• and then combines them with equal weight• and then combines them with equal weight
• Selection Combining (SC)
• selects the signal branch with the highest signal(SNR)
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Receiver diversity techniques
Receiver diversity techniques (combat multipath fading)
Equal gain combining (EGC)
phases signals on each branch
and then combines them with equal weightand then combines them with equal weight
selects the signal branch with the highest signal-to-noise ratio
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Receiver diversity techniques
• Maximal ratio combining (MRC)
• MRC outputs the weighted sum of all the branches
• Weights = complex conjugate of the channel gain coefficients
• MRC is optimal in terms of SNR• MRC is optimal in terms of SNR
• but complex to implement
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Receiver diversity techniques
Maximal ratio combining (MRC)
MRC outputs the weighted sum of all the branches
Weights = complex conjugate of the channel gain coefficients
MRC is optimal in terms of SNRMRC is optimal in terms of SNR
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Single-user MIMO
Fig. 3 Multiple-input, Single-output (MISO) system Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Single User: Point
output (MISO) system (NT≥2 and NR =1)Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
User 1
Single User: Point-to-point MIMO communicatio
Multi-user MIMO
In cellular communication:
• Multiple users with
• Single antenna
• Base station with multiple antennas• Base station with multiple antennas
+H. Huang, C. B. Papadias and S. Venkatesan, MIMO communication for cellular
networks, Springer, 2012.Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Base station with multiple antennasBase station with multiple antennas
MIMO communication for cellular
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Multi-user MIMO
Fig. 4 Multi-user MIMO (1 BS with NRakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
user MIMO (1 BS with NT antennas & K users)Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO systems
Fig. 5 Point-to-point NT × NR multiple-input multiple
(NT transmitting antennas and NR receiving antennas)
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
input multiple-output (MIMO) system
receiving antennas)
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
SISO system capacity
CSISO=BW log2 (1+SNR)+
In order to increase data rate either• BW
• Signal to noise ratio (SNR)
should increaseshould increase
BW is precious, almost always fixed for different applications
Signal power increases• Device’s battery life time decreases
• causes higher interference
• needs expensive RF amplifiers
+ T. M. Cover and J. A. Thomas, Elements of Information Theory
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
In order to increase data rate either
BW is precious, almost always fixed for different applications
Elements of Information Theory, Wiley, 1999.
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO capacity
MIMO (pronounced “My-Moe”)+
• capacity boosters for wireless channels
• without penalty in bandwidth and power
• In a rich Rayleigh scattering environment • In a rich Rayleigh scattering environment
• capacity increases linearly with the minimum N
• CMIMO= m CSISO
+J. G. Andrews, A. Ghosh and R. Muhamed, Fundamentals of WIMAX
2007.Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
capacity boosters for wireless channels
without penalty in bandwidth and power
In a rich Rayleigh scattering environment In a rich Rayleigh scattering environment
capacity increases linearly with the minimum NT or NR=m
Fundamentals of WIMAX, Prentice Hall,
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO advantages over SISO
Basically, two gains of MIMO over SISO systems
• Multiplexing (rate) gain
• Diversity gain
• For example, for 3×3 MIMO system• Rate gain =3
• Diversity gain= 9
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO advantages over SISO
Basically, two gains of MIMO over SISO systems
3 MIMO system
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Diversity gain in MIMO systems
Fig. 6 3×3 MIMO system
1. E. Biglieri et. al, MIMO wireless communications
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Diversity gain in MIMO systems
3 MIMO system1
MIMO wireless communications, Cambridge University Press, 2007
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Multiplexing gain in MIMO systems
001 1
0
Fig. 7 3×3 MIMO system
0
0
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Multiplexing gain in MIMO systems
1
0
001
3 MIMO system
0
0
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Advantages of Diversity gain in MIMO systems
0
0
0
Fig. 8 3×3 MIMO system
0
0
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Advantages of Diversity gain in MIMO systems
0
0
0
3 MIMO system
0
0
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO advantages over SISO
• Spatially multiplexed MIMO systems• 3 times data rate than that of SISO system for a 3
• Different message bits are sent in para
• Increases the capacity linearly with the number of antennas • Increases the capacity linearly with the number of antennas
• MIMO for diversity gain• Same message bits are sent from all the 3 transmitting antennas
• If any link is broken or down, receiver can decode message bit from the remaining working links
• It minimizes the probability of error in detection
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO advantages over SISO
Spatially multiplexed MIMO systems3 times data rate than that of SISO system for a 3×3 MIMO system
parallel from the 3 transmitting antennas
Increases the capacity linearly with the number of antennas Increases the capacity linearly with the number of antennas
Same message bits are sent from all the 3 transmitting antennas
If any link is broken or down, receiver can decode message bit from the
It minimizes the probability of error in detection
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Diversity Multiplexing trade-
• Instead of using all the antennas for rate gain or diversity gain
• We may employ some antennas for rate and diversity gain
• If we use more antennas for div• If we use more antennas for divmay be used rate gain, hence, a trade
• Diversity multiplexing trade-off
• doptimal=(NR-r) (NT-r); 0 ≤ r ≤ min
• Implies d increase, r decreases +L. Zheng and D. N. Tse, “Diversity and multiplexing: A fundamental trade
antenna channels,” IEEE Trans. Information Theory
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
-off
Instead of using all the antennas for rate gain or diversity
We may employ some antennas for rate and diversity gain
r diversity gain then less antennas r diversity gain then less antennas may be used rate gain, hence, a trade-off
off
≤ r ≤ minNR, NT
Implies d increase, r decreases , “Diversity and multiplexing: A fundamental trade-off in multiple
IEEE Trans. Information Theory, pp. 1073-96, May 2003.
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Diversity Multiplexing trade-off: Case study I
Fig. 8 3×5 MIMO system (r = minN
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
off: Case study I
5 MIMO system (r = minNR,NT=3)
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Diversity Multiplexing trade-off: Case study II
Fig. 9 5×5 MIMO system (r=2, Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
off: Case study II
5 MIMO system (r=2, doptimal=9)Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Diversity Multiplexing trade-off: Case study III
Fig. 10 5×5 MIMO system (r=3, Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
off: Case study III
5 MIMO system (r=3, doptimal=4)Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Applications
3G, 4G LTE, one of the proponents for 5G
IEEE 802.11n
IEEE 802.16m
WiMAXWiMAX
WiFi
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
3G, 4G LTE, one of the proponents for 5G
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Review questions
Review question 1.1: What is coherence bandwidth of the channel?
Review question 1.2: What is coherence time of the channel?
Review question 1.7: What are non
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Review question 1.1: What is coherence bandwidth of the channel?
Review question 1.2: What is coherence time of the channel?
Review question 1.7: What are non-coherent and coherent systems?
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Example 1.1
Assume that the multiplexing gain (diversity-multiplexing trade-off
( )(NrNd Topt −=
for SNR∞.
Assume MIMO system with an SNR of 10dB, one needs a spectral efficiency of R=16 bps per Hertz.
Find the supreme diversity gain such MIMO system can achieve.
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Assume that the multiplexing gain (r) and diversity gain (d) satisfy the
)rNR −
Assume MIMO system with an SNR of 10dB, one needs a spectral
Find the supreme diversity gain such MIMO system can achieve.
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Example 1.1
Given SNR =10 dB, R=16bps
r=4.8165
Therefore five antennas may be use
( ) RSNRr =2log
Therefore five antennas may be use(7-2) two antennas may used for diversity
The maximum diversity gain can be calculated as
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )(−= NrNd Topt
used for multiplexing and remaining used for multiplexing and remaining 2) two antennas may used for diversity
The maximum diversity gain can be calculated as
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
) ( )( ) 45757 =−−=− rN R
Exercises
Exercise 1.3
What are close loop, open loop and blind MIMO systems?
Exercise 1.4
Which diversity was left aside for many years? Why?Which diversity was left aside for many years? Why?
Exercise 1.5
What are frequency flat, frequency selective, fast and slow fading channels?
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
What are close loop, open loop and blind MIMO systems?
Which diversity was left aside for many years? Why?Which diversity was left aside for many years? Why?
What are frequency flat, frequency selective, fast and slow fading
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Course webpage & quick revision of SISO fading channel
http://www.iitg.ac.in/engfac/krs/public_html/lectures/ee634/
A quick review of SISO wireless systems over fading channels
What is a wireless fading channel?
A brief mention on large-scale fading (PL, shadowing) and smallA brief mention on large-scale fading (PL, shadowing) and smallfading (multipath)
How do we model it?
It is modelled as a multiplicative term to the transmitted signal
What are its performance metrics of wireless fading channels?
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Course webpage & quick revision of SISO fading channel
http://www.iitg.ac.in/engfac/krs/public_html/lectures/ee634/
A quick review of SISO wireless systems over fading channels
What is a wireless fading channel?
scale fading (PL, shadowing) and small-scale scale fading (PL, shadowing) and small-scale
It is modelled as a multiplicative term to the transmitted signal
What are its performance metrics of wireless fading channels?
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Random SISO channel
SISO fading channel:
For a SISO channel, the I-O relationship can be expressed as
where y is the received signal,
nhxy +=
where y is the received signal,
x is the transmitted signal and
n is the AWGN
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
O relationship can be expressed as
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
SISO fading channel
We will consider two performance metrics for random channel
• Average or Ergodic capacity
• Outage probability
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
We will consider two performance metrics for random channel h
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Capacity of random SISO channel
Average or Ergodic capacity for SISO fading channel
• Average of instantaneous capacity
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) 0
2 log1log WSNRWEC =+= ∫∞
α
Capacity of random SISO channel
Average or Ergodic capacity for SISO fading channel
Average of instantaneous capacity
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) ( ) 2
2 ;1log hdpSNR =+ αααα α
Capacity of random SISO channel
The outage probability denoted as P
• probability that the channel capacity C drops below a certain threshold information rate R
• the probability that the rate R is • the probability that the rate R is C(h)
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) ( ) (=<= WobRCobRPout logPrPr
Capacity of random SISO channel
denoted as Pout is the
probability that the channel capacity C drops below a certain
is greater than the channel capacity is greater than the channel capacity
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) )
−
<=<+SNR
obRSNRW
R
12Pr1log 2 αα
Capacity of random SISO channel
Average or Ergodic capacity is the average of the instantaneous capacity of the random channel
It is found out by taking the expectaover the probability density function (PDF) of over the probability density function (PDF) of
where h is the random channel gain coefficient
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
h=α
Capacity of random SISO channel
is the average of the instantaneous
ectation of the instantaneous capacity over the probability density function (PDF) of over the probability density function (PDF) of
is the random channel gain coefficient
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
2h
Capacity of random SISO channel
Outage probability can be obtained function (CDF) of the random variable
( ) ( ) αα dpxP
x
∫=
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) ( ) αααα dpxP ∫=
0
SNR
ex
W
R
12ln −=
Capacity of random SISO channel
ned from the cumulative distribution function (CDF) of the random variable α
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Capacity of random SISO channel
For example
Rayleigh fading
is exponential i.e., it has PDF2
h=α is exponential i.e., it has PDF
where α0 is the mean value of RV α
u(α) is the unit step function
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
h=α
( ) ( )αα
α
ααα up
−=
00
exp1
Capacity of random SISO channel
is exponential i.e., it has PDFis exponential i.e., it has PDF
α and
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Capacity of random SISO channel
The average or ergodic capacity is given by
( )( )2
0 00
1log 1 expC W SNR d
αα α
α α
∞ = + −
∫
The outage probability can be obtained from
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
0 00
( ) ( ) 01α
αα αα
xx
edpxP−
∞−
−== ∫ P
Capacity of random SISO channel
capacity is given by
0 0
C W SNR dα
α αα α
= + −
The outage probability can be obtained from
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
0 0
( ) 0
2ln 1
1αSNR
e
out
W
R
eRP
−−
−=
Capacity of random SISO channel
• there cannot be any reliable transzero outage probability
• regardless of the value of the bandwidth (BW) and
• transmit power for a Rayleigh fading channeltransmit power for a Rayleigh fading channel
• From outage probability,
• we may express data rate in terms of
• SNR and
• Outage probability as
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
(WSNR
⇒
−
2ln
1 α
Capacity of random SISO channel
ransmission at any rate guaranteeing a
regardless of the value of the bandwidth (BW) and
transmit power for a Rayleigh fading channel2ln 1eW
R
−transmit power for a Rayleigh fading channel
we may express data rate in terms of
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )( )) ( )( )( ) RRPSNR
eRP
out
W
R
out
=−−
=−
1ln1ln2
1ln
0
2ln0
α
α
( ) 0
2ln 1
1αSNR
e
out
W
eRP
−−
−=
Capacity of random SISO channel
We may express spectral efficiency in bps per Hertz as
If we want to have a zero outage probability,
( SNRW
R−=⇒ 1log 02 α
If we want to have a zero outage probability,
• Pout=0
• we obtain data rate, R=0+
Hence zero outage probability is an impossibility even for Rayleigh fading channel, the mostly widely us
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
+ S. Barbarossa, Multiantenna Wireless Communication Systems
Capacity of random SISO channel
We may express spectral efficiency in bps per Hertz as
If we want to have a zero outage probability,
( )( ))RPout−1ln0
If we want to have a zero outage probability,
Hence zero outage probability is an impossibility even for Rayleigh ly used wireless fading channel model
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Wireless Communication Systems, Artech House, 2003.
SER of various modulation schemes for SISO system over various fading channels
One may employ different modulation schemes
• BPSK, QPSK, QAM, PSK, etc at thebits to symbols to transmit it over the channel
Besides AWGN (additive term), Besides AWGN (additive term),
• we also have different wireless fading channel models (multiplicative term) like Rayleigh, channel.
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
SER of various modulation schemes for SISO system over
modulation schemes like
t the transmitter side to convert the bits to symbols to transmit it over the channel
we also have different wireless fading channel models ) like Rayleigh, Rician, Nakagami, etc for the
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
SER of various modulation schemes for SISO system over various fading channels
Assume Maximum likelihood decoding
• (Nearest neighbourhood rule) at the receiver side
How do we find the bit error rate or symbol error rate the receiver side?side?
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
SER of various modulation schemes for SISO system over
Maximum likelihood decoding
(Nearest neighbourhood rule) at the receiver side
How do we find the bit error rate or symbol error rate the receiver
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
SER of various modulation schemes for SISO system over various fading channels
Monte Carlo simulation
• Generate a stream of bits, convert it to symbols, simulate the channel, add AWGN
• Find the number of bits in error a• Find the number of bits in error abits sent which gives the bit error rate (BER)
Analytical
• Closed form formula of the BER
• Helps in designing the system
• Benchmark is the simulation results
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
SER of various modulation schemes for SISO system over
Generate a stream of bits, convert it to symbols, simulate the
ror and divide by the total number of ror and divide by the total number of bits sent which gives the bit error rate (BER)
Closed form formula of the BER
Benchmark is the simulation results
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
SER of various modulation schemes for SISO system over various fading channels
For single antenna case,
• total transmit power is P
Hence,
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) ( )σσ
==2
2
2
2PThEh
SNRs
SISO
SER of various modulation schemes for SISO system over
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
γ=PT
SER of various modulation schemes for SISO system over various fading channels
For SER analysis, there are two basic steps:
a) First, find the conditional error probability (CEP) for the specific modulation scheme modulation scheme
• error probability over AWGN channel (y=
b) Second, average it over the pdfaverage symbol error rate (SER)
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
SER of various modulation schemes for SISO system over
For SER analysis, there are two basic steps:
a) First, find the conditional error probability (CEP) for the specific
error probability over AWGN channel (y=x+n)
pdf of the received SNR to obtain
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
SER of various modulation schemes for SISO system over various fading channels
In the moment generating function (MGF) based approach
we may express the SER as function of the MGF of the particular fading channel
For example,For example,
BER for BPSK over Rayleigh fading channel
For single antenna case, conditionalgiven by
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) (γ 2| SNRQEPb =
+ M. K. Simon and M.-S. Alouini, Digital communications over fading channels
2005.
SER of various modulation schemes for SISO system over
moment generating function (MGF) based approach+
we may express the SER as function of the MGF of the particular
BER for BPSK over Rayleigh fading channel
onal error probability (CEP) for BPSK is
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
) ( )γ2QSNR =
Digital communications over fading channels, Wiley,
SER of various modulation schemes for SISO system over various fading channels
Q function is the tail probability of the standard normal distribution
Then average bit error rate (BER) cathe pdf of received SNR γ
( ) ( )[ ] ( γγ QEPEEP bb ∫∞
== 2|
where E is the expectation operator
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) ( )[ ] ( γγ QEPEEP bb ∫==0
2|
( ) ;sin2
exp1
2
0
2
2
−= ∫ d
xxQ θ
θπ
π
Q
J. Craig, “A new, simple and exact result for calculating the probability of error for two
constellations”, in Proc. IEEE MILCOM, pp. 25.5.1-25.5.5, Boston, MA, 1991. (
SER of various modulation schemes for SISO system over
Q function is the tail probability of the standard normal distribution
) can be obtained by averaging over
) ( ) γγγ dp
where E is the expectation operator
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
) ( ) γγγ dp
0; ≥x
J. Craig, “A new, simple and exact result for calculating the probability of error for two-dimensional signal
25.5.5, Boston, MA, 1991. (826 citations as of July 26, 2019)
SER of various modulation schemes for SISO system over various fading channels
we can obtain the average BER as
( )θ
γ
π
π
EPb ∫ ∫∞
−=
2
2sin2
2exp
1
Integrating w.r.t. γ first, we have, the average BER of BPSK for SISO over any fading channel as
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
θπ∫ ∫ 0 0
sin2
( ) (γθ
γ
π
π
γpEPb
−= ∫ ∫
∞2
0 0
2sinexp
1
SER of various modulation schemes for SISO system over
( ) γγθθ
γ dpd
. γ first, we have, the average BER of BPSK for SISO
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
θ
) θθπ
θγγ
π
γ ddd
−Μ=
∫ 2
2
0sin
11
SER of various modulation schemes for SISO system over various fading channels
where Mϒ is the moment generating function (MGF) of SNR γ
What are advantages?
• Converted two indefinite integrations to a definite integration
For example,For example,
For Rayleigh fading, the SNR (γ) is exponentially distributed
Hence, MGF of γ is given by
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) ( )γ
γγ
γ
γγγ dss =
−=Μ
∞∞
∫∫ exp1
exp1
exp
00
(sM X
SER of various modulation schemes for SISO system over
is the moment generating function (MGF) of SNR γ
Converted two indefinite integrations to a definite integration
For Rayleigh fading, the SNR (γ) is exponentially distributed
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
γ
γ
γ
γγ
γγ
γ
γγ
ss
s
ds−
=
−
−
=
−
∞
1
1
1
exp1
exp
0
) ( )[ ]sXEs exp=
SER of various modulation schemes for SISO system over various fading channels
where
Hence, the BER of BPSK for SISO casgiven by
Hence, the BER of BPSK for SISO cas
( )γγ E=
Hence, the BER of BPSK for SISO casgiven by
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )π
θ
γθ
π
ππ
dEPb ∫∫ =
+
=
2
0
2
2
02
sin
sin1
sin
11
11
SER of various modulation schemes for SISO system over
case over Rayleigh fading channel is
case over Rayleigh fading channel is case over Rayleigh fading channel is
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
θγθ
θd
+2
2sin
SISO fading channels
Performance metrics for wireless communications
• Average capacity
• Expectation of instantaneous capacity over
• Outage probability
• Can be obtained from CDF of
• BER/SER
• Can be obtained from MGF of • Can be obtained from MGF of
• CEP for various modulation schemes
For any SISO fading channel, in order to obtain the above three performance metrics,
• MGF, pdf and cdf of ϒ
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Kulkarni, L. Choudhary, B. Kumbhani and R. S. KshetrimayumTAS/MRC and OSTBC in Equicorrelated Rayleigh Fading
2014, pp. 1850-1858.
Performance metrics for wireless communications
Expectation of instantaneous capacity over pdf of α
Can be obtained from CDF of α
Can be obtained from MGF of ϒ
( ) ( )γ
σσ===
2
2
2
2PThEh
SNRs
SISO
Can be obtained from MGF of ϒ
CEP for various modulation schemes+
For any SISO fading channel, in order to obtain the above three
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Kshetrimayum, Performance Analysis Comparison oFading MIMO Channels, IET Communications, vol. 8, Is
3 important parameters for a random channel
For any random variable X
PDF:
CDF: The cdf of a RV X is defined as
MGF: The mgf of a RV X is defined as
( )Xp x
MGF: The mgf of a RV X is defined as
From mgf we can obtain characteristic function (
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) ( )[ ]sXEsM X exp= M s sx p x dx
+A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes
Tata McGraw Hill, 2002.
3 important parameters for a random channel
is defined as
is defined as
( ) [ ] ( )x
X UP x P X x p u du
−∞
= ≤ = ∫is defined as
we can obtain characteristic function (cf) by putting s=jω
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
−∞
( ) ( ) ( )expX XM s sx p x dx
∞
−∞
= ∫
Probability, Random Variables and Stochastic Processes,
Generalized fading distributions
These are three recent generalized fading distributions viz.,
k-μ,
α-μ and
η-μ fading distributionsη-μ fading distributions
In these fading distributions,
fading is generally considered as composed of many clusters of multipaths or rays
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
+M. D. Yacoub, “The k-μ distribution and the η-μ
Mag., vol. 49, no. 1, pp. 68-81, Feb. 2007.
Generalized fading distributions
These are three recent generalized fading distributions viz.,
fading is generally considered as composed of many clusters of
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
distribution,” IEEE Antennas Propagat.
Generalized fading distributions
Fig. 10 Typical Power Delay Profile (illustration of clusters and rays)
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Generalized fading distributions
Fig. 10 Typical Power Delay Profile (illustration of clusters and rays)
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Generalized fading distributions
Note that multipath waves within any one cluster the phases of scattered waves are random and
• have similar delay times with
delay-time spreads of different clusters being relatively largedelay-time spreads of different clusters being relatively large
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Generalized fading distributions
Note that multipath waves within any one cluster the phases of
time spreads of different clusters being relatively largetime spreads of different clusters being relatively large
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Generalized fading distributions
How are different clusters formed?
Along the path of signals from the transmitter to receiver,
• Let us assume there are different group of disturbances
• Each group of disturbances will form different clusters• Each group of disturbances will form different clusters
Every multipath will have a different amplitude and phase
• Hence it is a complex RV
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Generalized fading distributions
How are different clusters formed?
Along the path of signals from the transmitter to receiver,
Let us assume there are different group of disturbances
Each group of disturbances will form different clustersEach group of disturbances will form different clusters
Every multipath will have a different amplitude and phase
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Generalized fading distributions
How do I find its magnitude?
Usually take square root of the
• real part (in-phase) and
• imaginary part (quadrature component)• imaginary part (quadrature component)
Generally, all RV are Gaussian in nature
Both in-phase and quadrature component may assumed Gaussian distributed
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Generalized fading distributions
component)component)
Generally, all RV are Gaussian in nature
component may assumed Gaussian
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
k-μ fading distributions
In such a model,
where several clusters (say n) of many
the representation of envelope, X of the fading signal is
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )2 2
2
1
n
i i i ii
X I p Q q=
= + + + ∑
where several clusters (say n) of many multipaths or rays are formed
the representation of envelope, X of the fading signal is
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )2 2
i i i iX I p Q q
= + + +
k-μ fading distributions
where n is the number of clusters in the received signal (usually denoted by µ in the literature)
(Ii+pi) and (Qi+qi) are respectively the component of the resultant signal of component of the resultant signal of
Both Ii and Qi are mutually independent and Gaussian distributed with
• zero mean, i.e. E(Ii) = E(Qi) = 0 and
• equal variance, i.e.
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) (2 2 2
i iE I E Q= =
is the number of clusters in the received signal (usually
) are respectively the in-phase and quadrature phase of the resultant signal of ith clusterof the resultant signal of i cluster
are mutually independent and Gaussian distributed
) = 0 and
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
)2 2 2
i iE I E Q σ= =
k-μ fading distributions
pi and qi are the respective means
• in-phase and quadrature components of
ith cluster in the received signal
The non zero mean of in-phase and The non zero mean of in-phase and reveal the presence of a
• dominant component in the clusters of the received signal
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
of
components of
phase and quadrature phase components phase and quadrature phase components
dominant component in the clusters of the received signal
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
k-μ fading distributions
The pdf of k-μ distributed random variable (RV)
( )( )+
−+
ek αµ µµ
12 2
1
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )( )
Ω
+=
+−−
lk
ll
ek
ekp
k
αµα
µµ
µ
µ
α µ
12
22
1
2
distributed random variable (RV) αl is given by
( )
( )
+
Ω
+−
k
kkl
l
αµ
1
1 2
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )
Ω
+−+
Ω
l
l
kkI
l
αµµ1
21
2
1
k-μ fading distributions
In the above equation, k > 0 and μ > 0 distribution,
That’s why the name k-μ fading distributions
• k is the ratio of the total power dthe total power due to scattered waves and
• μ represents the number of clusters• μ represents the number of clusters
By varying these two parameters, one can obtain various fading channels
Iυ(・) represents the υth order modified Bessel function of the first kind (MATLAB function is besseli(nu,Z
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )2ll E α=Ω
k > 0 and μ > 0 are the main parameters of the
fading distributions
er due to dominant components to the total power due to scattered waves and
μ represents the number of clustersμ represents the number of clusters
By varying these two parameters, one can obtain various fading
order modified Bessel function of the first nu,Z), where nu is order)
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
k-μ fading distributions
Special cases:
Rice fading distribution
Limit the number of clusters in the received signal to 1 which represents μ = 1represents μ = 1
Rice (μ = 1 and k = K),
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) (2 2
X I p Q q= + + +
( ) ( +=
== l
Kp
Kkαα µ
121,
Limit the number of clusters in the received signal to 1 which
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
)2 2
X I p Q q= + + +
)( )
( )
Ω
+
Ω
Ω
+−
−
lll
K
lK
KKI
eeKl
l
αα
α
120
1 2
k-μ fading distributions
Nakagami-m fading distribution
consider that the received signal arrives in clusters but the clusters
does not have any dominant component, i.e. put p
with each Ii and Qi being mutually independent and Gaussian distributed with zero mean and equal variance
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )2 2
2
1
n
i ii
X I Q=
= + ∑
consider that the received signal arrives in clusters but the clusters
does not have any dominant component, i.e. put pi = qi = 0
being mutually independent and Gaussian distributed with zero mean and equal variance
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )2 2
i iX I Q
= +
k-μ fading distributions
Nakagami-m (k →0 and μ = m),
For small arguments,
µ
−1y
( )(12 +µ k
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )( )µ
µΓ
≅−1
2yI
( )(
2
1
12−
+=
− µα
µα
µ l
k
kp
k
( ) ( )e
km
k
Limp
ml
mk
ml
mm
lmk ΓΩ
+
→=⇒
−
=→ ,0
212
0α
αα
µ
)( )
( )1
1
2
1
1
2−Ω
+−+
+
µαµ
µµ
α
k
kkekl
l
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
)
( )
( )
2
11
2 1+
Ω
Ω
+
ΓΩ
µµ
µ
αµ
µ
αl
l
lk
l kk
e
ek l
( )
( ) ( )m
em
m
eml
m
ml
m
kml
ll
l
ΓΩ=
Γ
Ω−
−Ω
+−
−
22
12
1
1 2αα
α
k-μ fading distributions
Rayleigh fading distribution
For μ = 1, we consider that there is n= 0, it reduces to
( ) ( )2 2
X I Q= +
For m=μ=1, in the above pdf we have the Rayleigh distribution
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) ( )X I Q= +
( ) 21,0
αα
α
αΩ
=Ω
−
=→l
llmk
ep
e is no dominant component, i.e. p = q
2 2
we have the Rayleigh distribution
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
2
2
2
2;
2
22
σσ
α σ
αα
=Ω=
−Ω
ll
l
l
l
e
k-μ fading distributions
For the k-µ distribution, the pdf of instantaneous SNR is
( ) ( )( )
γ
µµµ
µµ
γ
µµµ
γ µI
ek
exkxp
k
xk
k
+=
+−
+−−
+
−
1
2
1
2
1
1
2
1
2
1
The mgf of instantaneous SNR for k
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
γµek
k 22
( ) ( ) ( )ssdpeeEsM
kk γγγ
γ γµµ
∞
−− == ∫ −−
0
of instantaneous SNR is
( ) ( )γγγ
µ Exkk
=
+− ;
121
k -μ fading distribution is given by
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )( )
( )( )
ksk
kk
esk
kd
µγµ
µµ
γµ
µγ
−++
+−
++
+= 1
12
1
1
k-μ fading distributions
function x=kappa_mu_channel(kappa,mu,Nr,Nt
m = sqrt( kappa/((kappa+1))) ;
s = sqrt( 1/(2(kappa+1)) );
ni=0;
nq=0;nq=0;
for j=1:2*mu
• if mod(j,2)==1
• norm1=m+s*randn(Nr,Nt);
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
+ B. Kumbhani and R. S. Kshetrimayum, MIMO Wireless Communications over
Generalized Fading Channels, CRC Press, 2017
kappa,mu,Nr,Nt);
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO Wireless Communications over
k-μ fading distributions
• ni=ni+norm1.ˆ2;
• else
• norm2=s*randn(Nr,Nt);
• nq=nq+norm2.ˆ2;• nq=nq+norm2.ˆ2;
• end
end
h_abs=sqrt(ni+nq)/sqrt(mu);
theta = 2*pi*rand(Nr,Nt);
x=h_abs.*cos(theta)+sqrt(−1)*h_abs
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
h_abs.*sin(theta);
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Weibull fading distributions
Weibull fading distribution (one cluster)
Weibull fading statistics is best fit foin 800/900 MHzin 800/900 MHz
The envelope of the fading signal R is obtained as phase and quadrature components
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )2 2R X Yα= +
fading distribution (one cluster)
it for mobile radio systems operating
The envelope of the fading signal R is obtained as αth root of the in-components
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
α-μ fading distributions
Weibull distribution for μ = 1 (one cluster)
( )α
α
αβα α
α
er
rrf r
r
Rˆ
ˆ1
==−−
How about extending this classical ffading distribution?
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
αr
( ) ( )ˆ ˆ;r E R E R rα α αα= Ω = =
(one cluster)
αβα βαβ
α
rer
r
ˆ
1;1 =−−
cal fading distribution to generalized
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
αr
ˆ ˆr E R E R rα α α= Ω = =
α-μ fading distributions
α-μ distribution can be used to model fading channels in the environment characterized by
• non-homogeneous obstacles that may be nonlinear in nature
Suppose that such a non-linearity isSuppose that such a non-linearity isα > 0 thereby the emerging envelope R is
It is more general case as the usual cthis
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )∑=
+=n
i
ii YXR
1
22α
μ distribution can be used to model fading channels in the
homogeneous obstacles that may be nonlinear in nature
ty is expressed by a power parameter ty is expressed by a power parameter thereby the emerging envelope R is
ual case of α =2 is a particular case of
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
)
α-μ fading distributions
The probability density function of the follows
( )( )
α
α
µ
αµ
αµµ
µ
αµr
r
R er
rrf ˆ
1
ˆ
−−
Γ=
Special cases
Weibull fading distribution (one cluster)
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )µr Γ
( )2 2R X Yα= +
The probability density function of the α-μ signal is obtained as
fading distribution (one cluster)
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
α-μ fading distributions
Weibull distribution for μ = 1 (one cluster)
( )α
α
αβα α
α
er
rrf r
r
Rˆ
ˆ1
==−−
Weibull fading statistics is best fit foin 800/900 MHz
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
αr
( ) ( )ˆ ˆ;r E R E R rα α αα= Ω = =
(one cluster)
αβα βαβ
α
rer
r
ˆ
1;1 =−−
it for mobile radio systems operating
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
αr
ˆ ˆr E R E R rα α α= Ω = =
α-μ fading distributions
exponential fading distribution
considering α = 1 and μ = 1 in the expression of physical model described by
2 2R X Y= +
This represents exponential distribution
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) (r
erf
r
r
R expˆ
ˆ
−==
−
χ
2 2R X Y= +
in the expression of physical model
This represents exponential distribution
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
)r
rˆ
1; =χχ
α-μ fading distributions
Nakagami-m fading distribution
for α = 2 (note that n below is usually denoted by
( )2 2n
= +∑
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )2 2
2
1
n
i ii
X I Q=
= + ∑
( )( )
ˆ2
12
ˆ
2e
r
rrf r
r
RΓ
=−− µ
µ
µµ
µ
µ
α = 2 (note that n below is usually denoted by µ)
( )2 2 = +
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )2 2
i iX I Q
= +
( )2
12ˆ ˆ;
22
2
2
rer
r
r
r
=ΩΓΩ
= Ω−− µ
µ
µµ
µ
µ
α-μ fading distributions
Relation of α-μ fading distribution and
In MATLAB, Gamma RV can be generated using the function
( )2 2 2
1
n
i i Nakagamii
R X Y Xα
=
= + =∑
In MATLAB, Gamma RV can be generated using the function
shape parameter, k, and scale parameter,
k and θ are related to the Nakagami
k=m and respectively
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
m
γθ =
μ fading distribution and Nakagami-m distribution
In MATLAB, Gamma RV can be generated using the function gamrnd
2 2 2
i i NakagamiR X Y X
In MATLAB, Gamma RV can be generated using the function gamrnd
shape parameter, k, and scale parameter, θ, as input arguments
Nakagami-m fading parameter m as
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
α-μ fading distributions
MATLAB code to generate α-μ channel matrix
• n=gamrnd(mu,a_SNR/mu,Nr,Nt
• a_inv=1/alpha;
• phi=2*pi*rand(Nr,Nt);• phi=2*pi*rand(Nr,Nt);
• H=(n.ˆa_inv).*exp(j*phi);
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
+ B. Kumbhani and R. S. Kshetrimayum, MIMO Wireless Communications over
Generalized Fading Channels, CRC Press, 2017
μ channel matrix
mu,Nr,Nt);
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO Wireless Communications over
α-μ fading distributions
Rayleigh fading distribution
considering α = 2 and μ = 1 in the expression of physical model described by
( ) ( )2 2
X I Q= +
Note that in this case, Nakagami-m distribution with Rayleigh distribution
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) ( )X I Q= +
( )ˆ
2 ˆ2
2
2
er
rrf r
r
R =−
in the expression of physical model
2 2
m distribution with μ = 1 is like
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
2
ˆ;
222
2
2
2
2
2
re
rr
==−
γγ
γ
η-μ fading distributions
First case, k-μ fading distribution, extension of Rice fading channel
• consideration of μ clusters instead of single clusters
Second case, we have considered the case of
• where instead of taking square root of the inquadrature component of the fading signal
• we have taken α-th root, extension of • we have taken α-th root, extension of
One more extension is possible
• where we assume the variance of the incomponents are different
• It is the third case of η-µ fading distributionNakagami-m fading channel
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
extension of Rice fading channel
clusters instead of single clusters
Second case, we have considered the case of α-μ fading distribution
where instead of taking square root of the in-phase and component of the fading signal
extension of Weibull fading channelextension of Weibull fading channel
where we assume the variance of the in-phase and quadrature
fading distribution, extension of
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
η-μ fading distributions
μ is number of clusters
η is defined as the
• ratio of power of the in-phase component to power of phase component of the receivephase component of the receivephase and quadrature components are uncorrelated)
• Correlation coefficient of the incomponents in format II (assuming incomponents are correlated)
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
phase component to power of quadratureeived signals in format I (assuming in-eived signals in format I (assuming in-
components are uncorrelated)
Correlation coefficient of the in-phase and quadraturecomponents in format II (assuming in-phase and quadrature
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
η-μ fading distributions
η-µ fading distribution is similar to
• it is assumed that the multi-pathare in the form of several clusters and
the clusters does not have any domithe clusters does not have any domidistribution
However, the parameter η makes it different from as mentioned before
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )2 2
2
1
n
i ii
X I Q=
= + ∑
similar to Nakagami-m fading model,
path components in received signal are in the form of several clusters and
omina`ng or LOS component in η − μ omina`ng or LOS component in η − μ
However, the parameter η makes it different from Nakagami-m fading
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )2 2
i iX I Q
= +
η-μ fading distributions
In η −μ fading, the varia`on from Nakagami
that the variance which is same as the power content of Idifferent
( ) ( )2 2 2 2;i I i Q
E I E Qσ σ= =
The probability density function (pdf
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) ( );i i
i I i QE I E Qσ σ= =
( )
( ) ΩΓ
=−
+
−
2
1
22
1
4 ll
H
hp
µ
αµπα
µ
µµµ
α µη
Nakagami-m fading is
that the variance which is same as the power content of Ii and Qi is
2 2 2 2
i I i Qσ σ= =
pdf) of η-μ distributed RV is given by
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
i ii I i Q
σ σ= =
ΩΩ
−+
Ω−
2
2
1
2
1
2
2
2
ll
l
h
HI
el
l
αµ
µµ
αµ
η-μ fading distributions
H and h are the functions of fading parameter η
• which is the fading parameter defined in two ways and
• thus there are two formats for η − μ
Format IFormat I
In this format, the in-phase component and component of the resultant signal in each cluster are
• assumed to be independent and with different powers
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
are the functions of fading parameter η
defined in two ways and
η − μ fading channels
phase component and quadrature phase component of the resultant signal in each cluster are
assumed to be independent and with different powers
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
η-μ fading distributions
η is defined as the ratio of power of in phase component to the power of quadrature component, i.e.
∈The parameter η ∈ (0, ∞) is also assfor all the clusters in the received signal
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )η
η
η
η −=
+=
4
1;
4
12
Hh
η is defined as the ratio of power of in phase component to the component, i.e. ( )
( )
2 2
22
i
i
i I
Qi
E I
E Q
ση
σ= =
assumed that this ratio is constant for all the clusters in the received signal
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )ii
η
η
η
η
+
−=
1
1;
2
h
H
η-μ fading distributions
Since
The pdf has Iʋ which is function of H and it is symmetric for H=0
( ) ( ) ( )1I z I zν
ν ν− = − ( )I zν
=
(−
p αα µη
Hence the distribution is symmetric around
• therefore power distribution maythe regions
It can be shown that the values of h= 1, i.e.
• the values of H and h for 0 < η ≤ 1
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
which is function of H and it is symmetric for H=0
( )
2
0
1
21!
i
i
z
i i
ν
ν
+∞
=
=
Γ + + ∑
)
( )
ΩΩΓ
=−+−
Ω−+
2
2
1
2
1
2
1
2
22
1
24
2
ll
l
h
ll
HI
H
ehl
l
αµ
µ
αµπα
µµµ
αµ
µµµ
Hence the distribution is symmetric around η=1 since for η=1, H=0,
may be considered only within one of
of h and H are symmetrical around η
0 < η ≤ 1 are same for the range 1 ≤ η < ∞
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
η-μ fading distributions
Format II
assumption made is that the in-phase and components of MPCs within each cluster are
• correlated and
∈
• correlated and
• have same powers
The parameter η ∈ (-1, 1) is the corrcomponents
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )( )
(2 2
, ,i i i i
i i
E I Q E I Q
E Q E Iη = =
phase and quadrature phase components of MPCs within each cluster are
correlation coefficient between these
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
)( )2 2
, ,i i i i
i i
E I Q E I Q
E Q E I
η-μ fading distributions
It is also assumed that the
• correlation coefficient between in phase component and the quadrature component is same
• for all the clusters in the received signal• for all the clusters in the received signal
It can be shown that the values of h= 0 i.e. H=0
• (0 ≤ η < 1 or −1 < η ≤ 0 needs to be considered)
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
21
1
η−=h
21 η
η
−=H
H
hη=
correlation coefficient between in phase component and the component is same
for all the clusters in the received signalfor all the clusters in the received signal
of h and H are symmetrical around η
needs to be considered)
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
η-μ fading distributions
Relation between format I and format II can be obtained by
equating the ratio H/h of both the formats
1 ηη
−=
Special cases
η−μ distribution for format I, differs from in only one parameter
the different variance of in-phase components and components of resultant of each cluster in the received signal
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
1formatI
ηη
=+
Relation between format I and format II can be obtained by
equating the ratio H/h of both the formats
formatIIη−
distribution for format I, differs from Nakagami-m fading model
phase components and quadrature phase components of resultant of each cluster in the received signal
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
formatII
formatIIη+
η-μ fading distributions
μ = n/2 but it constraints the values of
• μ to be discrete on the account of discrete values of n
For η=1 and μ=m/2, it gives the Nakagami
( )
μ=0.5 and η=1 gives Rayleigh distribution
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )( )
2 2
22
i
i
i I
Qi
E I
E Q
ση
σ= = =
( ) ( )2 2
X I Q= +
n/2 but it constraints the values of
μ to be discrete on the account of discrete values of n
Nakagami-m fading distribution
=1 gives Rayleigh distribution
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
1= = =
)2 2
X I Q
η-μ fading distributions
Hoyt fading distribution (η= q2 and fading
( ) (2 2
X I qQ= +
The pdf of the instantaneous signalη-µ distribution is
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) (
( )
( )µ
µπ
µ
µµ
γ µη
H
hxp
Γ
=
+
−
2 2
1
and μ = 0.5) also called as Nakagami-q
)2 2
X I qQ
of the instantaneous signal-to-noise ratio (SNR) of the
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
)2lαγ =
( )γγγ
µ
γµµµ
γ
µµ
µ
EHx
Iex
xh
=
−+−
−−
;2
2
1
2
1
2
1
2
2
1
η-μ fading distributions
The mgf of instantaneous SNR for η
Special cases
( ) ( ) ( )γγγ
γ γγµηµη
=== ∫∞
−−
−−dpeeEsM
ss
0
Special cases
Rayleigh fading
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )5.0,1 == µη
( )γµηγ
ssM
+=
== 1
15.0,1
+N. Ermolova, “Moment generating functions of the generalized
their applications to performance evaluations of communication systems,”
Letters, vol. 12, no. 7, pp. 502-504, 2008.
η -μ fading distribution+ is given by
( )( ) ( )( )
µ
γµγµ
µ
+++−=
sHhsHh
h
22
42
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
, “Moment generating functions of the generalized η-μ and k-μ distributions and
their applications to performance evaluations of communication systems,” IEEE Communications
η-μ fading distributions
Nakagami-m
• For format 1, η 1, implies
==
2,1
mµη
( )
• For format 2, η 0 implies
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )4
1;1
4
122
−==
+=
η
η
η
ηHh
01
;11
122
=−
==−
=η
η
ηHh
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
0=
0
η-μ fading distributions
Nakagami-q
( )m
sm
msM
m
+=
== γµηγ
2,1
( )5.0,2 == µη qNakagami-q
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )5.0,2 == µη q
( )( )( ) ( )(5.0,
+++−=
= γµηγHhsHh
hsM
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
)
5.0
γs
η-μ fading distributions
For format 1,
( ) ( ) 2
2
222
4
1;
4
1
4
1H
q
qh =
−=
+=
+=
η
η
η
η
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )
( )
2 0 5
22
2
2 2
2
1
4
1 1
2 2
, .q
q
qM s
q qs s
q
η µγ
γ γ= =
+
= = + +
+ +
2
22
2
4
2
1;
2
1;
4
1
q
qHh
qHh
q
q +=+
+=−
−=
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )( )( )
0 5
02
2
2 2 2
1
1 2 1 2
.
q
q s q q ss s
γ γγ γ
+ = = + + + + + +
η-μ fading distributions
It is more suitable for NLOS signal propagation
Hoyt fading (or Nakagami-q) is best fit for satellite links subject to strong atmospheric scintillation
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )(15.0,2
+
=∴== µη
γ
q
sMq
Scintillation effects are because of arbitrary refraction
small-scale fluctuations in air density
gradients
It is more suitable for NLOS signal propagation
q) is best fit for satellite links subject to
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
) ( )5.0
22222
2
421
1
++
+
γγ sqsq
q
Scintillation effects are because of arbitrary refraction caused by
air density due to temperature
η-μ fading distributions
function x=eta_mu_channel(eta,mu,Nr,Nt
coeff=sqrt(eta);
ni=0;
nq=0;nq=0;
for j=1:2*mu
• norm1=randn(Nr,Nt);
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
+ B. Kumbhani and R. S. Kshetrimayum, MIMO Wireless Communications over
Generalized Fading Channels, CRC Press, 2017
eta,mu,Nr,Nt);
1,m
Nakagami η µ
= =
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO Wireless Communications over
12
,m
Nakagami η µ
= =
η-μ fading distributions
• ni=ni+norm1.ˆ2;
• norm2=coeff*randn(Nr,Nt);
• nq=nq+norm2.ˆ2;
end
h_abs=(sqrt(ni+nq))/sqrt((2*mu*(1+eta)));
theta = 2*pi*rand(Nr,Nt);
x=h_abs.*cos(theta)+sqrt(−1)*h_abs
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )( )
2 2
22
i
i
i I
Qi
E I
E Q
ση
σ= =
((2*mu*(1+eta)));
h_abs.*sin(theta);
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO I-O system model
Fig. 11 2×2 MIMO systemRakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
2 MIMO system, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO I-O system model
Received signal at antenna 1
y1=h11x1+h12x2+n1
Received signal at antenna 2
y =h x +h x +ny1=h21x1+h22x2+n2
In matrix form,1 11 12 1
2 21 22 2
y h h x
y h h x
=
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
1 11 12 1
2 21 22 2
y h h x
y h h x
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO I-O system model
For NT ×NR MIMO systems, I-O model is given by
y=Hx+n
where
y n h h h x 1 1 11 12 1 1
2 2 21 22 2 2
1 2
; ; ;
R R R R R T TN N N N N N N
y n h h h x
y n h h h x
y n h h h x
= = = =
y n H xM M M O N M M
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
O model is given by
y n h h h x K1 1 11 12 1 1
2 2 21 22 2 2
1 2
; ; ;
T
T
R R R R R T T
N
N
N N N N N N N
y n h h h x
y n h h h x
y n h h h x
= = = =
y n H x
K
L
M M M O N M M
L
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO I-O system model
Notation:
• hij, i is the receiver antenna indexindex
• Bold face small letters are used for representing vectors• Bold face small letters are used for representing vectors
• Bold face capital letters are used for representing matrices
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
ndex and j is the transmitter antenna
Bold face small letters are used for representing vectorsBold face small letters are used for representing vectors
Bold face capital letters are used for representing matrices
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Analytical MIMO channel models
Some of the analytical MIMO channel models are:
• iid (uncorrelated) MIMO channel model
• fully correlated MIMO channel model
• separately correlated MIMO channel model• separately correlated MIMO channel model
• Uncorrelated keyhole MIMO channel model
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Analytical MIMO channel models
Some of the analytical MIMO channel models are:
(uncorrelated) MIMO channel model
fully correlated MIMO channel model
separately correlated MIMO channel modelseparately correlated MIMO channel model
Uncorrelated keyhole MIMO channel model
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
In MIMO communication, the channel
• whose elements are hij, i=1,2,..,N
A complex RV Z=X+jY, a pair of real RVs X and Y
The pdf of a complex RV, the joint pdfThe pdf of a complex RV, the joint pdf
pdf of a complex normal RV
A complex normal RV (Z=X+jY) is a complex RV
• whose real (X) and imaginary (Y) parts are mean and variance ½
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
In MIMO communication, the channel H is a matrix
=1,2,..,NR, j=1,2,…,NT complex RVs
, a pair of real RVs X and Y
pdf of its real and complex partspdf of its real and complex parts
) is a complex RV
whose real (X) and imaginary (Y) parts are i.i.d. Gaussian with zero
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
From probability theory:
Independent and non-identically distributed (
RVs X1,X2,…,XN are iid if for all x1,x2,…,
( ), , 1 1 2, ,X X N X X X Np x x p x p x p x=L L L
Independent and identically distributed (
RVs X1,X2,…,XN are iid if for all x1,x2,…,
( )1 , , 1 1 2, ,
NX X N X X X Np x x p x p x p x=L L L
( )1 1 2, , 1 1 2, ,
N NX X N X X X Np x x p x p x p x=L L L
+A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes
Tata McGraw Hill, 2002.
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
identically distributed (i.i.n.d.) RVs:
,…,xN
( ) ( ) ( ), , 1 1 2X X N X X X Np x x p x p x p xL L
Independent and identically distributed (i.i.d.) RVs+:
,…,xN
( ) ( ) ( ), , 1 1 2X X N X X X Np x x p x p x p xL L
( ) ( ) ( )1 1 2, , 1 1 2N NX X N X X X Np x x p x p x p xL L
Probability, Random Variables and Stochastic Processes,
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
For example, hij, i=1,2,..,NR, j=1,2,…,N
( ) 1
/
/
; ~ ,real imag real imag
ij ij ij ij
real imag
h h jh h N
p h e
= +
=
Rayleigh distributed
( ) 1
12
2
/real imag
ijp h e
π
=
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( real imag
ij ij ijh h h= +
, j=1,2,…,NT are complex normal RV
( )2
12
2
10
2
/
/; ~ ,
real imagij
real imag real imag
h
h h jh h N
p h e
−
22p h e
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
) ( )2 2
real imag
ij ij ijh h h= +
iid MIMO channel model
( )0 1~ ,ij C
h N
complex normal RV hij has pdf as
( ) ( ) ( )
( )( ) ( )
2 2
1 1
2 2
1 1real imagij ij
real imag
ij ij ij
h h
ij
p h p h p h e e
p h e e
π π
π π
− +
= =
= =
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) ( )2 2
real imagij ij
h h( ) ( )
2
1 12 2
2 21 1
1 12 2
2 2
ij ij
ij
h h
h
p h p h p h e e
p h e e
π π
− −
−
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
• A random matrix is a matrix whose elements are RVs
• If the elements of random matrix are complex RVs, then it is a complex random matrix
• A random matrix can have joint pdf
• For iid MIMO channel model, all elements of the channel matrix i=1,2,..,NR, j=1,2,…,NT) are iid complex normal RVs also called as Rayleigh MIMO fading channel
• For example, a complex normal matrixelements are complex normal RVs
+T. W. Anderson, An introduction to multivariate statistical analysis
3rd edition, 2003.Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
A random matrix is a matrix whose elements are RVs
If the elements of random matrix are complex RVs, then it is a
pdf of its elements+
MIMO channel model, all elements of the channel matrix H (hij
complex normal RVs also called as iid
complex normal matrix H is a random matrix whose elements are complex normal RVs
An introduction to multivariate statistical analysis, John Wiley & Sons,
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
For H an iid normal matrix, the pdfcomplex normal RVs, hij, i=1,2,..,NR
( )2 2
1 1 1, ,
R T R TN N N N
h h− −= = =∏ ∏
Note that trace for a square matrix is equal to the sum total of its diagonal elements
HHH is a square matrix whose trace is
( )1 1
1 1 1, ,
, ,
R T R T
ij ij
R T R T
h h
N N N Ni j i j
p e e eπ π π
− −
= =
= = =∏ ∏HH
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
pdf is the multiplication of pdfs of
R, j=1,2,…,NT
22 2
1 1 1
,
,
, ,
N NR T
R T R T ijN N N N h
h h−
− − ∑= = =∏ ∏
Note that trace for a square matrix is equal to the sum total of its
is a square matrix whose trace is
1
1 1
1 1 1
,
,
, ,
, ,
R T R T ijij ij i j
R T R T
hh h
N N N Ni j i j
p e e eπ π
=
−− −
= =
∑= = =∏ ∏
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
( )
11 12 1 11 21 1
21 22 2 12 22 2
1 2 1 2
22 2
T R
T R
R R R T T T R T
H
N N
N N
N N N N N N N N
trace
h h h h h h
h h h h h htrace
h h h h h h
=
+ + +
HH
L L
L L
M O O M M O O M
L L
22 2
11 12 1
2 2
21 22 2
TNh h h
h h htrace
+ + +
+ + +=
L L L L
L L L L
M O O M
L L L
,2
,
, 1
R TN N
i j
i j
h
=
= ∑Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
*
11 12 1 11 21 1
21 22 2 12 22 2
T R
T R
R R R T T T R T
N N
N N
N N N N N N N N
h h h h h h
h h h h h h
h h h h h h
L L
L L
M O O M M O O M
L L
22 2
21 22 2
2
1
T
R
N
N N
h h h
h h
+ + +
+
L L L L
L L L L
M O O M
L L L2 2
2R R TN Nh
+ +
L
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
Hence the pdf of the normal matrix
( ) ((1exp
R TN Np Trace
π= −H H HH
Short hand notation of exponential (trace) =
( ) (1
R TN Np etri
π= −H H HH
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
of the normal matrix H is
( ))Hp TraceH HH
Short hand notation of exponential (trace) =etri
)H= −H HH
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
All the components of H, hij are assu
Assume that the path gains are identically distributed RV
From probability theory
The covariance and correlation of two RVs X and Y is defined asThe covariance and correlation of two RVs X and Y is defined as
( ) ( )(, X YCov X Y E X Yµ µ = − −
( ) [ ],Cor X Y E XY=
+R. D. Yates and D. J. Goodman, Probability and Stochastic Processes
Sons, 2005.
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
assumed independent (uncorrelated)
Assume that the path gains are identically distributed RV
The covariance and correlation of two RVs X and Y is defined as+The covariance and correlation of two RVs X and Y is defined as+
)X Yµ µ = − −
( ) ( ), , X YCov X Y Cor X Y µ µ= −
Probability and Stochastic Processes, John Wiley and
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
For zero mean RVs X and Y,
The two RVs X and Y are
orthogonal if
( ) (, ,Cov X Y Cor X Y=
orthogonal if
uncorrelated if
Note that uncorrelated means covariance is zero
( ), 0Cor X Y =
( ), 0Cov X Y =
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
), ,Cov X Y Cor X Y
Note that uncorrelated means covariance is zero
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
If X and Y are independent, then
which implies that
Cor X Y
( ), , 0Cov X Y Cor X Y= − =
A Gaussian random vector X has independent components covariance matrix is diagonal matrix
A normal Gaussian random vector covariance matrix is an identity matrix
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ), X YCor X Y µ µ=
( ), , 0X YCov X Y Cor X Y µ µ= − =
has independent components iffcovariance matrix is diagonal matrix
A normal Gaussian random vector X has independent components iffcovariance matrix is an identity matrix
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
For iid MIMO channel matrix H, the covariance matrix matrix
Example,
uncorrelated (i.i.d.) fading MIMO channel model for a 2uncorrelated (i.i.d.) fading MIMO channel model for a 2system
Show that the covariance matrix is
Let us consider a 2×2 MIMO systemchannel matrix is
=
2221
1211
hh
hhH
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
, the covariance matrix RH is a diagonal
.) fading MIMO channel model for a 2×2 MIMO .) fading MIMO channel model for a 2×2 MIMO
Show that the covariance matrix is I4.
tem (for illustration purpose) whose
22
12h
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
Assume that
we can find the covariance matrix
0; , 1,2ijE h i j= =
“vect” (vectorization) stacks all the cvector
“Unvec” converts back a vectorizedmatrix
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
we can find the covariance matrix RH as ; ( )HH E vect= =R hh h H
the columns of a matrix into a column
vectorized matrix into its corresponding
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
We can “vect” (vectorize) the above system and find the covariance matrix
h
( ) ( ) [ ]H
H
h
hE vect vect E h h h h
h
h
= =
R H H
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
) the above 2×2 H matrix for 2×2 MIMO system and find the covariance matrix RH as follows
h
[ ]
11
*21
11 21 12 22
12
22
h
hE vect vect E h h h h
h
h
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
2
11 11 21 11 12 11 22
* * *
21 11 21 21 12 21 22
[ ]H
E h E h h E h h E h h
E h h E h E h h E h h
E
= =R hh
* * *
12 11 12 21 12 12 22
* * *
22 11 22 21 22 12 22
[ ]H
HE
E h h E h h E h E h h
E h h E h h E h h E h
= =
R hh
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
* * *
11 11 21 11 12 11 22
2* * *
21 11 21 21 12 21 22
E h E h h E h h E h h
E h h E h E h h E h h
2* * *
12 11 12 21 12 12 22
2* * *
22 11 22 21 22 12 22
E h h E h h E h E h h
E h h E h h E h h E h
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
Note that h11, h12, h21 and h22 are all mutually independent and identically distributed (uncorrelated) RVs with zero mean
2
11E h
( ) ( )
11
0 0 0
[ ]
0 0 0
0 0 0
H
E h
E vect vect
=
H H
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
are all mutually independent and identically distributed (uncorrelated) RVs with zero mean
0 0 0
2
21
2
12
2
22
0 0 0
0 0 0
0 0 0
0 0 0
E h
E h
E h
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
If we choose
then
2 2 2 2
11 12 21 22E h E h E h E h = = = =
( ) ( )
1 0 0 0
0 1 0 0[ ]
H
E vect vect
=H H
This analysis can be easily done for any arbitrary Nand show that
( ) ( ) 0 1 0 0
[ ]0 0 1 0
0 0 0 1
H
E vect vect =
H H
T RH N N=R I
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
2 2 2 2
11 12 21 22 1E h E h E h E h = = = =
1 0 0 0
0 1 0 0
This analysis can be easily done for any arbitrary NT×NR MIMO system
0 1 0 0
0 0 1 0
0 0 0 1
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
An important matrix we will frequenis the random complex Wishart matrix which is defined as
≥
<=
H
RH
NN
NN
,
,
HH
HHQ
where H is the random MIMO channel matrix
From spectral theorem
Q is a random matrix
Λ is also a random matrix
≥=
RH
NN,HHQ
=Q U Λ
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
quently use in MIMO capacity analysis matrix which is defined as
T
N
N
is the random MIMO channel matrixTN
H
Λ U
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iid MIMO channel model
1
2
0 0
0 0
0 0m
λ
λ
λ
= =
Λ
L
L
M M O M
L
Marginal distribution of an eigenvaluematrix+
( ) ( ) ( )( )∑∑∑
−
= = =− +−
−=
1
0 0
2
021
!!!2
!211m
i
i
j
j
lli
l
jmnlj
j
mp λ
+E. Biglieri, Coding for Wireless Channels, Springer, 2005.
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
; min ,R T
m N N
= =
M M O M
,max TR NNn =
eigenvalue (λ1) of complex Wishart
( ) −−+
−
−+
−
−1
1
2
22222 mnle
lj
mnj
ji
ji λλ
; min ,
, Springer, 2005.
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
,max TR NNn =
iid MIMO channel model
Fig. 12 NT×NR MIMO systemRakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO system, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Fully correlated MIMO channel model
In practice, MIMO channel is never uncorrelated
In fully correlated MIMO channel model
we need to consider all the
• co- and cross-correlations between all the channel coefficients • co- and cross-correlations between all the channel coefficients
for various channel paths between the transmitting antennas and receiving antennas
For a given H matrix, the correlation matrix
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
+ N. Costa and S. Haykin, Multiple-input multiple
Sons, 2010.
; ( )HH E vect= =R hh h H
Fully correlated MIMO channel model
In practice, MIMO channel is never uncorrelated
In fully correlated MIMO channel model
correlations between all the channel coefficients correlations between all the channel coefficients
for various channel paths between the transmitting antennas and
matrix, the correlation matrix+ can be defined as
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
input multiple-output channel models, John Wiley &
; ( )E vect= =R hh h H
Fully correlated MIMO channel model
11
1
12 * * * * *11 1 12 2
2
* * * * *
NR
H NR N N NR R T
NR
N NR T
h
h
hE h h h h h
h
h
=
R
M
L L LM
M
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
* * * * *11 11 11 1 11 11 1112 2
* * *1 11 1 1 1 112
NR N N N
N N N N NR R R R R N
h h h h h h h h h h
h h h h h h h h
E=
L L L
M O M M O M O M
L L
* * * * *12 11 12 1 12 12 1212 2
* * * * *2 11 2 1 2 2 212 2
* * * * *11 1 12 2
NR N N N
N N N N N NR R R R R N R N N
N N N N N N N N N N NR T R T R R T R T N R T N N
h h h h h h h h h h
h h h h h h h h h h
h h h h h h h h h h
L L L
M O M M O M O M
L L L
M O M M O M O M
L L L
Fully correlated MIMO channel model
* * * * *
12 2
* * * * *
R N N NR R TE h h h h h
L L L
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
* * * * *11 11 11 1 11 11 1112 2
1 11 1 1 1 1
R N N NR R T
N N N N NR R R R R N
h h h h h h h h h h
h h h h h h h h
L L L
M O M M O M O M
* *12
* * * * *12 11 12 1 12 12 1212 2
* * * * *2 11 2 1 2 2 212 2
* * * * *
12 2
NR N NR R T
R N N NR R T
N N N N N NR R R R R N R N NR R T
N N N N N N N N N N NR T R T R R T R T N R T N NR R T
h h
h h h h h h h h h h
h h h h h h h h h h
h h h h h h h h h h
L
L L L
M O M M O M O M
L L L
M O M M O M O M
L L L
Fully correlated MIMO channel model
mean vector m and covariance matrix
How do you calculate them?
they can be estimated as
This covariance matrix size and consof the covariance matrix become prohibitively large
• as the number of transmitting and receiving antennas increase
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )( )∑=
=−−=N
i
H
iiN 1
ˆ;ˆˆ1ˆ mmxmxΦ
Fully correlated MIMO channel model
and covariance matrix Φ are unknown
consequently the number of elements of the covariance matrix become prohibitively large
as the number of transmitting and receiving antennas increase
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
∑=
N
i
iN 1
1x
Fully correlated MIMO channel model
Let us take an example to find out this
Consider a 2×3 MIMO system as follows:
Vectorize it 11
h
h
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
21
31
12
22
32
h
h
h
h
h
=
h
Fully correlated MIMO channel model
Let us take an example to find out this
3 MIMO system as follows: 11 12
21 22
31 32
h h
h h
h h
=
H
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
31 32
Fully correlated MIMO channel model
Covariance matrix
( )
11
21
31
12H
h
h
h
hE h h h h h h
h
=
hh
How many components we need tomatrix? 6×6
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )22
32
h
h
Fully correlated MIMO channel model
11 21 31 12 22 32
*
E h h h h h h
d to calculate for finding covariance
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
11 21 31 12 22 32
Fully correlated MIMO channel model
For NT×NR MIMO system,
Obviously the above correlation matrix will have
For example, you consider a slightly larger MIMO system
10×10 MIMO system, we will have 10,00010×10 MIMO system, we will have 10,000
Not manageable
How do one reduce this number of components calculation in the covariance matrix?
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Fully correlated MIMO channel model
Obviously the above correlation matrix will have (NRNT)2 components
For example, you consider a slightly larger MIMO system
10 MIMO system, we will have 10,000 components10 MIMO system, we will have 10,000 components
How do one reduce this number of components calculation in the
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Finite correlation exist between ante
• But transmitter and Receiver are at very far distance
• We may assume receiver antenntransmitter antenna correlation and vice versatransmitter antenna correlation and vice versa
We can separate the transmitter andwrite
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
X X
H T
T RE vect= = ⊗ =
HR hh R R h H
Kronecker) MIMO channel model
antennas because of limited spacing
But transmitter and Receiver are at very far distance
enna correlation is independent of transmitter antenna correlation and vice versatransmitter antenna correlation and vice versa
r and receiver antenna correlation and
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
; ( )X XT R
E vect= = ⊗ =R hh R R h H
Separately correlated (Kronecker
Kronecker product:
Let A be an N×M matrix and B be an L
The Kronecker product of A and Bmatrixmatrix
It can be obtained as
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
=⊗
B
B
BA
NA
A
L
OM
L
1
11
Kronecker) MIMO channel model
be an L×K matrix
represented by is an NL×MK ⊗A B
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
B
B
NM
M
A
A
M
1
Separately correlated (Kronecker
11 12
21 22
a a
a a
=
A
11 12
21 22
b b
b b
=
B
a b a b a b a b
Example:
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
11 11 11 12 12 11 12 12
11 12 11 21 11 22 12 21 12 22
21 22 21 11 21 12 22 11 22 12
21 21 21 22 22 21 22 22
a b a b a b a b
a a a b a b a b a b
a a a b a b a b a b
a b a b a b a b
⊗ = =
B B
B BA B
Kronecker) MIMO channel model
11 12
21 22
b b
b b
a b a b a b a b
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
11 11 11 12 12 11 12 12
11 12 11 21 11 22 12 21 12 22
21 22 21 11 21 12 22 11 22 12
21 21 21 22 22 21 22 22
a b a b a b a b
a a a b a b a b a b
a a a b a b a b a b
a b a b a b a b
Separately correlated (Kronecker
How to calculate receiver and transmitter correlation matrices?
The correlation matrices at the transmitter and the receiver are calculated as
( ) ( )T
Example:
Let us consider a 2×2 MIMO system for illustration purpose whose channel matrix is
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) ( )* ;T
H T HE E E
= = = X XT RR H H H H R HH
=
2221
1211
hh
hhH
Kronecker) MIMO channel model
How to calculate receiver and transmitter correlation matrices?
The correlation matrices at the transmitter and the receiver are
( )
2 MIMO system for illustration purpose whose
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( );H T HE E E= = =
X XT RR H H H H R HH
Separately correlated (Kronecker
The transmitter correlation matrix (of H) is given by
*
11 21 11 12[ ]
T
H T h h h hE E
h h h h
= = = XTR H H
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
12 22 21 22
2 2 * *11 21 12 11 22 21
*11 12
h h h h
E h h E h h h h
E h h h
+ + =
+
XT
2 2*21 22 12 22h E h h
+
Kronecker) MIMO channel model
The transmitter correlation matrix (correlation between the columns
2 2 * *11 21 11 12 21 22
2 2* *
TT
E h h E h h h h
E h h h h E h h
+ + = = = + +
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
2 2* *12 11 22 21 12 22E h h h h E h h
+ +
Separately correlated (Kronecker
The receiver correlation matrix (correlation between the rows of given by
11 12 11 21[ ]H h h h h
E Eh h h h
= = = XRR HH
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
21 22 12 22
[ ]E Eh h h h
= = =
XRR HH
Kronecker) MIMO channel model
correlation between the rows of H) is
2 2 * **
11 12 11 21 12 22E h h E h h h h + + = = =
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
2 2* *21 11 22 12 21 22E h h h h E h h
= = = + +
Separately correlated (Kronecker
In total in order to find
we require to find elements only
For example, 10×10 MIMO system, we will have 200
How to introduce correlation in an otherwise
X X
T
T R= ⊗HR R R
( )2 2R TN N+
How to introduce correlation in an otherwise channel matrix Hw?
Define
What happens to covariance matrix
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( H wunvec=H R h
Kronecker) MIMO channel model
In total in order to find
elements only
10 MIMO system, we will have 200 components only
How to introduce correlation in an otherwise iid or spatially white
X XT RR R R
How to introduce correlation in an otherwise iid or spatially white
trix when we multiply to ?
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
)H wH R h
HR w
h
Separately correlated (Kronecker
Hermitian square root for the square root of the matrix last operation
R T
H H
H H w w H
H
H N N H H H H
E E = =
= = =
R hh R h h R
R I R R R R
last operation
Correlation matrix of iid hw has identity matrix
But h now has covariance matrix as
We have introduced correlation matrix
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Kronecker) MIMO channel model
square root for the square root of the matrix RH due to the
( )1/2 /2
HH H
H H w w H
H
H N N H H H H
E E
= = =
R hh R h h R
R I R R R R
has identity matrix
now has covariance matrix as RH
We have introduced correlation matrix RH into h from hw
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
We have introduced the correlation in an otherwise white channel matrix Hw
It means that the correlation matrix for It means that the correlation matrix for matrix for h becomes RH
What happens to channel matrix H
• For the correlation matrix RH above,
• let us find the expression for correlated channel matrix
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Kronecker) MIMO channel model
We have introduced the correlation in an otherwise iid or spatially
It means that the correlation matrix for hw is I but the correlation It means that the correlation matrix for hw is I but the correlation
H?
above,
let us find the expression for correlated channel matrix H
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
in terms of
• receiver correlation,
• transmitter correlation and
• spatially white channel matrix spatially white channel matrix
Using the identity
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) == wH unvecunvec hRH
(( 2 1 2/ /
x x
T
T R wunvec vect= ⊗H R R H
( ) ( )vect vect⊗ =A B C BCA
Kronecker) MIMO channel model
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
⊗ wR
TT Xx
hRR
))T R wH R R H
( )Tvect vect⊗ =A B C BCA
Separately correlated (Kronecker
Take
channel matrix H for Kronecker channel
( 1 2 1 2 1 2 1 2/ / / /
x x x xR w T R w T
unvec vect= =H R H R R H R
/ 2 1/2, ,x x
T
T R w= = =A R B R C H
channel matrix H for Kronecker channel
where ()1/2 denotes the Hermitian
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
12/1
X TR RHRH w=
+ J. P. Kermoal, L. Schumacher, K. I. Pederson, P. E.
Stochastic MIMO Radio Channel Model With Experimental Validation,”
Selected Areas in Communications, vol. 20, no. 6, Aug. 2002, pp. 1211
Kronecker) MIMO channel model
channel+ model can be written as
)1 2 1 2 1 2 1 2/ / / /
x x x xR w T R w T
= =H R H R R H R
channel+ model can be written as
square root of matrix
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
2/1
XT
, L. Schumacher, K. I. Pederson, P. E. Mogensen and F. Frederiksen, “A
Stochastic MIMO Radio Channel Model With Experimental Validation,” IEEE Journal on
vol. 20, no. 6, Aug. 2002, pp. 1211-26.
Separately correlated (Kronecker) MIMO channel model
How do we obtain Kronecker or sepmodel from iid channel model?
• In the iid MIMO channel model
• Premultiply by Hermitian square• Premultiply by Hermitian square
• Postmultiply by Hermitian square root of correlation at the transmitter
How to find the pdf of such matrices?
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
) MIMO channel model
separately correlated MIMO channel
MIMO channel model
are root of correlation at the receiverare root of correlation at the receiver
square root of correlation at the
of such matrices?
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
All matrices can be vectorized, better find the
A complex Gaussian random vector its mean (m=E(z)) and covariance matrix [
Once we have mean and covariance matrices, we can write its Once we have mean and covariance matrices, we can write its
When x is real we write and its
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
+ H. Wymeersch, Iterative receiver design, Cambridge University Press, 2007*G. A. F. Seber and A. J. Lee, Linear Regression Analysis
(~ ,nRNx m
( )( ) ( )
(1 1
exp2
2 detn
p
π
= − − −
x x m
Φ
Kronecker) MIMO channel model
, better find the pdf for random vectors
A complex Gaussian random vector z+ is completely characterized by )) and covariance matrix [Φ=(E(z-m)(z-m)H)]
Once we have mean and covariance matrices, we can write its pdfOnce we have mean and covariance matrices, we can write its pdf
is real we write and its pdf* is given by
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
, Cambridge University Press, 2007.
Linear Regression Analysis, John Wiley & Sons, 2003
)~ ,x m Φ
) ( )1T − = − − −
x x m Φ x m
Separately correlated (Kronecker
For example
For n=1, mean=0 and variance =1
For a complex n-multivariate Gaussian distribution with mean For a complex n-multivariate Gaussian distribution with mean and covariance matrix is denoted by
• Note that the subscript c denote
• superscript n means that it is an n
• N means it is normal or Gaussian distribution
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
nnC
×∈Φ
Kronecker) MIMO channel model
multivariate Gaussian distribution with mean
( )2
1exp
22
xp x
π
= −
n∈mmultivariate Gaussian distribution with mean and covariance matrix is denoted by
otes that it is a complex distribution
superscript n means that it is an n-multivariate distribution and
N means it is normal or Gaussian distribution
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
nC∈m
( )~ ,n
CNz m Φ
Separately correlated (Kronecker
It is basically a complex z
• with independent imaginary andmatrix ½ Φ
( )1
Its pdf+ is given by
For example
For n=1, mean=0 and variance=1
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
+ A. van den Bos, “The Multivariate Complex Normal Distribution
IEEE Trans. Inform. Theory, vol. 41, no. 2, Mar. 1995, pp. 537
( )( ) (
1
detn
pπ
= − − −z z mΦ
Kronecker) MIMO channel model
and real parts with same covariance
( ) ( )( )1H −
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
, “The Multivariate Complex Normal Distribution- A Generalization,”
, vol. 41, no. 2, Mar. 1995, pp. 537-539.
)( ) ( )( )1exp
H −= − − −z z m Φ z mΦ
( ) ( )21expp z z
π= −
Separately correlated (Kronecker
For NR×NT MIMO wireless channel, when we
It gives a vector with NR×NT components, , its
( )( ) ( )
1
detT RN N
H
pπ
×= −h h R h
R
For example, NR=NT=2, iid case RH=I
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
11
11 12 21
21 22 1211 21 12 22
22
* * * *; ; H
h
h h h
h h h h h h h
h
= = =
H h h
Kronecker) MIMO channel model
MIMO wireless channel, when we vectorize it
components, , its pdf is( )~ 0,R TN N
C HN×
h R
( ) ( )( )1expH
H−= −h h R h
=I4
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
11 21 12 22
* * * *h h h h
Separately correlated (Kronecker
Kronecker MIMO channel model+ used for
( )( )
( ) ( )( ) (4 4
1 1exp exp
Hp h h h h
ππ= − = − − − −h h h
Kronecker MIMO channel model+ used for
• IEEE 802.11n and
• IEEE 802.20 (Mobile Broadband Wireless Access)
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Kronecker) MIMO channel model
used for
)2 2 2 2
11 21 12 22exp expp h h h h= − = − − − −
used for
IEEE 802.20 (Mobile Broadband Wireless Access)
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Based on the antenna array geomettypes
• Constant
• Circular• Circular
• Exponential
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
+ K.-L. Du and M. N. S. Swamy, Wireless Communications
2010.
Kronecker) MIMO channel model
metry, correlation could be of various
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Wireless Communications, Cambridge University Press,
Separately correlated (Kronecker
Constant correlation model is the worst case scenario
• suitable for antenna array of three antennas placed on an equilateral triangle or
• for closely spaced antennas other than linear arrays• for closely spaced antennas other than linear arrays
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
1
1
1tancons t
x x
x x
x x
= < <
R
L
L
M M M M
L
Kronecker) MIMO channel model
Constant correlation model is the worst case scenario
suitable for antenna array of three antennas placed on an
for closely spaced antennas other than linear arraysfor closely spaced antennas other than linear arrays
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
0 1,
x x
x x
x
= < <
M M M M
1 0 3 0 3 0 3
0 3 1 0 3 0 3
0 3 0 3 1 0 3
0 3 0 3 0 3 1
. . .
. . .
. . .
. . .X
T
=
R
Separately correlated (Kronecker
Circular correlation model is appropriate for
• antennas lying on a circle, or
• four antennas placed on a square
1 L xx
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
=−−
−
1
1
1
*2
*1
2*
1
11
L
MMMM
L
L
xx
xx
xx
nn
n
circularR
Kronecker) MIMO channel model
Circular correlation model is appropriate for
four antennas placed on a square
1 0 1 0 2 0 3. . .
. . .
. . .
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
2
1 0 3 1 0 1 0 2
0 2 0 3 1 0 1
0 1 0 2 0 3 1
. . .
. . .
. . .
. . .X
R
=
R
Separately correlated (Kronecker
Exponential correlation model is suitable model for
• equally spaced linear antenna array
−1 1
Ln
xx
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) ( )
=
−−
−
1
1
1
2*1*
2*
exp
L
MMMM
L
L
nn
n
onential
xx
xx
xx
R
Kronecker) MIMO channel model
Exponential correlation model is suitable model for
equally spaced linear antenna array
2 3
2
1 0 2 0 2 0 2
0 2 1 0 2 0 2
. . .
. . .
. . .
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
2
2
3 2
0 2 1 0 2 0 2
0 2 0 2 1 0 2
0 2 0 2 0 2 1
. . .
. . .
. . .
. . .X
R
=
R
Keyhole MIMO channel model
Such model is appropriate for indoor wireless communication through
• corridor or
• underpass or • underpass or
• subway
Cooperative communication
• employing the amplify-and-forward protocol
may be considered as keyhole channels
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Such model is appropriate for indoor wireless communication
forward protocol
may be considered as keyhole channels
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Keyhole MIMO channel model
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Fig. 13 3×3 keyhole MIMO channel, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
3 keyhole MIMO channel
Keyhole MIMO channel model
Let us assume that the transmitted signal vector is
= 2
1
x
x
x
x
The signal incident at the keyhole is given by
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
3x
[ ]1
1 2 3 2
3
left
x
y h h h x
x
= =
h x
Let us assume that the transmitted signal vector is
The signal incident at the keyhole is given by
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
1
1 2 3 2
3
x
y h h h x
x
Keyhole MIMO channel model
The signal at the other side of the keyhole is given by
The signal vector at the receive antennas on the right side of the
1y yα=
The signal vector at the receive antennas on the right side of the keyhole is given by
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
1right right lefty α= =r h h h x
The signal at the other side of the keyhole is given by
The signal vector at the receive antennas on the right side of the The signal vector at the receive antennas on the right side of the
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
right right leftα= =r h h h x
Keyhole MIMO channel model
The equivalent channel matrix can be represented as
4 1 4 2 4 3 1 4 1
5 1 5 2 5 3 2 5 1 2 3 2
6 1 6 2 6 3 3 6 3
h h h h h h x h x
h h h h h h x h h h h x
h h h h h h x h x
α α
= = =
r Hx
The rank of this channel matrix is onmultiplexing gain (multiplexing gain is equal to rank of channel
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
4 1 4 2 4 3
5 1 5 2 5 3
6 1 6 2 6 3
h h h h h h
h h h h h h
h h h h h h
α
=
H
The equivalent channel matrix can be represented as
[ ]4 1 4 2 4 3 1 4 1
5 1 5 2 5 3 2 5 1 2 3 2
6 1 6 2 6 3 3 6 3
h h h h h h x h x
h h h h h h x h h h h x
h h h h h h x h x
is one which implies that there is no multiplexing gain (multiplexing gain is equal to rank of channel H)
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Keyhole MIMO channel model
Let us do the analysis for NR×NT keyhole MIMO channel
Assume α=hleft is for the equivalent (1
1 2 Nα α α L
Assume β=hright is for the equivalent (
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
1 2 Nα α α L
keyhole MIMO channel
is for the equivalent (1×NT) MISO channel
β Nα α α
is for the equivalent (NR×1) SIMO channel
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
1
2
RN
β
β
β
M
TNα α α
Keyhole MIMO channel model
The channel matrix H for keyhole M
== T
T
N
N
Tβαβαβα
βαβαβα
MOMM
L
L
22221
11211
βαH
For example for 2×2 keyhole MIMO channel
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
==
TRR NNNN βαβαβα L
MOMM
21
βαH
1 1 2 1
1 2 2 2
α β α β
α β α β
=
H
le MIMO channels can be obtained as
2
1
2 keyhole MIMO channel
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
RN
Keyhole MIMO channel model
Note that α is a row vector with NT
NC(0,1)
Hence α row vector is distributed as Hence α row vector is distributed as
and similarly β is an equivalent SIMO channel (column vector with elements which are distributed as N
Hence β is distributed as
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
(RNCNβ ,0~
T elements which are distributed as
row vector is distributed as ( )T
NN
N Iα ,0~row vector is distributed as
is an equivalent SIMO channel (column vector with NR
elements which are distributed as NC(0,1))
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )T
T
NN
CN Iα ,0~
)RNI,
Keyhole MIMO channel model
Now we can calculate the Z as
where
( )2 2
*H
H T T T HZ UV= = = = =HH βα βα βα α β α β
where
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
22 2 2
1 2T
NU α α α= = + + +α L
22 2 2
1 2R
NV β β β= = + + +β L
2 2
Z UV= = = = =βα βα βα α β α β
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Keyhole MIMO channel model
What is the distribution of U and V?
• Distribution of
• is exponential (square of Rayleigh distribution)
2
1 2, , , ,i T
i Nα = L
• is exponential (square of Rayleigh distribution)
• U and V are the sums of NT and Nrespectively
• hence they are central Chi-square distributed with 2 Ndegrees of freedom
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
What is the distribution of U and V?
is exponential (square of Rayleigh distribution)
, , , ,i T
i N2
1 2, , , ,j R
j Nβ = L
is exponential (square of Rayleigh distribution)
and NR independent exponential RVs
square distributed with 2 NT and 2NR
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Keyhole MIMO channel model
Note that pdf of Chi-square RV
• which is the sum of squares of i.i.dcommon variance σ2
with 2N degrees of freedom is given bywith 2N degrees of freedom is given by
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )( )
21 21
; 01 !
Np e
N
χ
σχ χ χ χ
−−= >
−
+ A. Paulraj, R. Nabar and D. Gore, Introduction to Space
Communications, Cambridge University Press, 2003
i.i.d. zero mean Gaussian RVs with
degrees of freedom is given bydegrees of freedom is given by
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
; 0= >
Introduction to Space-Time Wireless
, Cambridge University Press, 2003.
Keyhole MIMO channel model
Hence, for our case σ2=1/2 , N=NT
have,( )
( )11
1 !TN
U
T
p u u e uN
−= >−
1
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )( )
11
1 !RN
V
R
p v v eN
−= >−
for RV U and N=NR for RV V, we
1; 0u
p u u e u− −= >
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
1; 0v
p v v e ν−= >
Keyhole MIMO channel model
let us consider the transformation of functions of two RVs
as Z=XY and W=Y,
then Jacobian for this transformation is
( )
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )
( )1 1
,
,
xy xyz z
x y x yz w w w y yJ y wx y x y x y
∂ ∂∂ ∂
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= = = = = ∂ ∂ ∂ ∂
let us consider the transformation of functions of two RVs X and Y
for this transformation is
( )
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )
( ) 0 1
xy xyy x
x y x yy yJ y w
x y
∂ ∂
∂ ∂ ∂ ∂∂ ∂= = = = =
∂ ∂
Keyhole MIMO channel model
The joint pdf of Z and W after transfX and Y
( ), ,
,
, ,
,,
,
X Y X Y
Z W
z zp w p w
w wp z w
z w
= =
For the marginal density w.r.t. z, we have,
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
1 1
,
, ,
,,
,
z wJ
x y
( )w
zp
wzp YXZ ∫
∞
∞−
=
1,
ansformation of functions of two RVs
, ,, ,
X Y X Y
z zp w p w
w w
w
Z=XY and W=Y
, we have,
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
, ,
w
dwww
,
Keyhole MIMO channel model
For our case, the variables are Y=U, X=V, Z=UV
The transformed variables are w=y=u
Hence, the pdf of Z when U and V are independent RVs is given
( ) dwww
zp
wzp YXZ ∫
∞
= ,
1,
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) ( )1
Z U V
zp z p u p du
u u
∞
−∞
=
∫+ H. Shin and J. H. Lee, “Effect of keyholes on the symbol error rate of space
codes,” IEEE Comm. Lett., vol. 7, pp. 27-29, Jan. 2003.
( ) dwww
pw
zp YXZ ∫∞−
= ,,
Y=U, X=V, Z=UV
w=y=u and z=xy=uv
are independent RVs is given
dw
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
zp z p u p du
u u
H. Shin and J. H. Lee, “Effect of keyholes on the symbol error rate of space-time block
29, Jan. 2003.
dw
Keyhole MIMO channel model
Putting the pdf of U and V, we have,
( )( )
1
0
1 1 1
1 ! 1 !TN u
Z
T R
p z u e e duu N N u
∞
− −=− −∫
Expressing the terms of z in terms of
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )
( ) ( )( )
0
1
0
1 1
R T
T R
N NT R
z e duN N u
∞
− +=
Γ Γ ∫
of U and V, we have,
( )
11 1 1
1 ! 1 !
R zNu u
T R
zp z u e e du
u N N u
−−
−
− −
Expressing the terms of z in terms of
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )
)1R
T R
zuN uz e du
− −−
z
Keyhole MIMO channel model
If we assume that
1 1∞
( )( ) ( )
( )zuNN
zpN
NN
RT
z
R
TR
2
0
1
11 −∞
+−∫ΓΓ=
zx
ut ==2
,
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )( ) ( ) ( )0
1 1
R TZ N N
T R
p z e dtN N t
∞
− +=
Γ Γ ∫
( )( ) ( )
2 1 1
2 2 2
R T T R TN N N N N
Z
T R
x xp z e dt
N N
− + − −
⇒ = Γ Γ
2 ( ) 22 2
1 1 R T T xN N Ntx
− + −− −
( )due u
zu
2
2 −−
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )2 2
41
1 1
2
R T T
R T
xN N Nt
tN N
xp z e dt
− + −− −
− +
( )
22 2
41
0
2 1 1
2 2 2
R T T R T
R T
xN N N N Nt
tN N
x xp z e dt
t
∞− + − −− −
− +
∫
Keyhole MIMO channel model
The nth order modified Hankel function is expressed as
MATLAB command is besselh(nu,Z)
( )2
1
0
1 1exp ; arg , Re 0
2 2 4 2
n
n n
x xK x t dt x x
tt
∞
+
= − − < >
∫
MATLAB command is besselh(nu,Z)
For our case, n=NR-NT and
Hence, the pdf of Z is
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )( ) ( )
(1
222 ; 0
T R
R T
N N
Z N N
T R
zp z K z z
N N
+−
−= ≥Γ Γ
x z=
function is expressed as
), where nu is order
( )2exp ; arg , Re 02 2 4 2
K x t dt x xπ
= − − < >
), where nu is order
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
)2 ; 0p z K z z= ≥
2x z=
MIMO channel parallel decomposition
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Fig. 14 Parallel decomposition of a MIMO channel using
shaping
MIMO channel parallel decomposition
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Fig. 14 Parallel decomposition of a MIMO channel using precoding and
MIMO channel parallel decomposition
What is precoding?
• In precoding, the input x to the ainto the input vector
; =x = Vx x V x% %
What is receiver shaping?
• In receiver shaping, we multiply the channel output
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
; =x = Vx x V x% %
UyUy == H~
MIMO channel parallel decomposition
he antennas is linearly transformed
H=x = Vx x V x
In receiver shaping, we multiply the channel output y by UH
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
=x = Vx x V x
( )nHxU +H
MIMO channel parallel decomposition
From Singular Value Decomposition (SVD) of the channel matrix have,
Hence,
HVUH ∑=
Hence,
Since U and V are unitary matrices (
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( VΣUUy =⇒ ~ H
nxΣy ~~~ +=⇒
MIMO channel parallel decomposition
From Singular Value Decomposition (SVD) of the channel matrix H, we
are unitary matrices (UHU=VHV=I)
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
)nxVV +~H
MIMO channel parallel decomposition
y
y
000
000~
~
2
1
2
1
σ
σ
It may be good to write the above matrices component
in order to interpret clearly
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
=
+
H
R
H
H R
N
R
R
y
y
y
y
0000
0000
000
000
000
~
~
~
~2
1
2
OMMMM
O
M
M
σ
σ
MIMO channel parallel decomposition
n
n
x
x~
~
~
~
00
00
2
1
2
1
It may be good to write the above matrices component-wise
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
+
++
R
H
H
T
H
H
N
R
R
N
R
R
n
n
n
n
x
x
x
x
~
~
~
~
~
~
~
~
00
00
00
00
00
1
2
1
2
M
M
M
M
MO
MIMO channel parallel decomposition
HHHH RRRR nxy
nxy
nxy
~~
~~~
~~~
~~~
2222
1111
+=
+=
+=
MOM
σ
σ
σ
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
RR
HH
NN
RR
ny
ny
~~
~~11
=
= ++
MOM
We can use RH parallel Gaussian channels, R
channel matrix
MIMO channel parallel decomposition
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
parallel Gaussian channels, RH is the rank of the
MIMO channel parallel decomposition
Hence in order to get U and V matrices,
• we need the channel state informshaping) and transmitter (for transmitter
• Such MIMO systems are called Closed loop MIMO system • Such MIMO systems are called Closed loop MIMO system
It may be noted that the product of vector does not modify the noise distribution
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO channel parallel decomposition
matrices,
formation at the receiver (for receiver shaping) and transmitter (for transmitter precoding)
Closed loop MIMO system Closed loop MIMO system
t of a unitary matrix with the noise vector does not modify the noise distribution
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO channel parallel decomposition
How to find the singular matrix of H
• find the eigenvalues λi of HHH
• take the square root of the eigenvalues
• Put those singular values in descending order in a diagonal matrix• Put those singular values in descending order in a diagonal matrixwhich gives singular matrix Σ=diag
How to find the U and V matrices?
• Columns of U are the eigenvectors of
• Columns of V are the eigenvectors of
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO channel parallel decomposition
H?
eigenvalues gives the singular values σiPut those singular values in descending order in a diagonal matrixPut those singular values in descending order in a diagonal matrix
diag(σi)
are the eigenvectors of HHH
are the eigenvectors of HHH
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO channel parallel decomposition
Find the SVD of a MIMO channel given as
MATLAB command “[V D]=eig(H)” will give the
++
++=
ii
ii
5443
3221H
MATLAB command “[V D]=eig(H)” will give the
• diagonal matrix D with eigenvalues
• V matrix whose columns are the eigenvectors.
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
−−
+=
8844.00283.05899.0
0482.04642.08070.0
i
iV D
MIMO channel parallel decomposition
the SVD of a MIMO channel given as
(H)” will give the (H)” will give the
eigenvalues and
matrix whose columns are the eigenvectors.
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
+
−−=
i
i
2631.73567.50
02631.03567.0
MIMO channel parallel decomposition
We can see that the
• eigenvalues and eigenvectors
of a complex H matrix may be also complex
For our case, eigenvalues of HHH can be obtained as For our case, eigenvalues of HHH can be obtained as
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
=
0.88710.4615-
0.0539i + 0.45830.1037i + 0.8811V
MIMO channel parallel decomposition
matrix may be also complex
can be obtained as can be obtained as
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
0.0539i
=
83.80910
00.1909D
MIMO channel parallel decomposition
Taking the square root and keeping in descending order of eigenvalues, we get,
=09.1547
Σ
We can also find the SVD decomposition directly
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
=0.43690
Σ
HVUH ∑=
MIMO channel parallel decomposition
Taking the square root and keeping in descending order of
We can also find the SVD decomposition directly
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO channel parallel decomposition
The SVD of H for our example is
=
9.1547
0.2469i - 0.3899-0.7160i - 0.5238-
0.5589i + 0.68890.4017i - 0.2271-H
where U and V are unitary matrices
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
0.2469i - 0.3899-0.7160i - 0.5238-
MIMO channel parallel decomposition
0.0298i - 0.59630.8022-
0.0401i + 0.8012-0.5971-
0.43690
09.1547
are unitary matrices
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
0.0298i - 0.59630.8022-0.43690
Power allocation in MIMO systems
SISO
• we allocate all the power to the single transmit antenna
MIMO
• We have numerous antennas at the transmitter • We have numerous antennas at the transmitter
• The fundamental question is howtransmit antennas
• Note that power allocation plays a significant role in deciding MIMO capacity (this will be discussed
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Power allocation in MIMO systems
we allocate all the power to the single transmit antenna
We have numerous antennas at the transmitter We have numerous antennas at the transmitter
how much power we allocate to each
Note that power allocation plays a significant role in deciding MIMO capacity (this will be discussed in later)
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Power allocation in MIMO systems
Open loop MIMO system:
• CSI is available at the receiver but not at the transmitter
• Uniform (equal) power allocation is employed
Closed loop MIMO system: Closed loop MIMO system:
• CSI is available at the transmitter as well as at the receiver
• we may allocate more power to better channels than the bad channels
• adaptive power allocation
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Power allocation in MIMO systems
CSI is available at the receiver but not at the transmitter
Uniform (equal) power allocation is employed
CSI is available at the transmitter as well as at the receiver
we may allocate more power to better channels than the bad
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Uniform power allocation
From parallel decomposition of MIMO channels, we have,
Using the Shannon capacity formula fo
Hiiii Rinxy ,,2,1;~~~L=+= σ
Using the Shannon capacity formula fochannel capacity for equal power allocation is
where W is the bandwidth of the channel, P is the total power, each antenna will receive P/NT power for equal power allocation
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
∑∑==
+=
+=
HH
i
R
i
R
i
rW
PWC
1
2
122 1log1log
σ
From parallel decomposition of MIMO channels, we have,
la for parallel Gaussian channels, the la for parallel Gaussian channels, the channel capacity for equal power allocation is
where W is the bandwidth of the channel, P is the total power, each power for equal power allocation
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
∏=
+=
HR
i T
i
T
i
N
PW
N
P
1222
1logσ
λ
σ
λ
Uniform power allocation
Since λi are eigenvalues of Q matrix
where λ are the eigenvectors for Q
Hiii Ri ,,2,1; L== xQx λ
where λi are the eigenvectors for Q
eigenvalues (rank of Q matrix is RH
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
i
T
i
T N
P
N
P22
=
xxQ λ
σσ
matrix
Q and Q has R non-zero
≥
<=
TRH
TRH
NN
NN
,
,
HH
HHQ
Q and Q has RH non-zero
H)
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Hi Ri ,,2,1; L=x
Uniform power allocation
Since the identity matrix has all its
T
i
T
RN
P
N
PH
12
+=
+ xQI
σσ
We also know that determinant of aof its eigenvalues
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
TT NN σσ
+=
+∏
=12
det1σ
λ
T
R
R
i T
i
N
P
N
P
H
H
I
Since the identity matrix has all its eigenvalues equal as 1
Hii Ri ,,2,1;2
L=
xλ
σ
of a matrix equals the multiplication
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
σ
2σT
PQ
Uniform power allocation
Therefore the capacity formula+
+= detlogWC I
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
+= 2 detlog RWC
HI
+ B. Vucetic and J. Yuan, Space-time coding, John Wiley and Sons, 2003.
+PQ
2 21
1logH
R
i
i T
PC W
N
λ
σ=
= +
∏
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
+
2σTN
PQ
, John Wiley and Sons, 2003.
Adaptive power allocation
Usually the channel state information is available at the receiver (CSIR) using pilot signals
• If the receiver sends the CSI to thchannel, channel,
• then, the channel state information is also available at the transmitter (CSIT)
we may distribute power adaptivelyboost the spectral efficiency
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Usually the channel state information is available at the receiver
to the transmitter through a feedback
then, the channel state information is also available at the
ively to individual transmit antenna to
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Adaptive power allocation
the channel capacity may be expressed as
∑=
+=
HR
i
ii PWC
122 1log
σ
λ
where Pi is the transmit power at the
We need to maximize C by choosing P
Water-filling algorithm can be utilizethe ensuing power constraint
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
= i 1 σ
∑=
TN
i 1
the channel capacity may be expressed as
is the transmit power at the ith transmit antenna
We need to maximize C by choosing Pi properly
tilized in obtaining the capacity under
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
=i PP
Adaptive power allocation
Hence the capacity can be written as
Using the method of Lagrange multipliers
2 22 21 1
1 1log log ;H H
R R
i i i i i
i i
P P P PC W W
P
λ γ λ
σ σ= =
= + = + =
∑ ∑
Using the method of Lagrange multipliersor objective function as
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
log log ;
+
+=∑
=
HR
i
iiP
P
PF
1
2 1log ζγ
+ G. B. Arfken and H. J. Weber, Mathematical methods for physicists
2005.
Hence the capacity can be written as
Using the method of Lagrange multipliers+, let us introduce the cost
2 22 21 1
1 1log log ;H H
R R
i i i i i
ii i
P P P PC W W
P
λ γ λγ
σ σ= =
= + = + =
∑ ∑
Using the method of Lagrange multipliers+, let us introduce the cost
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
log log ;
−∑
=
HR
i
iP
1
Mathematical methods for physicists, Academic Press,
Adaptive power allocation
where is the Lagrange multiplier
The unknown transmit power Pi are determined
• by setting the partial derivative oto zero
ζ
to zero
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
0=idP
dF
1log 2
+
⇒i
ii
dP
P
Pd
γ
where is the Lagrange multiplier
are determined
ive of the cost or objective function F
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
0=
−
iPζ
Adaptive power allocation
Change the log2 to natural loge
( )
11
2
log
ln
i i
e i
Pd P
P
dP
γ + −
⇒
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )1
1
2ln
1=−
+
⇒ ζγ
γ P
P
P
i
ii
( ) 02ln1
=−
+
⇒ ζ
γi
i
PP
( )2
log
ln idP
0
i i
e i
Pd P
Pζ
+ −
=
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
0=
0
Adaptive power allocation
( )1 1
2ln
iP
P P γζ⇒ = −
(12ln
i
i
PP
ζ
γ
⇒ =
+
Since power allocated should be greater than or equal to zero ( Pwe have,
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )2lnP P γζ
−=⇒ i
P
P
γ
1
0
1 1
iγ
= −i
i
P
P
γγ
11
0
−=⇒
)2ln( )
1
2lni
i
PP
γ ζ⇒ + =
Since power allocated should be greater than or equal to zero ( Pi ≥ 0),
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
iγ
iP γγ 0
+
iγ
1
Adaptive power allocation
where the notation
The MIMO channel capacity may be rewritten as follows
[ ]
≤
>=
+
00
0,
k
kkk
The MIMO channel capacity may be rewritten as follows
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
∑∑==
=
+=
HH R
i
R
i
ii WP
PWC
1
2
1
2 1log1logγ
The MIMO channel capacity may be rewritten as follows
0
2
0:
logH
i
R
i
i
C Wγ γ
γ
γ
+
≥
=
∑
The MIMO channel capacity may be rewritten as follows
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
∑≥
++
=
−+
H
i
R
i
i
i
i W
0: 02
0
log11
1
γγγ
γ
γγγ
Adaptive power allocation
Find the spectral efficiency and optimal power distribution for the MIMO channel
++
++=
ii
ii
5443
3221H
assuming and BW=1 Hz
Solution:
The SVD of is given by
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
++
=ii 5443
H
dBP
52
==σ
γ
HVUH ∑=
Find the spectral efficiency and optimal power distribution for the
and BW=1 Hz
The SVD of is given by
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Adaptive power allocation
The singular values of the channel are
=
9.1547
0.2469i - 0.3899-0.7160i - 0.5238-
0.5589i + 0.68890.4017i - 0.2271-H
The singular values of the channel are
Hence,
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
1547.91 =λ 4369.02 =λ
iiii
Pλλ
σγλγ 1623.3
2=== γ
The singular values of the channel are
0.0298i - 0.59630.8022-
0.0401i + 0.8012-0.5971-
0.43690
09.1547
The singular values of the channel are
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
0276.2651 =γ 6037.02 =γ
Adaptive power allocation
Considering that power is distributepower constraint becomes
2.66021
12
111
22
∑∑ =+=⇒=
−
the second channel is not allocated any power
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
2.66021
12
111
101 0∑∑
==
=+=⇒=
−i ii i γγγγ
uted to the two parallel channels, the
2.6602
the second channel is not allocated any power
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
2.6602
751.002 =< γγ
Adaptive power allocation
Then the power constraint yields
1.00381
11
111
1010
=+=⇒=−γγγγ
The capacity is given by
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
99624.001 => γγ
0.99624
265.0276loglog 2
0
12
=
=
γ
γC
1.0038
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
zbits/sec/H8.05540.99624
265.0276=
Interpretation on log2 (1+SNR) curve
Interpretation on log2 (1+SNR) curve
Low SNR regions
( )2 2 21 2 7183 1 4427log logSNR SNR e+ ≈ ≈ ≈
High SNR regions
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )2 2 21 2 7183 1 4427log logSNR SNR e+ ≈ ≈ ≈
( ) SNRSNR 22 log1log ≈+
(1+SNR) curve
(1+SNR) curve
( )2 2 21 2 7183 1 4427log log . . ( )e SNR SNR+ ≈ ≈ ≈
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )2 2 21 2 7183 1 4427log log . . ( )e SNR SNR+ ≈ ≈ ≈
Near optimal power allocation
High SNR
• noise power level is much lower than the threshold
• it is advantageous to distribute equal power to all subwith the non-zero eigenvalues
How many non-zero eigenvalues?How many non-zero eigenvalues?
• rank decides this
condition number of the channel matrix also decides the performance
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( ) max
min
condσ
σ=H
noise power level is much lower than the threshold
it is advantageous to distribute equal power to all sub-channel
condition number of the channel matrix also decides the
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Near optimal power allocation
Capacity for RH parallel Gaussian channels
In adaptive power allocation
C W W= + = +
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
i
i
P
P
γγ
11
0
−=⇒i
Pγ
σ=
2
0
1i
i
P
P P
σ
γ λ⇒ = −
iP⇒ = −
0
;i i threshold i
PP N P N
γ λ⇒ + = = =
parallel Gaussian channels
2 22 21 1
1 1log logH H
R R
i i i
i i
i
P PC W W
λ
σ σ
λ
= =
= + = +
∑ ∑
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
2
iPλ
σ2
0 i
P σ
γ λ= −
2
;i i threshold i
i
P N P Nσ
γ λ+ = = =
log log
iλ
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Fig. 15 Waterfilling algorithm (adaptive power allocation)
most power into less noisy channels to make equal power +
noise in each channel
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
algorithm (adaptive power allocation) put
most power into less noisy channels to make equal power +
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Fig. 16 Waterfilling algorithm:
and recalculate
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
algorithm: If Pthreshold < N4 then set P4=0
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Fig. 17 Near optimal power allocation for high SNR (usually
signal power is much higher than effective noise power)
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Near optimal power allocation for high SNR (usually
signal power is much higher than effective noise power)
Near optimal power allocation
∑∑==
≈
+=
HH R
i
R
i
iiW
PWC
122 1log
σ
λ
Equal power allocation
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
∑=
=
≈⇒
HR
i
H
H
i WRR
PWC
122 loglog
σ
λ
Equal power allocation
∑=
H
ii P
122log
σ
λ
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
∑=
+
HR
i H
i
RW
P
1
222 loglogλ
σ
Near optimal power allocation
Low SNR
most noise power level is high and the threshold
it is advantageous to supply power to the strongest it is advantageous to supply power to the strongest exclusively
We need to fill water of the deepest vessel (communication)
rank and condition number does not influence the performance
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
and will be equal to or greater than
it is advantageous to supply power to the strongest eigenmodeit is advantageous to supply power to the strongest eigenmode
We need to fill water of the deepest vessel (opportunistic
rank and condition number does not influence the performance
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Fig. 17 Near optimal power allocation for low SNR
(N3,N4>Pthreshold, N1=Pthreshold)
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Near optimal power allocation for low SNR
)
Near optimal power allocation
2 22 21 1
log 1 logH HR R
i i i i
i i
P PC W W e
λ λ
σ σ= =
= + ≈
∑ ∑
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
( )max
22log
PC W e
λ
σ⇒ ≈
( )2 22 21 1
log 1 logH HR R
i i i i
i i
P PC W W e
λ λ
σ σ= =
∑ ∑
, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
FFFFuuuunnnnddddaaaammmmeeeennnnttttaaaallllssss ooooffff MMMMIIIIMMMMOOOO WWWWiiiirrrrPart IPart IPart IPart I
Thanks
Any suggestions.
Email: [email protected]
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
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Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017