Fundamentals of MIMO Wireless

Communications
Part I
Prof. Rakhesh Singh Kshetrimayum
Fundamentals of MIMO Wireless Communications
Part I
It covers
Chapter 1: Introduction to MIMO systems
Chapter 2: Classical and generalized fading distributions
Chapter 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

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

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 (combat
( multipath fading)
• Equal gain combining (EGC)
• co-phases
phases signals on each branch
• and then combines them with equal weight
• Selection Combining (SC)
• selects the signal branch with the highest signal-to-noise
signal ratio
(SNR)

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
• but complex to implement

Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Single-user MIMO

User 1

Single User: Point-to-point

Point MIMO communicat

Fig. 3 Multiple-input, Single-output

output (MISO) system (NT≥2 and NR =1)
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Multi-user MIMO
n cellular communication:
• Multiple users with
• Single antenna
• 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
Multi-user MIMO

Fig. 4 Multi-user
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

input multiple-output
multiple (MIMO) system
(NT transmitting antennas and NR 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 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,
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
• capacity increases linearly with the minimum {NT or NR}=m
• CMIMO= m CSISO

+J.
G. Andrews, A. Ghosh and R. Muhamed, Fundamentals of WIMAX,
WIMAX Prentice Hall,
2007.
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 3×33 MIMO system
• Rate gain =3
• Diversity gain= 9

Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
 Diversity gain in MIMO systems

Fig. 6 3×33 MIMO system1

1. E. Biglieri et. al, MIMO wireless communications,
communications Cambridge University Press, 2007

Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
 Multiplexing gain in MIMO systems

001 1 1
001

0 0

0 0

Fig. 7 3×33 MIMO system

Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
 Advantages of Diversity gain in MIMO systems

0 0 0 0

0 0

0 0

Fig. 8 3×33 MIMO system

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 3×3
3 MIMO system
• Different message bits are sent in para
parallel from the 3 transmitting 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
Diversity Multiplexing trade--off
• 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 forr divdiversity gain then less antennas
may be used rate gain, hence, a trade-off
trade
• Diversity multiplexing trade-offoff
• doptimal=(NR-r) (NT-r); 0 ≤ r ≤ min{N
min{ R, NT}
• Implies d increase, r decreases
+L.Zheng and D. N. Tse,, “Diversity and multiplexing: A fundamental trade-off
trade in multiple
antenna channels,” IEEE Trans. Information Theory,
Theory pp. 1073-96, May 2003.
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Diversity Multiplexing trade-off:
off: Case study I

Fig. 8 3×55 MIMO system (r = min{NR,NT}=3)

Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Diversity Multiplexing trade-off:
off: Case study II

Fig. 9 5×55 MIMO system (r=2, doptimal=9)

Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Diversity Multiplexing trade-off:
off: Case study III

Fig. 10 5×55 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
WiMAX
WiFi

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-coherent
non and coherent systems?

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Example 1.1

Assume that the multiplexing gain (r)

( and diversity gain (d) satisfy the
diversity-multiplexing trade-off
d opt = (N T − r )(N R − r )

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
Example 1.1
Given SNR =10 dB, R=16bps
r log 2 (SNR ) = R
r=4.8165
Therefore five antennas may be used
use for multiplexing and remaining
(7-2)
2) two antennas may used for diversity
The maximum diversity gain can be calculated as

d opt = (N T − r )(N R − r ) = (7 − 5)(7 − 5) = 4

Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
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?
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
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
scale fading (PL, shadowing) and small-scale
small
fading (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
Random SISO channel

SISO fading channel:

For a SISO channel, the I-O
O relationship can be expressed as
y = hx + n
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
SISO fading channel
We will consider two performance metrics for random channel h
• Average or Ergodic capacity
• Outage probability

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

∫
2
C = E{W log 2 (1 + αSNR )} = W log 2 (1 + αSNR ) pα (α )dα ; α = h
0

Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Capacity of random SISO channel

The outage probability denoted as Pout is the

• probability that the channel capacity C drops below a certain
threshold information rate R
• the probability that the rate R is greater than the channel capacity
C(h)
 R
 2 W −1
Pout (R ) = Pr ob(C < R ) = Pr ob(W log 2 (1 + αSNR ) < R ) = Pr ob α <
 SNR


Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
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 expecta
ectation of the instantaneous capacity
over the probability density function (PDF) of
2
α= h

where h is the random channel gain coefficient

Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Capacity of random SISO channel

Outage probability can be obtained

ned from the cumulative distribution
function (CDF) of the random variable α
x
Pα (x ) = ∫
0
pα (α )dα

R
e W ln 2
−1
x=
SNR
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Capacity of random SISO channel

For example
Rayleigh fading

2
α = h is exponential i.e., it has PDF

1  α 
pα (α ) = exp − u (α )

α0  α0 
where α0 is the mean value of RV α and
u(α) is the unit step function
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

∞
 α 1 
C = ∫ W log 2 (1 + α ( SNR ) ) exp  −  dα
0
α0  α0 

The outage probability can be obtained from

R
x x e W ln 2 −1
− −
∫
α0
Pα (x ) = pα (α )dα = 1 − e Pout (R ) = 1 − e SNRα 0

−∞

Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Capacity of random SISO channel
• there cannot be any reliable trans
ransmission at any rate guaranteeing a
zero outage probability
• regardless of the value of the bandwidth (BW) and
• transmit power for a Rayleigh fading channel e −1
R
W ln 2
−
• From outage probability, Pout (R ) = 1 − e SNRα 0

• we may express data rate in terms of

• SNR and
R
• Outage probability as1 − SNRα ln(1 − P (R )) = e W ln 2
0 out
⇒ (W ln 2) ln(1 − SNRα 0 ln(1 − Pout (R ))) = R
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Capacity of random SISO channel

We may express spectral efficiency in bps per Hertz as

R
⇒ = log 2 (1 − SNRα 0 ln (1 − Pout (R )))
W
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
ly us
used wireless fading channel model
+ S. Barbarossa, Multiantenna Wireless Communication Systems,
Systems Artech House, 2003.
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
SER of various modulation schemes for SISO system over
various fading channels
One may employ different modulation schemes like
• BPSK, QPSK, QAM, PSK, etc att the transmitter side to convert the
bits to symbols to transmit it over the channel
Besides AWGN (additive term),
• we also have different wireless fading channel models
(multiplicative term)) like Rayleigh, Rician, Nakagami, etc for the
channel.

Rakhesh Singh Kshetrimayum,, 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?

Rakhesh Singh Kshetrimayum,, 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
ror aand divide by the total number of
bits 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
various fading channels
For single antenna case,
• total transmit power is P
Hence,

( h )E ( h )PT
2 2
s
SNR SISO = = =γ
σ2 σ2

Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
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
• error probability over AWGN channel (y=x+n)
(y=
b) Second, average it over the pdf of the received SNR to obtain
average 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
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,
BER for BPSK over Rayleigh fading channel
For single antenna case, conditional
onal error probability (CEP) for BPSK is
given by
(
Pb (E | γ ) = Q 2 SNR = Q 2γ ) ( )
+M. K. Simon and M.-S. Alouini, Digital communications over fading channels,
channels Wiley,
2005.
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
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)) can
ca be obtained by averaging over
the pdf of received SNR γ
∞

∫ (
Pb (E ) = E [Pb (E | γ )] = Q 2γ pγ (γ )dγ
0
)
where E is the expectation operator
π

1 
2
x 2 
Q Q (x ) =
π
0
∫
exp −
 2 sin 2 θ

dθ ; x ≥ 0


J. Craig, “A new, simple and exact result for calculating the probability of error for two-dimensional
two signal
constellations”, in Proc. IEEE MILCOM, pp. 25.5.1-25.5.5,
25.5.5, Boston, MA, 1991. (826 citations as of July 26, 2019)
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
SER of various modulation schemes for SISO system over
various fading channels
we can obtain the average BER as
π
∞ 2
1  2γ 
Pb (E ) =
0
∫ ∫
π
0
exp − 2
dθp γ (γ )dγ
 2 sin θ 

Integrating w.r.t.. γ first, we have, the average BER of BPSK for SISO
over any fading channel as
π π

2
1 
∞
 γ   2
 1 
dθ = 1
Pb (E ) = ∫∫
π 
00
exp −
 sin 2
θ


p γ (γ )dγ


π ∫
0
Μγ − 2
dθ
 sin θ 

Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
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 Rayleigh fading, the SNR (γ) is exponentially distributed
Hence, MGF of γ is given by M X (s ) = E [exp(sX )]  γ
∞

∞ ∞ exp sγ − 
1  γ 1  γ 1  γ 1
∫
Μ γ (s ) = exp(sγ ) exp − dγ =
0
γ  γ γ
0 
∫
exp sγ − dγ =
γ γ
s−
1
=
1 − sγ
γ
0
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
SER of various modulation schemes for SISO system over
various fading channels
where γ = E (γ )
Hence, the BER of BPSK for SISO cas
case over Rayleigh fading channel is
given by
Hence, the BER of BPSK for SISO cas
case over Rayleigh fading channel is
given by
π π
2 2
1 1 1 sin 2 θ
π ∫ π ∫ sin
Pb (E ) = dθ = 2
dθ
1 θ +γ
1+
0 γ 0
sin 2 θ

Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
SISO fading channels
Performance metrics for wireless communications
• Average capacity
• Expectation of instantaneous capacity over pdf of α
• Outage probability
• Can be obtained from CDF of α
SNR SISO =
2
h Es
=
2
h PT
=γ
( ) ( )
• BER/SER σ 2
σ 2
• Can be obtained from MGF of ϒ
• 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
Kulkarni, L. Choudhary, B. Kumbhani and R. S. Kshetrimayum,
Kshetrimayum Performance Analysis Comparison o
/MRC and OSTBC in Equicorrelated Rayleigh Fading MIMO Channels, IET Communications, vol. 8, I
2014, pp. 1850-1858.
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
3 important parameters for a random channel

For any random variable X

PDF: pX ( x ) x
CDF: The cdf of a RV X is defined as PX ( x ) = P [ X ≤ x ] = ∫ pU ( u ) du
−∞
MGF: The mgf of a RV X is defined as
∞

M X (s ) = E[exp(sX )] M X (s) = ∫ exp ( sx ) p

−∞
X ( x ) dx
From mgf we can obtain characteristic function (cf)
( by putting s=jω

+A.Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes,

Processes
Tata McGraw Hill, 2002.
Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Generalized fading distributions

These are three recent generalized fading distributions viz.,

k-μ,
α-μ and
η-μ fading distributions
In these fading distributions,
fading is generally considered as composed of many clusters of
multipaths or rays

+M.D. Yacoub, “The k-μ distribution and the η-μ distribution,” IEEE Antennas Propagat.
Mag., vol. 49, no. 1, pp. 68-81, Feb. 2007.
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)

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
scattered waves are random and
• have similar delay times with
delay-time
time spreads of different clusters being relatively large

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 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 do I find its magnitude?

Usually take square root of the
• real part (in-phase) and
• 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
k-μ fading distributions

In such a model,
where several clusters (say n) of many multipaths or rays are formed
the representation of envelope, X of the fading signal is

n
X = ∑  I i + pi 
2 2
2

i =1 
( ) ( + Qi + qi ) 

Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
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 in-phase and quadrature phase
component of the resultant signal of ith cluster
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. E I i( ) ( )

2
= E Qi
2
= σ 2

Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
k-μ fading distributions

pi and qi are the respective means of

• in-phase and quadrature components of
ith cluster in the received signal
The non zero mean of in-phase
phase and quadrature phase components
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
k-μ fading distributions

The pdf of k-μ distributed random variable (RV) αl is given by

µ (1+ k ) 2
µ +1 − αl
2µ (1 + k ) 2 α lµ e Ωl
 k (1 + k ) 
pα k − µ (α l ) = I µ −1  2µ αl 
µ −1 µ +1  Ω 
 l 
k 2 e µk Ω l 2

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
k-μ fading distributions
In the above equation, k > 0 and μ > 0 are the main parameters of the
distribution,
That’s why the name k-μ fading distributions
• k is the ratio of the total power
er d
due to dominant components to
the total power due to scattered waves and
• μ represents the number of clusters
By varying these two parameters, one can obtain various fading
channels
( )
Ω l = E α l2
Iυ(・) represents the υth order modified Bessel function of the first
kind (MATLAB function is besseli(nu,Z
nu,Z), where nu is order)
Rakhesh Singh Kshetrimayum,, 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 = 1 2 2
X= ( I + p ) + (Q + q )
Rice (μ = 1 and k = K),
−
(1+ K )α 2
l
2(1 + K )e −K
αl e Ωl  K (1 + K ) 
pα k = K ,µ =1 (α l ) = I0  2 α 
Ωl  Ω
l 
 l 
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
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 pi = qi = 0
n
X = ∑  I i 
2 2
2

i =1 
( ) ( ) + Qi 

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
k-μ fading distributions

Nakagami-m (k →0 and μ = m),

For small arguments,
µ −1 µ (1+ k )
− α l2
 y µ +1
 
µ −1
  2 µ (1 + k ) 2
µ
αl e Ωl
µ k (1 + k ) 
pα k − µ (α l ) = α
I µ −1 ( y ) ≅  
2 µ −1 µ +1  l 
 Ω 
Γ(µ ) k 2 e µk Ω l 2 Γ(µ )
l

m (1+ k ) 2 m 2
− αl − αl
Lim 2m m (1 + k )m α l2 m−1e Ωl
2m mα l2 m−1e Ωl
⇒ pα k →0 ,µ = m (α l ) = =
k →0 e mk Ω lm Γ(m ) Ω lm Γ(m )

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
k-μ fading distributions

Rayleigh fading distribution

For μ = 1, we consider that theree is n
no dominant component, i.e. p = q
= 0, it reduces to 2 2
X= ( I ) + (Q )
For m=μ=1, in the above pdf we have the Rayleigh distribution

α l2 α l2
− − 2
2α l e Ωl α l e 2σ
pα k →0,m=1 (α l ) = = ; Ω l = 2σ 2
Ωl σ2

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
k-μ fading distributions

For the k-µ distribution, the pdf of instantaneous SNR is

µ +1 µ −1 − µ (1+ k )x
µ (1 + k ) 2 x
2
e γ  k (1 + k )x 
pγ k − µ (x ) = 
I µ −1 2µ ; γ = E (γ )
µ −1 µ +1  γ 
2 e µk γ 2  
k

The mgf of instantaneous SNR for k -μ fading distribution is given by

µ − µ k (1+ k ) − µk
2
∞
 µ (1 + k ) 
( ) ∫
M γ k − µ (s ) = E e − sγ = e − sγ pγ k − µ (γ )dγ =   e µ (1+ k )+ sγ

0  µ (1 + k ) + sγ 

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
k-μ fading distributions
function x=kappa_mu_channel(kappa,mu,Nr,Nt
kappa,mu,Nr,Nt);
m = sqrt( kappa/((kappa+1))) ;
s = sqrt( 1/(2(kappa+1)) );
ni=0;
nq=0;
for j=1:2*mu
• if mod(j,2)==1
• norm1=m+s*randn(Nr,Nt);
+B. Kumbhani and R. S. Kshetrimayum, MIMO Wireless Communications over
Generalized Fading Channels, CRC Press, 2017

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
k-μ fading distributions

• ni=ni+norm1.ˆ2;
• else
• norm2=s*randn(Nr,Nt);
• 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
h_abs.*sin(theta);
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Weibull fading distributions

Weibull fading distribution (one cluster)

Weibull fading statistics is best fit
it fo
for mobile radio systems operating
in 800/900 MHz
The envelope of the fading signal R is obtained as αth root of the in-
phase and quadrature components

(
R = α X 2 +Y2 )

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
α-μ fading distributions

Weibull distribution for μ = 1 (one cluster)

α
r
αr α −1 − rˆ α α −1 − βr α 1
f R (r ) = e α
= αβ r e ;β = α
rr̂ˆ r̂rˆ

( )
rˆ = α E Rα ; Ω = E Rα = rˆα ( )
How about extending this classical
cal ffading distribution to generalized
fading distribution?

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
α-μ fading distributions

α-μ
μ distribution can be used to model fading channels in the
environment characterized by
• non-homogeneous
homogeneous obstacles that may be nonlinear in nature
Suppose that such a non-linearity
ty is expressed by a power parameter
α > 0 thereby the emerging envelope R is
n
Rα = ∑ (X
i =1
i
2
+ Yi 2 )
It is more general case as the usual
ual case
c of α =2 is a particular case of
this
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
α-μ fading distributions

The probability density function of the α-μ signal is obtained as

follows r α
µ αµ −1 − µ
αµ r
f R (r ) = αµ ˆ α
e r
r̂rˆ Γ(µ )
Special cases
Weibull fading distribution (one cluster)
(
R = α X 2 +Y2 )

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
α-μ fading distributions

Weibull distribution for μ = 1 (one cluster)

α
r
αr α −1 − rˆ α α −1 − βr α 1
f R (r ) = e α
= αβ r e ;β = α
rr̂ˆ r̂rˆ

( )
rˆ = α E Rα ; Ω = E Rα = rˆα ( )
Weibull fading statistics is best fit
it fo
for mobile radio systems operating
in 800/900 MHz

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
α-μ fading distributions

exponential fading distribution

considering α = 1 and μ = 1 in the expression of physical model
described by
R = X 2 +Y2

This represents exponential distribution

r
−
e rˆ 1
f R (r ) = = χ exp(− χr ); χ =
rˆ rˆ

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
α-μ fading distributions

Nakagami-m fading distribution

for α = 2 (note that n below is usually denoted by µ)

n
X = ∑  I i 
2 2
2

i =1 
( ) ( ) + Qi 
2
r r2
2µ µ r 2 µ −1 − µ rˆ 2 2µ r µ 2 µ −1 −µ
f R (r ) = e = e Ω ; Ω = rˆ 2
rˆ Γ(µ )
2µ
Ω µ Γ(µ )

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
α-μ fading distributions

Relation of α-μ
μ fading distribution and Nakagami-m distribution
n

( )
Rα = ∑ X i2 + Yi 2 = X Nakagami
i =1
2

In MATLAB, Gamma RV can be generated using the function gamrnd

shape parameter, k, and scale parameter, θ, as input arguments
k and θ are related to the Nakagami-m
Nakagami fading parameter m as
k=m and θ = γ respectively
m

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
α-μ fading distributions

MATLAB code to generate α-μ μ channel matrix

• n=gamrnd(mu,a_SNR/mu,Nr,Nt
mu,Nr,Nt);
• a_inv=1/alpha;
• phi=2*pi*rand(Nr,Nt);
• H=(n.ˆa_inv).*exp(j*phi);

+B. Kumbhani and R. S. Kshetrimayum, MIMO Wireless Communications over

Generalized Fading Channels, CRC Press, 2017
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
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

m distribution with μ = 1 is like
Rayleigh distribution r2 r2
2r − rˆ 2 r − 2γ 2 2 rˆ 2
f R (r ) = 2 e = 2e ;γ =
rˆ γ 2
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
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 α-μ fading distribution
• where instead of taking square root of the in-phase
in and
quadrature component of the fading signal
• we have taken α-th root, extension of Weibull fading channel
One more extension is possible
• where we assume the variance of the in-phase
in and quadrature
components are different
• It is the third case of η-µ fading distribution,
distribution extension of
Nakagami-m fading channel
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
phase component to power of quadrature
phase component of the receive
eived signals in format I (assuming in-
phase and quadrature components are uncorrelated)
• Correlation coefficient of the in-phase
in and quadrature
components in format II (assuming in-phase
in and quadrature
components are correlated)

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
η-μ fading distributions

η-µ fading distribution is similar to Nakagami-m fading model,

• it is assumed that the multi-path
path components in received signal
are in the form of several clusters and
the clusters does not have any domi omina`ng or LOS component in η − μ
distribution
However, the parameter η makes it different from Nakagami-m fading
as mentioned before
n
 2 2
X = ∑ ( I i ) + ( Qi ) 
2 
i =1  

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
η-μ fading distributions

In η −μ fading, the varia`on from Nakagami-m

Nakagami fading is
that the variance which is same as the power content of Ii and Qi is
different
( )
E I i2 = σ I2 ; E Qi2 = σ Q2
i
( ) i

The probability density function (pdf

pdf) of η-μ distributed RV is given by
1 2 µh 2
µ+ − αl
µ µ Ω
4 πµ 2h α e 2 l
 2µH 2 
pαη − µ (α l ) = 1
l
1
I 1  α l 
µ− µ+ µ −  Ωl 
Γ(µ ) H 2 Ωl 2 2

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
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 fading channels
Format I
In this format, the in-phase
phase component and quadrature phase
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
η-μ fading distributions

η is defined as the ratio of power of in phase component to the

power of quadrature component, i.e. 2 2

η=
( )
E Ii
=
σ Ii

E (Q ) σ 2 2
i Qi

The parameter η ∈ (0, ∞) is also assassumed that this ratio is constant

for all the clusters in the received signal

h=
(1 + η )2
;H =
1 −η 2 H
; =
1 −η
4η 4η h 1+η

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
 1 2 µh 2
µ+ − αl
µ µ Ω
4 2
π µ 2 h αl e l  2µH 2 
η-μ fading distributions pαη − µ (α l ) = 1 1
I 1  α l 
µ− µ+ µ −  Ωl 
( )
Γ µ H 2 Ωl 2 2

ν +2 i
ν
Since Iν ( − z ) = ( −1) Iν ( z ) ∞
1z
Iν z =∑
()  
(
i =0 i ! Γ i + ν + 1  2  )
The pdf has Iʋ which is function of H and it is symmetric for H=0
Hence the distribution is symmetric around η=1 since for η=1, H=0,
• therefore power distribution may be considered only within one o
the regions
It can be shown that the values of h and H are symmetrical around η
= 1, i.e.
• the values of H and h for 0 < η ≤ 1 are same for the range 1 ≤ η < ∞
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
η-μ fading distributions

Format II
assumption made is that the in-phase
phase and quadrature phase
components of MPCs within each cluster are
• correlated and
• have same powers
The parameter η ∈ (-1, 1) is the corr
correlation coefficient between these
components
η=
(
E I i , Qi ) = E ( I ,Q )
i i

E Qi2( ) ( )
E I i2

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
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
1 η H
h= H= =η
1 −η 2 1 −η 2 h
It can be shown that the values of h and H are symmetrical around η
= 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
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 − η formatII
η formatI =
1 + η formatII
Special cases
η−μ distribution for format I, differs from Nakagami-m fading model
in only one parameter
the different variance of in-phase
phase components and quadrature phase
components of resultant of each cluster in the received signal
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
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-m
Nakagami fading distribution

η=
( )
E I i2
=
σ I2
i
=1
E (Q ) σ 2 2
i Qi

μ=0.5 and η=1

=1 gives Rayleigh distribution
2 2
X= ( I ) + (Q )
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
η-μ fading distributions

Hoyt fading distribution (η= q2 and μ = 0.5) also called as Nakagami-q

fading
2 2
X= ( I ) + ( qQ )
2
The pdf of the instantaneous signal-to-noise
signal ratio (SNR) γ = α l of the
η-µ distribution is
1 1 2 µxh
µ+ µ− −
2 πµ 2 hµ x 2 e γ  2 µHx 
pγ η − µ ( x ) = 1 1
I 1  ; γ = E (γ )
µ− µ+ µ−  γ 
Γ(µ )H 2γ 2 2

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
η-μ fading distributions

The mgf of instantaneous SNR for η -μ fading distribution+ is given by

∞ µ
 µ 2 
( ) ∫
M γη − µ (s ) = E e −sγ = e −sγ 
pγη − µ (γ )dγ = 
4 h 

0  (2(h − H )µ + sγ )(2(h + H )µ + sγ ) 
Special cases
Rayleigh fading (η = 1, µ = 0.5)
1
M γ η =1, µ =0.5 (s ) =
1 + sγ
+N. Ermolova,, “Moment generating functions of the generalized η-μ and k-μ distributions and
their applications to performance evaluations of communication systems,” IEEE Communications
Letters, vol. 12, no. 7, pp. 502-504, 2008.
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
η-μ fading distributions

 m
Nakagami-m η = 1, µ = 
 2

• For format 1, η
1, implies

h=
(1+η )2
= 1; H =
1 −η 2
=0
4η 4η

• For format 2, η
0 implies

1 η
h= 2
= 1; H = 2
=0
1 −η 1 −η
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
η-μ fading distributions

m
 m 
Mγ (s ) =  
 m + sγ 
m
η =1, µ =
2

(
Nakagami-q η = q 2 , µ = 0.5 )
0.5
 h 
M γη ,µ =0.5 (s ) =  
 ((h − H ) + sγ )((h + H ) + sγ ) 

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
η-μ fading distributions

For format 1,

h=
(1+η ) 2
=
(
1+ q )
2 2
;H =
1 −η 2 1 − q 4
= ;h − H =
q2 +1
;h + H =
1+ q2
4η 4q 2 4η 4q 2
2 2q 2
0.5
 2

 1 +(q 2
) 
 2

0
 4 q 2   (
1 + q 2
) 
Mγ ()
s =  = 
η = q2 ,µ =0.5   q2 + 1
 
 1 + q 2
+ sγ  + sγ

 


1(+ q 2
+ 2 sγ )(
1 + q 2
+ 2 q 2
sγ ) 

 2
 2  2q 
 

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
η-μ fading distributions

It is more suitable for NLOS signal propagation

Hoyt fading (or Nakagami-q)q) is best fit for satellite links subject to
strong atmospheric scintillation

1+ q2
∴Mγ
η = q 2 , µ = 0.5
(s ) = 0.5
(
 1+ q2

) (1 + 2sγ ) + 4q s γ
2 2 2 2


Scintillation effects are because of arbitrary refraction caused by

small-scale fluctuations in air density due to temperature
gradients
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
η-μ fading distributions

function x=eta_mu_channel(eta,mu,Nr,Nt
eta,mu,Nr,Nt);
coeff=sqrt(eta);
ni=0;
 m
nq=0; Nakagami η = 1
1,, µ = 
 2
for j=1:2*mu
• norm1=randn(Nr,Nt);

+B. Kumbhani and R. S. Kshetrimayum, MIMO Wireless Communications over

Generalized Fading Channels, CRC Press, 2017
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
η-μ fading distributions

• ni=ni+norm1.ˆ2; η=
( )
E I i2
=
σ 2
Ii

E (Q ) σ 2 2

• norm2=coeff*randn(Nr,Nt); i Qi

• nq=nq+norm2.ˆ2;

end
h_abs=(sqrt(ni+nq))/sqrt((2*mu*(1+eta)));
((2*mu*(1+eta)));
theta = 2*pi*rand(Nr,Nt);
x=h_abs.*cos(theta)+sqrt(−1)*h_abs
h_abs.*sin(theta);

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
MIMO I-O system model

Fig. 11 2×22 MIMO system

Rakhesh Singh Kshetrimayum,, 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
y1=h21x1+h22x2+n2
In matrix form,  y   h h12   x1 
1 11
    
 y2  =  h21 h22   x2 
    
     

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
MIMO I-O system model
For NT ×NR MIMO systems, I-O
O model is given by
y=Hx+n
where
 y1   n1   h11 h12 K h1N   x1 
     T
  
 y2   n2   h21 h22 L h2 N   x2 
T
 M   M   M O N M  ;x =  M 
y=  ;n =   ; H =    
 yNR   nN R   hN R 1 hN 2 L hN N   xN 
       
R R T T

       
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO I-O system model

Notation:
• hij, i is the receiver antenna index
ndex and j is the transmitter antenna
index
• 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
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
• Uncorrelated keyhole MIMO channel model

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
id MIMO channel model

In MIMO communication, the channel H is a matrix

• whose elements are hij, i=1,2,..,N
=1,2,..,NR, j=1,2,…,NT complex RVs
A complex RV Z=X+jY,, a pair of real RVs X and Y
The pdf of a complex RV, the joint pdf of its real and complex parts
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 i.i.d. Gaussian with zero
mean and variance ½

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
id MIMO channel model
From probability theory:
Independent and non-identically
identically distributed (i.i.n.d.)
( RVs:
RVs X1,X2,…,XN are iid if for all x1,x2,…,x
,…, N
p X1 ,L, X N ( x1 , L , xN ) = p X1 ( x1 ) p X 2 ( x2 ) L p X N ( xN )

Independent and identically distributed (i.i.d.)

( RVs+:
RVs X1,X2,…,XN are iid if for all x1,x2,…,x
,…, N
p X1 ,L, X N ( x1 ,L, xN ) = p X ( x1 ) p X ( x2 ) L p X ( xN )
+A.Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes,
Processes
Tata McGraw Hill, 2002.
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
id MIMO channel model
For example, hij, i=1,2,..,NR, j=1,2,…,NT are complex normal RV
real imag real /imag  1
hij = hij
+ jh ij
;h ij
~ N  0, 
 2
2

−
( hijreal /imag )
1
1
( )
2
p hijreal /imag = e 2

1
2π
2
Rayleigh distributed
2 2
hij = (h ) + (h )
real
ij
imag
ij

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
id MIMO channel model
complex normal RV hij has pdf as

( )
hij ~ N C 0 ,1

2 2

−
( )
hijreal
−
( hijimag )
1 1
1 1
( ) ( ) ( )
2 2
p hij = p hijreal p hijimag = e 2
e 2

1 1
2π 2π
2 2

) 
2 2

1 ( ) (
− hijreal + hijimag
1 2

( )
p hij =
π
e 
=
π
e
− hij

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
id 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 of its elements+
• For iid MIMO channel model, all elements of the channel matrix H (hi
i=1,2,..,NR, j=1,2,…,NT) are iid complex normal RVs also called as iid
Rayleigh MIMO fading channel
• For example, a complex normal matrix H is a random matrix whose
elements are complex normal RVs
+T.W. Anderson, An introduction to multivariate statistical analysis,
analysis John Wiley & Sons,
3rd edition, 2003.
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
id MIMO channel model

For H an iid normal matrix, the pdf is the multiplication of pdfs of

complex normal RVs, hij, i=1,2,..,NR, j=1,2,…,NT
N R , NT
2
N R , NT
1 − hij
2
1
N R , NT
− hij
2
1 − ∑ hij

( ) ∏π
pH H = e =
π N R NT ∏ e =
πN R NT
e i , j =1

i , j =1 i , j =1

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
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
d MIMO channel model
trace HH H( )
 h h12 L h1NT   h11 h21 L hN R 1 
*
  11   
  h21 h22 L h2 NT   h12 h22 L hN R 2  
= trace     
 M O O M  M O O M  
 
  hN R 1 hN R 2 L hN R NT   h1NT h2 NT L hN R NT  
 
 2 2 2 
  11 + h12
h + L + h1NT L L L 
 2 2 2 
 L h21 + h22 + L + h2 NT L L 
= trace   
 M O O M 
 2 2 2
 L L L hN R 1 + hN R 2 + L + hN R NT 
 
N R , NT

∑
2
= hi , j
i , j =1
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
id MIMO channel model

Hence the pdf of the normal matrix H is

1
pH ( H ) =
π N R NT
(
exp −Trace HH H ( ))

Short hand notation of exponential (trace) =etri

=

1
pH ( H ) =
π N R NT (
etri −HH H )

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
id MIMO channel model

All the components of H, hij are assumed

assu independent (uncorrelated)
Assume that the path gains are identically distributed RV
From probability theory
The covariance and correlation of two RVs X and Y is defined as+

Cov ( X , Y ) = E ( X − µ X )(Y − µY )

Cor ( X , Y ) = E [ XY ] Cov ( X , Y ) = Cor ( X , Y ) − µ X µY
+R.D. Yates and D. J. Goodman, Probability and Stochastic Processes,
Processes John Wiley and
Sons, 2005.
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
id MIMO channel model

For zero mean RVs X and Y,

Cov ( X , Y ) = Cor ( X , Y )
The two RVs X and Y are
orthogonal if Cor ( X , Y ) = 0

uncorrelated if Cov ( X , Y ) = 0

Note that uncorrelated means covariance is zero

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
id MIMO channel model

If X and Y are independent, then Cor ( X , Y ) = µ X µY

which implies that Cov ( X , Y ) = Cor ( X , Y ) − µ X µY = 0

A Gaussian random vector X has independent components iff

covariance matrix is diagonal matrix
A normal Gaussian random vector X has independent components iff
covariance matrix is an identity matrix

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
id MIMO channel model

For iid MIMO channel matrix H,, the covariance matrix RH is a diagonal
matrix
Example,
uncorrelated (i.i.d.)
.) fading MIMO channel model for a 2×2 2 MIMO
system
Show that the covariance matrix is I4.
Let us consider a 2×2 MIMO system tem (for illustration purpose) whose
channel matrix is  h11 h12 
H=
h21 h22 

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
id MIMO channel model

{ }
Assume that E hij = 0; i, j = 1, 2

we can find the covariance matrix RH as R H = E hh H ; h = vect (H) { }

“vect” (vectorization) stacks all the ccolumns of a matrix into a column
vector
“Unvec” converts back a vectorized matrix into its corresponding
matrix

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
id MIMO channel model

We can “vect” (vectorize)) the above 2×2 H matrix for 2×2 MIMO
system and find the covariance matrix RH as follows

  h11  
  
H
  h21  *
R H = E[vect ( H ) {vect ( H )} ] = E
  h12  [ 11
h h21 h12 h22 ]

  
  h22  

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
id MIMO channel model

 Eh 2 E  h11h21
*
 E  h11h12* E  h11h22
*
 
  11  
 
E  h21 
2
  21 11 
 E  h21h12  E  h21h22  
* * *
E h h
 
R H = E[hh H ] =  
 E  h h*  E  h12 h21
*
 E  h12  E  h12 h22
2 *
 
  12 11    
 2 
  22 11 
E  h h *
E  h22 h21
*
 E  h22 h12  E h22  
 *
 
 

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
id MIMO channel model

Note that h11, h12, h21 and h22 are all mutually independent and
identically distributed (uncorrelated) RVs with zero mean

E  h 2  0 0 0 
  11  
 
E h21 
 2
H  0
 
0 0 
E[vect ( H ) {vect ( H )} ] =  
 0 0 E h12 
 2
0 
   
 2 
 0 0 0 E h22  

 

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
id MIMO channel model

E  h11  = E  h12  = E  h21  = E  h22  = 1

2 2 2 2
If we choose
       

1 0 0 0
then 0 1 0 0 
E[vect ( H ) {vect ( H )} ] = 
H

0 0 1 0
 
0 0 0 1

This analysis can be easily done for any arbitrary NT×NR MIMO system
and show that
R H = I NT NR

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
id MIMO channel model

An important matrix we will frequen

quently use in MIMO capacity analysis
is the random complex Wishart matrix which is defined as
HH H , N R < N T
Q= H
H H, N R ≥ N T
where H is the random MIMO channel matrix
From spectral theorem Q = U Λ U H
Q is a random matrix
Λ is also a random matrix

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
id MIMO channel model
λ1 0 L 0
 
0 λ2 L 0
 O M  ; m = min N , N
Λ=M M
 R T { }
0 0 L λm 
  n = max{N R , N T }
 
Marginal distribution of an eigenvalue (λ1) of complex Wishart
matrix+
m −1 i 2j
(− 1)l (2 j )!  2i − 2 j  2 j + 2n − 2m 
∑∑∑
1
p(λ1 ) = 
 i − j   
 (λ1 )l + n − m −λ
e 1

m i =0 j =0 l =0 2
2i − l
j! l! (n − m + j )!   2 j −l 
+E. Biglieri, Coding for Wireless Channels,, Springer, 2005.
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
id MIMO channel model

Fig. 12 NT×NR MIMO system

Rakhesh Singh Kshetrimayum,, 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
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+ can be defined as
{ }
R H = E hh H ; h = vect (H )
+N. Costa and S. Haykin, Multiple-input
input multiple-output
multiple channel models, John Wiley &
Sons, 2010.
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Fully correlated MIMO channel model
  h11  
  
 M  
  hN 1  
 R  
  h12  * 
* *
L h*N L h*N  
R H = E    h11 L hN R 1 h12 
 M 
R2 R NT

  hN 2  
 R  
 M  
h  
  N R NT  
  h h* L h11hN* R 1 *
h11h12 L h11h*N L h11h*N 
  11 11 R2 R NT 
 M O M M O M O M 
 
  hN 1h11 *
L hN R 1hN* R 1 *
hN R 1h12 L hN R 1h*N L hN R 1h*N 
 R R2 R NT 
  h h* L h12 hN* R 1 *
h12 h12 L h12 h*N L h12 h*N  
 12 11
= E  R2 R NT 
 M O M M O M O M 
  
  hN R 2 h11
*
L hN R 2 hN* R 1 *
hN R 2 h12 L hN R 2 h*N L hN R 2 h*N 
R2 R NT
 
 M O M M O M O M 
  * 
  hN R NT h11 L hN R NT hN* R 1 *
hN R NT h12 L hN R NT h*N L hN R NT h*N  
R2 R NT 
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Fully correlated MIMO channel model

mean vector m and covariance matrix Φ are unknown

How do you calculate them?
they can be estimated as
N N
1 1
ˆ ∑ (x i − m )(xi − m ) ; m = ∑x
H
Φ= ˆ ˆ ˆ i
N i =1 N i =1

This covariance matrix size and cons

consequently the number of elements
of 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
Fully correlated MIMO channel model
 
Let us take an example to find out this  
h h12 
Consider a 2×33 MIMO system as follows:  11 
H =  h21 h22 
Vectorize it  h11  h
  h32 
 31 
 h21   
h   
 31 
h =  h12 
h 
 22 
 h32 
 
 
 
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Fully correlated MIMO channel model
 h11 
Covariance matrix  
 h21  *
h   
 31   
(
E hh H ) =  h12   h11
h  
h21 h31 h12 h22 h32 

 22   
 h32   
 
 
 
How many components we need
d to calculate for finding covariance
matrix? 6×6

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Fully correlated MIMO channel model

For NT×NR MIMO system,

Obviously the above correlation matrix will have (NRNT)2 components
For example, you consider a slightly larger MIMO system
10×10
10 MIMO system, we will have 10,000 components
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
Separately correlated (Kronecker
Kronecker) MIMO channel model

Finite correlation exist between ante

antennas because of limited spacing
• But transmitter and Receiver are at very far distance
• We may assume receiver antenn
enna correlation is independent of
transmitter antenna correlation and vice versa
We can separate the transmitterr and receiver antenna correlation and
write
{ }
R H = E hh H = RTTX ⊗ R RX ; h = vect (H)

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model

Kronecker product:
Let A be an N×M matrix and B be an L×K
L matrix
The Kronecker product of A and B represented by A ⊗ B is an NL×MK
matrix
It can be obtained as

 A11B L A1M B 
A ⊗ B =  M O M 
 AN1B L ANM B

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model
 a11 a12   b11 b12 
Example:    
A =  a21 a22  b b
B =  21 22 
   
   

 a11b11 a11b12 a12 b11 a12 b12 

 
 a11B a12 B   a11b21 a11b22 a12 b21 a12 b22 
  
a
A ⊗ B =  21 B a B
22  =
a21b11 a21b12 a22 b11 a22 b12 
 
  a b a21b22 a22 b21 a22 b22 
   21 21 
 

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model

How to calculate receiver and transmitter correlation matrices?

The correlation matrices at the transmitter and the receiver are
calculated as
 H T
R TX ( ) ( )
= E  H H  = E H* HT ; R R X = E HH H
 
( )
Example:
Let us consider a 2×2
2 MIMO system for illustration purpose whose
channel matrix is  h11 h12 
H=
h21 h22 
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model

The transmitter correlation matrix (correlation

( between the columns
of H) is given by
T
 T   E  h11 2 + h21 2  E  h11
* *
+ h21 h22  
h12   h
*
R TX
h
= E[H H H ]T = E    11
h21   h11
  =     12 
 h h22   h21 
h22      2

   12   E  h12 h11 + h22 h21 
* * 2
 E  h12 + h22  
  
 E h 2 + h 2 E  h12
* *
h11 + h22 h21  
  11 21 
  
= 
 E  h11
* *
h12 + h21 h22   2
E  h12 + h22  
2
    

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model

The receiver correlation matrix (correlation

correlation between the rows of H) is
given by
 E h 2 + h 2 E  h11h21
* * 
h h12   h11 h21 
*
  11 12 
 
+ h12 h22

R RX = E[HH H ] = E   11  =
  h21 h22   h12 h22    E  h h* + h h*  E  h21 + h22  
2 2

 
  21 11 22 12   

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model

In total in order to findR H = RTTX ⊗ R RX

we require to find ( N R2 + NT2 ) elements only

For example, 10×10 10 MIMO system, we will have 200 components only
How to introduce correlation in an otherwise iid or spatially white
channel matrix Hw?
Define (
H = unvec R H h w )
What happens to covariance matrix
trix when we multiply R H to h w ?

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model
H
R H = E hh  = R H E h wh
H
( H
w ) RH
H
= R H I N R NT R H = R1/H2 R HH / 2 = R H
Hermitian square root for the square root of the matrix RH due to the
last operation
Correlation matrix of iid hw has identity matrix
But h now has covariance matrix as RH
We have introduced correlation matrix RH into h from hw

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model

We have introduced the correlation in an otherwise iid or spatially

white channel matrix Hw
It means that the correlation matrix for hw is I but the correlation
matrix for h becomes RH
What happens to channel matrix H? H
• For the correlation matrix RH above,
• let us find the expression for correlated channel matrix H

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model

in terms of
• receiver correlation,
• transmitter correlation and
• spatially white channel matrix
( )
H = unvec R H h w = unvec R TTx ⊗ R R X h w 
 

(
H = unvec R TT /2 ⊗ R1R/2 vect H w
x x
( ))
Using the identity ( A ⊗ B ) vect ( C ) = vect ( BCAT )

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model

Take A = RTT / 2 , B = R1/R 2 , C = H w

x x

{ (
H = unvec vect R1R/2 H w R1T/2
x x
)} = R1R/2 H w R1T/2
x x

channel matrix H for Kronecker channel+ model can be written as

H = R 1R/ X2 H w R 1T/X2

where ()1/2 denotes the Hermitian square root of matrix

+J. P. Kermoal,, L. Schumacher, K. I. Pederson, P. E. Mogensen and F. Frederiksen, “A
Stochastic MIMO Radio Channel Model With Experimental Validation,” IEEE Journal on
Selected Areas in Communications, vol. 20, no. 6, Aug. 2002, pp. 1211-26.
1211
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Separately correlated (Kronecker)) MIMO channel model

How do we obtain Kronecker or separately

sep correlated MIMO channel
model from iid channel model?
• In the iid MIMO channel model
• Premultiply by Hermitian square
are root of correlation at the receiver
• 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
Separately correlated (Kronecker
Kronecker) MIMO channel model

All matrices can be vectorized,, better find the pdf for random vectors
A complex Gaussian random vector z+ is completely characterized by
its mean (m=E(z)))) and covariance matrix [Φ=(E(z-m)(z-m)
[ H)]

Once we have mean and covariance matrices, we can write its pdf
When x is real we write x ~ N Rn ( m, Φ ) and its pdf* is given by
1  1 T 
p ( x) = exp  − ( x − m ) Φ −1 ( x − m ) 
( 2π )n det ( Φ )  2 

+H. Wymeersch, Iterative receiver design,, Cambridge University Press, 2007.

2007
*G. A. F. Seber and A. J. Lee, Linear Regression Analysis,
Analysis John Wiley & Sons, 2003

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model

For example
1  x2 
For n=1, mean=0 and variance =1 p ( x) = exp  −
 2 
2π  

multivariate Gaussian distribution with mean m ∈ C n

For a complex n-multivariate
and covariance matrix Φ ∈ C n×n is denoted by z ~ NCn ( m, Φ )
• Note that the subscript c denote
otes that it is a complex distribution
• superscript n means that it is an n-multivariate
n distribution and
• N means it is normal or Gaussian distribution

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model

It is basically a complex z
• with independent imaginary and real parts with same covariance
matrix ½ Φ

Its pdf+ is given by p (z) =

(π ) n
1
det ( Φ )
( H
exp − ( z − m ) Φ −1 ( z − m ) )
For example
For n=1, mean=0 and variance=1 p( z) =
1
π ( )
exp − z
2

+ A. van den Bos,, “The Multivariate Complex Normal Distribution-

Distribution A Generalization,”
IEEE Trans. Inform. Theory,, vol. 41, no. 2, Mar. 1995, pp. 537-539.
537

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model
For NR×NT MIMO wireless channel, when we vectorize it
It gives a vector with NR×NT components, h ~ NCN × N ( 0, R H ) , its pdf is R T

p (h ) =
(π )
1
NT × N R
det ( R H )
( H
exp − ( h ) R H −1 ( h ) )
For example, NR=NT=2, iid case RH=I4
 h11 
   
 h11 h12  h
 21 
   
H =  h21 h22  ;h =  h12  ;h H =  h * h21* h12* h22* 
   11 
   h22 
   
   
 
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model

p (h ) =
1
(π ) 4 (
exp − ( h )
H
(h )) =
1
π 4 ( 2 2
exp − h11 − h21 − h12 − h22
2 2
)
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
Separately correlated (Kronecker
Kronecker) MIMO channel model

Based on the antenna array geomet

metry, correlation could be of various
types
• Constant
• Circular
• Exponential

+K.-L. Du and M. N. S. Swamy, Wireless Communications,

Communications Cambridge University Press,
2010.

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model

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
1 x L x  1 0.3 0.3 0.3
   
x 1 L x 0.3 1 0.3 0.3
  
R cons tan t = M M M M  ,0 < x < 1 RT = 0.3 0.3 1 0.3
 X

x x L 1 0.3 0.3 0.3 1 

   
   

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model

Circular correlation model is appropriate for

• antennas lying on a circle, or
• four antennas placed on a square
 1 0.1 0.2 0.3
 1 x1 L x n −1  
x*   0.3 1 0.1 0.2
1 L x n−2  
R circular =  n −1 RR = 0.2 0.3 1 0.1
 M M M M  X

 *   0.1 0.2 0.3 1

 x1 x 2* L 1  


Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Separately correlated (Kronecker
Kronecker) MIMO channel model

Exponential correlation model is suitable model for

• equally spaced linear antenna array
 1 0.2 0.22 0.23 
 1 x L x n −1   2

 * n−2   0.2 1 0.2 0.2 
L x  0.22 0.2 
 x 1
R exp onential = RR = 0.2 1
 M M M M  X

 * n −1   0.23 0.22 0.2 1 
( )
 x (x )
* n−2
L 1   
 
 

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Keyhole MIMO channel model

Such model is appropriate for indoor wireless communication

through
• corridor or
• underpass or
• subway
Cooperative communication
• employing the amplify-and-forward
forward protocol
may be considered as keyhole channels

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Keyhole MIMO channel model

Fig. 13 3×33 keyhole MIMO channel

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Keyhole MIMO channel model

Let us assume that the transmitted signal vector is

 x1 
x =  x 2 
 x 3 

The signal incident at the keyhole is given by

 x1 
y = hleft x = [ h1 h2 h3 ]  x2 
 x3 

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Keyhole MIMO channel model

The signal at the other side of the keyhole is given by

y1 = α y

The signal vector at the receive antennas on the right side of the
keyhole is given by

r = h right y1 = α h right hleft x

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Keyhole MIMO channel model
 h4 h1 h4 h2 h4 h3   x1   h4   x1 
r = Hx = α  h5 h1 h5 h2 h5 h3   x2  = α  h5  [ h1 h2 h3 ]  x2 
 h6 h1 h6 h2 h6 h3   x3   h6   x3 
The equivalent channel matrix can be represented as
 
 
h h h4 h2 h4 h3 
 4 1 
H = α  h5 h1 h5 h2 h5 h3 
h h h6 h2 h6 h3 
 6 1 
 
 
The rank of this channel matrix is on
one which implies that there is no
multiplexing gain (multiplexing gain is equal to rank of channel H)
Rakhesh Singh Kshetrimayum,, 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×N
(1 T) MISO channel

α1 α 2 L α NT   β1 
   
( R×1) SIMO channel  β 2 
Assume β=hright is for the equivalent (N
 M 
 
 β NR 
 
 

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Keyhole MIMO channel model

The channel matrix H for keyhole

le MIMO
M channels can be obtained as
 α 1 β1 α 2 β1 L α NT β 1 
 
T  α1β 2 α2β2 L α NT β 2 
H = βα =  
M M O M
 
α 1 β N α 2 β NR L α NT β N R 
 R 

For example for 2×22 keyhole MIMO channel

 α1 β1 α 2 β1 
 
α β
H= 1 2 α β
2 2
 
 
 
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Keyhole MIMO channel model

Note that α is a row vector with NT elements which are distributed as

NC(0,1)
C (0, I NT )
NT
Hence α row vector is distributed as α ~ N
and similarly β is an equivalent SIMO channel (column vector with NR
elements which are distributed as NC(0,1))
Hence β is distributed as
(
β ~ N CN R 0, I N R )
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Keyhole MIMO channel model

Now we can calculate the Z as

H
( )
2 2
H T T T * H
Z = HH = βα βα = βα α β = α β = UV

where
2 2 2 2
U = α = α1 + α 2 + L + α N
T

2 2 2 2
V = β = β1 + β 2 + L + β N
R

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Keyhole MIMO channel model

What is the distribution of U and V?

• Distribution of 2 2
α i , i = 1, 2,L , NT β j , j = 1, 2,L , N R

• is exponential (square of Rayleigh distribution)

• U and V are the sums of NT and NR independent exponential RVs
respectively
• hence they are central Chi-square
square distributed with 2 NT and 2NR
degrees of freedom

Rakhesh Singh Kshetrimayum,, 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.d.
i.i.d zero mean Gaussian RVs with
common variance σ2
with 2N degrees of freedom is given by
χ
1 −
pχ ( χ ) = χ N −1e 2σ 2 ;χ > 0
( N − 1)!
+A. Paulraj, R. Nabar and D. Gore, Introduction to Space-Time
Space Wireless
Communications,, Cambridge University Press, 2003.
2003

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Keyhole MIMO channel model

Hence, for our case σ2=1/2 , N=NT for RV U and N=NR for RV V, we
have, 1
pU ( u ) = u NT −1e−u ; u > 0
( NT − 1)!
1
pV ( v ) = v N R −1e−v ;ν > 0
( N R − 1)!

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Keyhole MIMO channel model

let us consider the transformation of functions of two RVs X and Y

as Z=XY and W=Y,
then Jacobian for this transformation is

∂z ∂z ∂ xy ( ) ∂ xy( ) y x
 z , w  ∂x ∂y ∂x ∂y
J
 x ,y  = ∂w ∂w = ∂ y ( ) ∂ y ( ) = 0 1 = y=w
 1 1  ∂x ∂y ∂x ∂y

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Keyhole MIMO channel model

The joint pdf of Z and W after transf

ansformation of functions of two RVs
X and Y z  z 
p X ,Y  , w  p X ,Y  , w 
w  w  Z=XY and W=Y
( )
pZ ,W z , w =
 z,w 
=
w
J
 x , y 
 1 1
For the marginal density w.r.t. z,, we have,
∞
1 z 
p Z (z ) = ∫
−∞
w
p X ,Y  , w dw
w 

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Keyhole MIMO channel model
For our case, the variables are Y=U, X=V, Z=UV
The transformed variables are w=y=u and z=xy=uv
Hence, the pdf of Z when U and V are independent RVs is given
∞
1 z 
p Z (z ) = ∫
−∞
w
p X ,Y  , w dw
w 
∞
1 z
pZ ( z ) = ∫
−∞
u
pU ( u ) pV   du
u
+H. Shin and J. H. Lee, “Effect of keyholes on the symbol error rate of space-time
space block
codes,” IEEE Comm. Lett., vol. 7, pp. 27-29,
29, Jan. 2003.

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Keyhole MIMO channel model

Putting the pdf of U and V, we have,

∞ N R −1 z
1 1 1 z −
∫
NT −1 − u
pZ ( z ) = u e u e u du
u ( NT − 1) ! ( R ) 
N − 1 !
0
∞ z
1 1 − u −
∫
N R −1
= ( z) e u du
Γ ( NT ) Γ ( N R ) u N R − NT +1
0

Expressing the terms of z in terms of z

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Keyhole MIMO channel model
∞ ( z)
2

pz (z ) =
1 1
Γ(N T )Γ(N R ) ∫0 u N R − NT +1
( z) 2 N R −2
e
−u −
u
du

x
If we assume that t = u, = z
2 ∞ 2( N R − NT + NT ) − 2 x2
1 1 x −t −
pZ ( z ) =
Γ ( NT ) Γ ( N R ) ∫
0 (t )
N R − NT +1  
2
e 4t dt

N R − NT + 2 NT − 2 N R − NT ∞ x2
2 x 1 x 1 −t −
⇒ pZ ( z ) =
Γ ( NT ) Γ ( N R )  2  2  2  ∫ (t )
0
N R − NT +1
e 4t dt

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Keyhole MIMO channel model

The nth order modified Hankel function is expressed as

n ∞
1 x 1  x2  π
Kn ( x ) =  
22 ∫0 t n +1  4t 
exp −t − dt ; arg x <
2
, Re x 2
>0 ( )
MATLAB command is besselh(nu,Z)), where nu is order
For our case, n=NR-NT and x=2 z
Hence, the pdf of Z is
NT + N R
−1
2z 2
pZ ( z ) =
Γ ( NT ) Γ ( N R )
(
K N R − NT 2 z ; z ≥ 0 )
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO channel parallel decomposition

ig. 14 Parallel decomposition of a MIMO channel using precoding and

haping
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO channel parallel decomposition

What is precoding?
• In precoding, the input x to the
he aantennas is linearly transformed
into the input vector
H
x = Vx% ; x% = V x
What is receiver shaping?
• In receiver shaping, we multiply the channel output y by UH

~
y = U H y = U H (Hx + n )

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
MIMO channel parallel decomposition

From Singular Value Decomposition (SVD) of the channel matrix H, we

have, H
H=U ∑ V

Hence,
⇒~ (
y = U H U Σ V H V~
x +n )
( HU=VHV=I)
Since U and V are unitary matrices (U
⇒~
y = Σ~ ~
x+n

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
MIMO channel parallel decomposition
It may be good to write the above matrices component-wise
component
in order to interpret clearly

 ~ y1  σ 1 0 0 0 0
~
0  x1   n1 
~
 ~     ~   n~ 
 2   0 σ2
y 0 0 0 x
0  2   2 
 M  0 0 O 0 0 0  M   M 
 ~ =  ~  +  ~ 
y
 RH   0 0 0 σ RH 0 0   x R H   n RH 
~y RH +1   0 0 0 0 0
~ ~
0  x RH +1  n RH +1 
      
 ~M   M M M M O M  M   M 
~ ~
 y N R   0 0 0 0 0 0  x NT   n N R 

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
MIMO channel parallel decomposition
~
y1 = σ 1 ~
x1 + n~1
~
y2 = σ 2 ~x 2 + n~2
M O M
~y RH = σ RH ~
x RH + n~RH
~y = n~
RH +1 RH +1

M O M
~y N R = n~N R

We can use RH parallel Gaussian channels, RH is the rank of the

channel matrix
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
MIMO channel parallel decomposition

Hence in order to get U and V matrices,

• we need the channel state inform
formation at the receiver (for receiver
shaping) and transmitter (for transmitter precoding)
• Such MIMO systems are called Closed loop MIMO system
It may be noted that the productt of a unitary matrix with the noise
vector does not modify the noise distribution

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
MIMO channel parallel decomposition

How to find the singular matrix of H?

H
• find the eigenvalues λi of HHH
• take the square root of the eigenvalues gives the singular values σi
• Put those singular values in descending order in a diagonal matrix
which gives singular matrix Σ=diag
diag(σi)
How to find the U and V matrices?
• Columns of U are the eigenvectors of HHH
• Columns of V are the eigenvectors of HHH

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
MIMO channel parallel decomposition

Find the SVD of a MIMO channel given as

1 + 2i 2 + 3i 
H= 
3 + 4i 4 + 5i 
MATLAB command “[V D]=eig(H)”(H)” will give the
• diagonal matrix D with eigenvalues and
• V matrix whose columns are the eigenvectors.

 0.8070 0.4642 + 0.0482i  − 0.3567 − 0.2631i 0 

V=  D= 
− 0.5899 − 0.0283i 0.8844   0 5 . 3567 + 7 . 2631i 

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
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

0.8811 + 0.1037i 0.4583 + 0.0539i  0.1909 0 

V=  D= 
 - 0.4615 0.8871   0 83.8091

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
MIMO channel parallel decomposition

Taking the square root and keeping in descending order of

eigenvalues, we get,

9.1547 0 
Σ= 
 0 0.4369 

We can also find the SVD decomposition directly

H = U ∑ VH

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
MIMO channel parallel decomposition

The SVD of H for our example is

 - 0.2271 - 0.4017i 0.6889 + 0.5589i  9.1547 0   - 0.5971 - 0.8012 + 0.0401i 

H=  0  - 0.8022 0.5963 - 0.0298i 
 - 0.5238 - 0.7160i - 0.3899 - 0.2469i  0.4369  

where U and V are unitary matrices

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Power allocation in MIMO systems

SISO
• we allocate all the power to the single transmit antenna
MIMO
• We have numerous antennas at the transmitter
• The fundamental question is how much power we allocate to each
transmit antennas
• Note that power allocation plays a significant role in deciding
MIMO capacity (this will be discussed in later)

Rakhesh Singh Kshetrimayum,, 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:
• 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
Uniform power allocation

From parallel decomposition of MIMO channels, we have,

~
yi = σ i ~
x i + n~i ; i = 1,2, L , R H
Using the Shannon capacity formula
la fo
for parallel Gaussian channels, the
channel capacity for equal power allocation is

RH
 Pri  RH
 λ P  RH
 λ P 
C =W ∑
i =1
log 2 1 + 2
 σ

 =W


∑
i =1
log 2 1 + i
 N σ2 
 T
 =W log 2

∏
i =1
1 + i
 N σ2 
 T


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
Uniform power allocation

Since λi are eigenvalues of Q matrix

HH H , N R < N T
Qx i = λ i x i ; i = 1, 2, L , R H Q= H
H H, N R ≥ N T

where λi are the eigenvectors for Q and Q has RH non-zero

eigenvalues (rank of Q matrix is RH)

 P   P 
 Q x i =  λ x i ; i = 1,2, L , R H
N σ2  N σ2 i
 T   T 

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Uniform power allocation

Since the identity matrix has all its eigenvalues equal as 1

 P   P 
I R + Q  x i = 1 + λ x i ; i = 1,2, L , R H
 H
N σ 2   N σ2  i
 T   T 

We also know that determinant of a matrix equals the multiplication

of its eigenvalues
RH
   
1 + Pλ i  = det I R + PQ
P 
∏i =1
 N σ2 
 T 
 H N σ2
 T



Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Uniform power allocation

Therefore the capacity formula+  λi P 

RH

C = W log 2 ∏ 1 + 

i =1 
2 
NT σ 

  PQ 

C = W log 2 det I RH + 

  N σ 2 
T 

+ B. Vucetic and J. Yuan, Space-time coding,, John Wiley and Sons, 2003.

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Adaptive power allocation

Usually the channel state information is available at the receiver

(CSIR) using pilot signals
• If the receiver sends the CSI to the
th transmitter through a feedback
channel,
• then, the channel state information is also available at the
transmitter (CSIT)
we may distribute power adaptivelyively to individual transmit antenna to
boost the spectral efficiency

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Adaptive power allocation

the channel capacity may be expressed as

RH
 λi Pi 
C =W ∑
i =1
log 2 1 + 2
 σ



where Pi is the transmit power at the ith transmit antenna

We need to maximize C by choosing Pi properly
Water-filling algorithm can be utilize
tilized in obtaining the capacity under
the ensuing power constraint N T

∑P = P
i =1
i

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Adaptive power allocation

Hence the capacity can be written as

RH
 Pλi Pi  RH
 γ i Pi  Pλi
C = W ∑ log 2 1 +  = W ∑ log 2 1 + ;γ i = 2
 Pσ 2
P σ
i =1   i =1  
Using the method of Lagrange multipliers+, let us introduce the cost
or objective function as
RH
 γ i Pi   RH 
F= ∑ log 2 1 +
 P
 + ζ  P −
  ∑ Pi 

i =1  i =1 
+G. B. Arfken and H. J. Weber, Mathematical methods for physicists,
physicists Academic Press,
2005.
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Adaptive power allocation

where ζ is the Lagrange multiplier

The unknown transmit power Pi are determined
• by setting the partial derivative
ive o
of the cost or objective function F
to zero
dF
=0
dPi

  γ i Pi  
d log 2 1 +  − ζPi 
  P  
⇒ =0
dPi

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Adaptive power allocation

Change the log2 to natural loge

  γ i Pi  
d log e 1 +  − ζ Pi 
 P
1    
⇒ =0
ln 2() dPi
1 1 γi
⇒ −ζ = 0
ln (2 ) γ i Pi P
1+
P
1
⇒ − ζ ln (2 ) = 0
P
+ Pi
γi
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Adaptive power allocation
1 P 1
⇒
P
= ζ ln 2 () ⇒
γi
+ Pi =
ζ ln ( 2 )
+ Pi
γi
Pi 1 1 Pi 1 1
⇒ = − ⇒ = −
P ζ P ln ( 2 ) γ i P γ0 γi

Since power allocated should be greater than or equal to zero ( Pi ≥ 0)

we have,
+
Pi  1 1
⇒ = − 
P γ 0 γ i 

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Adaptive power allocation
+
 γi
RH

where the notation C = W ∑ log 2  
γ
i:γ i ≥γ 0  0 
k , k > 0
[k ]+
=
0 k ≤ 0
The MIMO channel capacity may be rewritten as follows
RH
 γ i Pi 
RH   
+ RH
 γi 
+
 
∑ ∑ ∑
1 1
C =W log 2 1 +  = W log 2 1 + γ i  −  = W log 2  

 
i =1  P  i =1  γ0 γi   i:γ i ≥γ 0 γ0 

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Adaptive power allocation

Find the spectral efficiency and optimal power distribution for the
MIMO channel
1 + 2i 2 + 3i 
H= 
3 + 4i 4 + 5i 

P
assuming γ = 2 = 5dB and BW=1 Hz
σ
Solution:
The SVD of H = U ∑ V H is given by

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Adaptive power allocation

 - 0.2271 - 0.4017i 0.6889 + 0.5589i  9.1547 0   - 0.5971 - 0.8012 + 0.0401i

H=  0  - 0.8022 0.5963 - 0.0298i 
 - 0.5238 - 0.7160i - 0.3899 - 0.2469i 0.4369 

The singular values of the channel are

λ1 = 9.1547 λ2 = 0.4369

Hence,
P γ 2 = 0.6037
γ i = γλi = λi = 3.1623λi γ 1 = 265 .0276
σ2
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Adaptive power allocation

Considering that power is distribute

uted to the two parallel channels, the
power constraint becomes
2 2
 1 1
∑ ∑γ
2 1
 −  = 1 ⇒ = 1+ =2.6602
γ
i =1  0
γi  γ0 i =1 i
γ 2 < γ 0 = 0.751

the second channel is not allocated any power

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Adaptive power allocation

Then the power constraint yields

1 1 1 1
− =1⇒ =1+ = 1.0038
γ0 γ1 γ0 γ1

γ 1 > γ 0 = 0.99624

The capacity is given by

 γ1   265.0276 
 
C = log 2   = log 2   = 8.0554 bits/sec/Hz
 γ0   0.99624 
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
nterpretation on log2 (1+SNR) curve

Interpretation on log2 (1+SNR) curve

Low SNR regions

( )
log 2 1 + SNR ≈ SNR log 2 e ≈ SNR log 2 2.7183 ≈ 1.4427( SNR ) ( )
High SNR regions
log 2 (1 + SNR ) ≈ log 2 SNR

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Near optimal power allocation

High SNR
• noise power level is much lower than the threshold
• it is advantageous to distribute equal power to all sub-channel
sub
with the non-zero eigenvalues
How many non-zero eigenvalues?
• rank decides this
condition number of the channel matrix also decides the
performance
σ max
cond ( H ) =
σ min

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Near optimal power allocation

Capacity for RH parallel Gaussian channels  

 
RH
 λi Pi  RH
 Pi 
C = W ∑ log 2 1 + 2  = W ∑ log 2 1 + 2
 σ   σ 
In adaptive power allocation i =1   i =1
 
 λ 
 i 
Pi 1 1 Pλi
⇒ = − γi = 2
P γ0 γi σ
Pi 1 σ 2 P σ2
⇒ = − ⇒ Pi = −
P γ 0 Pλi γ 0 λi
P σ2
⇒ Pi + Ni = = Pthreshold ; Ni =
γ0 λi
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Fig. 15 Waterfilling algorithm (adaptive power allocation) put
most power into less noisy channels to make equal power +
noise in each channel
Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless
Communications, Cambridge University Press, 2017
Fig. 16 Waterfilling algorithm: If Pthreshold < N4 then set P4=0
and recalculate

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)

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Near optimal power allocation

RH RH
 λ i Pi   λ i Pi 
C =W ∑
i =1
log 2 1 + 2  ≈ W
 σ 
∑i =1
log 2  2
σ


Equal power allocation
RH
 λi P  RH
 λi 
 = WR H log 2  P  + W
⇒ C ≈W ∑
i =1
log 2  2
σ R
 H

  σ 2

∑i =1
log 2 
 RH



Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Near optimal power allocation

Low SNR
most noise power level is high and will be equal to or greater than
the threshold
it is advantageous to supply power to the strongest eigenmode
exclusively
We need to fill water of the deepest vessel (opportunistic
(
communication)
rank and condition number does not influence the performance

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)

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Near optimal power allocation

RH
 λi Pi  RH
λi Pi
C = W ∑ log 2 1 + 2  ≈ W ∑ 2 log 2 ( e )
i =1  σ  i =1 σ

λmax P
⇒ C ≈W 2
log 2 ( e )
σ

Rakhesh Singh Kshetrimayum,, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017
Fundamentals of MIMO Wireless Communications
Part I
Thanks
Any suggestions.
Email: rakhesh@comsoc.org

Rakhesh Singh Kshetrimayum, Fundamentals of MIMO Wireless

Communications, Cambridge University Press, 2017