[PPT] - Fundamentals of MIMO W Wireless Communications Pa art II Prof. PowerPoint Presentation

SLIDE 1

Fundamentals of MIMO W Pa

Prof. Rakhesh Sing

Wireless Communications art II

Singh Kshetrimayum

SLIDE 2

Fundamentals of MIMO W Part II

It covers
Chapter 5: Channel capacity of simp
Chapter 6: MIMO channel capacity

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

Wireless Communications

implified MIMO channels ity

, Fundamentals of MIMO Wireless ridge University Press, 2017 2

SLIDE 3

Channel Capacity of Determi

SISO channel:
NT=NR=1, h=1
+

=

2 2 1

log σ P W C

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

where P is the signal power and
is the variance of the noise
SIMO channel:
Since SIMO channel is a vector ( RH
2

2

σ

inistic MIMO Channels

3 , Fundamentals of MIMO Wireless ridge University Press, 2017

H=1)

1 2

R

N

h h h

=
h

SLIDE 4

Channel Capacity of Determi

its SVD will have a single singular va
equals to the Frobenius norm of
There is only one transmitting anten
If the channel coefficients are equal

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

+

=

2 2 2 1

log h σ P W C

2 2 2 1 2

1

R

N

h h h = = = =

inistic MIMO Channels

r value

f the vector

ntenna and therefore SNR is P/ qual and normalized

, Fundamentals of MIMO Wireless ridge University Press, 2017 4

SLIDE 5

Channel Capacity of Determi

the capacity becomes

There is some power gain over SISO

2 2 2 2 2 1

log 1 log 1

R

N R i i

N P P C W h W σ σ

=

=

+ = +

There is some power gain over SISO
MISO:
there are NT transmit antennas and

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

[

T

N

h h h

2

1

= h

inistic MIMO Channels

ISO case

P

ISO case

and only one receive antenna (NR=1)

, Fundamentals of MIMO Wireless ridge University Press, 2017 5

]

T

SLIDE 6

Channel Capacity of Determi

for MIMO channel,

where

+

=

2 2 det

log σ

T R

N P W C

H

Q I

where
for MISO channel

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

≥

< =

T R H T R H

N N N N , , H H HH Q

=

= =

T

N j j H

h

1 2

hh Q

R

I

inistic MIMO Channels

2

, Fundamentals of MIMO Wireless ridge University Press, 2017 6

1 =

H

R

SLIDE 7

Channel Capacity of Determi

Hence, we get the capacity for MISO

allocation as

=
+

=

2 2

log 1 log

H

W P W C hh

If the channel coefficients are equal

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

=
+

=

2 2 2

log 1 log σ

T

W N W C

=

= = = 1

2 2 2 2 1

T

N

h h h

inistic MIMO Channels

ISO channel for equal power

+

2

1

N j

P h

T

qual and normalized

, Fundamentals of MIMO Wireless ridge University Press, 2017 7

+

= 2 1

1 σ

T j j

N h

SLIDE 8

Channel Capacity of Determi

the capacity becomes

The capacity doesn’t increase with t

+

=

2 2 1

log σ P W Cequal

The capacity doesn’t increase with t
This is the case when we allot the po

antennas

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

inistic MIMO Channels

ith the number of transmit antennas ith the number of transmit antennas e power equally for all transmitting

, Fundamentals of MIMO Wireless ridge University Press, 2017 8

SLIDE 9

Channel Capacity of Determi

If we assume CSI is available at the t
we can apply waterfilling algorith
Since the rank of the vector channel
there is only one nonzero eigenv
there is only one nonzero eigenv
is given by
So we get the capacity for equal and

as

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

=

+

=

2 2 2

log 1 log σ λ W P W C

ng waterfilli

inistic MIMO Channels

he transmitter

rithm

nnel matrix is one, envalue of hhH and

T

N 2

envalue of hhH and l and normalized channel coefficients

, Fundamentals of MIMO Wireless ridge University Press, 2017 9

=

=

T

j j

h

1 2

λ

+

=

+

= 2 2 2 1 2 2

1 log 1 σ σ P N W P h

T N j j

T

SLIDE 10

Channel Capacity of Determi

Here we see that there is a power ga
when the power is allotted using the
but no MUX gain
MIMO channel with unity channel m
High spatial interference
High spatial interference
Its SVD is

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

=
=

R R R

N N N 1 1 1 1 1 1 1 1 1 1 1 1

H

inistic MIMO Channels

er gain for MISO channel g the waterfilling algorithm el matrix :

, Fundamentals of MIMO Wireless ridge University Press, 2017 10

[ ]

T

T T T R

N N N N N 1 1 1

SLIDE 11

Channel Capacity of Determi

Since there is only one singular value
We can allot all the power P to single c

1 =

H

R

T R N

N =

1

λ

We can allot all the power P to single c
with non-zero eigenvalue yielding t
there is power gain from proper co
but no rate or MUX gain

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

+

=

2 2 1

log σ P N N W C

T R

inistic MIMO Channels

gle channel gle channel ing the channel capacity r combination of the received signals

, Fundamentals of MIMO Wireless ridge University Press, 2017 11

SLIDE 12

Channel Capacity of Determi

MIMO channel with identity channe
there is no spatial interference here
transmission occurs over parallel AW
Since singular values and eigenvalue
Channel gains for each path also eq
Thus for parallel Gaussian channels

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

inistic MIMO Channels

nnel matrix: ere and el AWGN channels each with SNR

H

R

I

alues of identity matrix equals 1

equal 1

nels,

, Fundamentals of MIMO Wireless ridge University Press, 2017 12

2

σ

H

R P

+

=

2 2 1

log σ

H H

R P W R C

SLIDE 13

Channel Capacity of Determi

asymptotic capacity

( )

2 2

1 1 log log

x

Lim e x x

+

=

→ ∞
1/19/2018

Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

R P Lim

P R H P R

H H

2 2

log 1 log

2 2

=

+

∞ → σ σ

σ

log log

inistic MIMO Channels

+

=

2 2 1

log σ

H H

R P W R C

2

1 log

H

R P

P P C W

σ

=

+

, Fundamentals of MIMO Wireless

ridge University Press, 2017 13

e

2

g

e W P C

2 2

log σ =

2 2 2

1 log

H

C W R σ σ = +

SLIDE 14

Capacity of random MIMO ch

MIMO channel are usually random
Hence we need to find the capacity
rather than the deterministic MIM
We will find the SIMO and MISO ran
We will find the SIMO and MISO ran
move on to random MIMO chann

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

channel

m

city of random MIMO channel MIMO channels random MIMO channel and random MIMO channel and hannels

, Fundamentals of MIMO Wireless ridge University Press, 2017 14

SLIDE 15

Basics of Information Theory

For an outcome with probability p,
the Shannon information content (S

( ) ( ) ( )

2 ln ln log2 p p − = −

The outcomes of tossing a coin are e

probability p=1/2

and their SIC equals 1 bit

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

2 ln

2

p, nt (SIC) is defined as are either head or tail with equal

, Fundamentals of MIMO Wireless ridge University Press, 2017 15

SLIDE 16

Basics of Information Theory

Is it your birthday?
There are two possible answers yes
SIC: 8.512
or no with probabilities 364/365 and
or no with probabilities 364/365 and
SIC: 0.004 bits
An important observation is that un

information

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

yes with probabilities 1/365 and and t unlikely outcomes give more

, Fundamentals of MIMO Wireless ridge University Press, 2017 16

SLIDE 17

Basics of Information Theory

Entropy: H(X) is defined as the avera

where E denotes the expectation op

( ) ( ) ( ) ( )

x p E X H

X 2

log − =

where E denotes the expectation op
If you throw a dice, the possible out

probabilities

is also known as probability

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

=

6 1 , 6 1 , 6 1 , 6 1 , 6 1 x p X

( )

x pX

verage SIC of a RV X n operator n operator

utcomes are X={1,2,3,4,5,6} with

ility mass function of X

, Fundamentals of MIMO Wireless ridge University Press, 2017 17

6

1 ,

SLIDE 18

Basics of Information Theory

If g(x) is a function defined on a disc

( ) ( ) ( )

=

X

x g x p x g E ) (

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

∈X

x X

( )

x E x E = + = = = 1 . 15 ; 5 . 3

2 2 2

µ σ µ

discrete RV X, we have,

, Fundamentals of MIMO Wireless ridge University Press, 2017 18

( ) ( ) ( ) ( )

X H x p E

X

= = − 58 . 2 log ; 17

2

SLIDE 19

Basics of Information Theory

For a Bernoulli RV,
the possible outcomes are X={0,1}

( ) {

x p X 1 =

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) {

x p X 1 =

( ) ( ) ( ) ( )

p x p E X H

X

− = − = l log2

with corresponding probabilities

{ }

p p, 1−

, Fundamentals of MIMO Wireless ridge University Press, 2017 19

{ }

p p, 1−

( ) ( ) ( )

p H p p p = − − − 1 log 1 log

2 2

SLIDE 20

Basics of Information Theory

H(p) is purely a concave function
It is maximum when p=1/2 (supr
and it is zero for p=1 or 0 (uncert
Entropy has a key role in information
Entropy has a key role in information
Differential entropy:

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( ) ( ) ( )

x f x f X h

X X 2

log − =

∞ ∞ −

supreme uncertainty) certainty is minimum) ation theory ation theory

, Fundamentals of MIMO Wireless ridge University Press, 2017 20

) ( ) ( ) ( )

x f E dx

X 2

log − =

SLIDE 21

Basics of Information Theory

Find the entropy of complex multiva
A zero mean multivariate complex G

covariance matrix R has the followin

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

1

1 exp

H N

ϕ π

−

= − = x x R x R

ltivariate Gaussian distribution ex Gaussian distribution with

wing pdf

, Fundamentals of MIMO Wireless ridge University Press, 2017 21

( )

1 1

exp

H

π

− −

= − R x R x

SLIDE 22

( ) ( )

( )

(

( )

( ) ( )

( )

2 2 1 2 2 1 2 ,

log log log ln log log ln

f f f H f f i j ij i j

h E e E e E e e E x x ϕ π π

− −

= − = − = + =

=

+ =

x

x R x R x R R

1/19/2018

Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( ) (

)

1 2 ,

log ln

ji ij i j

e π

−

=

+

R

R R

( ( ) ( ) ( ) (

( )

2 2 , 2

l log ln log log

jj i j

e e e π π

=
=

+ =

=
R

I R

( )

)

( )

( )( )

1 1 , 1 2 ,

ln ln log ln

H f i j ij i j f j i ij i j

e E x x e E x x π π π

− − −

− −

+
=

+

R

x R x R R R R

, Fundamentals of MIMO Wireless

ridge University Press, 2017 22

( )

( ) ( )

( ) (

)

1 2 , 2

log ln ln log ln

jj i j

e N e e π π π

−

+
+

=

R

RR R R

SLIDE 23

Basics of Information Theory

Mutual Information:
the decrement in the
uncertainty (entropy) of X

because of knowledge of Y

( ( (

E E Y X I l l ; − = − =

because of knowledge of Y

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ( (Y

H Y I X H E , = = =

−

=

) ( ) ( ) ( ) ( )) ( ) ( ) ( ) ( ) ( )) ( ) ( )

Y P Y X p E X p Y X p E X p Y X H X H Y , log log / log log |

2 2 2 2

−

− − − − =

, Fundamentals of MIMO Wireless ridge University Press, 2017 23

( ) ( ) ( ) ( ) ) ( ) ( ) ) ) ( )

X Y H Y X Y X H Y H X Y X p Y P X P Y P / , , log 2 − − +

SLIDE 24

Basics of Information Theory

Note that conditioning cut downs en

( ) ( ) ( ) ( ) ( )

Y H X Y H X Y H Y H Y X I ≤

−

= ≤ / / ;

In the above expression,
H(Y) is the differential entropy of
H(Y/X) is the conditional differen
The equality is possible for independ

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( )

ns entropy y of random variable (RV) Y and rential entropy pendent Y and X

, Fundamentals of MIMO Wireless ridge University Press, 2017 24

SLIDE 25

Basics of Information

Convex and Concave functions:
Definition: f(x) is strictly convex ove
(

) ( ) ( ) (1

1 − + < − + λ λ λ u f v u f

In other words, each chord in f(x) lie
For convex function f(x), -f(x) is a co

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( ) ( ) (1

1 − + < − + λ λ λ u f v u f

n Theory

ver (a,b) if

) ( ) ( )

1 , , < < ∈ ≠ ∀ λ λ b a v u v f lies above f(x) a concave

, Fundamentals of MIMO Wireless ridge University Press, 2017 25

) ( ) ( )

1 , , < < ∈ ≠ ∀ λ λ b a v u v f

SLIDE 26

Basics of Information Theory

x2, x4, ex and xlog(x) (x≥0) are strictly
log(x), √x are strictly concave functi
x is a concave and convex function

Jensen’s inequality:

Jensen’s inequality:
For an arbitrary convex function f(

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( ) ( (

X E f X f E ≥

rictly convex function nction

n

) and any random variable X,

, Fundamentals of MIMO Wireless ridge University Press, 2017 26

))

X

SLIDE 27

Basics of Information Theory

For strictly convex function f(x)
For example, f(x)=x2 is a strictly conv

Let us say the possible outcomes are

( ) ( ) ( ) ( )

X E f X f E >

Let us say the possible outcomes are

probabilities of p={1/2,1/2}

Then E(X)=0, f(E(X))=0, but, E(f(X))=1
Hence E(f(X))>f(E(X))

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

convex function s are X={-1,+1} with equal s are X={-1,+1} with equal ))=1

, Fundamentals of MIMO Wireless ridge University Press, 2017 27

SLIDE 28

Basics of Information Theory

Kullback-Leibler distance:
Relative differential entropy of two

( ) ( ) ( ) ( )

=
=

dx x g x f x f g f D

2

log ||

Information inequality:
Proof:
Define

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

x

g

( )

|| ≥ g f D

( ) { }

: > = x f x S

wo pdf f and g is expressed as

( ) ( ) ( ) ( )

− − X g E X h

f f 2

log

, Fundamentals of MIMO Wireless ridge University Press, 2017 28

SLIDE 29

Basics of Information Theory

In the above, we have used Jensen’s

concave function

( ) ( ) ( ) ( )

=

−

= −

E

dx x f x g x f g f D

2

log ||

concave function

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( ) ( ) ( )

log ||

2

=

≤

−

S

dx x f x g x f g f D

en’s inequality and log being a

( ) ( ) ( ) ( )

≤
X

f X g E X f X g E

f f 2 2

log log

, Fundamentals of MIMO Wireless ridge University Press, 2017 29

( ) ( )

1 log log

2 2

= = =

S

dx x g

SLIDE 30

Basics of Information Theory

Find the entropy maximizing distribu
Assume f(x) is a distribution over th
We have,

( ) ( ) ( ( (

A uniform distribution (
maximizes entropy over the interval

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( ) ( ( (

u E X h u f D

f f

− − = ≤

2

log ||

( ) ( )

a b X h f − ≤

2

log

( )

a b x u − = 1

tribution over the interval (a,b) r the interval (a,b)

( ))) ( ) ( )

) rval

, Fundamentals of MIMO Wireless ridge University Press, 2017 30

( ))) ( ) ( )

a b X h X

f

− + − =

2

log

SLIDE 31

Basics of Information Theory

For a given covariance matrix K, find

maximizing distribution over the inf

Answer:
A real multivariate Gaussian distribu
A real multivariate Gaussian distribu
Proof:

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

−

=

−

x K x

2 1

2 1 exp 2π φ

( ) ( )

X ||

f

E h u f D − − = ≤

find the zero mean entropy infinite interval tribution with the pdf

( )n

∞ ∞ − ,

tribution with the pdf

, Fundamentals of MIMO Wireless ridge University Press, 2017 31

− x

K x

1 T

( ) ( ) ( )

X φ

2

log

f

SLIDE 32

Basics of Information Theory ( ) (

(

( )

2 2

1 2 2 log log ln log ln

f f f

h E e E ϕ π

≤
= −

−

X

X K

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( )

(

( )

( ) (

)

2 2 2

2 1 2 2 1 2 2 1 2 2 log ln log ln log ln log ln

f f

e E e E e E π π π

=

+

=
=

+

K

K K

)))

1

1 2

T

π

−

−
X

K x K x

, Fundamentals of MIMO Wireless ridge University Press, 2017 32

) ) ( ) ( )

1 1 1

2

, , T f i j ij i j f i j ij i j

E x x E x x

− − −

+
+
x K x

K K

SLIDE 33

( ) (

)

( ) (

)

( ( ) (

) (

( ) (

)

(

2 2 2

1 2 2 1 2 2 1 2 2 1 2

, , , ,

log ln log ln log ln log ln log ln

i j i j i j

e E e e e π π π π

=

+

=

+

=

+

=

+

K

K K K K K I

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) (

)

( ( ) (

)

( ) (

)

2 2 2

2 2 1 2 2 1 2 2

, ,

log ln log ln log ln log ln

i j

e e n e e π π π = +

=

+

=
K

I K K

( )

2

1 2 2 log e h

ϕ

π

=
=

K X

( )( )

) (

) )

)

1 1 1 f j i ij ji ij jj

E x x

− − −

K

K K KK I

, Fundamentals of MIMO Wireless ridge University Press, 2017 33

) jj

I

SLIDE 34

Basics of Information Theory

Capacity of a parallel Gaussian chan
Let us consider n independent Gaus

the ith channel as

i i i

N X Y + =

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

where are zero mean Gaussian
with the power constraint

i i i

N X Y + =

i

N

E n

n i

=

− 1 1

hannel aussian channel with I-O relation for

, Fundamentals of MIMO Wireless ridge University Press, 2017 34

ssian i.i.d.

( )

P X E

i

≤

2

( )

2

, ~ σ N Ni

SLIDE 35

Basics of Information Theory

Let us analyze and find the capacity
Assumption: Xi and Ni are independ
We may express information capacit

( ) (

2 2 2

2 + + = N X N X E Y E

i i i i

We may express information capacit
Maximization of mutual information

power constraint

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( )

( )

P X E Y X I f C

i i i X i

≤

=

2

; max

city of the ith Gaussian channel first endent RV with zero mean pacity as

)

2

σ + = P Ni pacity as tion is w.r.t. pdf of Xi subject to the

, Fundamentals of MIMO Wireless ridge University Press, 2017 35

SLIDE 36

Basics of Information Theory

We know that optimal input (Xi) is G

(Yi) is also Gaussian distributed and distributed

Hence,

( ) ( ) ( ) ( ) (

i i i i i i i

X h Y h X Y h Y h Y X I − = − = | ;

(

)

( ) ( )

+ ≤

2

1 σ π

Hence,
In the above case, we have conside

there is a half factor in the capacity

If we consider complex RV, the capa

will get added up and the half facto

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

( ) ( )

+ ≤

2 2 2

log 2 1 ; σ π P e Y X I

i i

) is Gaussian distributed, hence output and the noise (Ni) is also Gaussian

) ( ) ( ) ( ) ( )

i i i i i i i i

N h Y h X N h Y h X N − = − = + | |

) ( )

+

= −

2

1 1 σ π P

sidered Xi, Yi and Ni are real RV, hence, city expression capacity for real and imaginary part actor becomes 1

, Fundamentals of MIMO Wireless ridge University Press, 2017 36

) ( )

+

= −

2 2 2 2

1 log 2 1 2 log 2 1 σ σ π P e

SLIDE 37

Basics of Information Theory

The I-O relation can be represented
y=x+n
Let us find the capacity for this case

( ) ( ) ( ) ( ) ( +

− = − =

where the real random vectors are

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( ) ( ) ( ) (x

y x y y y x h h h h I + − = − = | ;

=
=
=

n n n

N N N X X X Y Y Y

2

1 2 1 2 1

; ; n x y

ted in the vector form as ase

) ( ) ( ) ( ) ( )

− = − =

are

, Fundamentals of MIMO Wireless ridge University Press, 2017 37

) ( ) ( ) ( ) ( )

n y x n y x n h h h h − = − = | |

SLIDE 38

Basics of Information Theory

Note that mutual information is max

Gaussian with zero mean

Hence, for optimal input Xi, we have

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( ) (

=

− = ≤

n i i

N h Y h C I

1

;Y X

+ T. M. Cover and J. A. Thomas, Elements of Inform

maximum when Ni and Xi are i.i.d. have

, Fundamentals of MIMO Wireless ridge University Press, 2017 38

)

=

+

=

n i i i i

P N

1 2 2 1

log 2 1 σ

rmation Theory, Wiley, 2006.

SLIDE 39

Capacity of random SIMO ch

For a time slot m, the received signa

( ) ( )

) ( ) ( m m x m m n h y + =

Dropping the time index m, we can

system as

y=hx+n

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

( ) ~ 0,

R

N C

m N h I

+ S. Barbarossa, Multiantenna Wireless Communic

(0

R

N C

N

( ) ~

m n

hannel

ignal+ can be written as can rewrite the I-O relation of SIMO

, Fundamentals of MIMO Wireless ridge University Press, 2017 39

unication Systems, Artech House, 2003.

)

2

0,σ I

SLIDE 40

Capacity of random SIMO ch

where

1 1 1 2 2 2

; ;

R R R

N N N

y n h y n h y n h

=

= =

y

n h

The covariance of the received signa
Note that we are assuming the chan

instant of time

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

R R R

[

]

( )( )

[ ]

H H

E x x E E n h n h yy Ryy = + + = =

hannel

[

]

P x E =

2

ignal vector can be calculated as channel is deterministic at a particular

, Fundamentals of MIMO Wireless ridge University Press, 2017 40

(

) ( )

R

N H H H H

P E xx E I hh nn hh

2

σ + = +

SLIDE 41

Capacity of random SIMO ch

Note that we have assumed that x
Then, mutual information

( ) ( ) ( ) ( )

y y y y h x h h x I − = − = | ;

due to translation invariance of the

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( ) ( )

n h x y x h h = + = / x |

hannel

and n are independent ( )

n h −

the entropy (h) and independence

, Fundamentals of MIMO Wireless ridge University Press, 2017 41

( ) ( )

n n h x h = /

SLIDE 42

Capacity of random SIMO ch

Since jointly proper Gaussian random

differential entropy

Hence,
We next use the upper bound on th

the capacity of the channel as

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( )

2 2

log ; log h e h e π π = =

yy

y R n R

hannel

ndom vectors maximize the n the mutual information by rewriting

, Fundamentals of MIMO Wireless ridge University Press, 2017 42

( )

2 2 2 2

log log

R R

N N

e e π σ π σ = =

nn

R I

SLIDE 43

Capacity of random SIMO ch

( ) ( ) ( )

( (

( ) ( )

+

= ≤ − =

N H

P e e h h x I

R

I hh n y y

2 2

det log det log ; σ π π

For any two matrices M×N matrix A

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

=

N

e

R

2 2

log σ π

( ) ( )

BA I AB I + = +

N M

det det

hannel

))

( )

+

=

−

H N yy

P e e

R

hh I R

2 2

det log log σ π

A and N×M matrix B, we have,

, Fundamentals of MIMO Wireless ridge University Press, 2017 43

+

=

NR

hh I

2 2 det

log σ

SLIDE 44

Capacity of random S

Hence, for SIMO system, using the a

Therefore, instantaneous capacity

( )

+

≤

2 2 2

1 det log ; h y σ P x I

Therefore, instantaneous capacity
where is the square

norm of vector h

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

+

=

2 2 2 1

log h σ P W C

2 2 1

R

N H i i

h

=

= = h h h

SIMO channel

he above identity, we have, is is uare of Frobenius or Euclidean or L2-

, Fundamentals of MIMO Wireless ridge University Press, 2017 44

SLIDE 45

Capacity of random S

The average capacity of this channe

Assume iid Rayleigh fading SIMO ch

+

=

2 2 2 1

log h σ P E C

Assume iid Rayleigh fading SIMO ch
is a sum of the squ
hence it is Chi-square distributed wi

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

2 2 1

R

N H i i

h

=

= = h h h

SIMO channel

nnel is given by channel

channel

square of independent Gaussian RVs d with 2NR degrees of freedom

, Fundamentals of MIMO Wireless ridge University Press, 2017 45

SLIDE 46

Capacity of random S

Its pdf is

Therefore, the average capacity of t

( ) ( )

1

! 1 2 1

2

N R N

e x N x f

R R

− −

− =

h

Therefore, the average capacity of t

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

P N C

R NR

∞
+

− =

2 2 1

log ! 1 2 1 σ

SIMO channel

f this channel is given by

2 x −

f this channel is given by

, Fundamentals of MIMO Wireless ridge University Press, 2017 46

dx e x x P

x NR − −

2

1 2

SLIDE 47

Capacity of random S

Average capacity of random SIMO c

For high SNR case

+

=

2 2 2 1

log h σ

R R

N P N E C

For high SNR case

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

≈

R R

N P N E C

2 2 2

log h σ

2 2

log

R

N P C E σ

≈

+

SIMO channel

O channel

2

, Fundamentals of MIMO Wireless ridge University Press, 2017 47

+
=
R

R

N E P N E

2 2 2 2

log log h σ

2 2

log

R

N

h

SLIDE 48

Capacity of random S

Note that in high SNR region,
the average capacity of the i.i.d
equal to that of AWGN having an
with an additional term which reduc
with an additional term which reduc
The second tends to zero as
since the PDF of
approaches a Dirac delta function ce

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

→

R

N

R

N

2

h

SIMO channel

.i.d. Rayleigh channel is g an effective SNR of educes capacity

2

σ P N R

educes capacity n centered at 1

, Fundamentals of MIMO Wireless ridge University Press, 2017 48

σ

∞ →

SLIDE 49

Capacity of random S

Approximate Outage probability:
For a threshold or target rate of R (b
outage probability is given by

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

2 2 2

Pr Pr log 1 P

b R
b

σ

=

+

h

SIMO channel

(bits/s/Hz),

, Fundamentals of MIMO Wireless ridge University Press, 2017 49

2 2

2 1 Pr

R

b

P σ

−
<

= <

h

SLIDE 50

Capacity of random S

Hence the corresponding threshold
Let us compute the outage probabil

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) (

−

=

2

1 2

2 1

σ P R N

ut

R R N

R P

SIMO channel

ld on is

ability as follows

2

h

2

2 1 /

R

P σ −

, Fundamentals of MIMO Wireless ridge University Press, 2017 50

)

− −

−

2 1

! 1

x N R

R

dx e x

SLIDE 51

Capacity of random S

Substituting , we have,

2 x y =

( )

−

2

1 2 2 1

1

σ P N

R

For a high SNR, y tends to zero

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( )

−

= ! 1 1

N R

ut

R

y N R P

2 2 y <

SIMO channel

− −1 y

, Fundamentals of MIMO Wireless ridge University Press, 2017 51

− −1 ydy

e

2

2 1 σ P

R −

1 ≈

− y

e

SLIDE 52

Capacity of random S

( ) ( ) ( )

! 1 ! 1 1

1 2 2 1 1

2

P N

ut

N dx y N R P

R R

= − ≈

− −

σ
Hence there is diversity gain of NR

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( )

! ! 1

R R

N N −

SIMO channel

)

( )

1 2 2 1 !

1 2 2 1

2

N N R N P N

P y

R R R R R

−

=

−

σ

R with respect to (w.r.t.) SISO case.

, Fundamentals of MIMO Wireless ridge University Press, 2017 52

) ( )

! 2 !

2 R N N

N P

R R

σ

SLIDE 53

Capacity of random S

Exact Outage Probability:

2

Pr log Pr

R

H N T

b

R

N

γ

+

< =

I

hh

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

T

N

(

) (

2

r log 1 Pr 1 2

H H R

b

R

b

γ γ + < = + < h h h h

SIMO channel

( )

2 2

r log 1 ;

H

P

b

R γ γ σ + < = h h

, Fundamentals of MIMO Wireless ridge University Press, 2017 53

σ

)

( )

2 1 2 Pr 2 1 Pr

R R H R H

b
b

γ γ

−

= < − = <

h h

h h

SLIDE 54

Capacity of random S

Hence

( ) ( )

−

− =

2

1 2

! 1 2 1

σ P R N

ut

R R

x N R P

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( )

R R R

ut

N N R P Γ

−

= ∴ γ γ 1 2 ,

( )

− ! 1 2

R

N

SIMO channel

− − 2 1 x NR

dx e x

, Fundamentals of MIMO Wireless ridge University Press, 2017 54

1

SLIDE 55

Capacity of random S

where is the incomplete ga

( )

, z a γ

( )

1 1

,

z low a t a inc

a z t e dt a z γ

− − −

= =

and

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

,

2

1 2 , ,

R R

x a N z t γ − = = =

SIMO channel

e gamma function

( )

1 1

1 ; ;

a

z F a a z + −

, Fundamentals of MIMO Wireless ridge University Press, 2017 55

( )

; ;

SLIDE 56

Capacity of random S

Hypergeometric functions
Pochhammer symbol defined as

( ) ( ) (

a n a a a Γ + = = +

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( ) ( ) (

n

a a a a = = + Γ

( ) ( ) ( )

, , 1

1

a a a a = =

SIMO channel

) ( )

1 1 a a n + + −

, Fundamentals of MIMO Wireless

ridge University Press, 2017 56

) ( )

1 1 a a n + + −

)

( )

,

1

2 2

a a a a + = + =

SLIDE 57

Capacity of random S

The Hypergeometric function is defi

and single variable z

{ }

1 2

, , ,

p

a a a = a

{ 1

2

, , ,

q

b b b = b

and single variable z
Note that a is a vector of p elements
b is a vector of q elements
that’s why the Hypergeometric func

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

SIMO channel

defined for two complex vectors

}

, ,

q

b

ents and function is denoted as

, Fundamentals of MIMO Wireless ridge University Press, 2017 57

q p F

SLIDE 58

Capacity of random S

It is defined as

( ) ( ) ( ) ( )

; ;

1 2 1

b a a z F

k k q p

∞

= b a

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

1

b

k

=

SIMO channel

( )

( ) ( )

) ( )

( )

!

2 1 3

k z b b a a a

k q k k k p k p k k −

, Fundamentals of MIMO Wireless

ridge University Press, 2017 58

) ( )

( )

!

2

k b b

k q k k

SLIDE 59

Capacity of random S

For integer a

( ) ( )

1 1 , !

low inc

a z a e γ

=

− −

1/19/2018

Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

SIMO channel

1

! !

k a z k

z e k

− − =

−
, Fundamentals of MIMO Wireless

ridge University Press, 2017 59

SLIDE 60

Capacity of random M

For a time slot m, the received signa

( ) ( ) ( ) (

y m m m n = + h x

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( )

~ 0,

T

N C

m N h I

( ) ~

T

N C

m N x

( )

(

m E

2

x

MISO channel

ignal can be written as

( )

n m

, Fundamentals of MIMO Wireless

ridge University Press, 2017 60

0,

T

N T

P N

I

( )

2

~ 0,

C

n m N σ

)

P ≤

2

SLIDE 61

Capacity of random M

If we assume that channel is not kno
we can have equal power allocation
each transmitting antenna will send
The instantaneous capacity for unifo

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

2

log 1

uniform T

P C N σ

=

+

MISO channel

t known at the transmitter, tion and therefore, end signal with power of

T

N P

niform power allocation is given as

, Fundamentals of MIMO Wireless ridge University Press, 2017 61

T

N

2 2

P σ

h

SLIDE 62

Capacity of random M

The average capacity of this channe
+

=

2 2 1

log

h T

N P E C σ

the PDF of this RV is

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

T

( ) (

2 1

2

T N

N x f

T

− =

h

2 2 1

T

N j j

h

=

= h

MISO channel

nnel is given by

2

h

, Fundamentals of MIMO Wireless ridge University Press, 2017 62

)

2 1

! 1

x N

e x

T

− −

SLIDE 63

Capacity of random M

Therefore, the approximate average
for high SNR case is given by

P

One point to be noticed is that there

first term of the average capacity

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

2 2 2

log log P C E σ

≈

+

MISO channel

rage capacity of this channel

2

h

here is no power or array gain in the

, Fundamentals of MIMO Wireless ridge University Press, 2017 63

2

g

T

h N

SLIDE 64

Capacity of random M

Outage probability:

( )

<
+

=

2 2 2 1

log Pr σ R N P

b

R Pout h

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

2

σ NT

( )

2 2

2 1

R

ut

T

P R P N σ

−
=

<

h

MISO channel

( )

−

< =

2

1 2 Pr P N

b

R

R T

h

, Fundamentals of MIMO Wireless ridge University Press, 2017 64

2

σ P

SLIDE 65

Capacity of random M

Hence, at high SNR (similar analysis

( )

2 1

R

ut

P R −

≈
Hence there is diversity gain of w.r.t

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( )

2 !

T

ut

N T

P R N σ

≈
MISO channel

ysis with SIMO case above), we get,

)

1

T

N N

w.r.t. SISO case

, Fundamentals of MIMO Wireless ridge University Press, 2017 65 2

T

N T

P N σ

SLIDE 66

Capacity of i.i.d. Rayleigh fad

Average capacity
For equal power allocation:
+

=

H

R i P

W E C

2 1

log λ

where are the singular values o

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

+

=

=

i T

N W E C

1 2 1

log σ

i

λ

ading MIMO channels

P

es of the channel matrix

, Fundamentals of MIMO Wireless ridge University Press, 2017 66

2

σ

SLIDE 67

Capacity of i.i.d. Rayleigh fad

Alternatively we could also write the

fading channels in terms of determi

=

2 det

log

N

W E C

R

I

where Q is the Wishart matrix defin

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

R

ading MIMO channels

e the mean MIMO capacity for ergodic rminant of matrices as

+

2

σ N P

R

Q

efined as

, Fundamentals of MIMO Wireless ridge University Press, 2017 67

2

σ

T

N

R

≥

< =

T R H T R H

N N N N , , H H HH Q

SLIDE 68

Capacity of i.i.d. Rayleigh fad

Assume W=1 for brevity of the analy
=

2 det

log

N

E C

R

I

We will use a more convenient nota
Assuming the channel matrix is full r

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

ading MIMO channels

nalysis

+

2

σ

T

N PQ

tation of P/σ2 as γ

full rank, then,

, Fundamentals of MIMO Wireless ridge University Press, 2017 68

σ

T

N

{ }

R T N

N m , min =

SLIDE 69

Capacity of i.i.d. Rayleigh fad

It is equivalent to m times (m is the
finding the expectation of an arbitra

the Q matrix

( )

γ

m

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

+

=

=

λ γ

k k T

N E e C 1 ln log

1 2

ading MIMO channels

the rank of the full rank matrix H) bitrary and unordered eigenvalue of

( )

γ

, Fundamentals of MIMO Wireless ridge University Press, 2017 69

( )

+

=

λ

γ

T

N E e m 1 ln log 2

SLIDE 70

Capacity of i.i.d. Rayleigh fad

we have the marginal PDF of an uno

( ) ( ) ( ) ( )

2 1 2

1 2 ! 1 2 ! ! !

l j m i i l i j l

j p m j l n m j λ

− − = = =

− = − +

1/19/2018

Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

0 2

! ! !

i j l

m j l n m j

= = =

− +

{N

n max =

{N

m min =

ading MIMO channels

unordered is given by

( )

2 2 2 2 2 2 !

l n m

i j j n m e i j j l

λ

+ − −

− + −

−

−

, Fundamentals of MIMO Wireless

ridge University Press, 2017 70

2 ! i j j l − −

}

R T N

N ,

}

R T N

N ,

SLIDE 71

Capacity of i.i.d. Rayleigh fad

( ) ( ) ( ) ( ) ( ) ( )

2 2 1 2 2

log ln 1 1 2 ! 2 2 log

T l j m i i l

C m e p d N j i e i j γ λ λ λ

∞ − −

=

+

−

−

=
−

− +

1/19/2018

Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( )

2 2

log 2 ! ! !

i l i j l

e i j j l n m j

− = = =

=

−

− +

ading MIMO channels

( )

2 2 2 2 ln 1 2

l n m

j j n m e d j j l

λ

γ λ λ λ

∞ + − −

+ −

+
−
, Fundamentals of MIMO Wireless

ridge University Press, 2017 71

( )

ln 1 2

T

e d j j l N λ λ λ +

−

SLIDE 72

Capacity of i.i.d. Rayleigh fad

In order to calculate
The complementary incomplete gam

γ N I

T ∞

+

= 1 ln

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

,

x q k

q k e x

υ

∞ − − + −

Γ − + =

ading MIMO channels

gamma function is given by

( )

λ λ λ

λ d

e

m n l T − − +

, Fundamentals of MIMO Wireless

ridge University Press, 2017 72

1

;

k dx

q k

−

− + >

SLIDE 73

Capacity of i.i.d. Rayleigh fad

( )

λ λ λ γ

λ d

e N I

m n l T − − + ∞

+

= 1 ln γ

T

N x =

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( ) (

∞

− −

− = + =

x q q

q dx e x x I

1

1 1 ln ν

υ

λ =

+ M.-S. Alouini and A. J. Goldsmith, “Capacity of Ray

transmission and diversity-combining techniques,” 1999.

ading MIMO channels

γ υ λ γ

T T

N m n l q = − + = − ; 1 ; N dλ

, Fundamentals of MIMO Wireless ridge University Press, 2017 73

) ( )

=

+ − Γ

q k k

k q e

1

, ! 1 υ υ

υ

T

N d x x dx λ λ ν γ ν = =

=

Rayleigh fading channels under different adaptive s,” IEEE Trans. Veh. Technol., vol. 48, pp. 1165–1181, Jul

SLIDE 74

Capacity of i.i.d. Rayleigh fad

( ) ( )( ) (

2 1

ln 1

q q x

I x x e dx

υ

∞ − − −

∴ = + =

( )

2 q q

I I ν ν

−

=

1/19/2018

Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( ) (

2 1

1 ! ,

q k q k

q e q k q

υ

υ υ

− + − =

= − Γ − + =

ading MIMO channels

( ) ( ) ( )

2

, 1 !

q q k

q k q eυ υ υ υ

−

Γ − + = −

, Fundamentals of MIMO Wireless

ridge University Press, 2017 74

) ( )

1 1 1

1 ! 1,

k k q k q k

q e q k

υ

υ υ υ

= − − + − =

− Γ − + +

SLIDE 75

Capacity of i.i.d. Rayleigh fad

The exponential integral function of

( )

1

; 0,

y r r

E e y dy r

υ

υ υ

∞ − −

= >

The exponential integral function is

complementary incomplete gamma

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

1

( )

1 , ;1

x r

r e x dx r

υ

∞ − −

Γ − = − >

ading MIMO channels

n of order r could be expressed as

, 0,1, r =

n is a particular case of the

ma function

, Fundamentals of MIMO Wireless ridge University Press, 2017 75

> ( )

1

, ;

x q k

q k e x dx q k

υ

∞ − − + −

Γ − + = − + >

SLIDE 76

Capacity of i.i.d. Rayleigh fad

Substituting x= ν y, dx= ν dy, we hav

( ) ( )

1 1 1

1 ,

r y r

r e y dy

ν

υ ν ν ν

∞ ∞ − − − +

Γ − = =

Therefore,

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

1 1

( ) ( )

1

1 ,

r r

E r υ υ υ

−

= Γ −

r

E

( )

1 ! I q e = −

ading MIMO channels

have,

1 y r

e y dy

ν ∞ − −

(

)

1 , ;1

x r

r e x dx r

υ

∞ − −

Γ − = − >

, Fundamentals of MIMO Wireless

ridge University Press, 2017 76

1

( )

1

; 0, 0,1,

y r r

e y dy r

υ

υ υ

∞ − −

= > =

(

)

1 1

1 ! ,

q k q k

e q k

ν

ν ν

− − + − =

Γ − + +

SLIDE 77

Capacity of i.i.d. Rayleigh fad

If we assume that r-1=q-k-1, then 1
Hence,

( ) ( )

−

− = ∴

1

! 1

q r

E e q I υ

υ

Putting back,

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

=0

k

n l q k q r + = − − = 1 ;

( )

−

+ =

− + =

m n l k N

e m n l I

T

!

γ

ading MIMO channels

n 1-r=-q+k+1

, Fundamentals of MIMO Wireless ridge University Press, 2017 77

γ υ

T

N m n = − ;

−

+ − +

m

T k m n l

N E

1

γ

SLIDE 78

Capacity of i.i.d. Rayleigh fad

Note that k=0, l+n-m+1-k= l+n-m+1
and
k=l+n-m, l+n-m+1-k=1

Hence it similar to k going from 1 to

Hence it similar to k going from 1 to
Or k+1 going from 0 to l+n-m
Hence, it can be further expressed a

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

−

+ =

− + =

m n l k N

e m n l I

T

!

γ

ading MIMO channels

+1 1 to l+n-m+1 1 to l+n-m+1 ed as

, Fundamentals of MIMO Wireless ridge University Press, 2017 78

+
m

T k

N E

1

γ

SLIDE 79

Capacity of i.i.d. Rayleigh fad

Hence the average capacity of i.i.d. R

( ) ( ) ( ) ( )

2 2 1

log ln 1 1 2 ! 2 2

T l j m i

C m e p d N j i j γ λ λ λ

∞ −

=

+

−

−

1/19/2018

Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

+ H. Shin and J. H. Lee, “Capacity of multiple-antenna

scattering and Keyhole,” IEEE Trans. Information The

( ) ( ) ( ) ( )

2 1 2 2

1 2 ! 2 2 log 2 ! ! !

l j m i i l i j l

j i j e i j j l n m j

− − = = =

− −

=
−

− +

( )

( ) ( ) ( (

−

= = = −

−

− − =

1 2 2 2 ,

! ! 2 ! 2 1 log

m i i j j l l i l N N N

n l j n j e e C

T R T

γ

ading MIMO channels

. Rayleigh fading MIMO channel

2 2 2 j j n m γ

∞

+ −

, Fundamentals of MIMO Wireless

ridge University Press, 2017 79

nna fading channels: Spatial fading correlation, Double Theory, 2003, pp. 2636-2647.

( )

2 2 2 ln 1 2

l n m T

j j n m e d j j l N

λ

γ λ λ λ

∞ + − −

+ −

+
−
)

)

+

− = +

−

− +

−

− + + −

1

2 2 2 2 2 2 ! !

l m n k T k

N E l j m n j j i j i j m l m γ

SLIDE 80

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg , Fundamentals of MIMO Wireless ridge University Press, 2017 80

SLIDE 81

Capacity of i.i.d. Rayleigh fad

Fig. Average capacity vs SNR (dB) of

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

ading MIMO channels

) of open loop MIMO system

, Fundamentals of MIMO Wireless ridge University Press, 2017 81

SLIDE 82

Capacity of i.i.d. Rayleigh fad

Example
The average capacity for i.i.d. Raylei

calculated as

N
where
(exponential integral function of ord

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( ) ( ) ( (

−

= = = −

−

− − =

1 2 2 2 ,

! ! 2 ! 2 1 log

m i i j j l l i l N N N

n l j n j e e C

T R T

γ

( )

∞ − −

=

1

dy y e x E

k xy k

ading MIMO channels

yleigh fading MIMO channel can be

f order k), and

, Fundamentals of MIMO Wireless ridge University Press, 2017 82

) )

+

− = +

−

− +

−

− + + −

1

2 2 2 2 2 2 ! !

l m n k T k

N E l j m n j j i j i j m l m γ

SLIDE 83

Capacity of i.i.d. Rayleigh fad

find the average capacity of
(a) MIMO channel with

and receiver

(b) MISO channel with antenna

N N

T R

= = N

(b) MISO channel with antenna

at the receiver

(c) SIMO channel with one antenna

antennas at the receiver

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

T

N

ading MIMO channels

antennas at the transmitter nnas at the transmitter and 1 antenna

N =

nnas at the transmitter and 1 antenna nna at the transmitter and

, Fundamentals of MIMO Wireless ridge University Press, 2017 83

R

N

SLIDE 84

Capacity of i.i.d. Rayleigh fad

(a) Given that
and hence,

N N N

T R

= =

{ }

N N N m

R T

= = , min

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

{ }

N N N n

R T

= = , max

R T

n m =

ading MIMO channels

, Fundamentals of MIMO Wireless ridge University Press, 2017 84

SLIDE 85

Capacity of i.i.d. Rayleigh fad

( ) ( ) ( ( ) ( )

−

− = = = −

−

=

−

=

1 2 1 2 2 2 ,

1 log ! 2 2 1 log

N i j l N N i i j j l l i l N N N

e e C l j j e e C

T T

γ γ

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( ) ( ) ( ) ( ) ( )

−

= = = − − = = = − = = = −

−

=

−

=

=
1

2 2 2 , 1 2 2 2 , 2 2 ,

2 1 log 2 1 log 2 log

N i i j j l l i l N N N N i i j j l l i l N N N i j l l i N N

e e C e e C e e C

T T

γ γ γ

ading MIMO channels

) ( ) ( ) ( ) ( ) ( )

=

+

−
−
−

−

1

! 2 2 2 ! ! 2 2 2 2 2 ! ! ! !

l T l k T k

N E j j i l j N E l j j j i j i j l l j γ

, Fundamentals of MIMO Wireless ridge University Press, 2017 85

( ) ( ) ( ) ( ) ( ) ( ) ( )

=

+ = + = +

−

−

−

−

−
−

1 1 1

2 2 2 2 2 2 2 ! ! ! 2 ! ! 2 ! ! !

l k T k l k T k l k T k l

N E l j j i j i j j N E l j j i j i j j j E l l j j i j l j γ γ γ

SLIDE 86

Capacity of i.i.d. Rayleigh fad

(b) Given that

and hence,

1 =

R

N

{ }

T T

N N n = = 1 , max

m

Therefore, i=j=l=0, we have,

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

− = +

=

1 2 1 ,

log

T T T

N k k N N

E e e C

γ

ading MIMO channels

{ }

1 1 , min = =

T

N m

, Fundamentals of MIMO Wireless ridge University Press, 2017 86

+

1

T

N γ

SLIDE 87

Capacity of i.i.d. Rayleigh fad

(c)

Given that

and hence,

1 =

T

N

{ }

and hence,
Therefore, i=j=l=0, we have,

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

{ }

R R

N N n = = , 1 max

( )

− =

=

1 2 1 , 1

log

R R

N k N

E e e C

γ

ading MIMO channels

{ }=

=

, Fundamentals of MIMO Wireless ridge University Press, 2017 87

R

{ }

1 , 1 min = =

R

N m

+

1

1

k

E γ

SLIDE 88

Capacity of i.i.d. Rayleigh fad

Example
Show that a simple upper bound on

fading MIMO channel is given as

1/19/2018

Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

+

≤

R

N C γ 1 log min

2

ading MIMO channels

d on the average capacity of Rayleigh

N γ

, Fundamentals of MIMO Wireless ridge University Press, 2017 88

)

+

T R T

N N N γ 1 log ,

2

SLIDE 89

Capacity of i.i.d. Rayleigh fad

Note that the log-det function is con

matrices

Therefore, applying Jensen’s inequa
In the above we have used the relat

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

γ ≤

+

= log log

2 2 H T N

N E C

R

HH I

( )

N T H

N E I HH =

ading MIMO channels

concave over the set of nonnegative quality, we have elation

, Fundamentals of MIMO Wireless ridge University Press, 2017 89

( )

γ γ + = + 1 log2

R H T N

N E N

R

HH I

R

N

SLIDE 90

Capacity of i.i.d. Rayleigh fad

( )

11 12 1 11 21 21 22 2 12 22 1 2 1 2

T T R R R T T

N N H N N N N N N

h h h h h h h h h h E E h h h h h =

HH
1/19/2018

Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) (

1 2 1 2 2 2 2 11 12 1 2 2 21 22

R R R T T T

N N N N N N N

E h h h E h h

+

+ + + + =

ading MIMO channels

* 21 1 22 2

R R T R T

N N N N N

h h h h h

, Fundamentals of MIMO Wireless

ridge University Press, 2017 90

)

2 2 2 2 1 2

T R T T R R R

N N N N N N N N

h E h h h + + + + +

(

)

2

T

SLIDE 91

Capacity of i.i.d. Rayleigh fad

In retrospect, the matrices HHH and

eigenvalues, therefore

≤

+

=

N H T N

N E C

T T

γ log log

2 2

I H H I

In the above we have used the relat
By combining the above two cases,

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

T

N

( )

+

≤

T R

N N C γ , 1 log min

2

ading MIMO channels

and HHH have identical nonzero

( )

+

= +

T R T H T

N N N E N γ γ 1 log2 H H

elation es, we can obtain the upper bound as

, Fundamentals of MIMO Wireless ridge University Press, 2017 91

T

T

N N

( )

T

N R H

N E I H H =

+

T R

N N γ 1 log2

SLIDE 92

Capacity of i.i.d. Rayleigh fad

Outage capacity of iid Rayleigh fadin

( )

+

= W

b

R

b

NR

I

2 det

log Pr Pr

It has been shown+ also that the ins

a Gaussian RV for all values of NT an

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

+ P. J. Smith and M. Shafi, “On a Gaussian approximat

ICC, 2002, New York, April 2002.

ading MIMO channels

ading MIMO channel

<
+

R N P

H T

HH

2

σ

instantaneous capacity Cinst leads to and NR

, Fundamentals of MIMO Wireless ridge University Press, 2017 92

NTσ

imation to the capacity of wireless MIMO systems,” IEEE

SLIDE 93

Capacity of i.i.d. Rayleigh fad

Therefore the outage probability ma

combination of NT and NR antenna

( )

−

≈

C

ut

Q R P µ

where R is the target data rate,
μC=E[Cinst],
σc is the square root of the variance
Q-function is tail integral of a Gauss

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

≈

C

ut

Q R P σ

ading MIMO channels

y may be nearly approximated for all nnas as

R

nce of the Cinst ussian pdf

, Fundamentals of MIMO Wireless ridge University Press, 2017 93

( )

dz e x Q

z x 2

2

2 1

− ∞

=

π

SLIDE 94

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg , Fundamentals of MIMO Wireless ridge University Press, 2017 94

SLIDE 95

Capacity of i.i.d. Rayleigh fad

Fig. CDF of open loop NT×NR MIMO c
For a 5×5 MIMO channel,
the 0.2 outage capacity is approx
7.5 bits/sec/Hz for SNR of 5 d
7.5 bits/sec/Hz for SNR of 5 d
Whereas, for a 7 ×7 MIMO channel,
the 0.2 outage capacity is approx
10.5 bits/sec/Hz for SNR of 5

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

ading MIMO channels

O channel capacity for SNR=5dB proximately 5 dB 5 dB nel, proximately f 5 dB

, Fundamentals of MIMO Wireless ridge University Press, 2017 95

SLIDE 96

Capacity of separately corr MIMO channel

Instantaneous capacity of separately

channel

=

+

=

2 2 det

log σ

T N

W N P W C

R

Q I

For separately correlated MIMO cha

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

σ

T

N

1/ 2 2

log det

R X

N R T

C P W N σ

=

+

I

R

rrelated Rayleigh fading

ately correlated Rayleigh fading MIMO

+

2 2 det

log σ

T H N

N P W

R

HH I

channel,

, Fundamentals of MIMO Wireless ridge University Press, 2017 96

σ

T

N

1/2 /2

X X X

H H R w T w R

H R

H R

1/2 1/2

X X

R w T

= H R H R

SLIDE 97

Capacity of separately corr MIMO channel

For NT=NR=N, and assuming that the
and

are full rank, we have

X

R

X

T

R

( ) ( )

BA I AB I + = + det det

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

{

1/2 2 2 2 2 2

log det log det log de

X

H H w w R R T H w w T

C P W N P N σ σ

=
=

+

H H R

R H H

( ) ( )

BA I AB I + = + det det

rrelated Rayleigh fading

t the matrices and have for high SNR case,

, Fundamentals of MIMO Wireless ridge University Press, 2017 97

( )} ( )

{ }

/2 2

et log det

X X X X

H R T R T

+

R R R

SLIDE 98

Capacity of separately corr MIMO channel

Hence the MIMO channel capacity h
and the amount of reduction in the

( )

{ }

2 2

log det log

X

R

+ R

Example
Show that

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

{ }

2 2

X

R

( )

{ } {

2 2

log det log de

X

R

+ R

rrelated Rayleigh fading

ity has been reduced (why reduced?) the capacity is given by

( )

{ }

2 det

X

T

R

is always negative.

, Fundamentals of MIMO Wireless ridge University Press, 2017 98

( )

{ }

2

X

T

( )}

det

X

T

R

SLIDE 99

Capacity of separately corr MIMO channel

Note that

The diagonal elements are 1 and

[ ]

T H T

E

X

H H R =

[ ]

H R

E

X

HH R =

The diagonal elements are 1 and
off-diagonal elements hold a valu
Hence

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

T T

N trace

X =

R

( )

R R

N trace

X =

R

rrelated Rayleigh fading

value between 0 and 1

, Fundamentals of MIMO Wireless ridge University Press, 2017 99

SLIDE 100

Capacity of separately corr MIMO channel

The geometric mean is bounded by

1 1 1

1

R R R

N N N i i i i R

N N λ λ

= =

≤

=

∏
Note that product of all eigenvalues

determinant of the matrix

Therefore

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

1 1 i i R

N N

= =

(

)

1

det 1

R X

N R i i

λ

=

= ≤

∏

R

rrelated Rayleigh fading

by the arithmetic mean

( )

1 1

X

R R

trace N = = R

lues of a matrix is equal to the

, Fundamentals of MIMO Wireless ridge University Press, 2017 100

R

N

SLIDE 101

Capacity of separately corr MIMO channel

In the similar way we can show that

Hence,

( )

{ } {

log det log d + R

Hence,

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

{ } {

2 2

log det log d

X

R

+ R

rrelated Rayleigh fading

that is always negative

( )

1

det 1

T X

N T i i

λ

=

= ≤

∏

R

( )}

det R is always negative

, Fundamentals of MIMO Wireless ridge University Press, 2017 101

( )}

det

X

T

R

SLIDE 102

Average capacity of equi-co MIMO channels

Average capacity for equi-correlated

1 1 ρ ρ

=

= R R

where correlation coefficient

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

1

X X

R T

ρ ρ ρ ρ ρ

=

=

R

R

= ρ

correlated Rayleigh fading

ated Rayleigh fading MIMO channel

1 ρ ρ ρ ρ ρ

, Fundamentals of MIMO Wireless

ridge University Press, 2017 102

1 1 1 ρ ρ ρ ρ ρ ρ

9

. , 7 . , 5 . , 3 .

SLIDE 103

1/19/2018 Rakhesh Singh Kshetrimayum, Fun Communications, Cambridg , Fundamentals of MIMO Wireless ridge University Press, 2017 103

SLIDE 104

Average capacity of equi-corr channels

Fig. Average capacity of 4 ×4 open loo

correlated Rayleigh fading MIMO ch

It can be observed that the correlat
Average capacity is highest for unco
and it keeps on decreasing for highe
Recall instantaneous capacity for co

channel

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

{

1/2 2 2 2 2 2

log det log det log

X

H w w R T H w w T

C P W N P N σ σ

=
=

+

H H R

H H

rrelated Rayleigh fading MIMO

n loop MIMO system for equi- O channel elation reduces the capacity ncorrelated case igher values of correlation coefficient r correlated Rayleigh fading MIMO

, Fundamentals of MIMO Wireless ridge University Press, 2017 104

( )

{ }

( )

{ }

/2 /2 2

det log det

X X X X X

H R T R T

+

R R R R

SLIDE 105

Average capacity of equi-co MIMO channels

For iid case,
For our case assume

N N N

T R

= =

( )

R

N T H

N E I HH =

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( )

2 2 2 2

log log log

X

H w w T R

P N E N σ

=

+ +

H H

R R

( )

2 2 2

log log log

X

asymptotic T

C N N N N γ

≈

+ +

R

R

CC

correlated Rayleigh fading

N

, Fundamentals of MIMO Wireless ridge University Press, 2017 105

( )

2 2 2

log log log

X X

H R w w T

N E N γ

=

+ +

H H

R

( )

2 2

log log

X X X

R T R

N γ = + R R R

SLIDE 106

Average capacity of equi-co MIMO channels

Hence the capacity increases linearl

with a term

( )

2 2

log log lo

asymptotic

C N N N N γ

≈

+ +

CC

with a term

which reduces the capacity i.e.

Example

Assume a constant and separately c

with receiver and transmitter correl

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

log

correlated Rayleigh fading

early with the number of antennas

( )

2 2 2

log log log

X X X X

T R T R

N γ = + R R R R

which is always negative ly correlated MIMO channel model rrelation matrices as given below

, Fundamentals of MIMO Wireless ridge University Press, 2017 106

2

g

X X

T R

R R

SLIDE 107

Average capacity of equi-co MIMO channels

1 1 1

X

R R R R R

ρ ρ ρ ρ ρ ρ

=
R
Find the approximate asymptotic av
Solution:

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

1

R R N N

ρ ρ

×

( )

2

log l

asymptotic

C N γ ≈ +

CC

correlated Rayleigh fading

1 1 1

X

T T T T T

ρ ρ ρ ρ ρ ρ

=
R
ic average capacity of such channel

, Fundamentals of MIMO Wireless ridge University Press, 2017 107

1

T T N N

ρ ρ

×

2

log

X X

T R

+ R R

SLIDE 108

Average capacity of equi-co MIMO channels

It has two eigenvalues

( ) (

1

1 1

X

N T T T T

N ρ ρ ρ

−

= − − + R

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( )

1

1 1

X

N R R R R

N ρ ρ ρ

−

= − − + R

correlated Rayleigh fading

)

T

, Fundamentals of MIMO Wireless ridge University Press, 2017 108

SLIDE 109

Average capacity of equi-co MIMO channels

Hence

( ) (

{ }

2 2 2

log log log

X X

asymptotic T R

C N N γ γ ≈ + = R R

CC

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( ) ( )

{ }

( ) ( ) ( ) (

1 2 2 2 2 2

log log 1 1 lo log 1 log 1 log 1

N T T T T T

N N N N N γ ρ ρ ρ γ ρ ρ

−

= + − − + + = + − − + − +

correlated Rayleigh fading

)

{ }

2 2

log log

X X

T R

γ + + R R

, Fundamentals of MIMO Wireless ridge University Press, 2017 109

( ) ( )

{ }

) ( ) ( ) ( )

1 2 2 2

log 1 1 1 log 1 log 1

N R R R T R R R

N N N N ρ ρ ρ ρ ρ ρ ρ

−

− − + + + − − + − +

SLIDE 110

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg , Fundamentals of MIMO Wireless ridge University Press, 2017 110

SLIDE 111

Outage capacity of equi-co MIMO channels

Fig. CDF of open loop 5 ×5 MIMO cha
5 ×5 MIMO channel,
the 0.2 outage capacity is approx
7.5 bits/sec/Hz
7.5 bits/sec/Hz
for SNR of 5 dB when (uncorrela

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

rrelated Rayleigh fading

channel capacity for SNR=5dB proximately related)

, Fundamentals of MIMO Wireless ridge University Press, 2017 111

SLIDE 112

Outage capacity of equi-co MIMO channels

Whereas, the 0.2 outage capacity is
6.5 bits/sec/Hz,
5.7bits/sec/Hz,

4.5bits/sec/Hz and

. = ρ

4.5bits/sec/Hz and
3bits/sec/Hz
Hence the 0.2 outage capacity decre
with increase in the correlation c

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

rrelated Rayleigh fading

ty is approximately

9 . , 7 . , 5 . , 3 .

ecreases

n coefficient

, Fundamentals of MIMO Wireless ridge University Press, 2017 112

SLIDE 113

Capacity of Keyhole Raylei

The average capacity of keyhole MIM

averaging the instantaneous capacit
over the pdf of Keyhole Rayleigh cha

( )

1 2

2

T R

N N

z p z K

+ −

=

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

( ) ( ) ( )

R

Z N T R

p z K N N

−

= Γ Γ

( )

2 2

log 1 log

Z Z T T

z z C p z dz p N N γ γ

∞ ∞

=

+ =

+ H. Shin and J. H. Lee, “Capacity of multiple-an

Double scattering and Keyhole,” IEEE Trans. Info

igh fading MIMO channel

IMO is obtained by acity expression channel model

( )

2 ; z z ≥

, Fundamentals of MIMO Wireless ridge University Press, 2017 113

( )

2 ;

T

N

z z

−

≥

( ) ( )

2 1 2

log 1

T Z Z

N p z dz p z dz I I z γ

∞

+

+ = +

antenna fading channels: Spatial fading correlation

Information Theory, 2003, pp. 2636-2647.

SLIDE 114

Capacity of Keyhole Raylei

Average capacity is derived as

( ) ( ) ( ) { }+

Ψ + Ψ +

=

log log

2 2 2 R T T

N N e N P C σ

where the Euler’s digamma function

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

T

N σ

( ) ( ) ( ) ( ) ( )

Γ Γ = Γ = Ψ

'

ln z z z dz d z

igh fading MIMO channel

( ) ( ) ( )

Γ

Γ + , 1 , , 1 , 1 log

2 2 , 3 4 , 2 2 T R T R T

N N N P G N N e σ

tion is given by

, Fundamentals of MIMO Wireless ridge University Press, 2017 114

( ) ( )

T

R T R T

N σ

( )

∞ =

+

+ − + − = 1 1 ln 1 ln

k

k x k x x

SLIDE 115

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg , Fundamentals of MIMO Wireless ridge University Press, 2017 115

SLIDE 116

Capacity of Keyhole Raylei

Fig. CDF of open loop MIMO channel

propagation

For a 5 ×5 MIMO channel,
the 0.2 outage capacity is approx

dB Whereas, for a 7 ×7 MIMO channel

Whereas, for a 7 ×7 MIMO channel
the 0.2 outage capacity is approx

5 dB

There is significant reduction in the
for the 5 ×5 and 7 ×7 MIMO chan

propagation

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

igh fading MIMO channel

nnel capacity for SNR=5dB for keyhole proximately 3 bits/sec/Hz for SNR of 5 nel nel proximately 3.5 bits/sec/Hz for SNR of the 0.2 outage capacity channel due to the keyhole

, Fundamentals of MIMO Wireless ridge University Press, 2017 116

SLIDE 117

Capacity of Keyhole Ray channel

Capacity could be expressed in Meij

(b

a a

m j j p n m

− Γ =

∏

=

1 , ,

1 1 ,

The integral looks like inverse Laplac

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

(

b i b b a a x G

q m j j q p n m q p

− Γ =

∏

+ = =

1 2 1 , , , ,

1 1 1 1 , ,

π

ayleigh fading MIMO

eijer’s G-function

) ( )

s a s

s n j j

≤ ≤ ≤ ≤ − − Γ − ∏

=

1

place transform

, Fundamentals of MIMO Wireless ridge University Press, 2017 117

) ( )

p n q m ds x s a s b

s p n j j j j

≤ ≤ ≤ ≤ − Γ + ∏

+ = =

, ;

1 1

SLIDE 118

Capacity of Keyhole MIMO channel

Meijer’s G-function

(b

a a

m j j p n m

− Γ =

∏

=

1 , ,

1 1 ,

complex numbers

1/19/2018 Rakhesh Singh Kshetrimayum, Fu Communications, Cambridg

(

b i b b a a x G

q m j j q p n m q p

− Γ =

∏

+ = =

1 2 1 , , , ,

1 1 1 1 , ,

π

le Rayleigh fading

) ( )

s a s

s n j j

≤ ≤ ≤ ≤ − − Γ − ∏

=

1

, Fundamentals of MIMO Wireless ridge University Press, 2017 118

) ( )

p n q m ds x s a s b

s p n j j j j

≤ ≤ ≤ ≤ − Γ + ∏

+ = =

, ;

1 1