ACT_3_SIMO_MISO_MIMO.pdf

EE401: Advanced Communication Theory
Professor A. Manikas
Chair of Communications and Array Processing
Imperial College London
Modelling of SIMO, MISO and MIMO Antenna Array
Prof. A. Manikas (Imperial College) EE.401: SIMO, MISO and MIMO v.19c3 1 / 87

Table of Contents
1 Notation 3
2 Common Symbols 6
3 The Concept of the Projection Operator 7
4 Space Selectivity in Wireless Channels 11
Space-Selective Fading 11
Small-Scale and Large-Scale fading 13
5 Scattering Function - Wireless Channel Analysis 15
Wavenumber Spectrum and Angle Spectrum 16
Correspondence between Frequ., Time & Space
Parameters 22
The Relationship between Coherence-Time and
Bandwidth 23
The Concept of the "Local Area" 26
6 Wireless SIMO Channels 27
Rx Antenna Array Modelling 28
Array Manifold Vector 29
Single-path SIMO Channel Modelling 31
Multi-path SIMO Channel Modelling 38
Modelling of the Received Vector-Signal x(t) 41
Multi-user SIMO 42
7 Wireless MISO Channels 44
Reciprocity Theorem 45
Single-path MISO Channel Modelling 46
Multi-path MISO Channel 49
Modelling of the Rx Scalar-Signal x(t) 51
Transmit Diversity 54
Transmit Diversity: "Close Loop"
UMTS 3GPP Standard (Close Loop)
Transmit Diversity: "Open Loop" (without geometric
information)
8 Wireless MIMO Channels 62
Single-path MIMO Channel Modelling 64
Multi-path MIMO Channel 66
Modelling of the Rx Vector-Signal x(t) 68
9 Multipath Clustering in SIMO, MISO and MIMO 71
SIMO Channels: Multipath Clustering 71
MISO Channels: Multipath Clustering 73
MIMO Channels: Multipath Clustering 75
Summary 78
10 Some Important Comments 80
MIMO Systems (without geometric information) 80
Capacity - General Expressions 83
11 Equivalence between MIMO and SIMO/MISO 85
Spatial Convolution and Virtual Antenna Array 85

Notation
Notation
a, A denotes a column vector
A(or A) denotes a matrix
IN N N identity matrix
1N vector of N ones
0N vector of N zeros
ON,M N M matrix of zeros
(.)T transpose
(.)H Hermitian transpose
A# pseudo-inverse of A
, Hadamard product, Hadamard division
Kronecker product
Khatri-Rao product (column by column Kronecker product)
exp(a), exp(A) element by element exponential
L[A] linear space/subspace spanned by the columns of A
L[A]?
complement subspace to L[A]
P[A] (or PA) projection operator on to L[A]
P[A]? (or P?
A) projection operator on to L[A]?

Notation
The expression "N-dimensional complex (or real) observation
space " is denoted by the symbol H and pictorially represented as
follows
Note that any vector in this space has N elements.

Notation
The expression "one-dimensional subspace spanned by the (N 1)
vector a" is mathematically denoted by L[a] and pictorially
represented by
L[a]
The expression "M-dimensional subspace (with M 2) spanned by
the columns of the (N M) matrix A" is mathematically denoted by
L[A] and pictorially represented by
L[A]
Note that any vector x 2 L[A] can be written as a linear
combination of the columns of the matrix A
i.e.
x = A.a (2)
where a is an (M 1) vector with elements that are the coe¢ cients
of this linear combination.

Common Symbols
Common Symbols
N number of Rx array elements
φ elevation angle (Direction-of-Arrival)
θ azimuth angle (Direction-of-Arrival)
u [cos θ cos φ, sin θ cos φ, sin φ]T
(3 1) real unit-vector pointing towards the direction (θ, φ)
uT u = 1
c velocity of light
Fc carrier frequency
λ wavelength
k wavenumber
k wavevector

The Concept of the Projection Operator
Consider an (N M) matrix A with M N
(i.e. the matrix has M columns)
Let the columns of A be linearly independent
(i.e. a column of A cannot be written as a linear combination of the
remaining M 1 columns)
Then the columns of A span a subspace L[A] of dimensionality M
(i.e. dim{L[A]}=M) lying in an N-dimensional space H
(observation space), and this is shown below:

Any vector x 2 H can be projected on to L[A] by using the concept
of the projection operator P[A] (or PA). That is:
P[A] = PA = projection operator on to L[A] (3)
= A(A
H
A) 1
AH
(4)
Properties of Projection Operator
PA :
8
>
<
>
:
(N N) matrix
PAPA = PA
PA= PH
A
(5)

L[A]? denotes the complement subspace to L[A]
dim (L[A]) = M (6)
dim L[A]?
= N M (7)
P?
A represents the projection operator of L[A]? and is de…ned as
P?
A = IN PA (8)

Any vector x 2 H can be projected on to L[A]? and L[A] as follows
Comment:
if x 2 L[A] then
PAx = x
P?
Ax = 0N M
(9)

Space Selectivity in Wireless Channels Space-Selective Fading
Space-Selective Fading
A wireless receiver is located (and moves) in our 3D real space.
In addition to delay-spread (causing frequency-selective fading) and
Doppler-spread (causing time-selective fading) there is also
angle-spread
Angle Spread causes Space-Selective fading
Note that a channel has
I space-selectivity if it varies (i.e. its transfer function varies) as a
function of space and
I spatial-coherence if its transfer function does not vary as a function
of space over a speci…ed distance (Dcoh) of interest
where
Dcoh = is known as coherence distance. (10)

Space Selectivity in Wireless Channels Space-Selective Fading
jH(r)j V0 for jr r0j
Dcoh
2
(11)
I In other words a wireless channel has spatial coherence if the envelope
of the carrier remains constant over a spatial displacement of the
receiver.
I Dcoh represents the largest distance that a wireless receiver can move
with the channel appearing to be static/constant.

Space Selectivity in Wireless Channels Small-Scale and Large-Scale fading
Small-Scale and Large-Scale fading
If the displacement of the receiver is greater than Dcoh then the
channel experiences small-scale fading (this is due to multipaths
added constructively or destructively).
If this displacement is very large (i.e. corresponds to a very large
number of wavelengths) then the channel experiences large-scale
fading (this is due to path loss over large distances and shadowing
by large objects - it is typically independent of frequency)

Space Selectivity in Wireless Channels Small-Scale and Large-Scale fading

Scattering Function - Wireless Channel Analysis
Scattering Function - Wireless Channel Analysis

Scattering Function - Wireless Channel Analysis Wavenumber Spectrum and Angle Spectrum
Wavenumber Spectrum
If the transfer function of the channel includes "space" then we have:
H(f , t, r)
| {z }
Transfer function
corr
=) ΦH (∆f , ∆t, ∆r)
| {z }
Autocorrelation function
(frequency,time,space)
#
#
#
ΦH (∆r)
| {z }
(space)
(known as spatial Autocor function)
FT
=)
#
#
#
#
#
#
FT
=)
SH (τ, f, k)
| {z }
Scattering function
#
#
#
SH (k)
| {z }
wavenumber
spectrum
(12)
and the scattering function SH (k) is known as the wavenumber spectrum
where
8
<
:
f frequency ; τ delay
t time ; f Doppler frequency
r location (x,y,z) ; k wavenumber vector = kkk .u(θ, φ)
9
=
;
(13)

Angle Spectrum
Angle Spectrum is the average power as a function of
direction-of-arrival (θ, φ) where θ represents the azimuth angle and φ
is the elevation angle (in the case of Tx-array this is a function of the
direction-of-departure).
Examples of Angle Spectrum for φ = 0 :
1. Angle Spectrum (Specular ):
2. Angle Spectrum (Di¤used ):
3. Angle Spectrum (Combination) :

For φ = 0, the mean angle of azimuth and rms angle spread σθ are
functions of angle spectrum p(θ) as follows:
mean : θmean ,
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
,
R θmax
0 θ.p(θ)dθ
R θmax
0 p(θ)dθ
= cont. case
,
L
∑
i=1
θi .p(θi )
L
∑
i=1
p(θi )
= discrete case
(14)
rms : σθ ,
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
,
rR θmax
0 (θ θmean)2
.p(θ)dθ
R θmax
0 p(θ)dθ
= cont. case
,
v
u
u
u
u
u
t
L
∑
i=1
(θ θmean)2
p(θ)
L
∑
i=1
p(θ)
= discrete case
(15)

NB:
I angle spectrum causes space selective fading =) signal amplitude
depends on spatial location of antennas/signals
I Coherence distance Dcoh is the spatial separation for which the
autocorr. coef. of spatial fading drops to 0.7
I Coherence distance Dcoh is inversely proportional to σθ, i.e.
Dcoh ∝
1
σθ
(16)

The angle spectrum, denoted by p(θ, φ), is related to the
wavenumber spectrum SH (k) via the "ring mapping":
H(f , t, r)
| {z }
Transfer function
corr
=) ΦH (∆f , ∆t, ∆r)
| {z }
(frequency,time,space)
#
#
#
ΦH (∆r)
| {z }
Spatial
FT
=)
#
#
#
#
#
#
FT
=)
SH (τ, f, k)
| {z }
Scattering function
#
#
#
SH (k)
| {z }
wavenumber
spectrum
!
ring
mapping
p(θ, φ)
| {z }
angle
spectrum
(17)
That is,
SH (k) =
(2π)3δ(kkk 2π
λ )
( 2π
λ )2 p(θ, φ) (18)

Remember that if ∆f = 0 and ∆t = 0 then
ΦH (∆r)
| {z }
Spatial
FT
=) SH (k)
| {z }
Scattering function
(wavenumber spectrum)
(19)
ΦH (∆r)
| {z }
scalar spatial
FT
=) SH (k)
| {z }
Scattering function
(scalar wavenumber spectrum)
(20)

Scattering Function - Wireless Channel Analysis Correspondence between Frequ., Time & Space Parameters
Correspondence between Frequ, Time & Space Parameters
Consider a wireless channel transfer function H(f , t, r)
Frequency Time Space
Dependency f t r
Coherence Bcoh
coherence bandwidth
Tcoh
coherence time
Dcoh
coherence distance
Spectral domain τ
delay
f
Doppler freq
k
wavevector
Spectral width Tspread
delay spread
BDop
Doppler spread
kspread
wavevector spread
N.B.: Remember the various types of coherence.
8
<
:
temporal coherence -coherence time Tcoh
frequency coherence -coherence bandwidth Bcoh
spatial coherence -coherence distance Dcoh

Scattering Function - Wireless Channel Analysis The Relationship between Coherence-Time and Bandwidth
The Relationship between Coherence-Time and Bandwidth
We have seen that one of the wireless channels classi…cation is "slow"
and "fast fading - as this is shown below:
A trade-o¤ between "slow" and "fast" fading is when Tcoh ' Tcs
(or, for CDMA, Tcoh ' Tc ). This implies
Tcoh B ' 1 (or, for CDMA: Tcoh Bss ' 1) (21)

An electromagnetic wave of frequency Fc (carrier frequency)
travelling for the time Tcoh with velocity c (speed of light, i.e.
3 108m/s) will cover a distance d, where
d = c.Tcoh (22)
= Fc λc
|{z}
=c
Tcoh =) Tcoh =
d
Fc λc
(23)
Using Equations 21 and 23 we have
d
Fc λc
B ' 1 =) d '
Fc
B
λc (=
c
B
) (24)
However, if an electromagnetic wave is travelling for the time Tcs
with velocity c will cover a distance dcs ,where
dcs = c.Tcs (25)

We have seen that in slow fading region we have Tcoh & Tcs
(or, for CDMA, Tcoh & Tc ). In this case:
Tcoh & Tcs =) cTcoh & cTcs =) d & dcs
i.e.
slow fading: dcs . d '
Fc
B
λc (=
c
B
) (26)
N.B. - In summary (for slow fading):
Tcoh & Tcs (i.e. Tcoh B & 1) =) dcs . d '
c
B
(27)

Scattering Function - Wireless Channel Analysis The Concept of the "Local Area"
The Concept of the "Local Area"
"Local Area" is the largest volume of free-space (i.e. 4
3 πd3) about a
speci…c point in space ro = [rx0 , ry0 , rz0 ]T (e.g. the reference point of
the Rx-array) in which the wireless channel can be modelled as the
summation of homogeneous planewaves, where
radius of local area . d = c
B (28)
Example: The radius of the "local area" for WiFi electromagnetic
waves (Fc = 2.4GHz, B = 5MHz) is:
radius .
c
B
=) radius .
3 108
5 106
= 60m

Wireless SIMO Channels
Wireless SIMO Channels

Wireless SIMO Channels Rx Antenna Array Modelling
Antenna Array
Consider a single path from Tx single antenna system to an array of
N antennas
An array system is a collection of N > 1 sensors (transducing
elements, receivers, antennas, etc) distributed in the 3-dimensional
Cartesian space, with a common reference point.
Consider an antenna-array Rx with locations given by the matrix
[r1, r2, ..., rN ] = rx , ry , rz
T
(3 N) (29)
where rk is a 3 1 real vector denoting the location of the kth sensor
8k = 1, 2, ..., N
and rx ,ry and rz are N 1 vectors with elements the x, y and z
coordinates of the N antennas
The region over which the sensors are distributed is called the aperture
of the array. In particular the array aperture is de…ned as follows
array aperture
.
= max
ij
ri rj (30)

Wireless SIMO Channels Array Manifold Vector
Array Manifold Vector
It is also know as Array Response Vector.
Modelling of Array Manifold Vectors (see Chapter-1 of my book):
S(θ, φ) = exp( j[r1, r2, . . . , rN ]T
k(θ, φ)) (31)
or
= exp( j[rx , ry , rz ]k(θ, φ)) (32)
= (N 1) complex vector
where
k(θ, φ) =
2πFc
c .u(θ, φ) = 2π
λc
.u(θ, φ) in meters
π.u(θ, φ) in units of halfwavelength
= wavenumber vector
u(θ, φ) = [cos θ cos φ, sin θ cos φ, sin φ]T
(33)
= (3 1) real unit-vector pointing towards the direction (θ, φ)
ku(θ, φ)k = 1 (34)

Wireless SIMO Channels Array Manifold Vector
In many cases the signals are assumed to be on the (x,y) plane (i.e.
φ = 0 ). In this case the manifold vector is simpli…ed to
S(θ) = exp( j[r1, r2, . . . , rN ]T
k(θ, 0 ))
= exp( jπ(rx cos θ + ry sin θ)) (35)
A popular class of arrays is that of linear arrays: ry = rz = 0N
in this case, Equation 31 is simpli…ed to
S(θ) = exp( jπrx cos θ) (36)
Summary: An array maps one or more real directional parameters p
or (p, q) to an (N 1) complex vector S(p) or S(p, q), known as
array manifold vector, or array response vector, or source position
vector. That is
p 2 R1
r
7 ! S(p) 2 CN
(37)
or (p, q) 2 R2
r
7 ! S(p, q) 2 CN
(38)
Note: Ideally Equations 37 and 38 should be an ‘
one-to-one’mapping

Wireless SIMO Channels Single-path SIMO Channel Modelling
Single-path SIMO Channel Modelling
Single Path/Ray of an Electromagnetic Wave:

Single-path SIMO Channel Modelling
Impulse Response (Including the Carrier):

With reference to the previous …gure, consider at the baseband-input a
delta-line δ(t)
Then the antenna will transmit the signal exp(j2πFc t).δ(t) , i.e.
at the Tx = exp (j2πFc t) δ(t) (39)
in the form of an electromagnetic wave that travels with the velocity of
light c for time τ and covers a distance d arriving at the Rx’
s reference
point (see …gure) and modelled as follows:
at the Rx’
s ref. point =
K
d
α
exp (jφ) exp j2πFc (t
d
c
) δ(t
d
c
)
=
K
d
α
exp (jφ) exp j2πFc
d
c
| {z }
,β
exp (j2πFc t) δ(t
d
c
)
2
4where
8
<
:
K : constant = c
4πFc
g (known)
with g = antenna gain factor (known)
α : path loss exponent; 1.5 α 4 (known)
3
5
(40)
= β. exp (j2πFc t) .δ(t
d
c
) (41)

If the k-th antenna is not at the reference point of the array but there
is a displacement rk (within the Rx "Local Area"), then the
electromagnetic wave will travel with the velocity of light c for a time
∆τk (+ve or -ve) which corresponds to the distance rT
k u(θ, φ)
between the ref. point and the k-th antenna’
s location.That is,
∆τk =
rT
k u(θ, φ)
c
(42)
Proof of 42:
With reference to the LHS
…gure (and u , u(θ, φ)) we
have:
∆τk =
q
rT
k u(uT u)
1
uT rk
c
=
p
rT
k u uT rk
c
=
q
(rT
k u)
2
c =
rT
k u
c

Note: rk 2 R3 1 represents the Cartesian coords of the k-th Rx-antenna
(in m)
Thus, at the o/p (kth Rx-antenna) we have
o/p
(baseband)
=
"
Equation-41
with displacement: ∆τk
#
exp ( j2πFc t)
= β exp (j2πFc (t ∆τk )) exp ( j2πFc t) .δ(t ∆τk
d
c
)
= β exp ( j2πFc ∆τk ) .δ(t ∆τk
d
c
)
= β exp j
2πFc
c
rT
k u(θ, φ) .δ(t
'd (as d>>>krk k)
z }| {
d + rT
k u(θ, φ)
c
)
= β exp j
2πFc
c
rT
k u(θ, φ) .δ(t
d
c
)
= β exp j
2πFc
c
rT
k u(θ, φ) .δ(t τ
"
= d
c
) (43)

That is, the Tx planewave exp (j2πFc t) δ(t) arrives at each antenna of the
array and produces, at time t = τ, a constant-amplitude voltage-vector as
follows:
2
6
6
6
4
1st ant.
2nd ant.
.
.
.
Nth ant.
3
7
7
7
5
=
2
6
6
6
4
Equ.43, for k = 1
Equ.43, for k = 2
.
.
.
Equ.43, for k = N
3
7
7
7
5
=
2
6
6
6
6
6
6
4
β exp j 2πFc
c rT
1 u(θ, φ) .δ(t τ)
β exp j 2πFc
c rT
2 u(θ, φ) .δ(t τ)
.
.
.
β exp j 2πFc
c rT
N u(θ, φ) .δ(t τ)
3
7
7
7
7
7
7
5
= β
2
6
6
6
6
6
6
4
exp j 2πFc
c rT
1 u(θ, φ)
exp j 2πFc
c rT
2 u(θ, φ)
.
.
.
exp j 2πFc
c rT
N u(θ, φ)
3
7
7
7
7
7
7
5
| {z }
,array manifold vector S(θ,φ)
δ(t τ)
= β.S(θ, φ).δ(t τ) (44)

i.e.
Non-parametric Model
,
2
6
6
6
4
βTx,1
βTx,2
.
.
.
βTx,N
3
7
7
7
5
| {z }
Non-parametric
= β.S(θ, φ)
| {z }
Parametric
(45)
where
S(θ, φ) =
2
6
6
6
6
6
6
6
6
6
6
4
exp j 2πFc
c rT
1 u(θ, φ)
exp j 2πFc
c rT
2 u(θ, φ)
....
exp j 2πFc
c rT
k u(θ, φ)
....
exp j 2πFc
c rT
N u(θ, φ)
3
7
7
7
7
7
7
7
7
7
7
5
(46)
= exp j
2πFc
c
r1, r2, ... rN
T
u(θ, φ) (47)
= exp j
2πFc
c
rx , ry , rz u(θ, φ) (48)

Wireless SIMO Channels Multi-path SIMO Channel Modelling
Multi-path SIMO Channel Modelling
Let us assume that the transmitted signal arrives at the reference
point of an array receiver via L paths (multipaths).
Consider that the `th path arrives at the array from direction (θ`, φ`)
with channel propagation parameters β` and τ` representing the
complex path gain and path-delay, respectively.
Note that θ` and φ` represent the azimuth and elevation angles
respectively associated with `-th path.
Let us assume that the L paths are arranged such that
τ1 τ2 . . . τ` ... τL (49)
Furthermore, the path coe¢ cients β` model the e¤ects of path losses
and shadowing, in addition to random phase shifts due to re‡ection;
they also encompass the e¤ects of the phase o¤set between the
modulating carrier at the transmitter and the demodulating carrier at
the receiver, as well as di¤erences in the transmitter powers.

The vector S` = S(θ`, φ`) 2 CN , is the array manifold vector of the
`-th path .
The impulse response (vector) of the SIMO multipath channel is
SIMO: h(t) =
L
∑
`=1
S(θ`, φ`).β`.δ(t τ`)
(50)

i.e. multipath SIMO channel
Non-parametric
,
2
6
6
6
4
βTx,1
βTx,2
.
.
.
βTx,N
3
7
7
7
5
| {z }
Non-parametric
=
L
∑
`=1
β`.S(θ`, φ`)
| {z }
Parametric
(51)
Remember:
SISO Multipath Channel
Modelling

Wireless SIMO Channels Modelling of the Received Vector-Signal x(t)
Modelling of the Received Vector-Signal
Consider a single Tx transmitting a baseband signal m(t) via an L-path
SIVO channel.
The received (N 1) vector-signal x(t) can be modelled as follows:
x(t) = h(t) m(t) + n(t)
=
L
∑
`=1
S(θ`, φ`).β`.δ(t τ`)
!
m(t) + n(t)
) x(t) =
L
∑
`=1
S(θ`, φ`).β`.m(t τ`) + n(t)
= Sm(t) + n(t) (52)
where
S = S1, S2, , SL , with S` , S(θ`, φ`), ` = 1, .., L(53)
m(t) = β1m(t τ1), β2m(t τ2), , βLm(t τL)
T
(54)
n(t) = n1(t), n2(t), , nN (t)
T
(55)

Wireless SIMO Channels Multi-user SIMO
Multi-user SIMO
Next consider an Rx array of N antennas operating the the presence
of M co-channel transmitters/users.
In this case we have added the subscript i to refer to the i-th Tx.
The received (N 1) vector-signal x(t) from all M
transmitters/users (transmitting at the same time on the same
frequency band) can be modelled as follows:.
x(t) =
M
∑
i=1
L
∑
`=1
Si`.βi`.mi (t τ`) + n(t)
= Sm(t) + n(t) (56)
with
Sil , S(θil , φil )
where S, m(t) and n(t) de…ned as follows:&&

Wireless SIMO Channels Multi-user SIMO
S ,
2
6
4
,S1
z }| {
S11, S12, ..., S1L ,
,S2
z }| {
S21, S22, ..., S2L ,...,
,SM
z }| {
SM1, SM2, ..., SML
3
7
5(57)
m(t) ,
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
β11m1(t τ11)
β12m1(t τ12)
.
.
.
β1Lm1(t τ1L)
9
>
>
>
=
>
>
>
;
, m1(t)
β21m2(t τ21)
.
.
.
.
.
.
βM1mM (t τM1)
βM2mM (t τM2)
.
.
.
βMLmM (t τML)
9
>
>
>
=
>
>
>
;
, mM (t)
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(58)
n(t) , n1(t), n2(t), ..., nN (t)
T
(59)

Wireless MISO Channels
Wireless MISO Channels

Wireless MISO Channels Reciprocity Theorem
Reciprocity Theorem
Antenna characteristics are independent of the direction of energy
‡ow.
I The impedance & radiation pattern are the same when the antenna
radiates a signal and when it receives it.
The Tx and Rx array-patterns are the same.
The Tx-array is an array of N elements/sensors/antennas with
locations r
r = [r1, r2, ..., rN ] = rx , ry ,rz
T
(3 N)
with rm denoting the location of the mth Tx-sensor 8m = 1, 2, ..., N
Notation:
I the bar at the top of a symbol, i.e. (.), denotes a Tx-parameter.

Wireless MISO Channels Single-path MISO Channel Modelling
Single-path MISO Channel Modelling

Single-path MISO Channel Modelling

That is, if the Tx is displaced at a speci…c point rm (within its local
area LA) and the direction of the planewave propagation is described
by the vector ui = u(θ, φ) where
ui (θ,φ) = cos θ cos φ, sin θ cos φ, sin φ
T
If the Tx employs an array of N elements/sensors/antennas with
locations r
then the channel impulse response (single path) is
h(t) = βS
H
δ(t
d
c
) (60)
I where
S(θ, φ) = exp(+j[r1, r2, . . . , rN ]T
k(θ, φ)) (61)
or
= exp(+j[rx , ry , rz ]k(θ, φ)) (62)
= (N 1) complex vector
k(θ, φ) =
(
2πFc
c .u(θ, φ) = 2π
λc
)
= wavenumber vector

Wireless MISO Channels Multi-path MISO Channel
Multi-path MISO Channel
Let us assume that the transmitted signal(s) from the Tx-array arrives
at the receiver via L resolvable paths (multipaths).
Consider that the `th path’
s direction-of-departure is (θ`, φ`) and
propagates to the Rx with channel propagation parameters β` and τ`
representing the complex path gain and path-delay, respectively.
τ1 τ2 . . . τ` ... τL (63)
remember that the path coe¢ cients β` model the e¤ects of path
losses and shadowing, in addition to random phase shifts due to
re‡ection; they also encompass the e¤ects of the phase o¤set between
the modulating carrier at the transmitter and the demodulating
carrier at the receiver, as well as di¤erences in the transmitter power

Wireless MISO Channels Multi-path MISO Channel
The vector S` = S(θ`, φ`) 2 CN is the array manifold vector of the
`-th path .
The impulse response of the MISO channel is
MISO: h(t) =
L
∑
`=1
β`.S
H
` δ(t τ`)
(64)

Wireless MISO Channels Modelling of the Rx Scalar-Signal x(t)
Modelling of the Rx Scalar-Signal x(t)
Consider a Tx-array of N antennas transmitting a baseband signal
m(t) via MISO multipath channel of L resolvable paths (frequency
selective MISO). We will consider the following two cases:
I Case-1: The m(t) is demultiplexed to N di¤erent signals (one signal
per antenna element) forming the vector m(t).
I Case-2: All Tx-array elements transmit the same signal m(t).
In both cases the transmitted signals may, or may not, be weighted.

Case-1: The received scalar-signal x(t) can be modelled as follows:
x(t) =
L
∑
`=1
β`S
H
(θ`, φ`). (δ(t τ`) ~ (w m(t)) + n(t)
=
L
∑
`=1
β`S
H
` . (w m(t τ`)) + n(t) (65)

Case-2: The signal is "copied" to each antenna and the received
scalar-signal x(t) can be modelled as follows:
x(t) =
L
∑
`=1
β`S
H
(θ`, φ`). (δ(t τ`) ~ (wm(t)) + n(t)
=
L
∑
`=1
β`S
H
` w.m(t τ`) + n(t) (66)

Wireless MISO Channels Transmit Diversity
Transmit Diversity
Provides diversity bene…ts to a mobile using base station antenna
array for frequency division duplexing (FDD) schemes. Cost is shared
among di¤erent users.
Order of diversity can be increased when used with other conventional
forms of diversity (e.g. multipath diversity).
Two main types of diversity combining techniques in 3G:
I Transmit diversity with feedback from receiver (close loop)
I Transmit diversity without feedback from receiver (open loop)

Transmit Diversity: "Close Loop"
The transmitter transmits some pilot signals
The mobile (based on this pilot signals) estimates the Channel State
Information (CSI), i.e. channel parameters.
The mobile transmits the CSI to the BS (uplink)
The base station generates the weights and transmits data to the
mobile.
I That is a beamformer is formed which steers its main lobe towards the

UMTS 3GPP Standard (Close Loop)
N = 2 Tx-antennas
where
I w1, w2 are adjusted such as jx(t)j2 is maximised, e.g.
I w1, w2 are adjusted based on the feedback information from the
receiver

Transmit Diversity: "Open Loop"
without spreader:
with spreader:

Example of STBC Downlink Equations (without spreader):
I STBC = Space-Time Block Code
I No geometric/space information is used.
I Receiver input (L unresolvable multipaths - ‡at fading):
8
>
>
>
>
<
>
>
>
>
:
1st interval: x1 =
L
∑
j=1
β1j,Rx m1 β2j,Rx m2 + n1
2nd interval: x2 =
L
∑
j=1
β1j,Rx m2 + β2j,Rx m1 + n2
(67)

)
x1 = β1,Rx m1 β2,Rx m2 + n1
x2 = β1,Rx m2 + β2,Rx m1 + n2
(68)
where
β1,Rx
L
∑
j=1
β1j,Rx and β2,Rx
L
∑
j=1
β2j,Rx (69)

I Equivalently,
x1
x2
=
m1, m2
m2, m1
β1,Rx
β2,Rx
| {z }
,β
Rx
+
n1
n2
(70)
or, in an alternative format:
x =
x1
x2
=
β1,Rx , β2,Rx
β2,Rx , β1,Rx
| {z }
,H
m1
m2
| {z }
,m
+
n1
n2
|{z}
,n
(71)
i.e.
x = Hm + n (72)
where
HH
H = β1,Rx
2
+ β2,Rx
2
| {z }
= β
Rx
2
I2

I Decoder (Rx): This is denoted by the matrix H
decoder’
s o/p : G = HH
x (73)
= HH
Hm + HH
n
| {z }
,e
n
(74)
I That is, the decision variables are
G = β
Rx
2
m + e
n (75)
i.e.
G1 = β
Rx
2
m1 + e
n1
G2 = β
Rx
2
m2 + e
n2
(76)
I Note:
1 the receiver needs to know (estimate) the channel weights β1,Rx and
β2,Rx but there is no need to send them back to the transmitter (i.e.
open loop)
2 β1,Rx and β2,Rx can be estimated by transmitting some pilot symbols
as m1 and m2 and, then, using Equation 70

Wireless MIMO Channels
Consider a single path from a Tx-array of N antennas to an Rx-array
of N antennas with locations given by the matrices
Tx-array: r = [r1, r2, ..., rN̄ ] = rx, ry, rz
T
(3 N)
Rx-array: r = [r1, r2, ..., rN ] = rx, ry, rz
T
(3 N)

If the direction-of-departure of this single path planewave propagation
is θ, φ and the direction-of-arrival is (θ, φ) then the impulse
response is
h(t) = βS(θ, φ).S
H
(θ, φ).δ(t
ρ
c
)
where
Tx: S = S θ, φ = exp +jrT
k(θ, φ) (77)
Rx: S = S (θ, φ) = exp jrT
k(θ, φ) (78)
with k(θ, φ) and k(θ, φ) denote the wavenumber vectors of the
Tx-array and Rx-array respectively
For instance k(θ, φ) is de…ned as
k(θ, φ) =
(
2πFc
c .u(θ, φ) = 2π
λc
)
u θ, φ = cos θ cos φ, sin θ cos φ sin φ

Wireless MIMO Channels Single-path MIMO Channel Modelling
Single-path MIMO Channel Modelling

Wireless MIMO Channels Multi-path MIMO Channel
Multipath MIMO Channel
Let us assume that the transmitted signal(s) from the Tx-array arrives
at the Rx-array via L resolvable paths (multipaths).
Consider that the `th path’
s direction-of-departure is (θ`, φ`) and
propagates to the Rx with channel propagation parameters β` and τ`
representing the complex path gain and path-delay, respectively.
τ1 τ2 . . . τ` ... τL (79)
once again, remember that the path coe¢ cients β` model the e¤ects
of path losses and shadowing, in addition to random phase shifts due
to re‡ection; they also encompass the e¤ects of the phase o¤set
between the modulating carrier at the transmitter and the
demodulating carrier at the receiver, as well as di¤erences in the
transmitter power.
In the following …gure the vectors S` = S(θ`, φ`) 2 CN and
S` = S(θ`, φ`) 2 CN are the Tx- and Rx- array manifold vectors of
the `-th path.

Wireless MIMO Channels Multi-path MIMO Channel
The impulse response of the MIMO channel is
MIMO: h(t) =
L
∑
`=1
β`.S`.S
H
` δ(t τ`)
(80)

Wireless MIMO Channels Modelling of the Rx Vector-Signal x(t)
Modelling of the Rx Vector-Signal x(t)
Consider a Tx-array of N antennas transmitting a baseband signal
m(t) via MIMO multipath channel of L resolvable paths (frequency
selective MIMO).
We will consider the following two cases:
I Case-1: The m(t) is demultiplexed to N di¤erent signals (one signal
per antenna element) forming the vector m(t).
I Case-2: All Tx-array elements transmit the same signal m(t).
In both cases the transmitted signals may, or may not, be weighted.
The Rx is also equipped with an array of N antennas.

Case-1:
The received vector-signal x(t) can be modelled as follows:
x(t) = ∑L
`=1 β`.S`.S
H
` . (δ(t τ`) ~ (w m(t)) + n(t)
= ∑L
`=1 β`.S`.S
H
` . (w m(t τ`)) + n(t) (81)

Case-2:
In this case the signal is "copied" to each antenna and may be
weighted by a complex weight.
The received vector-signal x(t) can be modelled as follows:
x(t) = ∑L
`=1 β`S`S
H
` . (δ(t τ`) ~ (wm(t))) + n(t)
= ∑L
`=1 β`S`S
H
` w.m(t τ`) + n(t) (82)

Multipath Clustering in SIMO, MISO and MIMO SIMO Channels: Multipath Clustering
SIMO Channels: Multipath Clusterring
Consider a scatterer (`-th scatterer, say) which can be seen as a large
number of paths (Lscat , say) around the direction (θ`, φ`). That is, the
direction-of-arrival of the k-th path satis…es the condition
(θ`, φ`)
(∆θ`, ∆φ`)
2
(θ`k , φ`k ) (θ`, φ`) +
(∆θ`, ∆φ`)
2
, 8k (83)
In this case the impulse response for this
scatterer can be written as follows:
`-th scatterer =
Lscat
∑
k=1
S(θ`k , φ`k ).β`k .δ(t τ`k )
(84)

Multipath Clustering in SIMO, MISO and MIMO SIMO Channels: Multipath Clustering
If ∆θ` = relatively small then
(θ`1, φ`1) ' (θ`2, φ`2) ' ... ' (θ`Lscat
, φ`Lscat
) , (θ`, φ`) (85)
τ`1 ' τ`k ' ... ' τ`Lscat
, τ` (86)
and, thus,
`-th scatterer =
Lscat
∑
k=1
S(θ`k , φ`k ).β`k .δ(t τ`k )
=
Lscat
∑
k=1
S(θ`, φ`).β`k .δ(t τ`)
= S(θ`, φ`).δ(t τ`)
Lscat
∑
k=1
β`k
| {z }
,β`
= S(θ`, φ`).β`.δ(t τ`)
= S`.β`.δ(t τ`) (87)

Multipath Clustering in SIMO, MISO and MIMO MISO Channels: Multipath Clustering
MISO Channels: Multipath Clustering
Consider a scatterer (`-th scatterer, say) which can be seen as a large
number of paths (L`, say) around the direction (θ`, φ`). That is, the
direction-of-arrival of the k-th path satis…es the condition
(θ`, φ`)
(∆θ`, ∆φ`)
2
(θ`k , φ`k ) (θ`, φ`) +
(∆θ`, ∆φ`)
2
, 8k (88)
L`= number of paths
(`th scatterer)
In this case the impulse
response for this scatterer
can be written as follows:
`-th scatterer =
L`
∑
k=1
β`k .S
H
(θ`k , φ`k ).δ(t τ`k )
(89)

Multipath Clustering in SIMO, MISO and MIMO MISO Channels: Multipath Clustering
If ∆θ` = relatively small then
(θ`1, φ`1) ' (θ`2, φ`2) ' ... ' (θ`L`
, φ`L`
) , (θ`, φ`) (90a)
τ`1 ' τ`k ' ... ' τ`L`
, τ` (90b)
and, thus,
`-th scatterer =
L`
∑
k=1
β`k S
H
(θ`k , φ`k ).δ(t τ`k )
=
L`
∑
k=1
β`k .S
H
(θ`, φ`).δ(t τ`)
= S
H
(θ`, φ`).δ(t τ`)
L`
∑
k=1
β`k
| {z }
,β`
= S
H
(θ`, φ`)δ(t τ`).β`
= β`.S
H
` .δ(t τ`) (91)

Multipath Clustering in SIMO, MISO and MIMO MIMO Channels: Multipath Clustering
MIMO Channels: Multipath Clustering
Consider a scatterer (`-th
scatterer, say) which can
be seen as a large number
of paths (L`, say) with
directions-of-departure
around the direction
(θ`, φ`) and
directions-of-arrival
around the direction
(θ`, φ`).

That is, the direction-of-departure and direction-of-arrival of the k-th
path satis…es the condition
(θ`, φ`)
(∆θ`, ∆φ`)
2
(θ`k , φ`k ) (θ`, φ`) +
(∆θ`, ∆φ`)
2
, 8k
(92)
(θ`, φ`)
(∆θ`, ∆φ`)
2
(θ`k , φ`k ) (θ`, φ`) +
(∆θ`, ∆φ`)
2
, 8k
(93)
In this case the impulse response for this scatterer can be written as
follows:
`-th scatterer =
L`
∑
k=1
β`k .S(θ`k , φ`k )S
H
(θ`k , φ`k ).δ(t τ`k ) (94)
If ∆θ` and ∆θ` are relatively small then
(θ`1, φ`1) ' (θ`2, φ`2) ' ... ' (θ`L`
, φ`L`
) , (θ`, φ`) (95)
(θ`1, φ`1) ' (θ`2, φ`2) ' ... ' (θ`L`
, φ`L`
) , (θ`, φ`) (96)
τ`1 ' τ`k ' ... ' τ`L`
, τ` (97)

Thus, if L` denotes the No. of paths of the `-th scatterer,
`-th scatterer =
L`
∑
k=1
β`k .S(θ`k , φ`k )S
H
(θ`k , φ`k ).δ(t τ`k )
=
L`
∑
k=1
β`k .S(θ`, φ`)S
H
(θ`, φ`).δ(t τ`)
= S(θ`, φ`)S
H
(θ`, φ`).δ(t τ`)
L`
∑
k=1
β`k
| {z }
,β`
(98)
= β`.S(θ`, φ`)S
H
(θ`, φ`).δ(t τ`)
= β`.S`.S
H
` .δ(t τ`) (99)

Multipath Clustering in SIMO, MISO and MIMO Summary
Summary
From Equations 87, 91 and 99 a scatterer can be seen as a single path
I (for SIMO) with direction of arrival (θ`, φ`) and fading coe¢ cient the
term β` that represents the addition/combination of the fading
coe¢ cients of all paths of this scatterer i.e. β` =
Lscat
∑
k=1
β`k .
I (for MISO) with direction of departure (θ`, φ`) and fading coe¢ cient
the term β` that represents the addition/combination of the fading
coe¢ cients of all paths of this scatterer i.e. β` =
L`
∑
k=1
β`k .
I (for MIMO) with directions of departure (θ`, φ`) and (θ`, φ`) as well
as with fading coe¢ cient the term β` that represents the
addition/combination of the fading coe¢ cients of all paths of this
scatterer i.e. β` =
L`
∑
k=1
β`k .

Multipath Clustering in SIMO, MISO and MIMO Summary
N.B.:
Another way to represent a scatterer is using Taylor Series Expansion.
This will involve:
I the manifold vector S`, its …rst derivative Ṡ and (∆θ`, ∆φ`), for SIMO;
I the manifold vector S`, its …rst derivative
.
S and (∆θ`, ∆φ`), for MISO;
I the two manifold vectors S`, S` and their …rst derivatives Ṡ,
.
S as well
as (∆θ`, ∆φ`) and (∆θ`, ∆φ`) for MIMO.

Some Important Comments MIMO Systems (without geometric information)
MIMO Systems (without geometric information)
Let us consider a comm. system with multiple antennas at both the
Tx and the Rx.

βi,j = gain from the jth Tx-antenna to the ith Rx-antenna
β
j,Tx
= gain-vector with its ith element the gain βij
I (i.e. from the j-th Tx antenna to all the Rx antennas)
If Rx is synchronised to the Tx then for the nth data symbol interval
then we have the following received vector-signal:
x[n] = Hm[n] + n[n] (N 1)
where
H =
h
β
1,Tx
, β
2,Tx
, ..., β
N,Tx
i
Second order statistics (covariance matrix) of x[n] is as follows:
Rxx = HRmmHH
+ Rnn
|{z}
σ2
nIN
(N N)

In a MIMO system it is assumed that the Matrix H is known
Note:
I The matrix H and the manifold vectors are related according to the
following expression
H =
L
∑
`=1
β`S`S
H
` (100)
= S
2
6
6
4
β1, 0, ..., 0
0, β2, ..., 0
..., ..., ..., ...
0, 0, ..., βL
3
7
7
5 S
H
(101)
where
S = S1, S2, ..., SL
S = S1, S2, ..., SL

Some Important Comments Capacity - General Expressions
Capacity - of MIMO Channels (without space info)
SISO Capacity:
C = B log2 (1 + SNIRin) bits/s (102)
General MIMO Capacity expression:
C = B log2
det (Rxx )
det (Rnn)
bits/s (103)

Some Important Comments Capacity - General Expressions
Based on Equation 103 it can be easily proven that
C = B log2 det IN +
1
σ2
n
HRmmHH
bits/s (104)
Furthermore, for independent parallel channels (i.e. using a
multiplexer at Tx, Case-1) :
Rmm = diagonal =
2
6
6
4
P1, 0, ..., 0
0, P2, ..., 0
..., ..., ..., ...
0, 0, ..., PN
3
7
7
5
and thus Equation 104 is simpli…ed to
C = B log2
N
∏
j=1
1 +
β
j,Tx
2
Pj
σ2
n
!!
(105)
= B
N
∑
j=1
log2 1 +
β
j,Tx
2
Pj
σ2
n
!
bits/sec (106)

Equivalence between MIMO and SIMO/MISO Spatial Convolution and Virtual Antenna Array
Equivalence between MIMO and SIMO
Spatial Convolution and Virtual Antenna Array
MIMO () virtual-SIMO
Tx antennas:N , Rx antennas:N N N Rx /Tx antennas

Equivalence between MIMO and SIMO (cont.)

Tx-array: r , [r1, r2, ..., rN ] = rx, ry, rz
T
(3 N)
Rx-array: r , [r1, r2, ..., rN ] = rx, ry, rz
T
(3 N)
virtual-array: rvirtual
, r 1T
N + 1T
N
r (107)
m
Svirtual , S S (108)
N.B.:
1 Equation 107 is known as "spatial convolution"
2 The concept of the "virtual array", is also applicable to MISO
3 A MIMO is equivalent to a "virtual-MISO" (by virtually transfering
the Rx antenna array to the Tx) or to a "virtual-SIMO" (by virtually
transfering the Tx antenna array to the Rx).

ACT_3_SIMO_MISO_MIMO.pdf

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