Lecture # 11

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Abstract—The presence of fading in radio communications is

the major impairment to reach the promised capacity. Diversity

is a way to limit the fading effects and MIMO is a technique to

exploit it. It is shown that in terrestrial mobile radio

communications MIMO is an effective techniques while for

satellite systems it requires further study to exploit possible

applications.

Index Terms—Channel Capacity, MIMO, MIMO Channel

Model, Multiple Access Channel, Multiuser, Satellite MIMO.

I. INTRODUCTION

ADIO Communication systems have been affected since

the beginning by fading, i.e. time fluctuations of the

received signal strength. The first attempts to solve the

problem was to design a receiving amplifier with variable gain

in order to compensate for the time signal fluctuations. In any

case to guarantee at the receiver a Signal-to-Noise Ratio

(SNR) greater than a given acceptable threshold for most of

the time, the transmitter emits a power much greater than

needed in the absence of fading. In 1931 Peterson, Beverage

and Moore [1,2] proposed a new solution based on diversity as

stated in [1]: Diversity is the principle whereby advantages taken of the fact that fading does not occur simultaneously on: A) parallel channel of different frequencies; B) antennas of different polarizations on one frequency; or C) on spaced antennas of one polarization on one frequency.

The two papers [1, 2] proposed two different solutions based

on space diversity for radiotelegraphy and radiotelephony. For

radiotelegraphy system, which uses a binary digital

transmission at 50 baud, signal combining was used, while for

radiotelephony systems, which uses an analog transmission

with a bandwidth of 10kHz, signal selection techniques was

used. The radio transmission was with a carrier at 20 MHz for

the link between Europe and New York and with a carrier of

20.455 MHz during the day and 8.809 MHz during the night

for the radio link between New York and Buenos Aires. At the

receiver the antenna spacing was 150 wavelengths. These

papers shown that a rough Multiple Output (MO) receiver for

both digital and analog radio transmission were available and

operating since 1931.

In 1959 Brennan [3] proposed a general performance analysis

of combining techniques for diversity reception of signals.

Next, Multiple Input (MI) systems were studied at MIT

Manuscript received July 18, 2012. This work was supported by the

Department of Information Engineering of the University of Padua.

S. Pupolin is with the Department of Information Engineering, University

of Padua, Padua, Italy. (e-mail: pupolin@dei.unipd.it).

Lincoln Lab. for radar applications and developed by

associated companies [4-8]. The adopted scheme was named

Phased Arrays radar. The main improvement with respect to

the receiver built by Beverage et al. was the antenna spacing

reduced to half a wavelength, called critical distance. A large

array was designed and installed at Millstone Hill, Westford,

Massachusetts. The system were composed of 5,000 UHF

antennas array at 900 MHz. Also it enables to use more than

one power amplifier to feed the antenna arrays. So that it had

a great power, great receiving aperture and rapid wide angle

scan capability because of electronic beamforming in place of

rotating antenna.

Years later the phased arrays proposed for radar have been

applied to the base stations of mobile communications systems

in order to dynamically change the sector coverage. The

presence of a dense multipath in mobile communications

forces to modify the design of phased arrays used for radar.

The new system was named smart antennas and operates in

order to make a dynamic sector coverage in the downlink and

to collect useful signals coming from different directions in

the uplink. Also in the uplink it tries to reduce the interference

from other users in the same band designing nulls in the

radiation pattern in correspondence of the Direction of Arrival

(DoA) of the interfering signals. This system was designed at

the base station for mobile terminals having only one antenna.

The rapid evolution toward mobile terminals with multiple

antennas stopped this intermediate solution.

Exploiting the full usage of multiple antennas both at

transmitter and receiver sites in terms of channel capacity was

done in the pioneering papers by Foschini and Gans [9] and

Telatar [10]. They showed that the capacity increases linearly

with the minimum number of transmit and receive antennas in

case of independent fading among all the channels involved in

the transmission.

As it could be seen by reading the papers by Beverage et al.

[1,2] the characterization of the radio channel is a key factor

for designing the architecture of radio systems. Some

parameters are needed as the average attenuation versus

distance, the distribution of fade duration, etc. The random

time varying characteristic of radio channel have been

modeled both in a general theoretical study of random time

varying channel impulse response and by a model

characterizing the behavior of attenuation vs. distance.

Bello [11] proposed a theoretical model to analyze the

behavior of the received signal when it passes through a

channel with random variations. He introduced the definitions

of terms like: Wide Sense Stationary (WSS), Uncorrelated

Scatterers (US), and Wide Sense Stationary Uncorrelated

Scatterers (WSSUS), channels, Power Delay Profile, Doppler

MIMO Systems

Silvano Pupolin, Senior Member

R

978-1-4673-4688-7/12/$31.00 ©2012 IEEE

Lecture # 11

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Power Density Spectrum, Delay Spread, and Doppler

Bandwidth, that was derived from the general theory

developed.

The pioneering report by Longley and Rice [12] addresses a

new method to characterize and model a radio link. The

channel model derived was applicable only to narrowband

communications in a frequency range going from 20 MHz to

1000 MHz and for the first time introduces the concept of

multiplicative fading due to i) large obstacles (shadowing)

which gives a constant attenuations by moving the terminals in

a range of tens of wavelengths, and due to ii) local scatterers

around the receiver (fading) which produces a attenuation

change within a fraction of wavelength. Also it gives an

expression for the average attenuation vs. distance. The model

has been later modified to introduce multipath in wideband

digital communication where frequency selective fading is

another dominant factor to design the radio system [13].

The application papers for the characterization of channel

models were mainly devoted to terrestrial radio links. Satellite

systems which uses highly directive antennas between fixed

points had a limited study. With the proposal to use land

mobile satellite systems the necessity to understand the

behavior of the satellite channel to permit a good Quality of

Service (QoS) becomes a must. The mobile terminal had a

dipole antenna and the effects of multipath need to be studied

[14].

The above studies have been boosted after the discovery of

Turbo Codes [15-16] and a revision with application of Low

Density Parity Check (LDPC) Codes [17-18] which enable to

get bit rates close to the Shannon Capacity Limit of a link.

In recent years Multiple Input Multiple Output (MIMO)

systems have been deeply analyzed in several different

operating conditions. Most papers were devoted to land

mobile communications, but satellite communications got also

benefits from MIMO systems.

The paper is organized as follows: Section II is devoted to the

general theory of MIMO systems showing that we could

increases the link capacity by increasing the number of

available antennas. The Section is split in three parts, the first

devoted to single user and the others to multiuser as Multiple

Access Channel (MAC) and Broadcast Channel (BC). Section

III will show some practical limits due to the fact that to reach

the capacity limit we need a perfect knowledge of the

Channel State Information (CSI). Here the effects of CSI is

taken into account for the multiuser environments.

Section IV is devoted to MIMO channel model while Section

V to MIMO for satellite users. Section VI concludes the paper.

Fig. 1. Multiuser MIMO Channel model

II. MIMO SYSTEM CAPACITY

The MIMO channel model that we will use hereafter is shown

in Fig.1 for the multiuser MAC case. Let Nt,i be the number of

transmit antennas of the user i, Nr the number of receiving

antennas, hi the Nt,i x Nr matrix representing the i-th channel

response connecting user i to the receiver, K the number of

users, w the Nrx1 additive white circularly symmetric complex

Gaussian noise vector. In Fig.1 ui, i!K, K={1, 2, …, K},

represent the Nt,ix1 source data vectors, vi, i!K, represent the

Nrx1received data vectors from user i, v is the only

measurable received Nrx1vector, and ûi, i!K, represent the

Nt,ix1 i-th estimated received data vector.

Here we assume the channel is stationary and in the presence

of flat frequency behavior, so that only one tap gain

characterizes the channel.

A. Single User

1. Deterministic Channel

We recall the performance, in terms of capacity, of MIMO

systems in the presence of known and constant channel matrix

h. The Output-Input relationship is:

v = hu + w (1)

Matrix h, by using the Singular Value Decomposition (SVD),

can be decomposed as1:

h = UΛVH, (2)

where U!CNrxNr and V!C

NtxNt are unitary matrices, and

Λ!RNrxNt is a rectangular matrix with diagonal entries real

non-negative numbers λ1≥ λ2≥….≥ λnm≥0, where Nm=min(Nt,

Nr). We assume u a zero mean vector with covariance

Q=E[uHu]. The mutual information is given by [10]:

I(u;v) = log det(INr + hQhH) (3)

and the channel capacity C is obtained by maximizing (3) with

respect to u to obtain:

(4)

where [x]+ =max (x;0), is diagonal with entries

, µ is chosen to satisfy , and P

is the total available power.

2. Rayleigh fading channel

We assume that the matrix h is random independent of both u

and v. hi,j∈CN(mi,j,σ2

i,j) are mutually independent. Also, h is

known at the receiver. The mutual information is given by:

I[u;(v,h)] = I(u;h) + I(u;v|h) = Eh{ I(u;v|h=h)} (5)

where I(u;h) = 0 being u and h mutually independent random

1 Hereafter the following notations will be adopted. The superscripts (•)T,

(•)*, (•)H, (•)-1 represent transpose, conjugate, transpose conjugate and matrix

inversion, respectively. Given a matrix A, det(A), tr(A), Ai,j denote the

determinat, the trace and the (i,j) entry of matrix A. The function vec(A)

converts matrix A in a column vector by stacking its columns one below the

others. C and R are the complex and real sets. CN(m,σ2) denotes circular

symmetric Gaussian random variable.

Transmitter 2

Channel

1

Channel

2

Channel K

u1

u2

uK

v1

v2

vK

Σ v

Receiver

û

1

û

2

û

Transmitter K

Transmitter 1

Lecture # 11

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vectors.

If u is constrained to have a covariance matrix Q subject to

tr(Q) ≤ P the maximum of I with respect to u is obtained

when Q=(P/Nt)INt. Let Note that

C(h) is a random variable with Cumulative Distribution

Function (CDF) FC(a). Here we could give two formulation

for the capacity [19]. The first named ergodic capacity, is the

average value C(h). The second, named capacity versus

outage, is the value Co such that FC(Co)=Pout , where Pout is the

outage probability, i.e.: the probability for which the channel

capacity is less than Co. Pout is established in order to

guarantee the final user has a sufficient capacity to guarantee

the QoS. Hereafter, because of limit of space, we limit

ourselves to consider only ergodic capacity. Then, the ergodic

channel capacity is given by:

C=E{log det[INr+(P/Nt)hhH)]}=E{log det[INt+(P/Nt)h

Hh)]} (6)

After tedious algebra [10], C is given by:

where NM=max(Nt,Nr) and are the associated Laguerre

polynomials In [10] it has been

shown that for Nt=1 the capacity for large Nr becomes

asymptotically equal to log(1+PNr), while for Nr=1 as Nt

increases the capacity approaches log(1+P), which is the

capacity of a deterministic channel.

B. Multiple Access Channel

We refer to Fig.1 for the analysis of MIMO-MAC systems.

1. Deterministic Channels

The ergodic Mutual Information Region (MIR) of Gaussian

MIMO-MAC, CMAC, was derived in [20] for a given set of

transmission powers P=[P1, P2, … , PK]. Let

CS=½ log det [I+ ] , S⊆K. (7)

CMAC(P,h)=

K}

(8)

The ergodic MIR we get is a K-dimensional polyhedron. The

constrained maximum value CSM of CS, vs. Qi with

is called Sum Mutual Information (SMI) or

shortly capacity.

2. Rayleigh Fading Channels

In the presence of fading channels and of perfect knowledge

of the channel at the receiver, the ergodic capacity region is

obtained as the expectation of (8) with respect to the pdf of the

channel matrices hi which elements are mutually independent

CNs. In [10] it has been shown that when the transmitters

have the same number of antennas, i.e.: Nt=Nt,i, i!K, CSM

increases linearly with min (Nr, K Nt). In case of a large

number of users K, the minimum value is Nt, so CSM increases

linearly with Nt. [21] presents a clear overview on MIMO

capacity.

C. Broadcast Channel

The broadcast channel (BC) is the dual of MAC. Indeed we

are in the presence of one transmitter and K receivers. Even in

this case there is a capacity region as in MAC. In [22] it has

been shown that the ergodic SMI of the BC equals the SMI of

the MAC with the same total average power constraint. Then,

all the results derived in paragraph B) above apply to BC.

III. MIMO SYSTEM WITH PARTIAL CSI

The estimate of CSI at receiver reduces the channel capacity

due to estimation errors. Hereafter we consider a training

based LMMSE channel estimation at the receiver in the

presence of flat block fading, and we limit our analysis to

MIMO MAC systems.

A. MIMO Multiple Access Channel

The i-th channel estimate is given by:

(9)

where, during the training, Pτ,i is the power, αi=Pτ,i/ the

SNR, and Ri=E[vec(hi) vec(hi)H] of user i. For the achievable

SMI only bounds are available. The SMI lower bound, which

is closed to the SMI, is [23-24]:

(10)

where, during data transmission, Pd,i is the power, βi=Pd,i/// is

the SNR, , is the channel

estimation error correlation matrix of user i.

B. Applications

We assume that the channel matrices hi be correlated and that

the correlation depends only from the transmit Rt,i, and receive

Rr antennas matrix correlations. Then, we get:

, where =CN(0,1) and the random

variables are mutually independent. In the examples below we

assume Rr(l,m)=φr|l-m|

and Rt,i(l,m)=φt|l-m|

. Fig. 2 emphasizes

the dependency of the SMI on the number of antennas and

users for Nt,i=Nt=Nr!{2, 4}, αi=α=βi=β, K!{2, 4, 8}. Note

that there is a gap in the SMI value reached

Lecture # 11

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Fig.2 SMI for different antenna elements, Nt, Nr, and number of users

K. Transmit and receive antenna correlation coefficients φr=φt=0.5.

Fig.3.SMI vs. α. β=10dB, antenna correlation coefficients φr=φt=0.

with respect to the perfect CSI case. In [24] it has been shown

that this gap vanishes if α<<βKNt/Tτ, where Tτ is the duration

of the training sequence per user in number of symbols. Fig. 3

shows this effect by plotting SMI versus the training SNR α

with the data SNR β as parameter.

IV. MIMO CHANNEL MODEL

Most of the results derived in Sections II and III rely on the

fact that MIMO channel exhibits only flat Rayleigh fading. In

order to verify the applicability of the results so obtained in a

broadband system is mandatory to model accurately the

MIMO channel [25]. The 2-D geometrical channel model,

depicted in Fig.4, is an extension of the classical model [13]

where Q paths modeled as CN mutually independent random

processes describe the broadband frequency selective

channels. In this new model Q clusters are present. The signal

transmitted is reflected/refracted by cluster i (i=1, 2, …, Q)

and received by the receiver. The Angle of Departure (AoD)

Φi, and Arrival (AoA) ϕi, are spread around their average

value with standard deviation of the AoD (AoA) σΦi ( σϕi). Each

cluster is the sum of several plane waves (rays), and is

characterized by common Large Scale parameters: mean

attenuation, shadow fading, average delay, Φi, σΦi, ϕi, σϕi. Each

ray within a cluster has a specific fading (Rice or Rayleigh

distributed) and small scale delay, which is typically much

shorter than large scale delay.

Each cluster is represented with a relative power with

respect to the first one, i.e.: the one with the shortest delay. An

exponential power delay profile is assumed vs. cluster average

Fig. 4. Geometrical MIMO Channel parameters.

delay. Within a cluster there are several paths with relative

powers with exponential power delay profile. The total power

is then normalized to 1. The effective power we receive is

scaled by the average link attenuation and the shadow fading.

This is important in the presence of multiuser since each

mobile terminal is located in different geographical position

having different average attenuation and shadow fading. The

Cluster Delay Line (CDL) channel impulse response is given

by: h(t,τ) = , where the entries of hi are given by:

x (11)

u=1, 2, …, Nr,i; s=1, 2, …, Nt.

where ( ) are the antenna element u field pattern

for vertical (horizontal) polarization, are the complex

gain of vertical-to-vertical (H stands for horizontal)

polarizations of ray (i,m); λ0 is the carrier wavelength, ϕi,m

(Φi,m) is the AoD (AoA) unit vector, ( ) is the

location vector of element u (s), and is the Doppler

frequency of ray (i,m). If the radio channel is dynamic all the

above parameters are function of time t. are

correlated CN, zero mean in case of Rayleigh fading.

V. MIMO FOR SATELLITE USERS

Satellite communications could be splitted in two families:

Geostationary Satellite (GS) systems with a fixed earth station

and Non Geostationary Satellite (NGS) systems with either a

mobile or fixed earth station.

A. Geostationary Satellite

Nowadays GS are operating at frequency bands above 10GHz

in a line of sight (LOS) environment. Link attenuation is given

by a free space attenuation plus extra attenuation due to either

the presence of water in the troposphere in its form as rain,

clouds, ice, etc. or hydrometeor scattering and absorption.

Because of the slow movement of clouds the fading induced is

slow. The spatial correlation of these phenomena, mainly rain,

is high for range up to 100km on the earth [26]. The use of

MIMO systems, which require almost uncorrelated links to

increase the capacity, needs earth stations spaced hundreds

Lecture # 11

5

kilometers and connected each other with high capacity links

in order to process the received signals. The dual solution uses

at least two GSs which should be spaced apart at least 90°[27].

The use of right and left circular polarized antennas does not

solve the problem of diversity because the rain attenuations on

both polarization are highly correlated. The conclusion is that

the use of MIMO for GS systems to increases the capacity of

single user needs a deeper analysis. On the other end multi-

user system could optimize the SMI of the system because of

the space distribution of the users in a large area, so that many

of the links are uncorrelated. In this way there is the

possibility to design appropriated Radio Resource

Management algorithm that take care of the status of the links

for scheduling the services and guarantee the QoS and fairness

among users. In this way it is possible either to guarantee or to

limit the out of service status of a user in the presence of

severe fading and to recover part of the damage by

transmitting data at higher rate when the severe fading is over.

B. Non Geostationary Satellite

NGS are typically used for Land Mobile Communications

(LMC) where the mobile terminals use omnidirectional

antennas. This fact rises a rich multipath near each earth

terminal. Also, in several cases the communication is done in

a non LOS, so shadowing and fading phenomena appears due

to the ground terminal only. Then, the use of multiple

antennas at the earth terminal is beneficial in terms on channel

capacity. Also, polarization antenna diversity at the satellite

transmitter permits a further diversity dimension. In any case,

as reported in the review paper [28], further studies are needed

to improve the satellite channel model to design appropriated

algorithms for MIMO satellite systems able to improve the

spectral efficiency [bit/s/Hz].

VI. CONCLUSION

MIMO systems have been rapidly evolved in the last decade

and are today able to reach bit rates of tens of bit/s/Hz in

terrestrial radio links. The theoretical SMI predicted by

Information Theory with perfect CSI is still far to be reached.

There is still a lot of job which should take care of several

constraints as: i) the rapidity of channel change, ii) the

duplexing technique, iii) QoS, etc. Channel changes require to

insert into the data stream pilot symbols to track it. As it has

been shown in Section III accurate channel estimate is

mandatory to reach the channel SMI. Then the density of pilot

symbols is proportional to the rapidity of channel changes. In

a multiuser scenario each user could have different rapidity of

channel changes so, in order to get more symbols for data, the

densities of pilot symbols per user should be different and

tailored to the user needs. Duplexing techniques could help for

CSI. Time Division Duplexing (TDD) helps CSI since the

same channel is used for both downlink and uplink

communications. The requirement is that the round trip delay

be shorter than the time needed for channel update. Frequency

Division Duplexing (FDD) uses two carriers for uplink and

downlink sufficiently spaced apart to make the fading

processes mutually independent. So, we need separated pilots

for both uplink and downlink. In today wireless systems pilots

have been designed for the maximum tolerable rapidity

channel changes, and they use 5-10% of channel capacity. In

case of bidirectional communications their insertion could be

done on request when the channel estimate is bad. Powerful

channel coding operating close to the channel capacity require

long codewords (several thousands of bits) and, for some

applications in which short packets are mandatory for QoS

limitation, they cannot be used and the system fall well below

the expected SMI.

The future development of MIMO systems is to permit to

satisfy the request of higher capacity from the users by

considering the strong limitation on the available bandwidth.

For terrestrial mobile communications we could either to

reduce the cell size or to increases the bit/s/Hz. As it is well

known for single antenna systems to increase the bit/s/Hz a

large alphabet is used which is contrary to the expectation of

green technology which is looking to reduce the energy/bit

spent. MIMO could help in this direction by increasing both

the spatial dimension (number of antennas) and e.m. field

polarization by keeping the alphabet size small. In this way we

rise the channel capacity without increasing the energy/bit.

This is a new challenges for the MIMO technology.

ACKNOWLEDGMENT

I would like to thank A. Assalini and R. Corvaja for helpful

comments and discussions.

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S. Pupolin (M’72–SM’83) received the Laurea degree in Electronic

Engineering from the University of Padova, Padova, Italy, in 1970.

Since then he joined the Department of Information Engineering,

University of Padova, where he is Full Professor of Electrical

Communications.

He was

• Chairman of the Faculty of Electronic Engineering (1990-1994),

• Chairman of the PhD Course in Electronics and

Telecommunications Engineering (1991-1997), (2003-04)

• Director of the PhD School in Information Engineering (2004-

2007),

• Chairman of the board of the Directors of the PhD Schools of the

University of Padova (2005-2007)

• Member of the programming and development committee of the

University of Padova (1997-2002),

• Member of Scientific Committee of the University of Padova

(1996-2001),

• Member of the budget Committee of the Faculty of Engineering of

the University of Padova (2003-2009)

• Member of the Board of Governor of the CNIT “Italian National

Interuniversity Consortium for Telecommunications” (1996-99),

(2004- 2007)

• Director of CNIT (2008 - 2010 )

• Director Dept. Quantum and Radio Communications of CNVR

(Consorzio Veneto di Ricerca) (2011- )

He was General Chair of the 9-th, 10-th and 18-th Tyrrhenian

International Workshop on Digital Communications devoted to

"Broadband Wireless Communications", "Multimedia

Communications", and “Wireless Communications”, respectively;

General Chair of the International Symposium “Wireless Personal

Multimedia Communications (WPMC’04)” Abano Terme, Padova ,

Italy, September 2004.

He spent the summer 1985 at AT&T Bell Laboratories on leave from

Padova, doing research on digital radio systems.

Actually, is actively engaged in researches on broadband mobile

communication systems, personal communication systems, MIMO

systems and applications.