MIMO
Ali Mohydeen
L'UNIVERSITE DE NANTES
COMUE UNIVERSITE BRETAGNE LOIRE
ECOLE DOCTORALE N° 601
Mathématiques et Sciences et Technologies
de l'Information et de la Communication
Spécialité : Électronique
Par
Ali MOHYDEEN
Contributions to MIMO channel parameter estimation
Composition du Jury :
First of all, I would like to thank my Ph.D supervisor, Professor Pascal Chargé, for
his guidance, support and encouragement. I am so grateful for all the valuable and
fruitful discussions we have had throughout the Ph.D period. I am also grateful to my
cosupervisors, Professor Yide Wang and Professor Oussama Bazzi, for their support
and encouragement.
I would also like to thank the administrative and technical staff of the IETR Labo
ratory for their administrative and technical support during the Ph.D period.
3
Contents
List of Figures 9
List of Abbreviations 13
Notations 17
General Introduction 19
5
6 Contents
Publications 127
Bibliography 129
List of Figures
4.1 RMSE of mean delay estimation of the modified SOMP method and the
classical SOMP method versus SNR; Pl = 20, K = 64, L = 3, vector
of chosen delays is given as t = [0.32 0.45 0.61]T , τlp are uniformly
distributed with 0.008T chosen as the standard deviation of the delay
spreading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
9
10 List of Figures
4.2 RMSE of mean delay estimation of the modified SOMP method and
the classical SOMP method versus standard deviation of delay spread
ing; Pl = 20, K = 64, L = 3, vector of chosen delays is given as
t = [0.32 0.45 0.61]T , τlp are uniformly distributed, SNR=15 dB. . . 94
4.3 RMSE of mean delay estimation of the proposed cost function for U = 1,
U = 2 and MUSIC versus SNR; Pl = 20, K = 64, τlp are uniformly
distributed with 0.005T chosen as the standard deviation of the delay
spreading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.4 RMSE of mean delay estimation of the proposed cost function for U = 1,
U = 2, U = 3, and MUSIC versus standard deviation of delay spreading;
Pl = 20, K = 64, SNR = 5 dB. . . . . . . . . . . . . . . . . . . . . . . 97
4.5 Mean value of the MDL criterion (mean M DLV ) versus standard devi
ation of delay spreading; Pl = 20, K = 64, SNR = 5 dB, mean M DLV
values are obtained from 500 independent simulations each. . . . . . . . 98
4.6 RMSE of mean delay estimation of the proposed cost function for U = 1,
U = 2 and U = 3, the proposed subspace tracking based method and
MUSIC versus standard deviation of delay spreading; Pl = 20, K = 64,
SNR = 5 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.7 RMSE of mean delay estimation of the proposed cost function for U = 1,
U = 2, and U = 3, the proposed subspace tracking based method and
MUSIC versus standard deviation of delay spreading; Pl = 20, K = 64,
SNR = 15 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.8 Mean M DLV versus standard deviation of delay spreading; Pl = 20,
K = 64, SNR = 15 dB . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.9 RMSE of mean delay estimation of the proposed subspace tracking based
method and MUSIC versus standard deviation of delay spreading; Pl =
20, K = 64, SNR = 5, 10, 15 dB. . . . . . . . . . . . . . . . . . . . . . . 102
4.10 RMSE of mean delay estimation of the proposed cost function for U =
1, U = 2 and U = 3, the proposed subspace tracking based method
and MUSIC versus standard deviation of delay spreading; L = 2, t =
[0.37 0.51]T , Pl = 20, K = 64, SNR = 15 dB. . . . . . . . . . . . . . . 103
4.11 RMSE of mean delay estimation of the proposed cost function for U =
1, U = 2 and U = 3, the proposed subspace tracking based method
and MUSIC versus standard deviation of delay spreading; L = 3, t =
[0.37 0.51 0.67]T , Pl = 20, K = 64, SNR = 15 dB. . . . . . . . . . . 104
4.12 RMSE of mean delay estimation of the modified SOMP method, the pro
posed subspace tracking based method, SOMP method and MUSIC ver
sus standard deviation of delay spreading; L = 3, t = [0.37 0.51 0.67]T ,
Pl = 20, K = 64, SNR = 15 dB. . . . . . . . . . . . . . . . . . . . . . . 105
List of Figures 11
4G Fourth Generation
5G Fifth Generation
AIC Akaike Information Criterion
AOA Angle of Arrival
AWGN Additive White Gaussian Noise
BER Bit Error Rate
CIR Channel Impulse Response
CS Compressive Sensing
CSI Channel State Information
CSIR Channel State Information at the Receiver
CSIT Channel State Information at the Transmitter
DFT Discrete Fourier Transform
EGC Equal Gain Combining
EM ElectroMagnetic
ESPRIT Estimation of Signal Parameter via Rotational Invariance
GSCM Geometrybased Stochastic Channel Model
ISI InterSymbol Interference
LS Least Squares
MDL Minimum Description Length
MIMO MultipleInput MultipleOutput
MISO MultipleInput SingleOutput
13
14 List of Abbreviations
ML Maximum Likelihood
MMSE Minimum Mean Square Error
MMV Multiple Measurement Vector
MMW Millimeter Wave
MP Matching Pursuit
MPC Multipath Component
MRC Maximum Ratio Combining
MUSIC MUltiple SIgnal Classification
MVDR Minimum Variance Distortionless Response
OFDM Orthogonal Frequency Division Multiplexing
OMP Orthogonal Matching Pursuit
PDP Power Delay Profile
RIP Restricted Isometry Property
RMS Root Mean Square
RMSE Root Mean Square Error
SC Selection Combining
SIC Successive Interference Cancellation
SIMO SingleInput MultipleOutput
SISO SingleInput SingleOutput
SMV Single Measurement Vector
SNR SignaltoNoise Ratio
SOMP Simultaneous Orthogonal Matching Pursuit
SVD Singular Value Decomposition
TOA Time of Arrival
US Uncorrelated Scattering
UWB UltraWideBand
VCR Virtual Channel Representation
WSS Wide Sense Stationary
15
x Scalar
x Vector
X Matrix
[.]T Transpose
[.]∗ Conjugate
[.]H Conjugate transpose
[.]−1 Inverse
[.]† Pseudo inverse
E[.] Expectation
. Absolute value
.p lp norm
C Set of complex numbers
diag{x1 . . . xn } n × n diagonal matrix with the elements x1 . . . xn on its diagonal
In n × n identity matrix
det[X] Determinant of X
Hadamard product
⊗ Kronecker product
17
General Introduction
In the ancient world, light and flags were used as a way of wireless communication. In
1867, James Clerk Maxwell predicted the existence of electromagnetic (EM) waves,
proposing an interrelation between electric and magnetic fields. In 1887, Heinrich
Rudolf Hertz confirmed the existence of EM waves traveling at the speed of light by
performing experiments in his laboratory. The waves he produced and received are
now called radio waves. A breakthrough came with Guglielmo Marconi who developed
the wireless telegraph in 1895. Since then, he succeeded in transmitting radio signals
through space, increasing the distance of communication gradually. In 1901, he estab
lished the first wireless communication across the ocean, by transmitting radio signals
across the Atlantic ocean. From then until today, different wireless technologies have
been developed, including radio and television broadcasting, radar communications,
satellite communications, wireless networking, mobile wireless communications, etc.
With the rise of big data era, along with the increasing demand of wireless data
services, the major goal of researchers over the years has been to support high data
rates to satisfy needs. A major obstacle to build a reliable high speed wireless commu
nication system is the wireless propagation medium. In wireless communication, the
signal propagating through the wireless channel is exposed to different types of fading,
especially multipath fading due to multipath propagation [1]. This impacts the relia
bility of the communication link and limits the data rate.
19
20 General Introduction
The benefits of MIMO are achieved through the exploitation of the spatial dimension
across multiple antennas at the transmitter and receiver, in addition to the time and
frequency dimensions already exploited in the conventional singleinput singleoutput
(SISO) systems [2].
MIMO, along with orthogonal frequency division multiplexing (OFDM), are key
technologies used in 4G (fourth generation) wireless networks. MIMO is a key technol
ogy for the next fifth generation (5G) wireless networks that employ massive antenna
arrays and millimeter wave (MMW) frequencies.
Knowledge of the wireless propagation channel characteristics is crucial for the re
liability of wireless communications, especially in MIMO communications in order to
fully benefit from the advantages provided by using multiple antennas at the transmit
ter and receiver sides. This information about the channel characteristics is referred
to as channel state information (CSI). CSI represents the information about the signal
propagation from the transmitter to the receiver, it represents wireless channel effects
such as power attenuation and time spreading of signals. CSI plays an important role in
the system performance, channel state information at the receiver (CSIR) can be used
for equalization purpose against intersymbol interference (ISI) caused by multipath
propagation, and channel state information at the transmitter (CSIT) can be used for
optimal transmission design. Hence a performant “ideal” MIMO communication sys
tem would require an exact knowledge of the MIMO channel or CSI. CSI estimation
approaches can be classified into parametric and nonparametric. In the nonparametric
approach, the channel matrix is estimated directly without referring to any underlying
physical propagation parameters. On the other hand, the parametric approach relies on
physical channel models to estimate channel parameters, such parameters are useful for
understanding the wireless channel, and can be utilized to improve the communication
system performance by adapting the transmission and reception designs according to
them.
The work in this thesis focuses on clustered MIMO channel parameter estimation,
21
specifically, time domain parameters. This thesis is divided into five chapters. The
first part of the first chapter represents the basic wireless channel propagation charac
teristics, especially, fading in wireless channels. Fading channel is classified into large
scale fading and small scale fading, large scale fading characterizes the channel behav
ior over large distances and incorporates path loss and shadowing. Small scale fading
characterizes the channel behavior over short time periods or travel distances, and is fur
ther classified into two categories based on multipath delay spread and doppler spread.
Based on multipath delay spread, fading is classified into flat fading and frequency se
lective fading. Based on doppler spread, fading is classified into slow fading and fast
fading. In the second part, we introduce diversity techniques in wirless communication
systems, and we focus on spatial diversity, by showing the benefits brought by using
multiple transmit or/and receive antennas.
Chapter 4 illustrates two proposed channel mean path (cluster) delays estimation
approaches. The first approach is based on the first order Taylor expansion around
the mean delay parameter, where a compressive sensing based method is proposed to
estimate the channel mean delays. And the second approach is based on higher order
Taylor expansion around the mean delay parameter, where a method for estimating
the channel mean delays is then proposed based on the subspace approach, and on the
tracking of the effective dimension of the signal subspace. The proposed methods are
validated through computer simulations and estimation performance is illustrated.
22 General Introduction
1.1 Introduction
Establishing a reliable wireless communication system requires deep understanding of
wireless channel propagation models and characteristics. Several factors are involved
in the process of determining the channel behavior: signal/channel bandwidth, envi
ronment or propagation medium, noise, etc. Due to these factors, a signal transmitted
through the environment exhibits fluctuation and attenuation in its level at the receiver
side. The phenomenon of fluctuation of the attenuation of the signal level, is referred
to as “fading”. In wireless channels, a signal emitted from a transmit antenna arrives
at the receive antenna through multiple paths (Figure 1.1) with different amplitudes,
phase shifts, and delays due to the reflection, diffraction or scattering of electromagnetic
waves in the environment. This results in a constructive and destructive interference of
signals from the different paths, leading to the fluctuation of the received signal level.
This phenomenon is the socalled “multipath fading”, and it has a significant impact
on the reliability and performance of wireless communication.
Mainly, two types of fading characterize a wireless channel: large scale fading and small
scale fading [1, 3–7]. Large scale fading refers to the signal power attenuation and fluc
tuation due to path loss and shadowing. Path loss refers to power loss due to the
propagation over large distances. Shadowing is when signals are interrupted or blocked
by large objects such as mountains and buildings over the propagation path between
the transmitter and the receiver. Shadowing results in relatively slow fluctuation in
23
Chapter 1 Basic wireless channel propagation characteristics and introduction to
24 multiple antenna systems
the signal level, its effect depends on the dimension of objects in the environment with
respect to the wavelength or the radio frequency of the electromagnetic waves. Three
basic phenomena, characterize wireless signal propagation: reflection, diffraction, and
scattering.
Small scale fading refers to the rapid fluctuation of the signal over short periods or
short travel distances. The small scale fading scheme can be divided into two categories.
The first one is related to the multipath delay spread, and the second is related to the
Doppler spread. Within each category, depending on the relation between the different
signal and channel parameters, different signals exhibit different types of fading. Figure
1.2 illustrates the different classifications of a wireless fading channel.
1.2 Flat fading 25
For small scale fading based on the multipath time delay spread, we have two types
of fading, flat fading and frequency selective fading.
L0
X
r(t) = Re{ γl sb (t − τl )ej2πfc (t−τl ) }
l=1
L0
X
= Re{( γl sb (t − τl )e−j2πfc τl )ej2πfc t } (1.3)
l=1
where sb (t) is the transmitted baseband signal, γl is the attenuation of path l and τl is
the corresponding time delay.
L0
X
rb (t) = γl sb (t − τl )e−j2πfc τl (1.4)
l=1
Assuming that we are dealing with narrowband signals, the narrowband assumption
states that sb (t−τl ) ≈ sb (t) for all l. This is related to the limited time resolution due to
signal’s narrow bandwith in comparison with the channel coherence bandwidth. Limited
time resolution means that signals from different paths cannot be distinguishable, as
this time resolution is inversely proportional to the signal bandwidth. According to
this, rb (t) is given as :
L0
X
rb (t) = sb (t) γl e−j2πfc τl (1.5)
l=1
Hence
L0
X
h= γl e−j2πfc τl (1.6)
l=1
is the complex channel coefficient. Depending on the values of the time delays τl , the
different propagation paths can add up constructively or destructively causing fading.
Constructive interference amplifies the signal amplitude at the receiver, while destruc
tive interference attenuates the signal amplitude. An example of this phenomenon is
illustrated in Figure 1.3, where it is assumed that at each time instant, the receive
antenna (for example mobile user) is changing its position, where the power of the
1.2 Flat fading 27
received fading signal versus time is shown. For each time instant, the channel coeffi
cient h is given as the constructive or destructive sum of a large number of multipath
components.
10
5
10
15
20
25
30
35
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Figure 1.3: Received signal power versus time, multipath fading channel
As we can see, for the case of flat fading, at a given instant t, the channel is seen as
a single coefficient (h), in other words, it is not described by its different propagation
path delays, the channel impulse response is seen as a single impulse. Due to this, the
channel over the given signal bandwidth looks flat in the frequency domain, that is, all
the frequency components of the signal will experience the same level of fading.
L0
X
h= γl e−j2πfc τl
l=1
L0
X L0
X
= γl cos(2πfc τl ) − j γl sin(2πfc τl )
l=1 l=1
= Xr + jXi (1.7)
PL0 PL0
where Xr = l=1 γl cos(2πfc τl ) and Xi = − l=1 γl sin(2πfc τl ).
p Xi
where γ = Xr2 + Xi2 is the magnitude and φ = tan−1 ( Xr
) is the phase.
Assuming that Xr and Xi are zero mean independent random variables with same
2
variance σX . It follows that γ is Rayleigh distributed with the following probability
density function
2
γ −γ2
fγ (γ) = 2 e 2σX (1.9)
σX
2
The Rayleigh probability density function is shown in Figure 1.4 for different σX .
1.3 Frequency selective fading 29
1.4
1.2
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
The Rayleigh fading is reasonable when there is no line of sight path between the
transmitter and receiver. When the line of sight path exists, Rician fading becomes a
more reasonable model.
If the channel is time invariant (static), the channel impulse response (CIR) is given
as:
L
X
h(τ ) = αl δ(τ − τl ) (1.11)
l=1
where L is the number of resolvable path delays, τl is the time delay of path l and αl
is the corresponding complex gain coefficient (including phase parameter).
The number of resolvable multipaths depends on the available time resolution (or
signal bandwidth). Consider Figure 1.5 that illustrates the signals scattering from
different boundaries (ellipses).
these are seen as one another path, and so on. Now if a sufficient number of unre
solvable multipath components falls within the intervals ([0, ∆τ ], [∆τ, 2∆τ ], ...), then
the envelope of each channel gain coefficient αl  within these intervals can be Rayleigh
distributed (assuming no line of sight) with uniformly distributed phase.
where E[αl 2 ] is the power associated to path delay τl , Rhh (τ ) is the channel autocor
relation function.
The main parameters characterizing the PDP are the mean delay, the root mean
square (RMS) delay spread, and the maximum excess delay (or maximum delay spread).
The mean delay is given as:
PL
τl Pdp (τl )
τ̄ = Pl=1
L
(1.13)
l=1 Pdp (τl )
τmax = τL − τ1 (1.15)
If the channel coefficients at any two distinct delays τ1 and τ2 are uncorrelated, such
that the channel autocorrelation function at these two delays is given as:
Similarly to what is done in the delay domain, the statistical properties of H(f )
can be obtained in the frequency domain, through the autocorrelation function. The
correlation between frequency domain channel components at two distinct frequencies
f1 and f2 is given as:
Let ∆f = f1 − f2 , we have
Z +∞
RHH (∆f ) = Rhh (τ )e−j2π∆f τ dτ (1.19)
−∞
which is the Fourier transform of the channel PDP.
RHH (∆f ) gives information about the range of frequencies over which the fading
pattern is highly correlated. This frequency range, over which the channel is considered
flat, is the coherence bandwidth (Bc ), that is, the frequency components of the signal
over this range, will experience the same fading pattern.
The value of Bc is inversely proportional to the channel delay spread στ , where gen
erally, the relation is given as Bc ≈ 1/στ . However there is no one universal definition.
There are two common definitions that are based on the least correlation value assigned
for the frequency correlation function RHH (∆f ). If the coherence bandwidth is defined
1.3 Frequency selective fading 33
as the frequency range over which RHH (∆f )/RHH (0) is at least 0.9, then it is given
as [3]:
1
Bc = (1.20)
50στ
However, if the coherence bandwidth is defined as the frequency range over which
RHH (∆f )/RHH (0) is at least 0.5, it is given as:
1
Bc = (1.21)
5στ
The main impairment introduced in the case of frequency selective fading is inter
symbol interference, that is, when the signal bandwidth is greater than the channel
coherence Bc or in other words, when the symbol period (Ts ≈ 1/Bs ) is significantly
less than the channel delay spread στ , multiple copies of symbols (with different gains)
arrive at different time delays causing different symbols to interfere with each other,
resulting in signal distortion. This is not the case for flat fading, where the symbol
period Ts is significantly greater than the channel delay spread στ such that different
multipath components arrive almost at the same time and the channel is modeled as a
single complex coefficient.
Figure 1.6 illustrates the flat fading and the frequency selective fading phenomena.
As shown, for the case of flat fading (Bs < Bc ), signal’s frequency components undergo
highly correlated fading pattern, while for frequency selective fading (Bs > Bc ), signal’s
frequency components undergo different fading patterns, causing signal distortion.
Chapter 1 Basic wireless channel propagation characteristics and introduction to
34 multiple antenna systems
fr = fc + fd (1.22)
where fc is the carrier frequency and fd the doppler frequency shift given as
ϑ cos θ
fd = fc (1.23)
c
where c is the speed of light and θ is the angle between the direction of motion and the
base station.
Consider again the flat fading case, if there is no motion between the transmitter
and the receiver, the channel coefficient is given as in (1.6). Now, as the mobile station
(transmitter) is moving toward the base station (receiver), after some time t has passed,
the distance decreases by (ϑ cos θ)t, the previous propagation delay (in the case where
there was no motion) for a given path l will also decrease as:
ϑ cos θl
τl (t) = τl − t (1.24)
c
Hence the time varying channel coefficient h(t) is given as:
L0
X
h(t) = γl e−j2πfc τl (t)
l=1
L0
X ϑ cos θl
= γl e−j2πfc (τl − c
t)
(1.25)
l=1
In fact, in the above equation, channel amplitudes, delays, and doppler shifts are
assumed to be approximately static (slow varying with time), however, the different
channel parameters γl , τl and θl themselves can vary with time. To avoid confusion
about time varying time delay, in (1.24), the time varying delay parameter will be
1.4 Doppler effect 35
0
written as τl (t) = τl (t) − ϑ coscθl (t) t, where τl (t) here (different from τl (t)) in (1.24)) is
the delay of the channel at time t (without doppler, due to the nature of the channel),
0
τl (t) is the overall delay at time t when doppler is included (this means that τl (t) will
0
be replaced by τl (t) in (1.25)), and the doppler frequency shift corresponding to path
l is fdl (t) = ϑ coscθl (t) fc . Channel coefficient h(t) can be generalized and written as:
L0
X
h(t) = γl (t)e−jφl (t) (1.26)
l=1
where φl (t) is the phase parameter including the contribution of doppler in addition to
the phase due to the propagation delay.
For the frequency selective case, the channel response is given as in (1.10), where
the phase of αl (t) contains also the contribution of doppler (each αl (t) is of the form of
h(t) in(1.26)).
The doppler spread is sometimes called the channel fading rate or fading bandwidth.
The key parameter related to doppler effect is the channel coherent time (Tc ). Tc is
the time interval over which the channel is almost invariant (static). Tc is inversely
proportional to fdmax and it can have different definitions when it is related to it. A
first approximative relationship between the coherence time and doppler spread is given
as:
1
Tc ≈ (1.27)
fdmax
where fdmax is the maximum doppler shift given as fdmax = ϑc fc .
Consider a frequency selective channel, the time varying channel complex coefficient
is given as:
αl (t) = αl (t)e−jφl (t) (1.28)
Assuming that the channel amplitudes, delays, and doppler shifts are approximately
static, hence we have αl (t) ≈ αl  and
Z π
1 −j2πfd max cosθl ∆t
Rαl (t, t + ∆t) = Pαl e dθl
0 π
= Pαl J0 (2πfd max ∆t) = Rαl (∆t) (1.31)
We can notice that the autocorrelation function depends on ∆t and not on t. Hence
the channel response is wide sense stationary (WSS). This WSS model combined with
the US model, the overall model is called wide sense stationary uncorrelated scattering
(WSSUS) model.
Now if Tc is defined as the time interval over which Rαl (∆t)/Rαl (0) is greater than
0.5, then Tc is given as [3]:
9
Tc ≈ (1.32)
16πfdmax
In the frequency domain, the doppler power spectrum of the channel is given as:
Z +∞
Rαl (∆t) −j2πf ∆t
S(f ) = e d∆t
−∞ Rαl (0)
r1 , if f  6 fdmax
f
πfdmax 1−( f )2
= dmax (1.33)
0 otherwise
hence the doppler spread is the frequency range over which S(f ) is nonzero. The above
doppler spectrum is referred to as the Jakes’ spectrum. Due to the doppler shift, the
propagating signals exhibit what is called “spectral broadening”.
1.4 Doppler effect 37
Based on the value of Tc relative to the symbol period Ts , we have two types of fad
ing: slow fading and fast fading. The fading phenomenon is referred to as slow fading
when the transmitted symbol period is less than the channel coherence time, that is
Ts < Tc (or Bs > fdmax ), while when Ts > Tc ( Bs < fdmax ), it is referred to as fast fading.
The channel coherence time has an important impact on the wireless communica
tion system design, since it affects the rate of channel estimation procedure that should
be done at the receiver.
Fading has a big impact on the amount of information that can be sent over a
channel, the maximum data rate that a communication channel can support with small
error probability is referred to as the channel capacity.
The capacity of an additive white Gaussian noise (AWGN) channel without fading
is given as
With the effect of fading, the channel capacity of a flat fading channel is given as
Diversity is defined as the scheme or technique that can be used to improve the com
munication reliability against fading problem in wireless communications. Diversity
techniques can be classified into three main categories: Time diversity, frequency di
versity and spatial diversity (or antenna diversity).
In time diversity, multiple copies of the same signal are transmitted on different
time slots seperated at least by the channel coherence time Tc , in order to assure that
the signals will exhibit uncorrelated fadings. The disadvantage of time diversity is that
different time slots are used to send the same data, hence resulting in lower data rate.
In frequency diversity, multiple copies of the same signal are transmitted on different
carrier frequencies seperated at least by the channel coherence bandwidth Bc to assure
uncorrelated fadings. The disadvantage of frequency diversity is that different frequency
bands are used to send the same data, hence requiring extra bandwidth and therefore
decreasing spectral efficiency. Another form of diversity is the spatial diversity, which is
also known as antenna diversity. Spatial diversity is a scheme that uses multiple anten
nas at the transmitter and/or receiver. The use of multiple antennas at the receiver is
referred to as “receive diversity”, and the use of multiple antennas at the transmitter is
referred to as “transmit diversity”. A system that consists of a single transmit antenna
and multiple receive antennas is referred to as singleinput multipleoutput (SIMO),
while a system consisting of multiple transmit antennas and single receive antenna is
referred to as multipleinput singleoutput (MISO). A system consisting of multiple
antennas at both the transmitter and receiver sides is referred to as MIMO.
1.5 Diversity and multiple antenna systems 39
Consider a flat fading channel, and a SIMO system with M receive antennas, the
M dimensional received vector is given as:
r = hs + z (1.36)
where s is the transmitted signal, r = [r1 . . . rM ]T such that rm is received by the mth
receive antenna, h = [h1 . . . hM ]T where hm is the channel coefficient between the trans
mit antenna and the mth receive antenna and z = [z1 . . . zM ]T where zm is an additive
white Gaussian noise at antenna m. We need to find an optimal weighting vector that
Chapter 1 Basic wireless channel propagation characteristics and introduction to
40 multiple antenna systems
r̃ = wH r (1.37)
where w = [w1 . . . wM ]T is a weighting vector. The operation in (1.37) is also known as
beamforming where w can be termed as beamformer.
r̃ = wH hs + wH z (1.38)
The received signal component is wH hs and the noise component is wH z. Hence
the output signal power is given by wH h2 σs2 where σs2 = E[s2 ] is the transmitted
signal power, and the output noise power is given as E[wH z2 ] = w2 σz2 .
Now the goal is to find w that maximizes the output SNR. The optimal w that
maximizes the ouput SNR is given as:
h
wout = (1.40)
h
The ouput SNR is then given as
h2 σs2
SNRout = (1.41)
σz2
this is also known as spatial matched filter.
This type of receive diversity is also referred to as maximum ratio combining (MRC).
Other receive diversity schemes are the selection combining (SC) and the equal gain
combining (EGC). In the SC scheme, the antenna with the highest received signal power
is only selected, ignoring the other received signals. In the EGC scheme, the signals at
the different receive antennas are coherently combined or cophased, where the magni
tudes of the beamforming weights are set to unity. SC and EGC schemes require lower
complexity than MRC, however MRC provides the best performance. In fact, in the SC
1.5 Diversity and multiple antenna systems 41
scheme, the channel state information is not required, simply the branch with highest
SNR is selected. The EGC scheme requires the knowledge of the phase shifts caused
by the fading channel. The MRC scheme requires full knowledge of the channel state
information, in addition, it requires the estimation of the noise power at each receive
antenna. When the noise power is not the same on the different receive antennas, the
above defined weighting coefficients at each antenna are divided by the corresponding
noise power. Generally, the weighting coefficients are approximated by considering the
channel coefficients (as derived above) obtained through channel estimation [8].
In Figure 1.9, the bit error rate (BER) versus energy per bit to noise power spectral
density ratio (Eb /N0 ) is shown for BPSK modulation, using a SIMO system with MRC,
for different number of receive antennas. As shown, the SIMO scheme provides better
performance (less BER) compared to the SISO scheme, it is also shown that as the
number of receive antennas increases, the BER decreases.
100
101
102
103
104
105
106
0 5 10 15 20 25 30
Figure 1.9: BER versus Eb /N 0 for BPSK modulation, using SIMO system with MRC.
Chapter 1 Basic wireless channel propagation characteristics and introduction to
42 multiple antenna systems
The capacity that can be achieved when using a SIMO system is given as:
h2 σs2
CSIMO = Blog2 (1 + ) (1.42)
σz2
Consider a MISO system having 2 transmit antennas, and assume Rayleigh flat
fading channel, the data symbols to be transmitted are given as:
s1 −s∗2
S= (1.43)
s2 s∗1
where in the first symbol period, data symbols s1 and s2 are transmitted from antenna
1 and antenna 2 respectively, then in the second symbol period, −s∗2 and s∗1 are trans
mitted from antenna 1 and antenna 2 respectively.
h1 h2
Let H̃= ∗ , the columns of H̃ are orthogonal where
h2 −h∗1
H h1 2 + h2 2 0
H̃H̃ = (1.48)
0 h1 2 + h2 2
Let w1 = [h1 h∗2 ]T , then at the first time instant, the first symbol s1 can be detected
as:
r(1)
w1H = (h1 2 + h2 2 )s1 + h∗1 z(1) + h2 z ∗ (2) (1.49)
r∗ (2)
σ2
The output SNR is given as SNRout1 = (h1 2 + h2 2 ) σs21 where σs21 is the power of
z
the signal transmitted from the first antenna.
Let w2 = [h2 − h∗1 ]T , then at the second time instant, the second symbol s2 can
be detected as:
r(1)
w2H = (h1 2 + h2 2 )s2 + h∗2 z(1) − h1 z ∗ (2) (1.50)
r∗ (2)
σ2
The output SNR is given as SNRout2 = (h1 2 + h2 2 ) σs22 , where σs22 is the power of
z
the signal transmitted from the second antenna.
As the transmission power is divided equally among the transmit antennas such
2
that σs21 =σs22 = σ2s . The output SNR for a given symbol is given as
h2 σs2
SNRout = (1.51)
2 σz2
As noticed, the Alamouti scheme can achieve a diversity gain of order 2, but results
in a 3 dB loss (half of the signal power) in SNR compared to the MRC scheme with 2
receive antennas. Hence the above transmit Alamouti scheme provides diversity gain
Chapter 1 Basic wireless channel propagation characteristics and introduction to
44 multiple antenna systems
without any array gain. This is the cost of achieving diversity in the absence of CSI at
the transmitter.
In Figure 1.10, the BER versus Eb /N0 is shown for BPSK modulation, using a MISO
system employing the Alamouti code, for different number of transmit antennas.
100
101
102
103
104
105
106
0 5 10 15 20 25 30
As shown, the MISO scheme allows to achieve less BER compared to the SISO
scheme, where this BER decreases as the number of transmit antennas increases.
We can notice from Figures 1.9 and 1.10 that the SIMO scheme employing MRC
provides better performance than the MISO scheme employing the Alamouti code. This
is better illustrated in Figure 1.11. In fact, in addition to the diversity gain provided
by the two schemes, the SIMO scheme (with MRC) can achieve array gain while the
MISOAlamouti scheme cannot.
1.5 Diversity and multiple antenna systems 45
100
101
102
103
104
105
106
0 5 10 15 20 25 30
Although MISO scheme does not provide any array gain, that is, there is no increase
2
in the input SNR ( σσs2 ), however due to the diversity gain, it results in an improvement
z
of the output SNR in comparison with the SISO case.
For a MISO system with N transmit antennas, the MISO channel capacity is given
as
h2 σs2
CMISO = Blog2 (1 + ) (1.52)
N σz2
Consider again a 2 × 1 MISO system, and a symbol s transmitted from the two
transmit antennas, but this time, assume that the transmitter knows the CSI, such
that the channel coefficients h1 and h2 are known to the transmitter. In this case, a
kind of precoding can be applied such that at the transmitter side, the symbol s is
h∗1 h∗1
multiplied by h and transmitted from the first antenna as s1 = h s, the symbol s is
Chapter 1 Basic wireless channel propagation characteristics and introduction to
46 multiple antenna systems
h∗ h∗2
also multiplied by h
2
and transmitted as s2 = h
s from the second transmit antenna.
At the receive antenna, we have:
h1 2 h2 2
r=( + )s + z = hs + z (1.53)
h h
The output SNR is then given as
σs2
SNRout = h2 (1.54)
σn2
which is the same as MRC. This can be referred to as transmit beamforming.
Consider a flat fading MIMO system with N transmit antennas and M receive
antennas, the MIMO system model is given as:
y = Hs + z (1.55)
where y = [y1 . . . yM ]T is the received signal vector, s = [s1 . . . sN ]T is the transmitted
signal vector, z = [z1 . . . zM ]T is the noise vector, and H is the MIMO channel matrix
1.5 Diversity and multiple antenna systems 47
given as:
h11 h12 . . . h1N
h21 h22 . . . h2N
. . . .
H=
.
(1.56)
. . .
. . . .
hM 1 hM 2 . . . hM N
where hmn is the channel coefficient between the mth receive antenna and nth transmit
antenna.
H
H = UH DH VH (1.57)
where DH is an M × N diagonal matrix, and the matrices UH and VH are M × M
and N × N unitary matrices, respectively. If H is a full rank matrix then its rank is
min(N, M ), which is also the number of spatial degrees of freedom. Let Nd =min(N, M ),
then H has Nd nonzero singular values on its diagonal such that µ1 ≥ µ2 ≥ · · · ≥ µNd .
Assuming the CSI is known at the transmitter side, the data symbols can be pre
coded as:
s̃ = VH s (1.58)
then s̃ is transmitted. Consequently,
ȳ = Hs̃ + z (1.59)
Now if we multiply at the receiver by UH
H , we have:
ỹ = UH H
H ȳ + UH z
= UH H
H UH DH VH VH s + z̃
= DH s + z̃ (1.60)
Hence we have:
when M > N (Nd = N ), then ỹm for m = 1, . . . N are the first N nonzero elements
of y, while when M < N (Nd = M ), only M symbols can be received, hence only M
nonzero data symbols can be transmitted.
Nd
X σs2m
CMIMO = Blog2 (1 + µ2m ) (1.62)
m=1
σz2
The MRC scheme, which is illustrated before for the SIMO case can be employed in
the MIMO case, where the received symbols are detected by multiplying the received
vector y by the conjugate transpose of the channel matrix H, as follows:
ŝ = HH y (1.63)
If the channel matrix H is known at the transmitter, a precoding scheme (transmit
beamforming) can be applied, by multiplying the data symbols to be transmitted by
the conjugate of the channel matrix.
Other detection schemes are zero forcing (ZF) and minimum mean square error
(MMSE) schemes. In the ZF scheme, the received signal vector is multiplied by the
pseudo inverse of the channel matrix, as follows:
ŝZF = H† y (1.64)
where H† = (HH H)−1 HH . ZF receiver attempts to minimize interantenna interfer
ence, but results in noise enhancement. ZF can be used as a precoding scheme. The
T
scheme involves multiplying the data symbols to be transmitted by (H† ) .
When the channel matrix H is known at the transmitter, MMSE can be used as a
precoding scheme. It involves multiplying the data symbols to be transmitted by
H∗ (HT H∗ + σz2 I)−1 .
The precoding/detection schemes described above are linear schemes. There exists
also a class of nonlinear schemes. Such schemes are more complex but can achieve
better performance. Among several nonlinear detection schemes, the vertical bell lab
layered space time (VBLAST) scheme [9] is popular . VBLAST relies on successive
interference cancellation (SIC), where symbols are detected in an iterative manner.
In the above, for the sake of simplification, we considered a flat fading channel to
illustrate the benefits achieved by using MIMO systems. Considering the frequency
selective fading case, a time varying frequency selective channel matrix for an N × M
MIMO system is given as:
h11 (t, τ ) h12 (t, τ ) . . . h1N (t, τ )
h21 (t, τ ) h22 (t, τ ) . . . h2N (t, τ )
. . . .
H(t, τ ) = (1.66)
. . . .
. . . .
hM 1 (t, τ ) hM 2 (t, τ ) . . . hM N (t, τ )
If the channel is time invariant, then hnm (t, τ ) = hnm (τ ) for all n and m. Consider
a frequency selective channel with L multipath components, the system model is then
written as:
L
X
y(t) = H(τl )s(t − τl ) + z(t) (1.67)
l=1
In the flat fading case, the narrowband assumption states that the different propoga
tion paths cannot be resolved temporally, which is not the case for frequency selective
fading. Based on that, flat fading or narrowband MIMO channel models are fully char
acterized by their spatial structure, while frequency selective or wideband channels are
characterized by their spatial and temporal (multipath) structure.
Chapter 1 Basic wireless channel propagation characteristics and introduction to
50 multiple antenna systems
1.6 Conclusion
This chapter provides a basic understanding of the fading characteristics in wireless
channels. Different types of fading are presented, and the effects of fading on commu
nication performance are addressed. Different diversity schemes used against fading
are then introduced, focusing on spatial diversity, where a brief review of the basics of
MIMO communication technology is provided.
Chapter 2
The design and performance of MIMO systems highly depend on the propagation envi
ronment [5, 10, 11]. A MIMO channel model that accurately describes the propagation
channel is important to exploit the advantages provided by MIMO systems.
MIMO channel models can be categorized into physical and nonphysical models. In
nonphysical models, the channel matrix coefficients are modeled statistically with re
spect to the correlation between them. On the other hand, physical models characterize
the MIMO channel by means of physical parameters related to the signal propagation
through the channel [12], such parameters can be the time of arrival (TOA), angle of
departure (AOD), angle of arrival (AOA).
51
52 Chapter 2 MIMO channel models
tion, etc., that occurs as a result of rays interaction with objects in the environment.
Although deterministic physical models have the ability to provide accurate channel
modeling, they are computationally intensive, which make statistical/stochastic physi
cal models more convenient to consider, as they are more computationally efficient.
In such models, the geometry of the channel is taken into account, where the physical
propagation parameters are modeled by defining statistical distributions to scatterers
in the environment, that is, the positions of the scatterers are modeled as random and
ruled by some predetermined statistical distributions. It follows that the values of the
physical parameters are influenced by these distributions.
The geometry of scatterers in the environment has an impact on the AOA and TOA
statistics, that is, the distributions of scatterer positions within clusters have an impact
on AOA and TOA statistical distributions. [18, 19].
54 Chapter 2 MIMO channel models
Pl
L X
X
h(t, Θt , Θr ) = βlp δ(t − Tl − τlp )δ(θT − θlT − θlp
T
)δ(θR − θlR − θlp
R
) (2.1)
l=1 p=1
The magnitude of βlp (βlp ) is assumed to be Rayleigh distributed and its phase is
uniform. The expected power of the pth ray in the lth cluster is given as
Assuming that TOA, AOD and AOA statistics are independent, the cluster and ray
arrival times follow a Poisson process, where the cluster arrival time is an exponentially
distributed random variable conditioned on the TOA of the previous cluster, given as:
T R
The distributions of the angles of arrival and departure θlp and θlp are assumed to
T R
be uniform over [0, 2π) (note that θl and θl are meant to be the mean AOD and mean
AOA respectively), and the rays within a cluster can be assumed to have a Laplacian
distribution, such that
√ T
θlp
T 1 − 2
σT

p(θlp )= √ Te θ (2.5)
2σθ
where σθT is the standard deviation of the angular spread corresponding to AOD. Sim
ilarly
√ R
θlp
R 1 − 2
σR

p(θlp ) = √ Re θ (2.6)
2σθ
where σθR is the standard deviation of the angular spread corresponding to AOA.
Zwick model
Zwick model treats the different MPCs separately, due to the reason that clustering
and multipath fading do not occur in an indoor channel in case of sufficiently large
sampling rate [12,25]. The MPCs are treated independently without amplitude fading,
while the phase changes of the different MPCs are involved in the model through geo
metric basis, characterizing the motion of the transmitter, receiver and scatterers.
In physical models, the channel between the transmitter and receiver is modeled on
the basis of electromagnetic waves propagation without referring to antenna configu
rations [12]. Alternatively, analytical models deal with MIMO channel matrix directly
by providing it a mathematical representation. Analytical models depend on antenna
configurations, and they can be classified based on correlation properties or through
the decomposition of the channel matrix in terms of propagation parameters. Analytic
models do not really rely on the physical propagation characteristics, however, when
the MIMO channel matrix is expressed mathematically in terms of some propagation
parameters, the model can be considered as a kind of propagation based analytical
model. Hence we choose to classify these kinds of models as physical models. The
other kind of analytical models are the correlation based models that totally ignore the
56 Chapter 2 MIMO channel models
propagation parameters.
In finite scatterer channel model [26], a finite number L of MPCs is considered, each
of them is characterized by a complex amplitude αl , AOD θlT , AOA θlR and delay τl .
Consider an N × M MIMO system, for the narrowband case, the delays can be
neglected, and the channel matrix is expressed as:
L
X
H= αl a(θlR )aT (θlT ) = AR ΞAT T (2.7)
l=1
where a(θlT ) and a(θlR ) are the transmit and receive steering vectors and
AR = [a(θ1R ) . . . a(θLR )]
AT = [a(θ1T ) . . . a(θLT )]
Ξ = diag{α1 . . . αL } (2.8)
For wideband systems, the delay is included in the model, the tapped delay line
representation of the channel matrix is then given as:
+∞
X
H(τ ) = Hκl δ(τ − κl Ts ) (2.9)
κl =−∞
1
where Ts = B
with B the system bandwidth, and Hκl is given as
L
X
Hκl = αl sinc(τl − κl Ts )a(θlR )aT (θlT )
l=1
= AR (Ξ Tκl )AT T (2.10)
The MPC parameters can be assigned some statistical distributions, and in some
environments these distributions can be guessed from measurements [12].
2.1 Physical models 57
In the previous finite scatterer model, the channel matrix is expressed in terms of the
actual physical AODs and AOAs. In virtual channel representation (VCR), the MIMO
channel is modeled in the beam space, through expressing the channel matrix in terms
of fixed spatial basis defined by fixed transmit and receive virtual angles associated
to virtual scatterers in the environment, such that the channel matrix H is expressed
as [27]:
H = FM (Ω̃ G)FH
N (2.11)
where FN and FM are the discrete Fourier transform (DFT) matrices (the fixed basis)
that consists of the steering vectors corresponding to N transmit and M receive vir
tual angles (or scatterers), respectively, and (Ω̃ G) is an M × N matrix representing
the environment in between the virtual transmit and receive scatterers. The matrix
Ω̃ is the elementwise square root of the coupling matrix Ω whose elements represent
the mean power coupled from the virtual transmit to the virtual receive angles (or
scatterers), and G is an M × N independent and identically distributed (i.i.d.) zero
mean complex Gaussian matrix. Hence the mnth element of (Ω̃ G) corresponds to a
pair of virtual transmit scatterer n and virtual receive scatterer m, if either the virtual
scatterer n or the virtual scatterer m does not exist, the nmth element of (Ω̃ G) will
be zero. In fact, the matrix (Ω̃ G) is a 2D DFT of H. The transmit and receive
virtual angles defined by the transmit and receive DFT matrices are determined by the
spatial resolution of the transmit and receive antenna arrays, which means that the
accuracy of model increases as the number of antennas increases. It is important to
mention that the virtual representation of the channel matrix in terms of DFT basis
is only convenient for uniform linear antenna arrays. The VCR model is useful for the
theoretical analysis of the capacity scaling in MIMO systems [12, 28].
Figure 2.2 illustrates both the physical channel model and the VCR model.
58 Chapter 2 MIMO channel models
Another propagation based analytical model is the Maximum entropy model, which
is based on the principle of maximum entropy, that exploits the available prior informa
tion about the propagation environment (such as time delays, AODs, AOAs, bandwidth,
etc.) to derive the model that expresses the available knowledge, that is, to find the
probability distribution of the channel that best fits the available data [12, 29].
The i.i.d model simply describes the MIMO channel matrix as random, with in
dependent and identically distributed elements. It is the spatially white MIMO case
that corresponds to a rich scattering environment. Referring to the VCR model, the
2.2 Nonphysical models 59
i.i.d model refers to the case where all the elements of Ω̃ are nonzero and identical.
The i.i.d model is generally used for theoretical analysis of MIMO systems such as
informationtheoretic analysis [12].
Kronecker model
The Kronecker model [30–33] assumes that the correlation between antennas at the
transmitter side is independent of the correlation between antennas at the receiver side,
such that the channel covariance matrix can be written as:
RH = RT ⊗ RR (2.12)
where RT = E[HH H] and RR = E[HHH ] are the transmit and receive covariance
matrices, respectively.
1/2 1/2
H = RR GRT (2.13)
where G is an M × N independent and identically distributed (i.i.d.) zero mean com
plex Gaussian matrix.
The Kronecker model results in poor estimation of channel capacity, due to the
assumption of independent transmit and receive correlations, which is not accurate in
the majority of physical channels [34]. Despite its popularity as a simple model, it has
been shown to be only applicable in specific scenarios.
Weichselberger model
The Weichselberger model [35] removes the separability condition imposed by the
Kronecker model, where it allows coupling between the transmit and receive eigen
modes. It replaces the transmit and receive DFT matrices in the VCR model with the
eigenbasis of the transmit and receive spatial covariance matrices, RT and RR . The
eigendecompositions of RT and RR are given as:
RT = UT ET UH
T
RR = UR ER UH
R (2.14)
60 Chapter 2 MIMO channel models
where the columns of UT and UR are the eigenvectors of RT and RR , respectively, and
ET and ER are diagonal matrices containing the corresponding eigenvalues. It follows
that H is given as:
H = UR (Ω̃ G)UTT (2.15)
where Ω̃ is the elementwise square root of the coupling matrix Ω whose elements
represent the mean power coupled from the transmit eigenvectors (or eigenmodes) to
the receive eigenvectors and G is an M × N independent and identically distributed
(i.i.d.) zero mean complex Gaussian matrix. The Weichselberger model is equivalent
to the VCR model when the DFT matrix turns into the eigenvector matrix (as the
number of antennas goes to infinity).
2.3 Sparsity
Recently, the issue of sparsity of the wireless channel has received a lot of attention. It is
a fact that, in many scenarios, wireless channels can be characterized by few significant
paths, this is mainly attributed to the limited number of dominant scatterers distributed
in the environment, and especially for large bandwidths [36–41]. This sparsity property
has been shown to be reasonable through physical investigations that have been reported
for some physical channels [42].
where δ(.) is the Dirac function; L is the total number of propagation paths (clus
ters), which is considered finite, with Pl contributing rays for the lth path (cluster),
(n,m) (n,m) (n,m)
tl + τlp is the pth contributing ray delay in the lth cluster with τlp a small de
(n,m) (n,m)
viation from the cluster mean delay tl and αlp is the corresponding complex gain.
In some existing works [40, 44], an exact common support condition is considered
such that different transmitreceive antenna pairs share exactly the same path delays.
However, as the bandwidth of the signal increases, the exact common support property
starts losing its validity. For this reason, in the above channel model (2.16), multiray
delays associated to a given cluster are not considered exactly the same. However, it
can be assumed that the multirays associated to a given scatterer l share the same
cluster mean delay at the different transmitreceive antenna pairs. This is a kind of
common support hypothesis. Accordingly, the CIR h(n,m) (t) between the nth transmit
antenna and mth receive antenna can be modeled as:
Pl
L X
X
(n,m) (n,m) (n,m)
h (t) = αlp δ(t − (tl + τlp )), 1 ≤ n ≤ N, 1 ≤ m ≤ M (2.17)
l=1 p=1
where tl is the mean delay associated to cluster l shared by all the transmitreceive
antenna pairs.
2.7 Conclusion
In this chapter, an overview of MIMO channel models is provided. The channel models
are divided into different categories based on different propagation scenarios under
different conditions. Sparsity and clustering properties in wireless channels are then
2.7 Conclusion 63
Channel estimation methods can be further divided into nonparametric and para
metric categories.
65
66 Chapter 3 Channel parameter estimation
Hence for this approach, we consider the wideband (or frequency selective) channel
model given in (1.11), where the channel is described in terms of different propagation
path delays and their corresponding gains. For an N × M MIMO system, the CIR
between the nth transmit antenna and the mth receive antenna is given as:
L
X
(n,m) (n,m) (n,m)
h (t) = αl δ(t − tl ), 1 ≤ n ≤ N, 1 ≤ m ≤ M (3.1)
l=1
(n,m)
where L is the total number of propagation paths, tl is the path delay between
(n,m)
the nth transmit antenna and mth receive antenna, and αl is the corresponding
complex gain.
3.2 Parametric approach (MIMO channel parameter estimation) 67
Assume that antennas at the transmitter side are transmitting pilot symbols on
different carriers in order to identify the channels associated with different transmit
receive antenna pairs. Consider the known pulse shape g(t) transmitted at the nth
transmit antenna at a constant rate 1/T through the medium,
For the mth receive antenna, the received baseband signal is:
L
X (n,m) (n,m)
x(n,m)
r (t) = αl g(t − tl ) + zr(n,m) (t) (3.2)
l=1
(n,m)
where zr (t) is an additive white Gaussian noise.
(n,m)
Assuming an exact common support [40, 41, 44], such that tl = tl . Hence the
above equation is written as:
L
X (n,m)
x(n,m)
r (t) = αl g(t − tl ) + zr(n,m) (t) (3.3)
l=1
Applying the DFT, the Fourier coefficients of the received signal are given by:
L
(n,m) −j 2π ktl
X
Xr(n,m) [k] = G[k] αl e T + Zr(n,m) [k] (3.4)
l=1
(n,m) (n,m)
where G[k] is the DFT of pulse g(t) and Zr [k] is the DFT of zr (t). Writing
(3.4) in the matrix form
x(n,m)
r = GVα(n,m) + z(n,m)
r (3.5)
T
x(n,m) = Xr(n,m) [0] . . . Xr(n,m) [K − 1]
r
V = [vd (t1 ) . . . vd (tL )]
2π 2π(K−1)
vd (tl ) = [1 e−j T tl . . . e−j T tl ]T
G = diag{G[0] . . . G[K − 1]}
(n,m) (n,m)
α(n,m) = [α1 . . . αL ]T
T
z(n,m) = Zr(n,m) [0] . . . Zr(n,m) [K − 1]
r (3.6)
68 Chapter 3 Channel parameter estimation
L
0 (n,m)
X (n,m)
yr(n,m) = αl vd (tl ) + z r (3.8)
l=1
The model in (3.8) is common in the array signal processing context where several
techniques have been proposed (or applied) to estimate the direction of arrival or time
of arrival of signals impinging on an array of sensors [60].
q (n,m) = wbf
H (n,m)
yr (3.9)
and then to measure the output power.
At a certain time instant, for the N × M antennas, the output power is measured
as
N M
1 X X (n,m) 2
P (wbf ) = q 
N M n=1 m=1
N M
1 X X H (n,m) (n,m) H
= w y yr wbf
N M n=1 m=1 bf r
H
= wbf R̂yr wbf (3.10)
where
N M
1 X X (n,m) (n,m) H
R̂yr = y yr (3.11)
N M n=1 m=1 r
For the classical or conventional beamformer (also known as bartlett beamformer [61]),
the weighting vector is chosen to be vd (t) (wB = vd (t)). Then the normalized output
of the beamformer is given by:
The conventional Bartlett beamformer is a simple one, at the cost of limited reso
lution and interference of closeby delays.
3.2.1.2 Capon
The Minimum variance distortionless response (MVDR) also known as Capon’s beam
former [62] was proposed to improve the resolution performance. It is based on mini
mizing the output power while keeping the signal of the desired delay undistorted:
H
min P (wbf ) s.t. wbf vd (t) = 1 (3.13)
wbf
R̂−1
yr vd (t)
wCAP = (3.14)
vd (t)H R̂−1
yr vd (t)
1
PCAP (t) = (3.15)
vd (t)H R̂−1
yr vd (t)
With the MVDR beamformer, the resolution is greatly enhanced compared to the
classical (Bartlett) beamformer.
70 Chapter 3 Channel parameter estimation
In the noise free case, the covariance matrix of the observation coefficient vectors
can be written as
Ryr is of size K ×K, and V of size K ×L where K > L. When V is full column rank
and the channel gain coefficients are uncorrelated, the eigenvectors associated with the
first L eigenvalues of Λ and column vectors of V will span the same subspace which is
the signal subspace, and the eigenvectors associated with the last K − L eigenvalues of
Λ will span the orthogonal subspace. Then U is defined as
U = [Us Un ] (3.19)
such that Us contains the L eigenvectors spanning the signal subspace and Un contains
the K − L eigenvectors spanning the orthogonal subspace.
of noise, the smallest K − L eigenvalues are zero and the rank of Ryr is L.
Ryr = Us Λss UH H
s + Un Λnn Un (3.20)
where Λss is an L × L diagonal matrix containing the L nonzero eigenvalues, and Λnn
is an (K − L) × (K − L) zero matrix (noisefree case).
Now the L largest eigenvalues of Ryr correspond to signal plus noise and the K − L
smallest eigenvalues correspond to noise only. By other words, the first L eigenvalues are
greater than σz2 and the last K − L eigenvalues are equal to σz2 . The problem is then to
define an algorithm to estimate the signal subspace, this can be done by comparing the
eigenvalues based on a threshold defined by σz2 or by using some information theoretic
criteria.
3.2.2.1 MUSIC
MUSIC is one of the subspace based methods which is based on the eigendecomposition
of the observation covariance matrix and relies on the orthogonality between the signal
and noise subspaces.
The expression of the observation covariance matrix as shown before is given as:
As mentioned before, when matrix Rα is a full rank matrix (channel gains are
uncorrelated), the columns of V and Us will span the signal subspace, and as the
signal and noise subspaces are orthogonal we have:
72 Chapter 3 Channel parameter estimation
UH
n vd (tl ) = 0 f or l = 1...L (3.23)
When V has a full column rank such that the delays are distinct, and as Rα has a
full rank, then we have L delays to estimate.
Given a dictionary of time delays, the algorithm used in MUSIC is to project vd (t)
for a given delay t chosen from the dictionary on the noise subspace in order to find
the true delay, if the delay chosen from the dictionary corresponds to the true one, the
projection will be zero. The MUSIC cost function is given by:
vd (t)H vd (t)
PM U SIC (t) = (3.24)
vd (t)H Ûn ÛH
n vd (t)
MUSIC
Figure 3.1: Power spectra of Bartlett, Capon and MUSIC, for a 12 × 12 MIMO system,
K = 32 and two delays chosen as t = [0.48 0.56]T with SNR = 10 dB.
Figure 3.1 shows the spectra of the classical beamformer, capon’s beamformer and
MUSIC. It is noticed that the Capon beamformer provides higher resolution compared
with the classical beamformer, and MUSIC outperforms both the classical and Capon
beamformers in terms of resolution.
3.2.2.2 ESPRIT
Another subspace based method is ESPRIT. It relies on the Vandermonde structure of
the matrix V and exploits its shift invariance property.
V2 = V1 Φ (3.25)
2π 2π
where Φ = diag{e−j T t1 , . . . , e−j T tL
}.
V = Us J (3.26)
where J is a non singular matrix.
Let Us1 and Us2 be the first K − 1 and the last K − 1 rows of Us respectively, we
have:
V1 = Us1 J
V2 = Us2 J (3.27)
T
t̂l = − arg(ψl ) (3.31)
2π
yo = Φs xs (3.32)
To achieve an efficient reconstruction, the sensing matrix Φs should have low coher
ence, where the coherence of a matrix is defined as the largest absolute inner product
between any two normalized columns of the matrix [67]. The lower the coherence of the
sensing matrix, the better the reconstruction quality can be achieved. The incoherence
property is related to the restricted isometry property (RIP) [68], which states that
for a given ks sparse vector xs , there is a high probability to recover xs from yo if the
sensing matrix Φs obeys:
In several cases, the signal is not sparse itself, but it can be sparsely represented in
some basis. Consider a nonsparse signal zs which is sparse in an orthogonal basis Ψs ,
such that zs = Ψs xs where xs is the sparse vector. The measurement vector yo is then
given as:
y o = Φ s Ψs xs (3.34)
For signal recovery, it is required that matrix Φs Ψs has low coherence.
Consider the case where the signal is itself sparse (3.32), the sparse vector xs can
be recovered through the following optimization problem:
76 Chapter 3 Channel parameter estimation
Although using l0 norm allows perfect signal reconstruction with high probability,
the above optimization problem is highly nonconvex and thus NPhard.
The methods used for signal recovery as a solution for the above nonconvex prob
lem are mainly classified into convex optimization (or relaxation) methods and greedy
pursuit methods.
3.2.3.1 L1 minimization
The convex relaxation methods replace the l0 norm with l1 norm, the optimization
problem turns out
is set as a residual, and at each iteration, the signal is correlated with the column vec
tors given in the sensing matrix, the contribution of the vector with highest correlation
is then removed from the residual.
Given (3.32), consider the problem of recovering xs from yo , with Φs the dictionary,
the steps of the MP algorithm are summarized as follows:
Matching Pursuit
2 Assume that all the columns of Φs are normalized, the algorithm selects the
column that has the highest correlation with yo by solving the following maxi
mization problem
where ζi is the index of the selected vector and Ns is the number of column vectors
in Φs .
4 Increment i and return to step 2, continue until the residual value ri 2 is below a
predefined threshold or stop after Ks iterations if Ks (number of nonzero elements
in xs ) is known a priori.
The OMP method provides better performance than the MP method with higher
complexity. In the OMP method, the basis vector is selected in the same manner as
78 Chapter 3 Channel parameter estimation
in the MP method, however for each iteration, the measurement vector is projected on
the subspace spanned by the current and previously selected vectors. The contribution
of these vectors is then removed. In this manner, the previously selected vectors will
not be selected again in the subsequent iterations. The steps of the OMP method are
summarized as follows:
Orthogonal Matching Pursuit
2 Let ϕi be the set containing the indices of vectors selected until the ith iteration
(where ϕ0 = Ø) and ζi the index of vector selected at the ith iteration by solving
the following maximization problem
3 Set ϕi = ϕi−1 ∪ ζi
5 The approximation of the signal and the residual are then given as:
y oi = Φ si c i
ri = yo − yoi (3.42)
6 Increment i and return to step 2, continue until the residual value ri 2 is below a
predefined threshold or stop after Ks iterations if Ks (number of nonzero elements
in xs ) is known a priori.
OMP deals with the case of single measurement vector (SMV). For the case of
multiple measurement vector (MMV), a method is proposed in [72] called simultaneous
orthogonal matching pursuit (SOMP), given that the sparse input vectors share the
3.2 Parametric approach (MIMO channel parameter estimation) 79
same location of nonzero elements. Consider again the model given in (3.5), the matrix
of Fourier coefficient vectors at the receive antenna m due to the N transmit antennas
is given as:
X(m)
r = GVΥ(m) + Z(m)
r (3.43)
(m) (1,m) (N,m) (m) (1,m) (N,m)
where Xr = [xr . . . xr ], Υ(m) = [α(1,m) . . . α(N,m) ], and Zr = [zr . . . zr ].
Arranging the Fourier coefficient matrices of the different receive antennas as follows:
Xr = [X(1) (M )
r . . . Xr ] (3.44)
we have
Xr = GVΥ + Zr (3.45)
(1) (M )
where Υ = [Υ(1) . . . Υ(M ) ] and Zr = [Zr . . . Zr ].
For sparse recovery, or for delays estimation, the steps of the SOMP method are
given as follows:
80 Chapter 3 Channel parameter estimation
2 Let ϕi be the set containing the indices of vectors selected from VQl until the ith
iteration (where ϕ0 = Ø) and ζi the index of vector selected at the ith iteration
by solving the following maximization problem
ζi = argmax RH
i−1 vd (tbql )1 (3.50)
ql =1...Ql
3 Set ϕi = ϕi−1 ∪ ζi
5 The approximation of the signal and the residual are then given as:
Yr i = Oi C i
Ri = Yr − Yri (3.52)
6 Increment i and return to step 2, continue until the residual value Ri F is below
a predefined threshold or stop after L iterations if L is known a priori.
3.3 Conclusion
In this chapter, an overview of several channel parameter estimation techniques is pro
vided. The estimation approaches are divided into parametric and nonparametric. The
nonparametric approach estimates the channel matrix without referring to any physical
propagation parameters. The parametric approach relies on physical channel models
to estimate channel parameters. The chapter divides the channel parameter estimation
methods into beamforming methods, subspace based methods and compressive sensing
based methods, focusing on the parametric approach for channel estimation.
Chapter 4
81
82 Chapter 4 Clustered MIMO channel delay estimation
Pl
L X
X
(n,m) (n,m) (n,m)
x (t) = αlp g(t − (tl + τlp )) + z (n,m) (t) (4.1)
l=1 p=1
Applying the DFT, the Fourier coefficients of the received signal are given by:
Pl
L X
(n,m)
(n,m) −j 2π
X
(n,m) k(tl +τlp )
X [k] = G[k] αlp e T + Z (n,m) [k] (4.2)
l=1 p=1
2π
Let vk (t) = G[k]e−j T kt ; (4.2) can be rewritten as:
Pl
L X
X
(n,m) (n,m) (n,m)
X [k] = αlp vk (tl + τlp ) + Z (n,m) [k] (4.3)
l=1 p=1
where
4.2 CS based channel delay estimation approach 83
T
x(n,m) = X (n,m) [−k/2 + 1] . . . X (n,m) [k/2]
(n,m) (n,m)
V(n,m) = [V1 . . . VL ]
(n,m) (n,m) (n,m)
Vl = [v(tl1 ) . . . v(tlP )]
(n,m) (n,m) (n,m) T
v(tlp ) = v−K/2+1 (tlp ) . . . vK/2 (tlp )
(n,m)T (n,m)T T
ᾱ(n,m) = [α1 . . . αL ]
(n,m) (n,m) (n,m)
αl = [αl1 . . . αlP ]T
T
z(n,m) = Z (n,m) [−k/2 + 1] . . . Z (n,m) [k/2]
(n,m) (n,m)
tlp = tl + τlp (4.5)
Arranging the Fourier coefficient vectors for the different receive antennas in matri
ces as X = [X(1) . . . X(M ) ] for X(m) = [x(1,m) . . . x(N,m) ], X can be written as
X = V̄Ā + Z (4.6)
where V̄ = [V(1) . . . V(M ) ] for V(m) = [V(1,m) . . . V(N,m) ].
The matrix ĀQi is a sparse matrix, since multiray delays associated to a scatterer
are not the same on different transmitreceive antenna pairs, the column vectors of ĀQi
do not share the same indices of the nonzero gain coefficients. However they share
the same indices of the gain coefficients associated to the cluster mean delays, as the
multirays associated to a given scatterer are assumed to share the same mean delay on
the different transmitreceive antenna pairs.
84 Chapter 4 Clustered MIMO channel delay estimation
(n,m)
Back to (4.3), as the deviations τlp are considered small, the first order Taylor
expansion of (4.3) gives:
Pl
L X
X
(n,m) (n,m) (n,m)
dk (tl ) + Z (n,m) [k]
X [k] ≈ αlp vk (tl ) + τlp (4.10)
l=1 p=1
L
X (n,m) (n,m)
X̃ (n,m) [k] = al vk (tl ) + bl dk (tl ) + Z (n,m) [k] (4.11)
l=1
Hence the vector x̃(n,m) of the Fourier coefficients can be written as:
L
X
(n,m) (n,m) (n,m)
x̃ = al v(tl ) + bl d(tl ) + z(n,m) (4.12)
l=1
where
Due to the common support assumption on mean delays. The vector v(tl ) is con
tained in all the Fourier coefficient vectors in X (4.7). The concept used in the SOMP
method can then be used to estimate the mean delays, with a modified selection step.
The steps of the proposed mean delay estimation method are given as:
4.2 CS based channel delay estimation approach 85
2. Let ϕi be the set containing the indices of vectors selected from VQi until the ith
iteration (where ϕ0 = Ø) and ζi the index of vector selected at the ith iteration
by solving the following maximization problem
ζi = argmax (RH H
i−1 v(tbqi )2 + Ri−1 d(tbqi )2 ) (4.14)
qi =1...Qi
3. Set ϕi = ϕi−1 ∪ ζi .
5. The approximation of the signal and the residual are then given as:
Xi = Oi C i
Ri = X − X i (4.16)
The proposed estimation method differs from the classical SOMP method in the
selection step, which is adapted according to the expression of the Fourier coefficient
vectors given in (4.12) by taking into account the vector v(tbqi ) and its derivative d(tbqi )
for selection.
86 Chapter 4 Clustered MIMO channel delay estimation
As several subspace based methods are high resolution methods, we find it interesting
to incorporate more details into the model rederived by means of Taylor expansion,
seeking for better estimation performance. To achieve this, we consider the U th order
Taylor expansion of (4.3), given as:
Pl
L X U (n,m)
(n,m)
X (n,m)
X (τlp )u (u) (n,m)
) + Z (n,m) [k] (4.17)
X [k] = αlp vk (tl ) + vk (tl ) + RU (τlp
l=1 p=1 u=1
u!
(u) (n,m)
where vk (tl ) is the uth order derivative of vk (tl ). In this last equation, RU (τlp ) is
the remaining term in the Taylor approximation, which is considered small, such that
its impact in the approximation can be neglected.
L X
X U
(n,m) (n,m) (u)
X [k] = G[k] al,u vk (tl ) + Z (n,m) [k] (4.18)
l=1 u=0
where
Pl (n,m) u
(n,m) (τlp )
(n,m)
X
al,u = αlp . (4.19)
p=1
u!
where
T
x(n,m) = X (n,m) [−K/2 + 1], . . . , X (n,m) [K/2]
Matrix W is the same for all the transmitreceive antenna pairs, as the mean de
lays of each cluster are assumed to be the same for all the transmitreceive antenna pairs.
The matrix of Fourier coefficient vectors at antenna m due to the N transmit an
tennas is given by:
X = WA + Z (4.25)
1
R̂X = XXH . (4.26)
NM
For the noisefree case:
RX = WRa WH , (4.27)
where the covariance matrix Ra is given as:
N M
1 X X (n,m) (n,m)H
Ra = a a (4.28)
N M n=1 m=1
Assuming that tl for l = 1, . . . , L are different from one another, with K > (U + 1)L
and N M > (U +1)L, it can be noted from the definition of matrix W that the dimension
of its column space is equal to (U + 1)L with a full rank property.
It follows that the dimension of the signal subspace is equal to the rank of Ra . In
the following we will show that Ra is a full rank matrix of rank equal to (U + 1)L,
which implies that the dimension of the signal subspace is (U + 1)L.
Proof Elements. Consider one cluster (L = 1) of mean delay t1 ; for the sake of simplifi
(n,m) (n,m)
cation, notations are abbreviated as t1 = t, P1 = P , α1,p = αp , τlp = τp , and a(n,m)
is a (U + 1)length random column vector which is replaced by a new vector a defined
as:
P P P
X X τpu X τpU T
a= αp αp · · · αp , (4.29)
p=1 p=1
u! p=1
U!
αp for p ∈ {1, . . . , P } are independent complex random variables such that ∀p, q ∈
{1, . . . , P }, E[αp ] = 0; E[αp αq∗ ] = σp2 if p = q, 0 otherwise; and Pp=1 E[αp 2 ] = 1.
P
τp for p ∈ {1, . . . , P } are independent real random variables having the same
distribution.
P P P P
X τpu X τqv 1 XX
[E[aaH ]](u+1,v+1) = E αp αq = E[αp αq∗ τpu τqv ]. (4.30)
p=1
u! q=1 v! u!v! p=1 q=1
It follows that:
P
1 X E[τpu+v ]
[E[aaH ]](u+1,v+1) = E[αp 2 ]E[τpu+v ] = . (4.31)
u!v! p=1 u!v!
According to the elements of matrix E[aaH ], it follows that its determinant is dif
ferent from zero, (det(E[aaH ]) 6= 0) and as it is a square matrix, then it is a full rank
matrix with rank equal to U + 1.
(U + 1)L, and hence (U + 2)L would be provided by the MDL criterion. However in
practice, in the presence of noise, the MDL algorithm will sometimes provide an integer
in the set {(U + 1)L + 1, (U + 1)L + 2, . . . (U + 2)L}. The proposed solution for this
case is to estimate U according to the following rule:
M DLV
U= −1 , (4.33)
L
where M DLV is the value obtained by the MDL criterion and b.c is the floor function.
1
Pss1 (t) = PU (4.34)
u=0 v(u) (t)H Ûn ÛH
nv
(u) (t)
where
(u) (u)
v(u) (t) = [v−K/2+1 (t), . . . , vK/2 (t)]T ,
(u) 2π
vk (t) = (−j k)u vk (t). (4.35)
T
The derived cost function can be written in the following form:
1
Pss1 (t) = PU (4.36)
v(t)H ( H H
u=0 Du Ûn Ûn Du )v(t)
92 Chapter 4 Clustered MIMO channel delay estimation
where
v(t) = v(0) (t) = [v−K/2+1 (t), . . . , vK/2 (t)]T
2π 2π
Du = diag{(−j (−K/2 + 1))u , . . . , (−j K/2)u }. (4.37)
T T
The proposed cost function turns to be the conventional MUSIC when setting U = 0,
which is the case for very small delay spreading (where it tends to be neglected). The
proposed technique can be seen as an extension of MUSIC in the delay spreading case.
4. Estimate the effective dimension of the signal subspace using the MDL criterion
combined with the proposed rule (4.33).
6. For each value of t, calculate the cost function Pss1 (t)(4.36). Then, the cluster
mean delays tl are estimated by searching for the L peaks of Pss1 (t).
The proposed cost function shows no major difference from the conventional MUSIC
in terms of the computational complexity as they share the same operations such as the
FFT, RX matrix construction, eigenvalue decomposition, and the spectral searching.
The proposed cost function requires a slight increase in the computational complexity
due to the addition of the terms corresponding to the delay spreading, which is the
projection of v(u) (t) for u = 0 . . . , U on the estimated noise subspace.
(n,m)
ated according to (4.3) with K = 64 Fourier coefficients. The delay deviations τlp
of multirays within each cluster l for each (n, m) transmitreceive antenna pair are
generated according to a uniform distribution and centered at tl . The correspond
(n,m)
ing gains αlp of the multirays are modeled as complex Gaussian random variables
generated in a way that the effective power of each cluster is normalized, such that
PPl (n,m) 2
p=1 E[αlp  ] = 1, ∀{l, n, m}. The added noise is modeled as complex white Gaus
sian noise. The estimation performance is assessed from Q = 500 independent sim
ulations. Theqroot mean square error (RMSE) of cluster mean delay estimation is
PQ PL
l=1 l (t̂ (i)−t )2
l
calculated as i=1
QT 2 L
, where t̂l (i) is the estimated mean delay for the ith
experiment and tl is the true mean delay.
Figure 4.1 shows the RMSE of mean delay estimation of the modified SOMP method
and the classical SOMP method versus SNR.
101
102
103
104
105
10 5 0 5 10 15 20 25
Figure 4.1: RMSE of mean delay estimation of the modified SOMP method and the
classical SOMP method versus SNR; Pl = 20, K = 64, L = 3, vector of chosen delays
is given as t = [0.32 0.45 0.61]T , τlp are uniformly distributed with 0.008T chosen
as the standard deviation of the delay spreading.
94 Chapter 4 Clustered MIMO channel delay estimation
As shown in the figure, the proposed modified SOMP method provides better mean
delay estimation in comparison with the classical SOMP method. By definition the
CRLB is the lowest bound of the variance of any unbiased estimator. However, as we
use the RMSE to evaluate the method, the CRLB in the figures represents the lower
bound of the standard deviation of any unbiased estimator.
Figure 4.2 shows the RMSE of the mean delay estimation versus standard deviation
of delay spreading for the modified SOMP method and the classical SOMP method.
100
101
102
103
104
105
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 4.2: RMSE of mean delay estimation of the modified SOMP method and the
classical SOMP method versus standard deviation of delay spreading; Pl = 20, K = 64,
L = 3, vector of chosen delays is given as t = [0.32 0.45 0.61]T , τlp are uniformly
distributed, SNR=15 dB.
(n,m)
For relatively small delay spreading, the coefficient bl in (4.12) is relatively small.
Hence it seems better to ignore the contribution of the vector d(tl ) where the classical
4.4 Simulation results 95
SOMP method can be used for estimation. However, as the standard deviation of delay
spreading increases, the contribution of d(tl ) increases, where as shown in the above
figure, the modified SOMP method provides better mean delay estimation in compari
son with the classical SOMP method.
102
103
104
105
5 0 5 10 15 20 25 30
Figure 4.3: RMSE of mean delay estimation of the proposed cost function for U = 1,
U = 2 and MUSIC versus SNR; Pl = 20, K = 64, τlp are uniformly distributed with
0.005T chosen as the standard deviation of the delay spreading.
Figure 4.3 shows that the proposed cost function (assuming U = 1) provides better
performance of the cluster mean delay estimation than the conventional MUSIC.
96 Chapter 4 Clustered MIMO channel delay estimation
According to the derivation of the model, in the noisefree case, it is better to choose
higher values for U to benefit from a better approximation, where small details can be
taken into account. On the other hand, with noisecontaminated data, the high order
terms in Taylor expansion (4.17) may be too small with respect to the noise power. In
fact, due to the property of Taylor expansion, it can be noted from (4.17) and (4.18),
that:
(n,m) (2)
Hence, for U = 2 the elements in vector al,2 v (tl ) will tend to be quite small.
(n,m)
Thus, for a high level of noise, the norm of the vector al,2 v(2) (tl ) is very small com
pared to noise. Hence, this part of the signal is dominated by the noise, and it would be
better not to consider this part of the signal to represent the signal subspace. For this
reason, for low SNR, as can be observed in Figure 4.3, it would be better to consider
U = 1 (K − 2L as dimension of noise subspace). However as SNR increases, it would
be worth considering vector v(2) (tl ) as a part of the signal subspace, and then U = 2
(K − 3L as dimension of noise subspace).
Figure 4.4 shows the comparison of RMSE of the cluster mean delay estimation
versus standard deviation of delay spreading.
We can observe that for very small delay spreading with relatively low SNR (SNR
= 5 dB), it would be better to consider that the dimension of the signal subspace is one
(U = 0), for the same reason discussed before, which is the situation of the conventional
MUSIC. However, as delay spreading increases, the proposed cost function with U = 1
outperforms MUSIC (U = 0). Again, as the delay spreading increases, the proposed
cost function for U = 3 outperforms the proposed cost function for U = 2. The increase
in the standard deviation of the delay spreading will give rise to an increase in the norm
(n,m)
of al,u v(u) (tl ). Then, this vector will have greater impact in formulating the signal
subspace, where higher values of U can be considered.
4.4 Simulation results 97
101
102
103
104
105
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 4.4: RMSE of mean delay estimation of the proposed cost function for U = 1,
U = 2, U = 3, and MUSIC versus standard deviation of delay spreading; Pl = 20,
K = 64, SNR = 5 dB.
We can also observe that as the delay spreading increases, the estimation is less
accurate, which is probably due to the approximation error of Taylor expansion.
5.5
4.5
3.5
2.5
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 4.5: Mean value of the MDL criterion (mean M DLV ) versus standard deviation
of delay spreading; Pl = 20, K = 64, SNR = 5 dB, mean M DLV values are obtained
from 500 independent simulations each.
Observing Figures 4.4 and 4.5 simultaneously, four flat regions can be distinguished.
When M DLV = 2, the best choice for the dimension of the signal subspace (U + 1)L
is 1 (U = 0). When M DLV = 3, 4, 5, the best dimensions of the signal subspace are 2
(U = 1), 3 (U = 2), and 4 (U = 3), respectively.
However, it can be noticed that the MDL criterion fails to estimate the effective
dimension of the signal subspace correctly in some regions, it can be also noticed that
flat regions are not always clearly distinguished. For example, when the delay spreading
is 0.004T , from the observed mean M DLV values, it seems that the MDL criterion is
providing the values 2 and 3 in different realizations. This shows that the MDL crite
rion, influenced by the random nature of scattering and the relatively high noise level,
is not always a foolproof indicator.
4.4 Simulation results 99
In Figure 4.6, the RMSE of mean delay estimation of the proposed subspace track
ing based method is shown. For each realization of received data, the decision about U
is obtained from the rule in (4.33), which is used to estimate the effective dimension of
the signal subspace. Then, the proposed cost function (4.36) is applied.
As shown in the figure, the proposed method allows for optimal selection of parameter
U and provides the best cluster mean delay estimation. However some minor failures of
the MDL criterion in systematically estimating the optimal effective dimension of the
signal subspace can be noted.
101
102
103
104
105
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 4.6: RMSE of mean delay estimation of the proposed cost function for U = 1,
U = 2 and U = 3, the proposed subspace tracking based method and MUSIC versus
standard deviation of delay spreading; Pl = 20, K = 64, SNR = 5 dB.
100 Chapter 4 Clustered MIMO channel delay estimation
Figures 4.7 and 4.8 show the RMSE of the mean delay estimation and the mean
M DLV versus standard deviation of delay spreading, respectively for SNR = 15 dB.
The obtained results show that the MDL criterion provides better estimations of the
different effective dimensions of the signal subspace for different standard deviations of
delay spreading, leading to an improvement in the mean delay estimation performance
of the proposed subspace tracking based method.
101
102
103
104
105
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 4.7: RMSE of mean delay estimation of the proposed cost function for U = 1,
U = 2, and U = 3, the proposed subspace tracking based method and MUSIC versus
standard deviation of delay spreading; Pl = 20, K = 64, SNR = 15 dB.
4.4 Simulation results 101
6.5
5.5
4.5
3.5
2.5
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 4.8: Mean M DLV versus standard deviation of delay spreading; Pl = 20,
K = 64, SNR = 15 dB
As mentioned before, the increase in the RMSE with respect to the delay spreading
may be due to the approximation error in Taylor expansion. However, as shown in
Figures 4.6 and 4.7, this increase is less significant for the proposed subspace tracking
based method; this is the main advantage of tracking the effective dimension of the
signal subspace that changes according to the value of the standard deviation of the
delay spreading and the noise level.
Figure 4.9 shows the RMSE of mean delay estimation of the proposed subspace
tracking based method and MUSIC versus standard deviation of delay spreading for
different SNR values.
102 Chapter 4 Clustered MIMO channel delay estimation
101
102
103
104
105
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 4.9: RMSE of mean delay estimation of the proposed subspace tracking based
method and MUSIC versus standard deviation of delay spreading; Pl = 20, K = 64,
SNR = 5, 10, 15 dB.
100
101
102
103
104
105
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 4.10: RMSE of mean delay estimation of the proposed cost function for U = 1,
U = 2 and U = 3, the proposed subspace tracking based method and MUSIC versus
standard deviation of delay spreading; L = 2, t = [0.37 0.51]T , Pl = 20, K = 64,
SNR = 15 dB.
104 Chapter 4 Clustered MIMO channel delay estimation
100
101
102
103
104
105
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 4.11: RMSE of mean delay estimation of the proposed cost function for U = 1,
U = 2 and U = 3, the proposed subspace tracking based method and MUSIC versus
standard deviation of delay spreading; L = 3, t = [0.37 0.51 0.67]T , Pl = 20,
K = 64, SNR = 15 dB.
Figure 4.12 shows the RMSE of mean delay estimation of the proposed modified
SOMP method, the proposed subspace tracking based method, SOMP method and MU
SIC versus standard deviation delay spreading. As shown in the figure, the modified
SOMP method provides better performance than the conventional MUSIC and SOMP
methods when the standard deviation of delay spreading is above a certain value. The
proposed subspace tracking based method provides the best performance.
4.5 Conclusion 105
100
101
102
103
104
105
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 4.12: RMSE of mean delay estimation of the modified SOMP method, the
proposed subspace tracking based method, SOMP method and MUSIC versus standard
deviation of delay spreading; L = 3, t = [0.37 0.51 0.67]T , Pl = 20, K = 64, SNR =
15 dB.
4.5 Conclusion
In this chapter, two methods for channel mean delays estimation are proposed. A de
terministic channel model is considered, and the DFT coefficients of the received signal
are rederived by means of Taylor expansion around the mean delay parameter. Based
on the first order Taylor expansion, a compressive sensing based method is proposed.
Then based on higher order Taylor approximation, a subspace based method is devel
oped based on the tracking of the effective dimension of the signal subspace, which
depends on the channel features. The two proposed schemes are applied to estimate
the channel mean delays. The proposed methods show better performance in compari
106 Chapter 4 Clustered MIMO channel delay estimation
son to the conventional methods. In comparison with the proposed compressive sensing
based method, the proposed subspace based method allows estimating the cluster mean
delays with more accuracy.
Chapter 5
A cluster of multirays can be characterized by its mean delay and its delay spreading.
In the previous chapter, we focus on estimating the mean delays of the different clusters
based on a deterministic channel model. For the work to be complete, we propose in
this chapter to estimate the standard deviation of the delay spreading of each cluster
based on a stochastic model, exploiting time delays distribution of the clustered signals.
Based on the stochastic model, a subspace based method is derived where both the
mean delay and the standard deviation can be joinlty estimated but through a two
dimensional expensive search. Instead, the estimation procedure is divided into two
steps. As a first step, the channel mean delays can be estimated using one of the
methods proposed in the previous chapter based on the deterministic channel model.
Then the associated standard deviations are estimated based on the stochastic model
using the mean delays already estimated in the first step.
In this approach, the channel delay parameter is modeled in a stochastic manner, assum
ing a predefined statistical distribution for multiray delays. The Fourier coefficients for
any transmitreceive antenna pair can be modeled with the following random function:
107
Chapter 5 Second order delay statistics estimation exploiting channel statistics  a
108 stochastic approach
L Z
X
X[k] = vk (t)αl (t; ξl )dt + Z[k] (5.1)
l=1 t∈T
where T is the interval in which the spreading for all clusters takes place; ξl = [tl , σl ]
is the parameter vector characterizing the channel, such that tl is the mean delay and
σl is the standard deviation of the delay spreading of cluster l; αl (t; ξl ) is the complex
gain in the cluster, where for a fixed ξl , αl (t; ξl ) is a random process with respect to the
delay variable t, and Z[k] is the additive noise modeled as a Gaussian random variable
with zero mean and variance σz2 . Since the gain coefficients are assumed to be identi
cally distributed for all the transmitreceive antenna pairs with the same distribution
of multiray delays, subscript (n, m) is omitted in the above equation.
The Fourier coefficients are concatenated to form the following random vector:
L Z
X
x= v(t)αl (t; ξl )dt + z (5.2)
l=1 t∈T
with x = [X[−K/2 + 1], . . . , X[K/2]]T , v(t) = [v−K/2+1 (t), . . . , vK/2 (t)]T and z =
[Z[−K/2 + 1], . . . , Z[K/2]]T .
RX = E[xxH ] =
L Z Z
0 0 0
X
E[αl (t; ξl )αl∗0 (t ; ξl0 )]v(t)v(t )H dtdt + σz2 I (5.3)
0
l,l =1 T T
0
where t, t ∈ T .
Assuming that the different clustered signals are uncorrelated, and the multirays
within each cluster are also uncorrelated. It comes that :
0
E[αl (t; ξl )αl∗0 (t ; ξl0 )] = δll0 δtt0 σα2 l wl (t; ξl ) (5.4)
where δpq is the Kronecker delta.
L
X
RX = R(tl , σl ) + σz2 I (5.5)
l=1
where
Z +∞
R(tl , σl ) = σα2 l wl (t; ξl )v(t)v(t)H dt (5.6)
−∞
is the covariance matrix of the lth received clustered signal, wl (t; ξl ) is the normalized
power delay function of the cluster and σα2 l is its total mean power.
Assuming that multiray delays in each cluster are uniformly distributed, we have:
1 √ √
wl (t; ξl ) = √ Rect(tl − 3σl , tl + 3σl ) (5.7)
2 3σl
It turns that:
RX = UΛUH = Us Λs UH H
s + Un Λn Un (5.10)
where the columns of Un are the K −(U +1)L eigenvectors spanning the noise subspace,
associated with the K − (U + 1)L smallest eigenvalues of RX .
For the clustered signal l, the signal has most of its energy concentrated in the
first few eigenvalues of the corresponding covariance matrix R(tl , σl ). The eigenvectors
Chapter 5 Second order delay statistics estimation exploiting channel statistics  a
110 stochastic approach
associated to these eigenvalues are orthogonal to the noise subspace, as the remaining
eigenvalues are considered so small, we have
R(tl , σl ) is given in (5.9) where it is expressed in terms of the mean delay of cluster
l and the corresponding standard deviation.
Hence practically, the mean delay and the standard deviation can be jointly esti
mated by searching for the peaks of the following 2D cost function:
1
Pss2 (t, σ) = (5.12)
R(t, σ)Ûn 2F
where Ûn is the matrix of the estimated noise subspace eigenvectors, and .F is the
Frobenius norm.
2π 0 2π 0 √
[R(t, σ)]k+K/2,k0 +K/2 = G[k]2 e−j T (k−k )t
sinc( (k − k ) 3σ) (5.13)
T
the corresponding standard deviation can then be estimated by (5.12) through a one
dimensional search over σ.
3 Estimate the effective dimension of the signal/noise subspace and construct matrix
Ûn .
4 Estimate the L mean delays using one of the proposed approaches in chapter 4.
5 For each estimated mean delay t̂l , substitute its value in R(t, σ) (5.13). Then the
corresponding standard deviation σl is estimated as:
1
σ̂l = argmax (5.14)
σ R(t̂l , σ)Ûn 2F
As the scope of this chapter is to estimate the standard deviation of the delay
spreading, we show in the first figure the RMSE of the standard deviation estimation
versus SNR, assuming that the mean delays are known.
103
104
105
0 5 10 15 20 25 30
Figure 5.1: RMSE of standard deviation estimation versus SNR of the proposed ap
proach with exact mean delay , σl = 0.005T for all l.
In Figure 5.2, we show the RMSE of standard deviation estimation versus SNR
provided by the proposed method (5.14) after substituting the mean delay already
estimated using (4.36).
5.3 Simulation results 113
102
103
104
105
0 5 10 15 20 25 30
Figure 5.2: RMSE of standard deviation estimation versus SNR, σl = 0.005T for all l.
As discussed in the previous chapter, two factors play the role in determining the
effective dimension of the signal subspace, the standard deviation of the delay spread
ing and the noise level. As we can see in Figure 5.2, for relatively small SNR, it is
better to use 2L as the dimension of the signal subspace (K − 2L as the dimension of
the noise subspace). However as SNR increases, it is better to increase the considered
dimension of the signal subspace to 3L (K −3L as the dimension of the noise subspace).
In fact for low SNR, small signal contributions are covered by noise, as a result, noise
seems to occupy higher dimension in the measurement space, and therefore the effec
tive dimension of the signal subspace decreases. On the other hand, as SNR increases,
the effective dimension of the signal subspace increases, hence it is better to consider
higher dimension for the signal subspace (or lower dimension for the noise subspace).
This is illustrated in the above figures. By observing the two figures, we can notice
the influence of the accuracy of the mean delay estimation on the standard deviation
Chapter 5 Second order delay statistics estimation exploiting channel statistics  a
114 stochastic approach
estimation. This is expected as the mean delay estimated using (4.36) is used for the
standard deviation estimation.
Figure 5.3 shows the RMSE of standard deviation estimation versus the standard
deviation of delay spreading at SNR=15 dB.
101
102
103
104
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 5.3: RMSE of standard deviation estimation versus standard deviation of delay
spreading, SNR =15 dB.
As noticed, the effective dimension of the signal subspace increases (or the effective
dimension of the noise subspace decreases) when the delay spreading increases. In fact,
with the same principle discussed before, as the delay spreading increases, the signal
contribution increases, subsequently, the effective dimension of the signal subspace in
creases, hence it is better to consider higher dimension for the signal subspace and lower
dimension for the noise subspace.
5.4 Conclusion 115
5.4 Conclusion
A method for estimating the standard deviation of delay spreading parameter of clus
tered MIMO channel is proposed in this chapter. The channel delay parameter is
treated as stochastic and the channel model is rederived and expressed in terms of the
mean delay and the standard deviation of delay spreading parameters, exploiting the
statistical distribution of multirays within clusters. A subspace based method that has
the ability to estimate the mean delay and the standard deviation of delay spreading
parameters jointly with twodimensional search is then derived. A more interesting
twostep approach is proposed where the mean delay parameter is estimated using one
of the methods proposed in chapter 4, this estimated mean delay is then utilized to es
timate the standard deviation of delay spreading parameter using the derived subspace
based method.
General conclusion and perspectives
Conclusion
The first chapter of the thesis provides an overview of the basic wireless propaga
tion characteristics and introduces MIMO technology. The second chapter provides
an overview of various MIMO channel models with different classifications based on
different situations. The chapter then introduces the sparsity and clustering properties
of several wireless channels, in addition to the common support property in MIMO
outdoor communication scenarios. A sparse clustered MIMO channel model with com
mon support hypothesis is then defined. The channel is considered sparse, such that
it contains limited number of multipath components, where each multipath component
is modeled as a cluster of multirays around a mean delay, and each cluster is param
eterized by its mean delay and delay spreading. For this considered outdoor channel
model, the multirays associated to the same scatterer are assumed to share the same de
lay parameters (mean and standard deviation) on the different transmitreceive antenna
pairs. Chapter 3 deals with the MIMO channel parameter estimation problem. The
estimation approaches are classified into parametric and nonparametric. The chapter
focuses on the parametric approach for estimation as it allows exploiting some proper
117
118 General conclusion and perspectives
Chapters 4 and 5 illustrate the methods proposed to estimate some channel parame
ters. The sparse clustered model defined in chapter 2 is considered. The work is focused
on estimating the mean delay and the standard deviation of delay spreading param
eters characterizing the clusters. Chapter 4 deals with mean delay estimation, where
two approaches are proposed, a compressive sensing based approach and a subspace
based approach, based on a deterministic channel model. The compressive sensing
based approach is based on the first order Taylor expansion of the observation Fourier
coefficients, where a modified SOMP method is proposed to estimate the cluster mean
delays. The proposed modified SOMP method shows a better estimate of the mean de
lay parameter in comparison with the conventional SOMP method. The second scheme
for estimation is based on the subspace approach. The expressions of the observation
Fourier coefficients are rederived based on higher order Taylor expansion. Then, a sub
space based method that exploits the rederived model is proposed to estimate the mean
delays. The proposed approach is based on the tracking of the effective dimension of the
signal subspace that changes depending on the standard deviation of delay spreading
and SNR. The proposed subspace based approach outperforms both the conventional
MUSIC method and the proposed compressive sensing based method (modified SOMP
method) in terms of cluster mean delays estimation. In chapter 5, we focus on clus
ter delay spreads estimation. A stochastic modeling of the channel delay parameter is
proposed, where the statistical distribution of multiray delays is exploited. A subspace
based method is then derived where the mean delay and the delay spreading parameters
can be estimated jointly through a twodimensional search. More interesting, a two
step approach is proposed. The mean delays are estimated using one of the approaches
proposed for mean delay estimation. The estimated mean delays are then exploited to
estimate the corresponding standard deviation of delay spreading using the subspace
approach.
Perspectives
The signal subspace tracking criterion proposed in chapter 4 does not seem to be effi
cient in some cases. As a future work, the problem of signal subspace tracking can be
addressed where other approaches can be developed to attain better performance.
A time domain channel model is considered in this work. However, it will be inter
119
esting to include some spatial domain parameters (angles), where different approaches
can be proposed to estimate both time and angular domain parameters.
Dans le monde antique, la lumière et les drapeaux étaient utilisés comme moyen de
communication sans fil. En 1867, James Clerk Maxwell prédit l’existence d’ondes
électromagnétiques (EM), proposant une interrelation entre les champs électriques et
magnétiques. En 1887, Heinrich Rudolf Hertz a confirmé l’existence d’ondes électromag
nétiques voyageant à la vitesse de la lumière en effectuant des expériences dans son
laboratoire. Les ondes qu’il a produites et reçues sont maintenant appelées ondes ra
dio. Guglielmo Marconi a fait une percée en mettant au point le télégraphe sans fil en
1895. Depuis lors, il a réussi à transmettre des signaux radio dans l’espace, augmen
tant progressivement la distance de communication. En 1901, il a établi la première
communication sans fil à travers l’océan, en transmettant des signaux radio à travers
l’océan Atlantique. Depuis lors jusqu’à aujourd’hui, différentes technologies sans fil ont
été développées, notamment la radiodiffusion et la télédiffusion, les communications
radar, les communications par satellite, les réseaux sans fil, les communications mobiles
sans fil, etc.
121
122 Résumé en francais
La connaissance des caractéristiques des canaux de propagation sans fil est cruciale
pour la fiabilité des communications sans fil, en particulier dans les communications
MIMO, afin de profiter pleinement des avantages offerts par l’utilisation de la tech
nologie antennes multiples du côté de l’émetteur et du récepteur. Ces informations
sur les caractéristiques du canal sont appelées informations sur l’état du canal (CSI).
CSI représente l’information sur la propagation du signal de l’émetteur au récepteur, il
représente les effets de canal sans fil tels que l’atténuation de puissance et l’étalement
dans le temps des signaux. La connaissance des canaux de propagation sans fil est
essentielle pour la conception des systèmes de communication MIMO. Les informa
tions sur l’état des canaux au niveau du récepteur (CSIR) peuvent être utilisées à
des fins d’égalisation contre les interférences intersymboles (ISI) causées par la prop
agation par trajets multiples, et les informations sur l’état des canaux au niveau de
l’émetteur (CSIT) peuvent être utilisées pour concevoir une transmission optimale. Par
conséquent, un système de communication MIMO “idéal” performant nécessiterait une
connaissance exacte du canal MIMO ou CSI. Les approches d’estimation CSI peuvent
être classées en deux catégories : paramétrique et non paramétrique. Dans l’approche
non paramétrique, la matrice des canaux est estimée directement sans référence à aucun
paramètre de propagation physique sousjacent. D’autre part, l’approche paramétrique
s’appuie sur des modèles de canaux physiques pour estimer les paramètres des canaux,
ces paramètres sont utiles pour comprendre le canal sans fil et peuvent être utilisés
pour améliorer les performances du système de communication en adaptant les concep
tions de transmission et de réception en fonction de cellesci. L’intérêt de l’approche
paramétrique est qu’elle permet d’exploiter certaines propriétés du canal telles que la
parcimonie et le support commun.
123
Le phénomène de regroupement dû aux diffuseurs est une propriété importante qui
caractérise plusieurs canaux sans fil, où, selon les différentes recherches sur les canaux
sans fil, les composantes de trajets multiples des canaux sont modélisées comme des
grappes (“clusters”) de rayons multiples. Par exemple, ce phénomène (regroupement
ou “clustering”) caractérise les canaux de communication à large bande/à bande ul
tralarge (UWB) et MMW. Par conséquent, la parcimonie et le regroupement sont deux
propriétés des canaux sans fil auxquelles les futurs systèmes de communication sans fil
devront faire face. Afin de permettre la communication sur de tels canaux, une con
naissance approfondie des caractéristiques et des paramètres du canal est nécessaire,
où les nouvelles caractéristiques du canal doivent être prises en compte dans les fu
tures techniques d’estimation du canal. Les travaux de cette thèse se concentrent sur
l’estimation de paramètres de canal MIMO en grappes, en particulier les paramètres
du domaine temporel. Le canal est caractérisé dans le domaine temporel, où différents
schémas sont proposés pour estimer certains paramètres de canal du domaine temporel.
Dans le deuxième chapitre, nous donnons un aperçu des différents modèles de canaux
MIMO. Les modèles de canaux sont classés en modèles physiques et non physiques. Les
modèles physiques sont ensuite classés en modèles déterministes et stochastiques, où les
modèles stochastiques sont classés en modèles géométriques, modèles non géométriques
et modèles analytiques fondés sur la propagation. Ensuite, nous introduisons les pro
124 Résumé en francais
priétés de parcimonie et de regroupement dans les canaux sans fil, en plus de la propriété
de support commune dans les canaux MIMO extérieurs. Enfin, nous introduisons un
modèle de canal MIMO en grappes parcimonieux avec un support commun, sur lequel
sont basées les méthodes d’estimation que nous proposons.
Les chapitres 4 et 5 illustrent les méthodes proposées pour estimer certains paramètres
de canal. Le modèle en grappes parcimonieux défini au chapitre 2 est considéré. Le
travail est axé sur l’estimation du retard moyen et de l’écarttype des paramètres
d’étalement du retard caractérisant les grappes. Le chapitre 4 traite de l’estimation du
retard moyen, où deux approches sont proposées, une approche fondée sur l’acquisition
comprimée et une approche fondée sur le sousespace, fondée sur un modèle de canal
déterministe. L’approche fondée sur l’acquisition comprimée est basée sur l’expansion
de Taylor du premier ordre autour du paramètre du retard moyen, où une méthode
SOMP modifiée est proposée pour estimer les retards moyens des grappes. La méthode
SOMP modifiée montre une meilleure estimation du retard moyen par rapport à la
méthode SOMP conventionnelle. L’approche sousespace est fondée sur l’expansion de
Taylor d’ordre supérieur autour du paramètre du retard moyen, où une méthode fondée
sur le sousespace est proposée pour estimer les retards moyens de la grappe. L’approche
proposée est fondée sur le suivi de la dimension effective du sousespace du signal qui
varie en fonction de l’écarttype de l’étalement du retard et du rapport signal/bruit.
L’approche basée sur les sousespaces proposée surpasse à la fois la méthode MUSIC con
ventionnelle et la méthode basée sur l’acquisition comprimée (méthode SOMP modifiée)
en termes d’estimation des retards moyens des grappes. Dans le chapitre 5, nous nous
concentrons sur l’estimation de l’écarttype de l’étalement du retard. Une modélisation
stochastique du paramètre de retard du canal est proposée, où la distribution statis
tique des retards multirayons au sein de chaque grappe est exploitée. Une méthode
basée sur le sousespace est dérivée où le retard moyen et les paramètres d’étalement
du retard peuvent être estimés conjointement par une recherche bidimensionnelle. Plus
intéressant, une approche en deux étapes est proposée. Les retards moyens sont es
timés à l’aide de l’une des approches proposées pour l’estimation du retard moyen. Les
125
retards moyens estimés sont ensuite exploités pour estimer l’écarttype de l’étalement
des retards en utilisant l’approche de sousespace.
Publications
Journal
A. Mohydeen, P. Chargé, Y. Wang, O. Bazzi, and Y. Ding, “Spatially correlated
sparse MIMO channel path delay estimation in scattering environments based on
signal subspace tracking,” Sensors (Basel, Switzerland),vol. 18, no. 5, 2018.
Conference
[1] A. Mohydeen, P. Chargé, Y. Wang, and O. Bazzi, “Estimation of clustered MIMO
channel parameters exploiting channel statistics”. Fifth SinoFrench Workshop
on Information and Communication Technologies (SIFWICT 2019), June 2019,
Nantes, France.
127
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Mots clés : MIMO, estimation des paramètres de canal, méthodes de sousespace, acquisition
comprimée, statistiques de canal