Optimized Power-Allocation for Multi-Antenna Systems

1
Optimized Power-Allocation for Multi-Antenna
Systems impaired by Multiple Access Interference
and Imperfect Channel-Estimation
Enzo Baccarelli, Mauro Biagi, Cristian Pelizzoni, Nicola Cordeschi
{enzobac, biagi, pelcris, cordeschi}@infocom.uniroma1.it
Abstract— This paper presents an optimized spatial signal
shaping for Multiple-Input Multiple-Output (MIMO) ”ad-hoc”like networks. It is adopted for maximizing the information
throughput of pilot-based Multi-Antenna systems affected by
spatially colored Multiple Access Interference (MAI) and channel
estimation errors. After deriving the architecture of the Minimum
Mean Square Error (MMSE) MIMO channel estimator, closed
form expressions for the maximum information throughput
sustained by the MAI-affected MIMO links are provided. Then,
we present a novel power allocation algorithm for achieving the
resulting link capacity. Several numerical results are provided
to compare the performance achieved by the proposed powerallocation algorithm with that of the corresponding MIMO
system working in MAI-free environments and equipped with
error-free (e.g., perfect channel-estimates). So doing, we are able
to give insight about the ultimate performance loss induced in
MIMO systems by spatially colored MAI and imperfect channel
estimates. Finally, we point out some implications about Space
Division Multiple Access strategies arising from the proposed
power allocation algorithm.
Index Terms— Multi-Antenna, MAI, imperfect channel estimation, signal-shaping, space-division multiple-access.
I. I NTRODUCTION AND G OALS
Due to the current fast increasing demand for highthroughput Personal Communication Services (PCSs) based
on small-size power-saving palmtops, the requirement for
”always on” mobile data access based on uncoordinated ”adhoc” and ”mesh” type networking architectures are expected to
dramatically increase within next few years [18,20,27,28]. In
order to increase the channel throughput, the spatial dimension
is viewed as lowest cost solution for wireless communication
systems. As a consequence, in these last years increasing
attention has been directed towards designing array-equipped
transceivers for wireless PCSs [25,27]. Moreover, such technological solution suitably addresses those energy-constrained
application scenarios in which wireless ad-hoc and mesh
networks are though to be applied, by providing adequate
diversity and coding gains. This is justified by considering that
both ad-hoc and mesh networks are typically characterized by
users equipped with battery-powered terminals. So the MIMO
capability to offer same performances of SISO systems with
Enzo Baccarelli, Mauro Biagi, Cristian Pelizzoni and Nicola Cordeschi are
with INFO-COM Dept., University of Rome ”Sapienza”, Via Eudossiana 18,
00184 Rome, Italy. Ph. no. +39 06 44585466 FAX no. +39 06 4873300.
This work has been partially supported by Italian National project: Wireless 8O2.16 Multi-antenna mEsh Networks (WOMEN) under grant number
2005093248.
a considerable gain in terms of power consumption, makes
the Multi-Antenna approach suitable for wireless ad-hoc and
mesh networks [30,31,32].
A. Related Works
In this respect, current literature mainly focuses on
transceivers working under the assumption of MIMO channel’s
perfect estimation. Specifically, in [1,2] the capacity of MIMO
systems under spatially colored MAI is evaluated when the
MIMO channel is assumed to be perfectly known at receive
and transmit sides, while in [23] the MAI is assumed still
spatially colored but the channel is assumed perfectly known
only at the receiver. The above assumptions may be considered reasonable when quasi-static application scenarios are
considered (e.g., Wireless Local Loop systems, [1]), but may
fall short when emerging applications for high-quality mobile
PCSs [9,10,27] are considered. Finally, some recent works
account for imperfect channel estimation [5,10], but they do
not analyze the effect of spatially colored MAI on the resulting
channel throughput.
B. Proposed Contributions
Therefore, motivated by the above considerations, in this
work we focus on the ultimate information throughput conveyed by pilot-based wireless MIMO systems impaired by
spatially colored MAI when imperfect channel estimates are
available at transmit and receive sides. Specifically, main
contributions of this work may be summarized as follows.
First, after developing the optimal MMSE channel estimator
for pilot-based MIMO systems impaired by spatially-colored
MAI, we derive the closed form expression for the resulting
sustained information throughput. Second, we propose an iterative algorithm for the optimized power allocation and signalshaping under the assumption of imperfect channel estimates at
transmit and receive sides. Third, we provide numerical results
and performance comparisons for testing the effectiveness of
the proposed spatial-shaping and power allocation algorithm
when ”ad-hoc” networking architectures are considered. Finally, we point out some (novel) guidelines about the optimized
design of Space Division Multiple-Access strategies arising
from the proposed power allocation algorithm.
2
C. Organization of the work
The remainder of this paper is organized as follows. The
system modelling is described in Sect.II and the MIMO channel MMSE estimator is developed in Sect.III. The information
throughput evaluation and the resulting optimal power allocation algorithm are presented in Sect.IV. In Sect.V a model for
the spatial MAI arising in Multi-Antenna ”ad-hoc” networks is
described. Numerical plots and performance comparisons for
testing the proposed power allocation algorithm are presented
in Sect.VI. Finally, Sect.VII is devoted to discuss some general
guidelines for the overall design of MAI-impaired MultiAntenna pilot-trained transceivers.
Before proceeding, let us spend few words about the adopted notation. Capital letters are for matrices, lower-case underlined symbols denote vectors, and characters with overlined
arrow → denote block-matrices and block-vectors. Apexes ∗ ,
T †
, are respectively meant as conjugation, transposition and
conjugate-transposition, while lower-case letters are used for
scalar values. In addition, det [A] and T ra[A] mean determinant and trace of matrix A , [a1 ... am ], and vect(A) denotes
the (block) vector obtained by stacking the A’s columns.
Finally, Im is the (m × m) identity matrix, ||A||E is the
Euclidean norm of the matrix A, A ⊗ B is the Kronecker
product of the matrix A by matrix B, 0m is the m-dimensional
zero-vector, 0m×n stands for (m × n) zero-matrix, lg denotes
natural logarithm and δ(m, n) is the (usual scalar) Kronecker
delta (e.g., δ(m, n) = 1 for m = n and δ(m, n) = 0 for
m 6= n).
Transmit and receive units are equipped with t and r antennas, respectively. The MIMO radio channel should be affected
by slow-variant Rayleigh flat fading1 and multiple access
interference. Path gains {hji } from i-th transmit antenna to
j-th receive one may be modelled as complex zero-mean unitvariance random variables (r.v.) [5,6,7,8], and they may be
assumed mutually uncorrelated when the antennas are properly
spaced2 .
Furthermore, when low-mobility applications are considered
(e.g., nomadic users over hot-spot cells), all path gains may
be assumed to change every T ≥ 1 signaling period at new
statistically independent values. The resulting ”block-fading”
model may be used to properly describe the main features
of interleaved frequency-hopping or interleaved packet-based
systems [7,18,19]. MAI affecting the link in Fig.1 depends on
the network topology [1,2,20]. Specifically, we suppose that
it is at least constant over a packet period. Anyway, {hji }
and MAI statistics may be different over temporally adjacent
packets, so that Tx and Rx nodes in Fig.1 do not exactly know
them at the beginning of any transmission period. Therefore,
according to Fig.2, the packet structure is composed by T ≥ 1
slots: the first TL ≥ 0 ones are used by Rx for learning the
MAI statistics (see Sect.II.A); the second Ttr ≥ 0 ones are
employed for estimating the (forward) MIMO channel path
gains {hji } (see Sect.II.B) and, finally, the last Tpay , T −
Ttr − TL ones are adopted to carry out payload data (see
Sect.II.C).
TL(learning)
Ttr(training)
Tpay(payload)
II. T HE S YSTEM M ODELING
The application scenario we consider refers to the emerging
local wireless ”ad-hoc” networks [18,20,27,28] where multiple
autonomous transmit-receive nodes are simultaneously active
over a limited-size hot-spot cell, so that all transmissions are
affected by MAI [18]. The (complex base-band equivalent)
radio channel from a transmit node Tx to the corresponding
receive one Rx is sketched in Fig 1.
Fig. 2.
The packet structure (T
As consequence, after denoting as RC (nats/slot) the spacetime information rate, the resulting system spectral efficiency
η (nats/sec/Hz) equates
η=
T
Space Time
Source Encoder and
Message Modulator
with
x
t
Antennas
1
H
Tpay RC
,
T ∆s B w
(1)
where ∆s (sec.) and Bw (Hz) denote the slot duration and RF
bandwidth of the radiated signals, respectively.
h11
h 21
2
1
2
Demodulator,
Channel
Detected
Estimator and
Message
Decoder with
r
Antennas
A. The Learning Phase
Rx
t
, TL + Ttr + Tpay )
hrt
r
MIMO FORWARD CHANNEL
Kd
During the learning phase (see Fig.2), Tx in Fig.1 is off
and Rx attempts to ”learn” the MAI statistics. Thus, all
receive antennas are now used to capture the interfering signals
H
Kd
FEEDBACK LINK
Fig. 1. Multi-Antenna system equipped with imperfect (forward) channel
estimates Ĥ and impaired by MAI with spatial covariance matrix Kd .
1 The flat fading assumption is valid when the radiated signal RF bandwidth
Bw is less than the MIMO forward channel coherence bandwidth Bc .
Furthermore, we anticipate that the effects of Ricean-distributed fading on
the system performance are accounted for and evaluated in the following
Sects. V, VI.
2 For hot-spot local area applications, proper antenna spacing may be
assumed of the order of λ/2 [15]. However, several measures and analytical
contributions estimate (very) limited throughput loss when the path gains’
correlation coefficient is less than 0.6 [4 and references therein].
3
emitted by the interfering transmit nodes3 . So, after denoting
by ẏ(n) , [ẏ1 (n)...ẏr (n)] the r-dimensional column vector of
the (sampled) signals received at the n-th ”learning” slot, this
last equates
ẏ(n) ≡ ḋ(n) , ẇ(n) + v̇(n), 1 ≤ n ≤ T.
(2)
The overall disturbance vector ḋ(n) , [d˙1 (n)...d˙r (n)]T in
(2) is composed by two mutually independent components,
which are denoted by ẇ(n) , [ẇ1 (n)...ẇr (n)]T and v̇(n) ,
[v̇1 (n)...v̇r (n)]T , respectively. The first component takes into
account for the receiver thermal noise and then it is modeled as
a zero-mean, spatially and temporally white Gaussian complex
r-variate sequence, with covariance matrix
n
o
E ẇ(n)(ẇ(m))† = N0 Ir δ(m, n),
(3)
where N0 (watt/Hz) is the power spectral density of the
thermal noise. The second component in (2) takes the MAI
into account. It is modelled as zero-mean, temporally white,
spatially colored Gaussian complex r-variate sequence, whose
covariance matrix
2
c11
n
o 6 c∗12
Kv , E v̇(n)(v̇(n))† ≡ 6
4 ..
.
c∗1r
...
...
..
.
...
c1r
c2r
..
.
crr
3
7
7,
5
B. The Training Phase
Based on the MAI covariance matrix Kd , Tx node can now
optimally shape the pilot streams {e
xi (n) ∈ C1 , TL + 1 ≤ n ≤
TL + Ttr }, 1 ≤ i ≤ t, which are used by Rx to estimate
the MIMO forward channel path gains {hji , j = 1, ..., r, i =
1, ..., t}. Specifically, when the pilot streams are transmitted,
the sampled signals {e
yj (n) ∈ C1 , TL +1 ≤ n ≤ TL +Ttr }, 1 ≤
j ≤ r, received at the output of j-th receive antenna are
t
(4)
1 X
hji x
ei (n) + dej (n), TL + 1 ≤ n ≤ TL + Ttr ,
yej (n) = √
t i=1
1 ≤ j ≤ r,
(7)
where the overall disturbances
is supposed to be constant over a packet transmission4 (at
least). Since its value may be different over temporally adjacent packets, we assume that both Tx and Rx nodes of Fig.1
do not exactly know the overall disturbance covariance matrix
n
o
Kd , E ḋ(n)(ḋ(n))† ≡ Kv + N0 Ir ,
(5)
at the beginning of any new packet transmission period. Since
the received signals {ẏ(n)} in (2) equate MAI {ḋ(n)} ones,
from the Law of Large Numbers [26] we obtain the following
unbiased and consistent (e.g, asymptotically exact) estimate
K̂d for the MAI covariance matrix: Kd
TL
1 X
ẏ(n)(ẏ(n))† .
K̂d =
TL n=1
mean square estimation errors under 10%. Furthermore, since
the numerical results in Sect.VI.D confirm that throughput
loss, due to imperfect MAI covariance matrix estimate, may
be neglected for TL exceeding 10, we assume that, at the end
of the learning phase (e.g., at step n = TL ), Kd is perfectly
estimated by Rx node and then it is transmitted back to Tx
via the ideal feedback link of Fig.15 . This assumption will be
relaxed in Sect.VI.C, when we will test the sensitivity of the
proposed signal-shaping algorithm to errors possibly affecting
the estimated K̂d .
(6)
Concerning the accuracy of the estimate in (6), analytical
results (see [3 and references therein]) show that the relative
square estimation error ||Kd − K̂d ||2E /||Kd ||2E vanishes as at
least 1/TL . So, in principle TL = 10 suffices for achieving
3 In principle, some system synchronization should be assumed to guarantee
that the learning procedure is carried out by only one user at time. However,
under the (milder) assumption that each user actives his learning procedure
at randomly selected times, it is likelihood to retain negligible the probability
that more users are simultaneously in the learning phase. Anyway, we
anticipate that the numerical results of Sect.VI.C support the conclusion that
the performance of the optimized power allocation algorithm we propose in
Sect.IV, is quite robust against errors possibly present in the estimate of actual
MAI covariance matrix Kd in (5).
4 The assumption of temporally white MAI sequence {v̇(n)} may be
considered reasonable when FEC coding and interleaving are employed [11].
In addition, by resorting to the Central Limit Theorem, the overall disturbance
{ḋ(n)} in (2) may be considered Gaussian distributed. Since the Gaussian pdf
maximizes the differential entropy [12], by fact we are considering a worstcase application scenario. Finally, since the network topology for serving
nomadic users is slow-variant [20], it can be reasonable to suppose Kv in
(4) to be constant (at least) over each packet transmission period.
dej (n) , vej (n) + w
ej (n), TL + 1 ≤ n ≤ TL + Ttr ,
1 ≤ j ≤ r,
(7.1)
are independent from the path gains {hji } and still described by (4) and (5). Hence, by assuming the (usual) power
constraint
t
1X
||e
xi (n)||2 = Pe, TL + 1 ≤ n ≤ TL + Ttr ,
(8)
t i=1
on the average transmitted power Pe, the resulting signal to
interference-plus-noise ratio (SINR) γ
ej at the output of j-th
receive antenna equates (see eqs.(7), (8))
γ
ej = Pe/(N0 +cjj ), 1 ≤ j ≤ r,
(8.1)
where N0 + cjj is j-th diagonal entry of Kd . All the
(complex) samples
ini (7) may be collected into the (Ttr × r)
h
e , y
f1 ...f
yr given by
matrix Y
e = √1 XH
e + D,
e
Y
t
(9)
e , [xe1 ...xet ] is the pilot matrix, H , [h1 ...hr ] is the
where X
e , [d
f...de ] is the (Ttr × r)
(t × r) channel matrix and D
1
r
disturbance matrix. Since the pilot streams are power limited
e becomes
(see eq.(8)), the resulting power constraint on X
†
eX
e ] = tTtr Pe.
T ra[X
(9.1)
5 We remark that Time-Division-Duplex (TDD) WLANs, designed for lowmobility applications, are usually equipped with (very) reliable duplex channels [15,18]. So the above assumption may be considered well met. Anyway,
the performance loss arising from noisy feedback channels is investigated in
Section VI.C.
4
e in (9) are
In Sect.III we detail how the training observations Y
employed by Rx in Fig.1 for computing the MMSE channel
e At the end of the training
estimates matrix Ĥ , E{H | Y}.
phase (e.g., at n = TL + Ttr ), Ĥ is transmitted by Rx back to
Tx through the (ideal) feedback link of Fig.1.
C. The Payload Phase
Based on Kd and Ĥ, Tx node in Fig.1 may properly shape
the (random) signal information streams {φi (n) ∈ C1 , TL +
Ttr + 1 ≤ n ≤ T }, 1 ≤ i ≤ t, to be radiated. After
their transmission, the resulting (sampled) signals {yj (n) ∈
C1 , TL + Ttr + 1 ≤ n ≤ T }, 1 ≤ j ≤ r, received by Rx are
t
1 X
hji φi (n) + dj (n), TL + Ttr + 1 ≤ n ≤ T,
yj (n) = √
t i=1
1 ≤ j ≤ r, (10)
where the disturbance sequences dj (n) , vj (n) + wj (n), 1 ≤
j ≤ r, are mutually independent from the channel coefficients
{hji } and the radiated information streams {φi }. As for the
pilot streams, the signals {φi (n)} radiated during the payload
phase are also assumed power-limited as in
¤T
£ T
y (TL + Ttr + 1) ...yT (T ) , we arrive at the following
final observation model:
1
→ −
→
T −
−
→
y = √ [IT pay ⊗ H] φ + d ,
(12)
t
→
−
where the (Tpay r × 1) (block) disturbance vector d ,
£ T
¤
T
d (TL + Ttr + 1) ...dT (T ) is Gaussian distributed, with
covariance matrix given by
n
o
−
→−
→
E d ( d )† = ITpay ⊗ Kd ,
and
the
(block)
signals
vector
h
iT
φT (TL + Ttr + 1) . . . φT (T )
is power
in
(12.1)
→
−
φ
,
limited
as
n † o
→
− →
−
E φ φ = Tpay tP.
(12.2)
III. MMSE MIMO C HANNEL E STIMATION UNDER
SPATIALLY COLORED MAI
Since in [9] it is proved that the MMSE matrix estimate
e of the MIMO channel matrix H
Ĥ ≡ [ĥ1 ...ĥr ] , E{H|Y}
(10.1) in (9) is a sufficient statistic for the ML detection of the
transmitted message M of Fig.1, we do not lose information
so that the SINR γj at the output of the j-th receive antenna by considering the receiver’s architecture composed by the
MIMO channel MMSE estimator cascaded to the ML detector
equates6 (see eqs.(5), (10))
of the transmitted message. Thus, before starting to develop
γj = P/(N0 +cjj ), 1 ≤ j ≤ r.
(10.2) the MMSE estimator, let us note that the Ỹ’s columns in (9) are
Now, from (10) we may express (r × 1) column vector mutually dependent, so that any estimated channel coefficient
y(n) , [y1 (n)...yr (n)]T of the observations received during ĥji is a function of the whole observed matrix Ỹ. However,
the j-th column ĥj of Ĥ can be computed via an application
n-th slot as
of the Orthogonal Projection Lemma as in (see the Appendix
A)
1
y(n) = √ HT φ(n) + d(n), TL + Ttr + 1 ≤ n ≤ T, (11)
t
ih 1 ³
´
i−1
1 h
†
†
−1/2
ĥj = √ eTj Kd
⊗ X̃
K−1
⊗ X̃X̃ + IrTtr
T
d
where {d(n) , [d1 (n)...dr (n)] , TL + Ttr + 1 ≤ n ≤ T }
t
t
³
´
is the temporally white Gaussian MAI vector with spatial
−1/2
· Kd
⊗ ITtr vect(Ỹ), 1 6 j 6 r.
(13)
covariance matrix still given by eq.(5), H is the previously
7
T
defined (t × r) channel matrix and φ(n) , [φ1 (n)...φt (n)]
r
collects the symbols transmitted by the t transmit antennas. In (13), ej denotes the j-th unit vector of R [13], vect(Ỹ) is
obtained via the ordered
Furthermore, after denoting as Rφ , E{φ(n)φ(n)† } the the rTtr -dimensional column vector−1/2
stacking
of
the
Ỹ’s
columns
while
K
is the positive square
d
spatial covariance matrix of φ(n) , [φ1 (n)...φt (n)]T , from root of K−1 [13]. Now, by denoting as ² ≡ [² ...² ] , H − Ĥ
1
r
d
(10.1) this last must meet the following power constraint:
the error matrix of the MMSE channel estimates, the cross
n
o
correlation among its columns may be evaluated as in
E φ(n)† φ(n) ≡ T ra[Rφ ] = tP,
½ ³ ´ ¾
o
n
†
†
TL + Ttr + 1 ≤ n ≤ T.
(11.1)
E ²j (²i ) = δ(j, i)It − E ĥj ĥi
t
o
1X n
E ||φi (n)||2 = P, TL +Ttr +1 ≤ n ≤ T,
t i=1
Finally, by stacking the Tpay observed vectors
→
,
in (11) into the (Tpay r × 1) block vector −
y
6 We point out that our model explicitly accounts for the different power
levels Pe and P that may be radiated by transmit antennas during the training
and payload phases, respectively.
7 We anticipate that the combined utilization of H in the model (9) and
HT in the relationship (11) simplifies the resulting expressions for the
MMSE
channel
estimates in (13) and the conveyed information throughput
−
→
→
I −
y ; |Ĥ in (24).
´† ³
´†
1³
−1/2
ej ⊗ It
Kd
⊗ X̃
t
· ³
´¸−1 ³
´³
´
1
†
−1/2
−1
Kd ⊗ X̃X̃ + IrTtr
· Kd
⊗ X̃ ei ⊗ It ,
·
t
= δ(j, i)It −
1 6 j, i 6 r.
(14)
Thus, the resulting total mean square error σtot , ||²||2E
equates
5
2
σtot
,
r
X
h
T ra
²j ²†j
i
= rt
j=1
·³
r
´† ³
´†
1X
−1/2
−
T ra ej ⊗ It
Kd
⊗ X̃
t j=1
³1 ³
´¸
´
´−1 ³
´³
†
−1/2
−1
·
Kd ⊗ X̃X̃ + IrTtr
Kd
⊗ X̃ ej ⊗ It .
t
(15)
A. Condition for the optimal training
2
Since the total mean square error σtot
in (15) depends on
the employed pilot streams via the training matrix X̃ in (9),
we are going to select it for minimizing (15) under the power
constraint (9.1). By properly applying the Cauchy inequality
[13], we provide the following condition for the design of the
optimal training matrix X̃ (see the Appendix B).
Proposition 1. The training matrix X̃, that minimizes the total
square error in (15) under the power constraint (9.1), must
meet the following relationship:
K−1
d
†
⊗ X̃ X̃ = aIrt ,
(16)
{ĥji } are uncorrelated and identically Gaussian distributed, so
that the pdf p(Ĥ) of the resulting estimated matrix Ĥ equates
(
)
´rt
³
†
1
1
exp −
T ra[Ĥ Ĥ] . (20)
p(Ĥ) =
π(1 − σε2 )
(1 − σε2 )
Furthermore, all the entries of the resulting MMSE error
matrix ² = H − Ĥ are mutually independent, identically
distributed and Gaussian, with variances given by eq.(19).
Finally, from (17) and (19), we may conclude that estimated
matrix Ĥ approaches the actual one H for σε2 → 0, while Ĥ
vanishes for σε2 → 1, so that we have the following limit
expressions:
lim Ĥ = H
σε2 →0
lim Ĥ = 0t×r .
(20.1);
σε2 →1
(20.2)
According to a current taxonomy, we refer to (20.1), (20.2) as
Perfect CSI (PCSI) and No CSI (NCSI) operating conditions,
while Imperfect CSI (ICSI) corresponds to 0 < σε2 < 1.
IV. C ONVEYED I NFORMATION T HROUGHPUT UNDER
CHANNEL ESTIMATION ERRORS AND SPATIALLY COLORED
MAI
where the positive scalar a equates
The MIMO block fading channel of Sect.II is information
stable, so that the resulting Shannon Capacity C is the corresponding maximum sustainable throughput. By following
Ttr P̃
a,
T ra[K−1
(16.1) quite standard approaches [14], this capacity may be expressed
d ].
r
as in
¨
Z
C
=
{C(
Ĥ)}
≡
C(Ĥ)p(Ĥ)dĤ, (nats/payload slot), (21)
Therefore, from (16) we deduce that the optimal X̃ depends on
the spatial coloration property of MAI via the corresponding
covariance matrix Kd . By fact, the practical implication of where p(Ĥ) is given by (20), and
´
the relationship (16) is that the pilot streams radiated by
1 ³→
−
→
C(Ĥ) ,
sup
I −
y ; φ |Ĥ ,
(22)
transmit antennas should be orthogonal after the whitening
Tpay
−
→ −
→† −
→
filter performed by the receiver. In the special case of Kd = It
φ :E{ φ φ }≤tTpay P
(e.g., when the MAI is spatially white), eq.(16) becomes
†
´ channel capacity conditioned on Ĥ. Furthermore,
X̃ X̃ = aIt , and the optimal X̃ matrix is the usual (para) is³the MIMO
−
→
→
I −
y ; φ |Ĥ in (22) is the mutual information conveyed by the
unitary one [9,10].
MIMO channel (12) when Ĥ is the channel estimate available
of Fig.1. Unfortunately, the optimal pdf
B. The MIMO channel MMSE Estimator for the optimal at Tx and Rx nodes
−
→
of
input
signals
φ
achieving the sup in (22) is currently
training
unknown, even in the case of spatially white MAI [1,2,4,5].
When X̃ meets the optimality condition in (16), eqs. (13), Anyway, in [7] it is shown that Gaussian distributed input
(14) assume the following (simpler) forms:
signals are the capacity-achieving ones for 0 ≤ σε2 ≤ 1 when
´
³
the payload phase length Tpay is largely greater than the
1 − σ2
†
ĥj = √ ε eTj K−1
vect(Ỹ), 1 ≤ j ≤ r, (17) number of transmit antennas (see [7] about this asymptotic
d ⊗ X̃
t
result). Therefore, in the sequel we directly consider Gaussian
and
distributed input signals. In this case, the Tpay components
n
o
n
o
t
Ttr + 1 ≤ n ≤ T } in (11) of the
2
†
†
E ²j (²i ) ≡ δ(j, i)It −E ĥj (ĥi ) ≡ σε It δ(j, i), 1 ≤ j, i ≤ r,{φ(n) ∈ C , TL + −
→
overall
signal
vector
φ
in (12) are modelled as uncorrelated
(18)
zero-mean
complex
Gaussian
vectors, with correlation matrix
where
†
}
constrained
as in (11.1).
R
,
E{φ(n)φ(n)
o ³ a ´−1
o
n
n
φ
2
2
2
Obviously,
the
MIMO
channel
information
throughput
σε , E ||εji || ≡ E ||hji −ĥji || = 1+
, 1 ≤ j, i ≤ r,
t
³
´
(19)
1
−
→
−
→
sup
I
y
;
φ
|
Ĥ
,
(23)
T
(
Ĥ)
,
G
is the mean square error estimation variance (which is the
£ ¤
Tpay
T
ra
R
≤P
t
same for all i and j). Furthermore, the estimated path gains
φ
6
is upper-bounded by C(Ĥ) in (22), so that, in general, we have
TG (Ĥ) ≤ C(Ĥ). Anyway, the equality is attained when the
above mentioned condition of ³
Tpay >> t´ is met.
−
→
−
About the computation of I →
y ; φ |Ĥ in (23), in general
it resists closed-form evaluation. However, in Appendix C we
prove the following result.
Proposition 2. Let us suppose X̃ ³
to meet eq.(16).
Then, the
´
→
−
→
conditional mutual information I −
y ; φ |Ĥ in (23) of the
MIMO channel (12) equates
³
´
−
→
→
I −
y ; φ |Ĥ = Tpay
¶¸
·µ
∗ −1/2
1 −1/2 T
+ σε2 P K−1
· lg det Ir + Kd Ĥ · Rφ Ĥ Kd
d
t
·µ
¶¸
σ 2 Tpay −1 ∗
− lg det Irt + ε
(Kd ) ⊗ Rφ ,
(24)
t
when (at least) one of following three conditions is verified :
a) both Tpay and t are large;
(24.1)
b) σε2 vanishes;
(24.2)
(1,1) to (s,s). Finally, let us introduce the following dummy
positions:
2
σ 2 Tpay
µm km
, 1 ≤ m ≤ s; βl , ε
, 1 ≤ l ≤ r.
2
t(µm + P σε )
tµl
(27)
Now, the optimized transmit powers {P ? (m), 1 ≤ m ≤ t}
achieving the sup in (23) may be obtained by applying the
Kuhn-Tucker conditions [14, eqs.(4.4.10), (4.4.11)]. They
are detailed by the following Proposition 3, proved in the
Appendix D.
αm ,
Proposition 3. Let us assume that at least one of the conditions (24.1), (24.2), (24.3) is met. Thus, for m = s+1, ..., t, the
optimal vanish, while for m = 1, ..., s they must be computed
according to the following two relationships:
´
³
σ 2 P ´³ t
2
+ σε2 T ra[K−1
P ? (m) = 0, when km
≤ 1+ ε
d ] ,
µm
ρ
(28)
½
·µ
¶
¸
1
1
r
βmin 1 −
ρ−
−1
2βmin
Tpay
αm
ý
¶
¸
¾2
·µ
1
r
ρ−
−1
+
βmin 1 −
Tpay
αm
µ
¶¶0.5 )
1
rρβmin
+4βmin ρ −
−
,
αm
αm Tpay
P ? (m) =
c) all the SINRs γj , 1 ≤ j ≤ r, in (10.2) vanish. (24.3)
¨
Several numerical results confirm that the condition (24.1) may
be considered virtually met when Tpay ≥ 6t , 7t and t ≥ 4, 5,
even for σε2 approaching 1 and SINRs of the order of 6dB-7dB.
A. Optimized Power allocation under colored MAI and
Channel Estimation errors
To evaluate the covariance matrix Rφ achieving the sup
in (23), let us begin with the Singular Value Decomposition
(SVD) of the covariance matrix Kd according to
Kd = Ud Λd U†d ,
(25)
where
³ σ 2 P ´³ t
´
2
when km
> 1+ ε
+σε2 T ra[K−1
d ] ,
µm
ρ
(29)
where βmin , min{βl , l = 1, .., r}. Furthermore, the nonnegative scalar parameter ρ, in (28), (29) must satisfy the
following relationship:
X
P ? (m) = P t;
(30)
m∈I(ρ)
½
³
σ2 P ´
2
m = 1, ..., s : km
> 1+ ε
µm
³t
´¾
−1
2
·
+ σε T ra[Kd ]
,
(30.1)
ρ
is the (ρ-depending) set of indexes fulfilling the inequality (29). Finally, the resulting optimized covariance matrix
Rφ (opt) of the radiated signals is given by
where
Λd , diag{µ1 , ..., µr },
(25.1)
denotes the (r × r) diagonal matrix of magnitude-ordered
singular values of Kd . Furthermore , we define by
∗
−1/2
A , Ĥ Kd
Ud ,
(26)
the (t × r) matrix which simultaneously accounts for the
effects of the imperfect channel estimate Ĥ and MAI spatial
coloration. The corresponding SVD is
A = UA DA V†A ,
where UA and VA are unitary matrices, and
·
¸
diag{k1 , ..., ks }
0s×r−s
DA ,
,
0t−s×s
0t−s×r−s
(26.1)
I(ρ) ,
Rφ (opt) = UA diag{P ? (1), ...P ? (s), 0t−s } U†A ,
(31)
so that the throughput in (23) may be directly computed as
in
TG (Ĥ) =
(26.2)
is the (t × r) matrix having the s , min{r, t} magnitudeordered singular-values k1 ≥ k2 ≥ ... ≥ ks > 0 of A along
the main diagonal of the sub-matrix starting from elements
m = 1, ..., s;
+
s
X
m=1
"
³
?
³
σ2 P ´
lg 1 + ε
µm
m=1
r
X
´
lg 1 + αm P (m) −
r
1 X
Tpay
#
´
lg 1 + βl P (m) .
³
?
l=1
¨ (32)
7
B. Some explicative remarks
Before proceeding, some explicative comments about the
meaning and practical application of eqs.(28), (29) are in order.
First, the derivation performed in the Appendix D leads to
the conclusion that the optimal covariance matrix in (31) must
be aligned along the eigenvectors of the matrix A in (26) that,
in turn, depend both on Ĥ and Kd . Therefore, A accounts both
for the MAI spatial coloration and errors possibly present in
the channel estimates Ĥ available at the receiver. Thus, matrix
A plays the key-role of ”effective” MIMO channel viewed by
the receiver.
√
Second, since for small x we have that 1 + x u 1 + 0.5x,
for vanishing σε2 we may rewrite (according to Taylor series
approximation) eqs.(28), (29) as follows:
n
t o
limσε2 →0 P ? (m) = max 0, ρ − 2 , m = 1, .., s. (33)
km
Thus, from (33), it follows that the proposed power allocation
algorithm reduces to the standard water filling one for vanishing σε2 .
Third, in the case of NCSI (e.g, when σε2 = 1), the channel
estimate Ĥ equates 0t×r (see(20.2)). As a consequence, the
resulting throughput TG (Ĥ) in (23) becomes
1.
2.
3.
4.
5.
Compute and order the eigenvalues of the MAI covariance matrix Kd ;
Compute the SVD of matrix A in (26.1) and order its singular values;
Set P ? (m) = 0, 1 ≤ m ≤ t;
Set ρ = 0 and I(ρ) = ∅;
Set thestep size ∆;
P
? (m) < P t
6. While
P
do
m∈I(ρ)
7.
Update ρ = ρ + ∆;
8.
Update the set I(ρ) via eq. (30.1);
9.
Compute the powers {P ? (m), m ∈ I(ρ)} via eq.(29);
10. End;
11. Compute the optimized powers {P ? (m), 1 ≤ m ≤ s} via eqs. (28), (29);
12. Compute the optimized shaping matrix R (opt.) ;
13. Compute the conveyed throughput G (Ĥ) via eq.(32).
T
TABLE I
A
PSEUDO - CODE FOR THE NUMERICAL IMPLEMENTATION OF THE
PROPOSED OPTIMIZED POWER ALLOCATION ALGORITHM .
V. A T OPOLOGY-BASED MAI MODEL FOR
M ULTI -A NTENNA ” AD - HOC ” N ETWORKS
To test the proposed power allocation algorithm, we consider the application scenario of Fig.3 that captures the keyfeatures of Multi-Antenna ”ad-hoc” networks impaired by
spatial MAI [15,18,20].
R x1
T x1

1+
P
µm
P Tpay
µm
θ d(1)


´1/Tpay  ,
T x2
l1
θ d(2)
R x2
...
m=1
1+

lg  ³
´
...
lim TG (Ĥ) , TG (0) =
σε2 →1
r
X
³
TxN
R xN
(nats/payload slot). (34)
Since this relationship is valid for large t and Tpay regardless
of employed power level P, the relationship (34) supports the
conjecture in [7] that for large Tpay the channel capacity is
attained by employing input signals with Gaussian pdf, even
when H is fully unknown at Rx. Thus, we conclude that,
for vanishing σε2 and/or small SINRs, the throughput TG (Ĥ)
approaches the MIMO channel capacity C(Ĥ) regardless of
Tpay and t values. Several numerical trials confirmed that, for
0 < σε2 ≤ 1, TG (Ĥ) in (32) is close to the capacity C(Ĥ)
when t ≥ 4 and Tpay ≥ 6t.
C. A Numerical Algorithm implementing the proposed Power
Allocation
The first step for computing (28), (29) is to properly set the
parameter ρ in order to meet the power constraint (30). For
this purpose, we note that the size of the set (30.1) vanishes at
ρ = 0 and grows for increasing values of ρ. As consequence,
for evaluating the ρ value meeting the relationship (30), we
may adopt the (very) simple iterative procedure which starts
by setting ρ = 0 and then increases ρ by using a properly
chosen step-size 8 of Table I.
8 Several
numerical trials confirmed that ∆ = 0.1P t is adequate for this
purpose. The iterative procedure of Table I is stopped when the summation
in (30) attains the power constraint.
θd( N )
l2
lN
θ a(1)
T x0
Rx0
θ a(2)
θ a( N )
l0
Fig. 3. A general scheme for an ”ad-hoc” network composed of (N+1)
point-to-point links active over the same hot-spot area.
Shortly, we assume that the network of Fig.3 is composed
of (N+1) no cooperative, mutually interfering, point-to-point
links Txf → Rxf , 0 ≤ f ≤ N . The signal received by the
reference node Rx0 is the combined effect of that transmitted
by Tx0 and those radiated by the other interfering transmitters
(Txf ,1 ≤ f ≤ N ). The transmit node Txf and the receive
node Rxf are equipped with tf and rf antennas, respectively.
Thus, after indicating as lf the Txf → Rx0 distance, then the
d(n) disturbance vector in (11) may be modelled as
s
N
³ l ´4 1
X
0
d(n) =
√ χf HTf φ(f ) (n) + w(n). (35)
lf
tf
f =1
The vector w(n) in (35) accounts for the thermal noise
(see (11)); the φ(f ) (n) term represents the tf -dimensional
(Gaussian) signal radiated by the Txf interfering transmitter;
8
(
χf accounts for the shadowing effects9 ; the matrix Hf models
the Ricean-distributed fast-fading affecting the interfering link
Txf → Rx0. Furthermore, according to the faded spatial
interference model recently proposed in [1,2], the channel
matrix Hf in (35) may be modeled as in
s
s
kf
1
(sp)
(sc)
Hf ≡
H
+
H , 1 ≤ f ≤ N, (36)
1 + kf f
1 + kf f
where kf ∈ [0, +∞) is the f -th Ricean-factor and all the (tf ×
(sc)
r0 ) terms of the matrix Hf are mutually independent, zeromean, unit-variance Gaussian distributed r.v.s, that account for
the scattering phenomena impairing the f -th interfering link
(sp)
Txf → Rx0. The (tf × r0 ) matrix Hf in (36) captures for
the specular components of the interfering signals and may be
modelled as in [1,2]
(sp)
Hf
´T
³
= a(f )b(f )T , 1 ≤ f ≤ N,
(36.1)
where a(f ) and b(f ) are (r0 × 1) and (tf × 1) column
vectors. They are used to model the specular array responses
at the receive node Rx0 and transmit node Txf , respectively
[1,2]. When isotropic regularly-spaced linear arrays are employed at the Txf and Rx0 nodes, the above vectors may be
evaluated as in [1,2,15]
h
³
´
a(f ) = 1, exp j2πν cos(θa(f ) ) ,
³
´iT
... exp j2πν(r0 − 1) cos(θa(f ) )
,
h
³
´
(f )
b(f ) = 1, exp j2πν cos(θd ) ,
³
´iT
(f )
... exp j2πν(tf − 1) cos(θd )
,
(f )
)
kf E{χ2f }
(f ) ∗
T
†
+
a(f )b (f )R b (f )a (f ) .
φ
lf 1 + kf tf
f =1
(38)
This relationship captures the MAI effects due to the topological and propagation features of the considered multi-antenna
ad-hoc network. Specifically, eq.(38) points out that MAI
interference may be considered spatially white when all the
interfering links’ Ricean factors may be neglected. On the
contrary, for high Ricean factors the MAI spatial coloration
is not negligible, as confirmed by the numerical results of the
next Sect.VI.
N ³
X
l0 ´4
B. A Worst-Case Application Scenario
Let us consider the hexagonal network of Fig.4. All transmit
and receive nodes have the same number of antennas (e.g.,
t0 = t1 = t2 = t and r0 = r1 = r2 = r) and all transmit
nodes radiate the same power level (e.g., P0 = P1 = P2 = P ).
We assume that the array elements are one-half wavelength
apart (e.g., ν=1/2), and all Ricean factors are equal (e.g.,
k1 = k2 = k). Furthermore, let us consider a worst-operating
scenario with all shadowing coefficients equal to unity (e.g.,
(1)
(2)
χ1 = χ2 = 1) and the correlation matrices R , R of the
φ
φ
signals radiated by the interfering transmit nodes Tx1, Tx2
equating P It [1,2]. Therefore, in this case eq.(38) becomes
2
n
2 P o
k P X
Kd = N0 +
Ir +{
a(f )bT (f )b∗ (f )a† (f )},
91+k
1+k 9
(36.2)
f =1
(39)
where
"
√
√ #T
3
3
a(1) = 1, exp(jπ
), ..., exp(jπ(r − 1)
) ,
(39.1)
2
2
(36.3)
(f )
where θa , θd are the arrival and departure angles of the
radiated signals (see Fig.3), while ν is the antenna spacing in
multiple of RF wavelengths10 .
A. The resulting model for the MAI Covariance Matrix
Therefore, after assuming the spatial covariance matrix
(f )
R
, E{φ(f ) (n)φ(f ) (n)† }, 1 ≤ f ≤ N , of signals
φ
radiated by the f −th transmit node Txf power-limited as (see
eq.(11.1))
(f )
T ra[R ] = tf P (f ) ,
(37)
φ
then the covariance matrix Kd of the MAI vector in (35)
equates
(
)
N ³
o
n
X
l0 ´4 E{χ2f } (f )
†
P
Ir0
Kd , E d(n)d(n) = N0 +
lf
1 + kf
f =1
9 Without loss of generality, we may assume χ to fall in the interval [0, 1].
f
When χf = 1 (worst case), MAI impairing effects arising from transmit
interfering node Txf are the largest.
10 Several tests show that rays impinging receive antennas may be
considered virtually uncorrelated when ν is of the order of 1/2 [15].
"
√ #T
3
3
), ..., exp(jπ(t − 1)
) .
b(2) = 1, exp(jπ
2
2
√
(39.2)
while b(1), a(2) are column vectors composed by t and r unit
entries respectively.
VI. N UMERICAL R ESULTS AND P ERFORMANCE
C OMPARISONS
Although the MIMO channel pdf in (20) is in closed form,
the corresponding throughput expectation
n
o
TG , E TG (Ĥ) ,
(40)
resists closed-form evaluations, even in the case of spatially
white MAI with vanishing σε2 [4,5,17 and references therein].
Thus, as in [1,2,4], we evaluate the expected throughput TG
in (40) by resorting to a Monte-Carlo approach based on the
generation of 10,000 independent samples of TG (Ĥ). All the
reported numerical plots refer to the hexagonal network of
Fig.4 with unit noise level N0 .
9
Tx2
Rx2
G
Tx1
(nats/slot)
Rx1
Tx0→ Rx0 of Fig.4 when the Ricean factor in (39) equates
10, σε2 = 0.01 and Tpay = 80.
Tx0
Fig. 4.
R x0
A hexagonal network with two interfering links.
A. Effect of the channel estimation errors
The first plots’ set of Fig.5 shows the sensitivity of the
throughput TG of reference link Tx0→ Rx0 on MIMO channel
estimation errors. All nodes are equipped with r = t = 8
antennas, all the Ricean factors in (39) are set to 10 and Tpay =
40. Fig.5 shows that throughput loss is at most 1% for σε2
values below 0.01.
Fig. 6. Sensitivity of the throughput conveyed by the reference link Tx0 →
Rx0 of Fig.4 on the number t=r of antennas (Tpay = 80, k=10,σε2 = 0.01 ).
An examination of these plots leads to the conclusion that,
by increasing the number of antennas, we are able to quickly
gain in terms of channel throughput.
G
(nats/slot)
C. Effect of Errors in the Estimation of the MAI covariance
matrix
2
σ ε =0.001
2
σ ε =0.01
2
σ ε =0.1
2
σ ε =0
2
σ ε =1 (eq.(34))
T
Fig. 5. Sensitivity of the throughput G conveyed by the reference link Tx0
→ Rx0 of Fig.4 on the squared error level σε2 affecting the available channel
estimates (Tpay = 40, k=10, r=t=8).
B. Effect of the number of transmit/receive antennas
The numerical plots drawn in Fig.6 allow us to evaluate the
effect on the throughput of the number r = t of antennas
equipping each node of the network of Fig.4. Specifically,
Fig.6 shows the average throughput (40) of the reference link
As anticipated in Sect.II.A, the estimation accuracy of K̂d
in (6) is mainly limited by the learning phase length TL , so it
can be of interest to test the sensitivity of the proposed power
allocation algorithm on errors possibly affecting the estimated
K̂d . For this purpose, we perturbed the actual Kd by using
a randomly generated (r × r) matrix N, composed by zeromean unit-variance independent Gaussian entries. Hence, the
(analytical) expression for the resulting perturbed K̂d is
r
||Kd ||2E √
δN,
(41)
K̂d = Kd +
r2
n
o2
where δ , E ||Kd − K̂d || /||Kd ||2E in (41) is a determinE
istic parameter which may be tuned so to obtain the desired
square estimation error. Thus, after replacing the Kd matrix by
the corresponding perturbed K̂d version, we have implemented
the proposed power allocation algorithm as dictated by the
relationships (28), (29). Finally, we evaluated the new value
of TG (Ĥ) according to eq.(32) and that we computed {αm }
and then {βl } according to (27) on the basis of the actual
MAI matrix Kd . The resulting average throughput is plotted
in Fig.7 for the reference link Tx0→ Rx0 of Fig.4 (Tpay = 40,
r = t = 8, σε2 = 0.015, k = 10). From these plots we may
conclude that throughput loss due to errors in the estimated of
K̂d may be neglected when the parameter δ is at most 0.01.
10
T
Fig. 7. Sensitivity of the throughput G conveyed by the reference link
Tx0 → Rx0 of Fig.4 on the estimation errors affecting the available MAI
covariance matrix (Tpay = 40, r=t=8, k=10 ,σε2 = 0.015 ).
D. Coordinated versus Uncoordinated Medium Access Strategies: some MAC considerations
Although in these last years the MAI-mitigation capability
of multi-antenna systems has been often claimed [8,15,18]. To
test there claims, it may be of interest we want to compare
the information throughput TG of the proposed power allocation algorithm with that of orthogonal MAI-free TDMA (or
FDMA)-based access techniques. Till now, it appears that none
of them definitively perform the best. In particular, this is true
when application scenarios as those of Fig.3 are considered,
where SINRs are usually low, so that multiuser detection
strategies based on iterative cancellation of the MAI do not
effectively work [22]. Therefore, on the basis of the above
considerations, we have
computed
the average information
n
o
throughput TG , E TG (Ĥ) (nats/ payload slot) conveyed
by the reference link Tx0→ Rx0 of Fig.4 when MAI-free
TDMA-based access is used11 . The numerical plots of Fig.8
for the network in Fig.4 have been obtained by setting Tpay =
80, σε2 = 0.1, k = 1000 and then by varying the number of
transmit/receive antennas from 4 to 12.
Although TG has been evaluated in the worst MAI case
(see (39) and related remarks), the plots of Fig.8 show how
much greater TG is than TT DM A , specially when low power
levels P are used and the transceivers are equipped with a
large number of transmit/receive antennas. This conclusion is
11 According
to [22, Sect.VI.C], the conditional information throughput
TT DM A (Ĥ) has been evaluated by fixing the estimation matrix Ĥ and by
running the algorithm of Table I under the following operating conditions:
i) all shadowing factors in (39) have been zeroed;
ii) the power level P in (39) has been replaced by 3P;
iii) the resulting throughput G (Ĥ) in (32) has been scaled by 1/3.
The condition i) is for modelling the MAI-free condition of the TDMA
technique , while the conditions ii) and iii) are due to the fact that the reference
link Tx0→ Rx0 of Fig.4 is in TDMA mode, and then it is active only over
1/3 of the overall transmission time.
T
Fig. 8. Throughput comparisons for the reference link Tx0 → Rx0 of Fig.4
for Tpay = 80, k=1000, σε2 = 0.1.
confirmed by the plots of Fig.9, which refer to Rayleigh-faded
application scenarios.
Fig. 9. Throughput comparisons for the reference link Tx0 → Rx0 of Fig.4
(Tpay = 80, k=0, σε2 = 0.1).
Therefore, from the outset we may conclude that when the
number of antennas increases, by using the spatial-shaping
algorithm of Table I we are able to achieve channel throughput
larger than those attained by conventional orthogonal access
methods.
VII. C ONCLUSIONS
The main contribution of this paper is the development of
an optimized spatial signal-shaping for multi-antenna systems
impaired by spatially colored MAI and channel estimation
11
i
h
i
h
i
h
(b)
†
T −1
errors. From our analysis we may draw three main con- = T ra eTj K−1
e
T
ra
X
X̃
≡
t
P̃
T
T
ra
e
K
e
tr
j
j
j , (B.4)
d
d
clusions. First, throughput loss induced by estimation errors
is not very critical, especially when the system operates at where (a) follows from an application of the propermedium/low SINRs. Second, the throughput comparisons of ty T ra [AB] = T ra [BA], (b) stems from the property
Sect.VI confirm the MAI-suppressing capability of multi- T ra [A ⊗ B] = T ra [A] T ra [A], while (c) arises from the
antenna transceivers, even in ”ad-hoc” operating scenarios. power constraint in (9.1). Hence, after inserting (B.4) into
Third, the plots of Figs.8,9 show the throughput improvement (B.2), this last may be equivalently rewritten as
attained by uncoordinated spatial-based multiple access techà r
!
r
h
i
niques respect to coordinated orthogonal ones (as, for example,
X
1 X
T −1
2
P̃
e
K
e
+
T
ra
Λ
(
X̃)
. (B.5)
σ
=
rt−T
tr
j
TDMA). Currently, we are going to test the validity of these tot
j
j
d
t2 j=1
j=1
conclusions in the mesh-like operating scenarios considered
by WOMEN project [27].
Now, our next task is to find the minimum value of the
A PPENDIX A - T HE MIMO C HANNEL MMSE E STIMATOR traces in the summation (B.5). For accomplishing this task,
we resort to a suitable application of the Cauchy inequality.
By using the following property [13]: vect(AB) = Specifically, after indicating by {λj (i), i = 1, .., t} the Λj (X̃)
[I ⊗ A] vect(B), we may rewrite (9) as
matrix eigenvalues, we have that

à t
!2 Ã t
! t
£
¤
1
Xq
X
X
(a)
vect(Ỹ) = √ Ir ⊗ X̃ vect(H) + vect(D̃). (A.1)
1
t
t2 =
λj (i) p
≤
λj (i)  (λj (i))−1 
λ
(i)
j
i=1
i=1
j=1
Therefore, since E{vect(D̃)(vect(D̃))† } = Kd ⊗ ITtr , and
t
³
´
†
X
−1
£
¤
E{vect(Ỹ)(vect(Ỹ))† } = 1t (Ir ⊗ X̃X̃ ) + (Kd ⊗ ITtr ), via
≡
λj (i)
T ra Λj (X̃) ,
(B.6)
an application of the Orthogonal Projection Lemma we obtain
j=1
eqs. (13), (14).
where (a) from an application p
of the Cauchy inequality
A PPENDIX B - O PTIMIZATION OF THE TRAINING MATRIX
[13, p.42] to the sequences { λj (i), i = 1, .., t} and
{(λj (i))−1/2 , i = 1, .., t}. Obviously, eq. (B.6) may be
Since [13,p.64]
rewritten as
µ ³
¶−1
´
1
†
−1
Kd ⊗ X̃X̃ + IrTtr
t
t
³X
´
£
¤
2
−1
·
¸−1
³
´
³
´
T
ra
Λ
(
X̃)
≥
t
/
λ
(i)
,
(B.7)
j
j
1
1
†
−1/2
Kd
⊗ X̃ Irt +
K−1
= IrTtr −
j=1
d ⊗ X̃ X̃
t
t
£
¤
³
´†
that gives arise to a lower bound on T ra Λj (X̃) . Further−1/2
· Kd
⊗ X̃ ,
(B.1) more, the Cauchy inequality also allows us to conclude that
the right-hand-side (r.h.s) of eq.(15) may be recast in the the lower bound (B.3) is attained when Λj (X̃) is equal to the
following diagonal matrix (see (B.3)):
following form:
2
σtot
= rt −
1
t
r
X
·³
ej ⊗ It
T ra
j=1
+
´† ³
´³
´¸
†
K−1
⊗
X̃
X̃
e
⊗
I
t
j
d
r
h
i
1 X
T
ra
Λ
(
X̃)
,
j
t2 j=1
(B.2)
where
³
Λj (X̃) , ej ⊗It
³
·
´† ³
K−1
d
¸−1
´·
1 −1
†
Irt + (Kd ⊗ X̃ X̃)
t
†
K−1
d ⊗ X̃ X̃
´³
´
⊗ X̃ X̃ ej ⊗ It , 1 ≤ j ≤ r,
†
(B.3)
is semidefinite positive and Hermitian. Now, traces present in
the first summation of (B.2) may be developed as
´† ³
´³
´i
h³
†
K−1
⊗
X̃
X̃
e
⊗
I
T ra ej ⊗ It
t
j
d
(a)
= T ra
h³
´
i
†
eTj K−1
d ej ⊗ X X̃
a2 t
It , 1 ≤ j ≤ r.
(B.8)
t+a
As a direct consequence, the condition (16) arises for the
optimal X̃.
Λj (X̃) =
A PPENDIX C - D ERIVATION OF T HROUGHPUT FORMULA IN
(24)
→
−
The whitening filter B of the (non singular) MAI covariance
matrix Kd is defined as
−
→
−1/2
B , (I⊗Kd )−1/2 = ITpay ⊗Kd .
(C.1)
It is a (rTpay × rTpay ) (non singular) block matrix, so that
−
→→
→
y constitute
the resulting transformed observations12 −
ω , B−
sufficient statistics for the detection of the transmitted message
M of Fig.1. On the basis of the above property, we may
directly write the following equality:
−
→
−
→
−
→
−
→
→
→
ω ; φ |Ĥ) , h(−
ω |Ĥ) − h(−
ω | φ , Ĥ), (C.2)
I(→
y ; φ |Ĥ) ≡ I(−
−
→
applying the linear transformation (C.1) to the disturbance vector d
−
→−
→−
→ −
→ †
in (12) we arrive at the following relationship E{ B d ( B d ) } = IrTpay .
So, according to our current taxonomy, we denote as ”spatial whitening filter”
−
→
the matrix B in (C.1).
12 By
12
where h(·) denotes the differential entropy operator. Furthermore, from the channel model in (12) and the linear transformation performed by the whitening filter in (C.1), it follows
−
→
→
that the conditional r.v. −
ω | φ , Ĥ is Gaussian distributed and
its covariance matrix is given by
(
¤T
σε2 ³ £
−
→
−
→
Cov( ω | φ , Ĥ) = IrTpay +
φ(TL + Ttr + 1)...φ(T )
t
)
¤´
£ ?
?
−1
· φ (TL + Ttr + 1)...φ (T ) ⊗ Kd ,
(C.3)
where σε2 in (C.3) arises from the fact that Ĥ = H − ²
and the elements {εji } of the MMSE estimation error matrix
² are uncorrelated zero-mean Gaussian r.v.s whose variances
E{kεji k2 } equate σε2 for any (j,i) indexes. Thus, being the
−
→
−
conditional r.v. →
ω | φ , Ĥ proper, complex and Gaussian distributed, its differential entropy in (C.2) may be directly
computed as in [29, Th.2]
h
h
ii
−
→
→
−
→
→
h(−
ω | φ , Ĥ) = lg (πe)rTpay det Cov(−
ω | φ , Ĥ) ,
(C.4)
→
−−
→
y equates (see channel
H = Ĥ + ², the r.v. −
ω , B→
model in (12))
i
1 h
→
−1/2 T −
−
→
ω = √ ITpay ⊗ Kd Ĥ φ
t
i
1 h
−
→ →
−1/2
+ √ ITpay ⊗ Kd ²T φ + −
w,
(C.7)
t
→
−→
−
→
where the zero-mean Gaussian r.v. −
w , B d is the
−
→
”whitened” version of the colored MAI d (see note 12). Thus,
→
the conditional r.v. −
ω |Ĥ
and the corresponding
³ is zero-mean
´
→
covariance matrix Cov −
ω |Ĥ may be developed as
³
´ (a) 1 h
i³
´
−1/2 T
→
Cov −
ω |Ĥ =
ITpay ⊗ Kd Ĥ
ITpay ⊗ Rφ
t
´
h
i 1 n³
∗ −1/2
−
→−
→†
−1/2
· ITpay ⊗ Ĥ Kd
+ E ITpay ⊗ Kd ²T φ φ
t
³
´
o
−1/2
· ITpay ⊗ ²∗ Kd
| Ĥ + IrTpay
(b)
= IrTpay +
that due to (C.3), may be developed as
(
·
σ2 ³
−
→
?
→
−
h( ω | φ , Ĥ) = rTpay lg(πe)+E lg det Irt + ε (K−1
d )
t
(c)
³
⊗
T
X
†
φ(n)φ(n)
´´
= IrTpay +
#)
,
(C.5)
n=TL +Ttr +1
where the expectation in (C.5) follows from definition of
conditional differential entropy [12].
−
→
Now, although the pdf of signal vector φ is assumed to
be Gaussian distributed too, for σε2 > 0 the corresponding
expectation in (C.5) cannot be put in closed-form, even in the
simplest case of spatially white MAI (see [6] and reference
therein). Anyway, by resorting to the Law of Large Numbers
[26, eqs.(8.95), (8.96)], we may conclude that for large Tpay =
T − TL − Ttr the summation in (C.5) converges (in the mean
square sense) to the expectation Tpay Rφ , so that the following
limit holds for large Tpay :
h
σ 2 Tpay ³ −1 ?
−
→
−
h(→
ω | φ , Ĥ) = rTpay lg(πe) + lg det Irt + ε
(Kd )
t
·
³
´
³
´† ¸
1
−1/2 T
−1/2 T
ITpay ⊗ Kd Ĥ Rφ Kd Ĥ
t
´o
1 n³
−1/2
−1/2
+ E ITpay ⊗ Kd ²T Rφ ²∗ Kd
t
·
³
´
³
´† ¸
1
−1/2 T
−1/2 T
ITpay ⊗ Kd Ĥ Rφ Kd Ĥ
t
´
1³
+ ITpay ⊗ σε2 tP K−1
d
t
³
´†
n
1 ³ −1/2 T ´
−1/2 T
= ITpay ⊗ Ir +
Kd Ĥ Rφ Kd Ĥ
t
o
+σε2 tP K−1
,
(C.9)
d
n
o
−
→−
→†
where (a) follows from E φ φ
= ITpay ⊗ Rφ , (b) arises
−
→
for the mutual independence
of o
the r.v.s φ , ², Ĥ, (c) stems
n
(d)
from the relationship E ²T Rφ ² = σε2 tP Ir , and, finally, (d)
exploits the property IrTpay = ITpay ⊗ Ir . Therefore, although
→
the differential entropy of −
ω |Ĥ is upper bounded as [29, th.2]
(
)
h
i
rTpay
−
→
−
→
h( ω |Ĥ) ≤ lg (πe)
det Cov( ω |Ĥ) ,
(C.10)
r X
t
´
³ σ2 T
X
ε pay P (m)
, (C.6)
⊗Rφ ≡ rTpay lg(πe)+
lg 1+
t
µl
nevertheless, the Limit Central Theorem guarantees that, for
l=1 m=1
−
large number t of transmit antennas, the r.v. →
ω |Ĥ becomes
where {P (m), 1 ≤ n ≤ t} in (C.7) are the eigenvalues of the
Gaussian, so that the upper bound in (C.10) may be assumed
signal correlation matrix Rφ , while {µl , l = 1, ..., r} are the
attained for large t. Furthermore, since for σε2 → 0 and/or
eigenvalues of the MAI covariance matrix Kd . Furthermore,
→
vanishing SINRs, Ĥ converges to H and the r.v. −
ω |Ĥ becomes
since the disturbance in (12) is Gaussian distributed, the
Gaussian distributed, then the upper bound in (C.10) can be
relationships (C.5), (C.6) are still valid, regardless of Tpay ,
attained regardless of t value. Hence, after inserting (C.5)
when σε2 vanishes and/or the SINRs {P (m)/µl , 1 ≤ m ≤
and (C.10) into (C.2), we directly obtain eq.(24).
t, 1 ≤ l ≤ r} in (C.6) approach zero.
2
−
→
About the differential entropy h( ω |Ĥ) in (C.2), for σ > 0
´i
ε
it cannot be expressed in closed-form, even in the simplest
case of r = t = 1 with white MAI [6]. However, since
A PPENDIX D - D ERIVATION OF THE P OWER A LLOCATION
FORMULAS IN (28), (29)
13
Since the eigenvalues of the Kronecker matrix product A ⊗ B
are given by the products of the eigenvalues of the matrix A
by those of B (see [13], Corollary 1, p.412], we may directly
express the second determinant in (24) as
·µ
¶¸
σε2 Tpay ¡ −1 ¢∗
lg det Irt +
Kd
⊗ Rφ
t
µ
¶
r X
t
X
σ 2 Tpay P (m)
=
lg 1 + ε
,
(D.1)
t
µl
m=1
l=1
where {µl } are the eigenvalues of Kd and {P (m)} are those
of Rφ (e.g., P (m) is the power allocated to the m-th mode
of the considered MIMO channel). Now, after introducing the
SVD in (25) of Kd into the first determinant in (24), we may
rewrite this last equation in the following equivalent form
·
¸
´
³
1 †
−
→
−1
2
→
−
I y ; φ |Ĥ = lg det Ir + σε P (Λ)d + A Rφ A
t
µ
¶
r
t
XX
σ 2 Tpay P (m)
−
lg 1 + ε
,
(D.2)
t
µl
m=1
{fm (P ? ), 1 ≤ m ≤ s}, in (D.4) are not positive over region
D of Rs given by
(
(P ? (1), ..., P ? (s)) : P ? (m)
)
)
(
p
√
βmax r − αm Tpay
£p
≥ max 0,
√ ¤ , m = 1, ..., s . (D.6)
αm βmax
Tpay − r
D,
Then,
we
conclude Tthat
the
sum-function
f (P ? (1), ..., P ? (s)) in (D.2) is −convex (at least) over D.
This region approaches the overall positive orthant Rs+ of Rs
when σε2 is vanishing and/or large Tpay is considered (see
eq. (27))13 . Thus, after applying the Kuhn-Tucker conditions
[14, eqs.(4.4.10), (4.4.11)] for carrying out the constrained
maximization of objective function (D.4) we arrive at the
following relationships:
−1 −1
(P ? (m) + αm
) −
+P
−
sup
f (P ? (1), ..., P ? (s))
P ? (m)≤P t
¶)
µ
σε2 Tpay P ? (m)
lg 1 +
, (D.3)
t
µl
t
m=1
t
r
X
X
1
Tpay
m=s+1 l=1
where
f (P ? (1), ..., P ? (s)) ,
fm (P ? (m))
m=1
"
s
X
=
lg(1 + αm P ? (m)) − (1/Tpay )
m=1
#
r
³
´
X
?
·
lg 1 + βl P (m) ,
+P
s
m=1
(D.7)
r
1 X ?
(P (m) + βl−1 )−1 = 1/ρ,
Tpay
l=1
for all m such that: P ? (m) > 0.
(D.8)
Now, while (D.7) directly gives arise to eq.(28), to prove
eq.(29) we need to rewrite (D.8) in the following form:
)
( r
h
³
´i
Y
−1
(P ? (m) + βl−1 ) ρ − P ? (m) + αm
Tpay
l=1
r h Y
r
³
´n X
io
−1
−ρ P ? (m)+αm
(P ? (m)+βj−1 ) = 0. (D.9)
j=1,j6=l
β1 = β2 = ... = βr = βmin ,
(D.4)
is an additive objective function which just depends on the
powers radiated by first s transmit antennas. Since the last
two-fold summation into brackets in (D.3) vanishes only when
P ? (s + 1) = ... = P ? (t) = 0, we may directly rewrite (D.3)
as
σ2 P ´
TG (Ĥ) =
lg 1 + ε
µm
( m=1
)
³
´
?
?
sup
f P (1), ..., P (s) .
l=1
Eq.(D.9) is an (r+1)-th order algebraic equation which
cannot generally be expressed in closed form as a function
of optimal power level P ? (m). Anyway, when
l=1
r
X
−1 −1
(P ? (m) + αm
) −
l=1
s
X
Tpay
for all m such that: P ? (m) = 0,
l=1
with A given by eq.(26). Therefore, after introducing in
(D.2) the SVD in (26.1) of A, an application of Hadamard
inequality [12] allows us to develop the constrained supremum
in (23) as
µ
¶
r
X
σ2 P
TG (Ĥ) =
lg 1 + ε
µm
( m=1
r
1 X ?
(P (m) + βl−1 )−1 ≤ 1/ρ,
then (D.9) reduces to the following 2nd-order algebraic
equation:
(
· ³
?
2
βmin P (m) + 1−βmin ρ 1 −
(D.5)
Now, after denoting by βmax , max{βl , l = 1, ..., r},
we see that the second derivatives of logarithmic functions
r ´
Tpay
¸)
−1
− αm
P ? (m)
³
rβmin ´
−1
−ραm
αm −ρ−1 −
= 0,
Tpay
³
P ? (m)≤P t
(D.10)
(D.11)
whose positive roots are given by (29). Thus, directly from
the relationship (27), it follows that the above condition (D.10)
T
practice, some sufficient conditions for the
−convexity og the
objective function
are:
p
2
2 ≥ (σ 2 rT
km
pay (µm + P σε ))/(µm µmin ), 1 ≤ m ≤ s,
ε
where µmin denotes the minimum eigenvalue of Kd .
13 In
14
is satisfied for vanishing σε2 and/or large Tpay and/or diagonal
Kd and/or low SINRs. Furthermore, when all above conditions
fall short, the worst-case application scenario is obtained when
the MAI covariance matrix Kd is equal to the diagonal one
µmax Ir , where µmax is the maximum eigenvalue of Kd . In
this case, the optimal power level P ? (m) is still given by
the positive root (29) of the algebraic equation (D.11). Thus,
we may conclude that, in any case, (29) represents the minmax solution of the constrained maximization of the objective
function in (D.4).
R EFERENCES
[1] F.R.Farrokhi, G.J.Foschini, A.Lozano, R.A.Valenzuela,”Link-Optimal
BLAST processing with Multiple-Access Interference”, VTC2002,
pp.87-91.
[2] F.R.Farrokhi, G.J.Foschini, A.Lozano, R.A.Valenzuela,”Link-Optimal
Space-Time Processing with Multiple Transmit and Receive Antennas”,
IEEE Comm. Letters, vol.5, no.3, pp.85-87, March 2001.
[3] S.L.Marple Jr., Digital Spectral Analysis with Applications, Prentice Hall,
1987.
[4] C.-N.Chuan, N.D .S.Tse, J.M.Kahn, R.A.Valenzuela,”Capacity-scaling in
MIMO wireless Systems under correlated Fading”, IEEE Trans. on Inform.
Theory, vol.48, no.3,pp.637-650, March 2002.
[5] E.Baccarelli, M.Biagi, ”Performance and Optimized Design of SpaceTime Codes for MIMO Wireless Systems with Imperfect ChannelEstimates ”, IEEE Trans. on Signal Proc., vol.52, no.10, pp.2911-2923,
October 2004.
[6] B.Hassibi, T.L.Marzetta, ”Multiple-Antennas and Isotropically Random
Unitary Inputs: the received Signal Density in closed form”, IEEE Trans.
on Inform. Theory, vol.48, no.6, pp.1473-1485, June 2002.
[7] T.L.Marzetta, B.M. Hochwald, ”Capacity of a mobile Multiple-Antenna
Communication link in Rayleigh flat fading”, IEEE Trans. on Inform.
Theory, vol.45, no.1, pp.139-157, January 1999.
[8] R.D.Murch, K.B.Letaief,”Antenna System for broadband Wireless
Access”, IEEE Comm. Mag., pp.637-650, March 2002.
[9] J.-C.Guey, M.P.Fitz, M.R.Bell, W.-Y.Kuo,”Signal Design for Transmitter
Diversity Wireless Communication System Over Rayleigh Flat fading
channels”, IEEE Trans. on Comm., vol.47, no.4, pp.527-537, April 1999.
[10] J.Baltarsee, G.Fock, H.Meyr,”Achievable Rate of MIMO Channels with
data-aided channel estimation and perfect interleaving”, IEEE Journ. on
Selected Areas in Comm., vol.19, no.12, pp.2358-2368, Dec.2001.
[11] L.Hanzo, C.H.Wong, M.S.Vee, Adaptive Wireless Transceivers, Wiley,
2002.
[12] T.M.Cover, J.A.Thomas, Elements of Information Theory, Wiley, 1991.
[13] P.Lancaster, M.Tismetesky, The Theory of Matrices, 2nd Ed., Academic
Press, 1985.
[14] R.G.Gallagher, Information Theory and Reliable Communication, Wiley,
1968.
[15] G.T.Okamoto, Smart Antennas Systems and Wireless LANs, Kluwert
2001.
[16] S.Verdu’, T.S.Han,”A general Formula for channel Capacity”, IEEE
Trans. on Inform. Theory, vol.40, no.6, pp.1147-1157, July 1994.
[17] D.W.Bliss, K.W.Forsythe, A.O.Hero, A.F.Yegulalp,”Environmental Issues for MIMO Capacity”, IEEE Trans. on Signal Proc., vol.50, no.9,
pp.2128-2142, Sept.2002.
[18] A.Santamaria, F.J.L.-Hernandez, Wireless LAN Standards and Applications, Artech House, 2001.
[19] H.Sampath, S.Talwar, J.Tellado, V.Erceg, A.Paulraj, ”A 4th Generation
MIMO-OFDM Broadband Wireless System: Design, Performance and
Field Trials results”, IEEE Comm.Mag., pp.143-149, Sept.2002.
[20] C.E.Perkins, Ad Hoc Networking, Addison Wesley, 2000.
[22] S.Verdu’, Multiuser Detection, Cambridge Uni. Press, 1998.
[23] A.Lozano, A.M.Tulino,”Capacity of Multiple-Trasmit Multiple-Receive
Antenna Architecture”, IEEE Trans. on Inform. Theory, vol.48, no.12,
pp.3117-3128, Dec.2002.
[24] E.Baccarelli, M.Biagi, C.Pelizzoni, ”On the Information Throughput and
Optimized Power Allocation for MIMO Wireless Systems with Imperfect
Channel Estimation”, IEEE Trans on Signal Proc., vol.53, no.7, pp.23352347, July 2005. Proc., accepted for publication.
[25] T.S.Rappaport, A.Annamalai, R.M. Buherer, W.H.Trenter,”Wireless
Communications: Past Events and future Perspective”, IEEE Comm. Mag.,
pp.148-161, May 2002.
[26] A.Papoulis, Probability, Random Variables and Stochastic processes,
McGraw-Hill, 1965.
[27] Wireless 8O2.16 Multi-antenna mEsh Networks (WOMEN) Website:
http://womenproject.altervista.org/
[28] I.Chlamtac, A.Gumaste, C.Szabo (Editors), Broadband Services : Business Models and Technologies for Community Networks, Chapter 14,
Wiley 2005.
[29] F.D.Neeser, J.L.Massey, ”Proper Complex Random Processes with applications to Information Theory”, IEEE Trans. on Inform. Theory, vol.39,
no.4, pp.1293-1302, July 1993.
[30] M.Hu, J.Zhang, ”MIMO ad hoc networks with spatial diversity: medium
access control and saturation throughput”, IEEE Conf. Decision and
Conbtrol, vol.3, pp.3301-3306, Dec. 2004.
[31] B.Chen, M.J.Gans, ”MIMO communications in ad hoc networks”, IEEE
Tr. on Sign. Proc., vol.54, pp.2773-2783, July 2006.
[32] T.Tang, R.W.Heath, ”A space-time receiver for MIMO-OFDM ad hoc
networks”, IEEE MILCOM Conf. 2005, vol.3, pp.1409-1413, Oct. 2005.
Enzo Baccarelli Enzo Baccarelli received the Laurea Degree summa cum laude in electronic engineering, the Ph.D. degree in communication theory and
systems, and the Post-doctorate Degree in information theory and application from the University of
Rome ”Sapienza” Rome, Italy in 1989, 1992, and
1995, respectively. He is currently with the University of Rome ”Sapienza”, where he was Researcher
Scientist from 1996 to 1998 and Associate Professor
in signal processing and radio communication from
1998 to 2003. Since 2003 he is Full Professor in data
communication and coding. He is also Dean of the Telecommunication Board,
and member of the Educational Board, both within the Faculty of Engineering.
From 1990 to 1995, he was Project Manager with SELTI ELETTRONICA
Corporation, where he worked on the design of high-speed modems for datatransmission. From 1996 to 1998 he attended the international project AC104 Mobile Communication Service for High-Speed Trains (MONSTRAIN),
where he worked on equalization and coding for fast-time varying radiomobile links. He is currently the Coordinator of the national Project Wireless
802.16 Multi-antenna mEsh Networks (WOMEN). He is author of more than
100 international IEEE publications and coauthor of two international patents
on adaptive equalization and turbo-decoding for high-speed wireless and
wired data-transmission systems licensed by international corporations. Dr.
Baccarelli is Associate Editor of the IEEE COMMUNICATION LETTERS,
and his biography isis listed in Who’s Who and Contemporary Who’s Who.
Mauro Biagi Mauro Biagi was born in Rome in
1974. He received his ”Laurea degree” in Telecommunication Engineering in 2001 from ”La Sapienza”
University of Rome. He obtained the Ph.D. on
information and communication theory in January
2005, at INFO-COM Dept. of the ”La Sapienza”
University of Rome and actually he covers the position of Assistant Professor in the same department.
His teaching activity deals with coding and statistical
signal processing. His research is focused on Wireless Communications (Multiple Antenna systems
and Ultra Wide Band transmission technology) mainly dealing with spacetime coding techniques and power allocation/ interference suppression in
MIMO-ad-hoc networks with special attention to game theory applications.
Concerning UWB his interests are focused on transceiver design for UWBMIMO applications. His research is focused also on Wireline Communications
and in particular bit loading algorithms and channel equalization for xDSL
systems and Power Line Communication and he is member of IEEE PLC committee and he joined several International Conferences as Technical Programm
Committee member. Actually he is involved in the Italian National Project
Wireless 8O2.16 Multi-antenna mEsh Networks (WOMEN) in research and
project managing activities.
15
Cristian Pelizzoni Cristian Pelizzoni was born in
Rome, Italy, in 1977. He received the Laurea Degree in Telecommunication Engineering from the
University of Rome ”Sapienza” in 2003. From 2003
to 2006 he was Ph.D student of Information and
Communication Engineering at Faculty of Engineering in University ”Sapienza”. Waiting for discussing
the final Ph.D thesis, related to optimization of
wireless transceivers for Multiple-Input MultipleOutput Ultra Wide Band (MIMO-UWB) systems,
he currently works as contractor researcher at the
INFOCOM dept. of the Faculty of Engineering (University ”Sapienza”).
He participates in the technical committee of the Italian Project ’Wireless
802.16 Multi-antenna mEsh Networks” (WOMEN). His research areas include
Project and Optimization of very high speed Wireless transceivers for the
emerging 4GWLANs, based on MIMO-UWB technology; Space-Time coding
for wireless (UWB-like) channels, affected by dense multipath, Space-Time
coding and game theory approach for power optimal allocation of wireless
ad-hoc networks; novel Physical and MAC layer solutions for Wireless Mesh
Networks.
Nicola Cordeschi Nicola Cordeschi was born in
Rome, Italy, in 1978. He received the Laurea Degree
(summa cum laude) in Telecommunication Engineering in 2004 from University of Rome ”Sapienza”. He is pursuing the Ph.D. at the INFOCOM
Department of the Engineering Faculty of ”Sapienza”. His research activity focuses on wireless communications, in particular dealing with the design
and optimization of high performance transmission
systems for wireless multimedia applications, both
in centralized and decentralized multiple antenna
scenarios.