Random Matrix Theory in Array Signal Processing: Application Examples

Random Matrix Theory in Signal Processing
Xavier Mestre
xavier.mestre@cttc.cat
Centre Tecnològic de Telecomunicacions de Catalunya (CTTC)
Klagenfurt University (Austria)
February 25, 2019

Centre Tecnològic de Telecomunicacions de Catalunya - CTTC
Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain, http://www.cttc.cat/
Outline
• Introduction to RMT: Convergence of spectral statistics of the sample covariance matrix.
• First Application: Subspace-based estimation of directions-of-arrival (DoA).
• Second Application: Detection tests of correlation and sphericity.
• Third Application: Large multivariate time series analysis
• Fourth Application: Outlier production characterization in Conditional/Unconditional Maximum
Likelihood parametric estimation.
Xavier Mestre: Random Matrix Theory in Signal Processing. 2/41

The sample covariance matrix
We assume that we collect  independent samples (snapshots) from an array of  antennas:
Consider the  ×  observation matrix Y = [y(1)     y()] and the sample covariance matrix
ˆR =
1

YY
 =
1

X
=1
y()y
()

Problem statement and objective of the talk
Typically, one expects ˆR to be a close approximation of R. Under conventional statistical
assumptions, we have
ˆR → R
almost surely when  → ∞ for a ﬁxed . Furthermore, if (·) is a reasonable function, we also
have 
³
ˆR
´
→  (R).
Unfortunately, when   have the same order of magnitude, this does not hold anymore: the Finite
Sample Size eﬀect appears.
In these situations, the regime where   → ∞ but  → , 0    ∞ becomes much more
relevant. RMT will help us in solving the following two problems in this regime:
• To what 
³
ˆR
´
does converge to?
• How do we design  (·) so that 
³
ˆR
´
→  (R).

Conditional versus unconditional model
Typically, the observations are superposition of some signals plus noise. For example, in array
processing the observation consists of the contribution from  signals:
Y = A (Θ) S + N
where:
• Matrix S ∈ C×
contains the contribution of the  signals (at each of its rows).
• Matrix A (Θ) ∈ C×
contains, at each of its columns, the spatial signature of each source,
namely
A (Θ) =
£
a (1) a (2) · · · a ()
¤
• Matrix N ∈ C×
contains the background noise samples. It is typically modeled as a matrix
with i.i.d. entries following a zero mean Gaussian distribution
{N} ∼ CN
¡
0 2
¢


Conditional versus unconditional model (II)
Depending on how the signals are modeled, we diﬀerentiate between the conditional and the
unconditional models. Let us denote
S = [s(1)     s()]
• Conditional Model: The entries of S are modelled as deterministic unknowns. In this case, the
observation can be described as
y() ∼ CN
¡
A (Θ) s() 2
I
¢

• Unconditional Model: The entries of S are modelled as random variables. Typically, we assume
that the column vectors s() are independent, circularly symmetric Gaussian Random variables, i.e.
s ∼ CN (0 P), P  0. In this case, we have
y() ∼ CN (0 R)  R = A (Θ) PA
(Θ) + 2
I

Conditional versus unconditional model (III)
Depending on the signal model, the structure of the sample covariance matrix model will be inherently
diﬀerent. Let X be an  × matrix of i.i.d. Gaussian standardized entries {X} ∼ CN (0 1).
The two most important models can be described as:
• Conditional Model (also known as Information plus Noise model), the SCM can be expressed as
ˆR =
1

(V + X) (V + X)
where V some deterministic matrix that contains the signal (information) contribution.
• Unconditional Model (also known as Single Side Correlation model), the SCM can be expressed as
ˆR = R
12

µ
1

XX

¶
R
12

where R
12
 is the positive Hermitian square root of R.
In many of the results obtained by RMT, the Gaussian assumption can be dropped.

Uncorrelated signals: the Marchenko-Pastur Law
Consider the simplest case where ˆR = 1
 XX
, where the entries of X are zero mean i.i.d. with
unit variance. Consider the eigenvalue distribution for diﬀerent  , but ﬁxed ratio .
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
2
4
6
8
10
12
Eigenvalues
Numberofeigenvalues
Histogram of the eigenvalues of the sample covariance matrix, M=80, N=800
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
2
4
6
8
10
12
14
Eigenvalues
Numberofeigenvalues

Uncorrelated signals: the Marchenko-Pastur Law (II)
It turns out, that when   → ∞,  → , 0    ∞, the empirical density of eigenvalues
converges to a deterministic measure.
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
2
4
6
8
10
12
14
Eigenvalues
Numberofeigenvalues
For   1 () = 1
2
q¡
 − −
¢ ¡
+
− 
¢
I[−
+
]() −
= (1 −
√
)
2
 +
= (1 +
√
)
2
.

The sample covariance matrix
We assume that we collect  independent samples (snapshots) from an array of  antennas:
ˆR = 1

P
=1 y()y
(). Example: R has 4 eigenvalues {1 2 3 7} with equal multiplicity.
0 1 2 3 4 5 6 7 8 9 10
0
5
10
15
20
25
lambda
Numberofeigenvalues
0 1 2 3 4 5 6 7 8 9 10
0
5
10
15
20
25
30
lambda
Numberofeigenvalues

The sample covariance matrix: asymptotic properties
When both   → ∞,  → , 0    ∞, the e.d.f. of the eigenvalues of ˆR tends to a
deterministic density function. Example: R has 4 eigenvalues {1 2 3 7} with equal multiplicity.
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Eigenvalues
Aysmptotic density of eigenvalues of the sample correlation matrix
c=0.01
c=0.1
c=1

1st Application: Subspace-based estimation of directions of arrival (DoA)

Introduction and signal model
We consider DoA detection based on subspace approaches (MUSIC), that exploit the orthogonality
between signal and noise subspaces.
Consider a set of  sources impinging on an array of  sensors/antennas. We work with a ﬁxed
number of snapshots ,
{y(1)     y()}
assumed i.i.d., with zero mean and covariance R.
The true spatial covariance matrix can be described as
R = A (Θ) ΦA (Θ)
+ 2
I
where A (Θ) is an  × matrix that contains the steering vectors corresponding to the  diﬀerent
sources,
A (Θ) =
£
a (1) a (2) · · · a ()
¤
and 2
is the background noise power.

Introduction and signal model (II)
The eigendecomposition of R allows us to differentiate between signal and noise subspaces:
R =
£
E E
¤
∙
Λ 0
0 2
I−
¸
£
E E
¤

It turns out that E
a () = 0,  = 1    .
Since R is unknown, one must work with the sample covariance matrix
ˆR =
1

X
=1
y()y
()
The MUSIC algorithm uses the sample noise eigenvectors, and searches for the deepest local minima
of the cost function
MUSIC () = a
() Ê
Ê
a () 
It is interesting to investigate the behavior of MUSIC () when   have the same order of magnitude.

Asymptotic behavior of MUSIC
The MUSIC algorithm suffers from the breakdown effect. The performance breaks down when
the number of samples or the SNR falls below a certain threshold. Cause: Ê is not a very good
estimator of E when   have the same order of magnitude.
The performance breakdown effect can be easily analyzed using random matrix theory, especially under
a noise eigenvalue separation assumption: |MUSIC () − ¯MUSIC ()| → 0
¯MUSIC () = s
()
Ã X
=1
()ee

!
s ()
() =
⎧
⎨
⎩
1 − 1
−
P
=−+1
³
2
−2 −
1
−1
´
 ≤  − 
2
−2 −
1
−1
   − 
where {  = 1     } are the solutions to 1

P
=1

− = 1


Asymptotic behavior of MUSIC: an example
We consider a scenario with two sources impinging on a ULA ( = 05,  = 20) from DoAs:
35◦
, 37◦
.
−100 −80 −60 −40 −20 0 20 40 60 80 100
−35
−30
−25
−20
−15
−10
−5
0
MUSIC asymptotic pseudospectrum, M=20, DoAs=[35,37]deg
Azimuth (deg)
32 34 36 38 40
−32
−30
−28
−26
−24
−22
−20
−18
N=25
N=15
SNR=12dB
SNR=17dB
2 4 6 8 10 12 14 16 18 20
25
30
35
40
45
50
SNR (dB)
Azimuth(deg)
Position of the two deepest local minima of the asymptotic MUSIC cost function
10 12 14
35
36
37
N=15
N=25

 -consistent subspace detection: G-MUSIC
We can derive an  -consistent estimator of the cost function  () = s
() EE
s ():
G-MUSIC () = s
()
Ã X
=1
()ˆeˆe

!
s ()
() =
⎧
⎨
⎩
1 +
P
=−+1
³
ˆ
ˆ−ˆ
− ˆ
ˆ−ˆ
´
 ≤  − 
−
P−
=1
³
ˆ
ˆ−ˆ
− ˆ
ˆ−ˆ
´
   − 
where now ˆ1     ˆ are the solutions to the equation
1

X
=1
ˆ
ˆ − ˆ
=
1



Performance evaluation MUSIC vs. G-MUSIC
Comparative evaluation of MUSIC and G-MUSIC via simulations. Scenarios with four
(−20◦
 −10◦
 35◦
, 37◦
) and two (35◦
, 37◦
) sources respectively, ULA ( = 20,  = 05).
−80 −60 −40 −20 0 20 40 60 80
10
−4
10
−3
10
−2
10
−1
10
0
Example of MUSIC and GMUSIC cost function, SNR=18dB, M=20, N=15, DoAs=35, 37, −10, −20 deg.
Angle of arrival (azimuth), degrees
MUSIC
GMUSIC
34 36 38
10
−4
10
−3
10
−2
5 10 15 20 25
10
−3
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
SNR (dB)
MSE
Mean Squared Error
MUSIC
GMUSIC
CRB
M=20, N=15
M=20, N=75

2nd Application: characterization of sphericity and correlation tests

Problem formulation
We consider two very important tests in signal processing, which try to establish the structure of the
covariance matrix of the received signal:
• Sphericity test: seeks to establish whether the received signal is spatio-temporal white noise:
H0 : R = 2
I
H1 : R 6= 2
I
• Correlation test: seeks to establish whether the signals received from multiple sensors is corre-
lated:
H0 : R = R ¯ I
H1 : R 6= R ¯ I
In both cases, the true covariance matrix is unknown, so one must work on the sampled version ˆR.

Generalized Maximum Likelihood Ratio Test (GLRT)
In order to address this binary hypothesis problem, one may resort to the Generalized Likelihood Ratio
Test (GLRT):
supR
Y
=1
Φ (y; R)
sup2
Y
=1
Φ (y; 2I)
H1
≷
H0

supR
Y
=1
Φ (y; R)
supD
Y
=1
Φ (y; D)
H1
≷
H0

where Φ (y; R) is the pdf of a complex Gaussian with zero mean and covariance R.
For  ≥ , the GLRT for sphericity and correlation respectively reject H0 for large values of
ˆsphr
 = log
∙
1

tr
³
ˆR
´¸
−
1

log det
³
ˆR
´
ˆcorr
 = log
∙
1

tr
³
ˆC
´¸
−
1

log det
³
ˆC
´
where ˆC =
³
ˆR ¯ I
´−12
ˆR
³
ˆR ¯ I
´−12
is the sample correlation/coherence matrix.

Frobenius norm test
Other more ad-hoc tests can be constructed using a more intuitive reasoning:
• Non-sphericity will manifest in ˆR being far from proportional to the identity.
• Correlation will lead to high absolute values of the oﬀ-diagonal elements of ˆC .
Therefore, it seems reasonable to design the test to reject H0 for large values of
ˆsphr
 =
1

°
°
°
°
ˆR −
1

tr
h
ˆR
i
I
°
°
°
°
2

ˆsphr
 =
1

°
°
°ˆC − I
°
°
°
2


In both cases, we have
ˆ
 =
1

X
=1
ˆ
2
 −
Ã
1

X
=1
ˆ
!2
which are LSS with () = 2
and () = .

General study of tests
It is generally diﬃcult to derive the distribution of these tests, so in practice the literature has focused
on the case  → ∞ for ﬁxed 
We would like to know the asymptotic behavior of these tests, for   having the same order of
magnitude, allowing for the possibility of    (undersampled regime).
Fortunately, there is a direct relationship between LSS and Stieltjes transform:
ˆ =
1

X
=1

³
ˆ
´
=
1
2 j
I
C−
() ˆ()
where
ˆ() =
1

X
=1
1
ˆ − 
and where the contour C−
enclosed all the positive eigenvalues and not zero.

First order convergence
By replacing ˆ() with the asymptotic equivalent, we obtain the almost sure asymptotic behavior
of the test ˆ, in the sense that |ˆ − ¯| → 0 where
¯ =
1
2 j
I
C−
() ¯()
Most of the times, we can carry out the integral and ﬁnd a closed form for ¯.
For example, for the ¯
 , we can establish
¯
 =
(

 + 
 + −
 log
¯
¯1 − 

¯
¯   

 + 
 − −
 log |∗| + 1

P
=1  log
¯
¯
¯

−∗
¯
¯
¯   
where ∗ ≤ 0 is a solution to a certain equation and 
 is the large- value of the GLRT

 = log
"
1

X
=1

#
−
1

X
=1
log 

Second order convergence
Using RMT tools that establish how ˆ() fluctuates around ¯(), we may establish a CLT on
these tests. Under certain statistical conditions, the LSS ˆ will asymptotically fluctuate as Gaussian
random variable, in the sense that
−1
 ( (ˆ − ¯) − )
L
−→ N (0 1) 
where
 =
1
2 j
I
C−

() () 
2
 =
−1
42
I
C−

I
C−

(1)(2)2
 (1 2) 12
where () is the original test function () after some change of variable, and where the mean
 () and variance 2
 (1 2) are different for the Sphericity and Correlation tests.
These integrals can be computed in closed form, and one can generally approximate ˆ ≈
N
¡
¯ +  2
2
¢


Numerical Results: correlation test
Simulations for 105
independent simulation runs (GLRT). Under H0, D takes uniform values between
0 and 1. Under H1, R = D + Ψ where {Ψ} = 09|−|
.
250 300 350 400 450 500 550 600
0
0.005
0.01
0.015
Density of the statistic under H
0
300 350 400 450 500 550 600
0
0.005
0.01
0.015
Density of the statistic under H1
Simulated
Theory (large M,N)
Theory (large N)
M=20,N=25
M=20,N=25
 = 20  = 25
250 300 350 400 450 500 550
0
0.005
0.01
0.015
Density of the statistic under H
0
, M=20, N=100
350 400 450 500 550 600 650 700
0
0.005
0.01
0.015
Density of the statistic under H1
, M=20, N=100
Simulated
Theory (large M,N)
Theory (large N)
 = 20  = 100

3rd Application: Large multi-variate time series

Introduction: testing independence of multiple time series
We consider an -variate zero-mean Gaussian time series
y() = [1()     ()]
where  = 1     , and ask ourselves whether the diﬀerent components of the series are independent.

Motivation
Consider a certain window of  samples and the extended random vector
y() =
⎡
⎣1()     1( +  − 1)
| {z }
 samples
     ()     ( +  − 1)
| {z }
 samples
⎤
⎦


and consider the second order statistics of this vector, namely
E
£
y()y
 ()
¤
= R =
⎡
⎢
⎢
⎢
⎣
R
(11)
 R
(12)
 · · · R
(1)

R
(21)
 R
(22)
 · · · R
(2)

... ... ... ...
R
(1)
 R
(2)
 · · · R
()

⎤
⎥
⎥
⎥
⎦
where R
(0
)
 has dimensions  × . If the diﬀerent time series are independent, R becomes block
diagonal and
 =
1

Ã
log det R −
X
=1
log det R
()

!
= 0

Spatio-temporal sample covariance matrix
In practice, R has to be estimated from the observations y(). Using the above formulation, we
can estimate the time covariance between series  and 0
as
ˆR
(0
)
 =
1

YY
0
where
Y =
⎡
⎢
⎢
⎢
⎣
 (1)  (2) · · ·  () · · ·  ()
 (2) ... · · · ... ...  ( + 1)
...  () ... ... · · · ...
 () · · ·  ()  ( + 1) · · ·  ( +  − 1)
⎤
⎥
⎥
⎥
⎦
has a Hankel structure. Under the null hypothesis (uncorrelation), and assuming stationarity
E
£
()∗
0(0
)
¤
=  ( − 0
) =0  () =
Z 1
0
S () e2i


Tests from the spatio-temporal sample covariance matrix
We can therefore build the test
ˆ =
1

Ã
log det ˆR −
X
=1
log det ˆR
()

!
and ask ourselves how to choose  (time window parameter) to make ˆ close to zero under the
uncorrelation hypothesis. There is some trade-oﬀ between choosing  small (so that ˆR is close to
R in spectral norm) and testing independence in large time lags  ar large as possible.
For this all this, it appears reasonable to investigate the behavior of the eigenvalues of ˆR when
 → ∞ and  → ∞ at the same rate, so that  = 
 → , 0    +∞. We will assume
that 4
→ 0, the spectral densities are uniformly bounded above and away from zero, and that
sup

X
∈Z
Ã
1

X
=1
| ()|2
!12
 +∞.

Tests from the spatio-temporal sample covariance matrix
Let Q(),  ∈ C+
, denote the resolvent matrix of ˆR, that is Q() =
³
ˆR − I
´−1
. Under the
above assumptions, Q() ³ T() (deterministic asymptotic equivalent), where T() is the unique
solution to
T() =
−1

µ
I + 
µ
−1

¡
I + Ψ (T())
¢
¶¶−1
in the class of matrix valued Stieltjes transforms, where Ψ : C×
→ C×
and Ψ : C×
→
C×
are the operators
Ψ(A) =
Z 1
0
d
 () Ad ()

¡
S () ⊗ d () d
 ()
¢

Ψ(B) =
1

X
=1
Z 1
0
S ()
d
 () B()
d ()

d () d
 () 
where d () =
£
1     ei(−1)
¤
and S () = diag (S1 ()      S ()).

4th Application: outlier characterization of Maximum Likelihood estimation

Considered scenario
• Consider an array of  sensors receiving the signal transmitted by  sources from parameters
¯ =
£
¯(1)     ¯()
¤
, where we assume   .
• Let y() denote an  × 1 complex vector containing the received samples. We model this obser-
vation vector as
y() = A(¯)s() + n()
where s() contains the signal transmitted by the    sources, n() contains the received
noise (assumed i.i.d. and CN(0 2
)) and
A(¯) =
£
a
¡
¯(1)
¢
   a
¡
¯()
¢ ¤
• We assume that a total of  snapshots are available, and that   .
• Problem: estimate the parameters ¯ from the observations {y()  = 1    }.
• We investigate the use of Maximum Likelihood approaches =⇒ Highest resolution at the cost of
increased computational complexity (multidimensional search)

Maximum likelihood
• The estimated angles are determined as ˆ = arg min∈Θ ˆ () where ˆ () is the negative (concen-
trated) log-likelihood function.
• The “conditional” (or deterministic) model: assumes that the signals s() are deterministic un-
knowns. In this situation, one must minimize
ˆ () =
1

tr
h
P⊥
()ˆR
i
where ˆR = 1

P
=1 y()y
() is the sample covariance matrix, P⊥
() = I − P(), and
P() = A()
¡
A
()A()
¢−1
A
()
is the orthogonal projection on the column space of A().
• The “unconditional” (or stochastic) model: assumes that the source signals are random variables,
typically s() ∼ CN(0 P) and i.i.d. in the time domain. In this situation,
ˆ () =
1

log det
h
ˆ () P⊥
() + P()ˆRP()
i


Breakdown eﬀect in maximum likelihood (I)
• General nonlinear parametric estimators exhibit a threshold eﬀect.
• At low SNR, or low , the MSE suddenly departs from the Cramér Rao Bound. The presence of
outliers is the main cause for this behavior.
MSE
Threshold
effect
B
CRB
d
SNR
dB

Breakdown eﬀect in maximum likelihood (II)
At low values of the SNR and , there exist realizations of the cost functions for which local minima
corresponding to outliers become deeper than the intended one.
UML, SNR=0dB, M=5, N=20, uncorrelated signals,DoA=[16,18]deg
θ1
(deg)θ2
(deg)
−80 −60 −40 −20 0 20 40 60 80
−80
−60
−40
−20
0
20
40
60
80
UML cost function
Local Minima
Intended Minimum
Selected Minimum

Probability of resolution
• We consider here the definition of [Athley, 05] of the resolution probability. If ˆ () is a generic cost
function that fluctuates around a deterministic ¯ (), which has  + 1 local minima at the values
¯ 1     , the probability of resolution can be defined as
 = P
" 
=1
©
ˆ ()  ˆ
¡
¯
¢ª
#

• It was shown in [Athley, 05] that this definition of  provides a very accurate description of both
the breakdown effect and the expected mean squared error (MSE) of the DoA estimation process.
• Unfortunately, in our ML setting,  is difficult to analyze for finite values of   due to the
complicated structure of the cost functions, especially ˆ ().
• We propose to use the asymptotic distributions (as   → ∞) instead of the actual ones. Very
accurate description of the actual probability, even for very low  .

First order behavior
When   → ∞, under several technical conditions, the two ML cost functions become (pointwise)
asymptotically close to two deterministic counterparts, namely
|ˆ () − ¯ ()| → 0 |ˆ () − ¯ ()| → 0
a.s. pointwise in  as   → ∞, where
¯ () =
1

tr
£
P⊥
()R
¤
and
¯ () =
1

log det
£
2
() P⊥
() + P()RP()
¤
+
 − 

log
µ

 − 
¶
−


respectively, where R is the true covariance matrix of the observations and 2
() =
1
− tr
£
P⊥
()R
¤
.

Second order behavior
Let 1      be a set of multidimensional points (e.g. local minima of ¯ () or ¯ ()). Let
ˆη = [ˆ (1)      ˆ ()]
and ¯η = [¯ (1)      ¯ ()]
and take the equivalent deﬁnitions for the UML cost function. Assume that y() ∼ CN (0 R).
Under certain technical conditions, as   → ∞ ,  → , 0    1, we have
Γ−1
 (ˆη − ¯η) → N (0 I) and Γ−1
 (ˆη − ¯η) → N (0 I)
for some covariance matrices Γ, Γ given by
{Γ} =
1

tr
£
P⊥
 P⊥

¤
and
{Γ} =
1
2
2

1

tr
£
P⊥
 P⊥

¤
+
1
2

1

tr
£
P⊥
Q
¤
+
1
2

1

tr
£
P⊥
 Q
¤
− log
¯
¯
¯
¯1 −
1

tr [QQ]
¯
¯
¯
¯
where P = R
12
 P
³

()

´
R
12
 ,P⊥
 = R − P, and Q = R
12
 A
£
A
 RA
¤−1
A
 R
12
 .

Application
• The previous asymptotic result is basically stating that for sufficiently large  , we are able
to approximate the finite-dimensional distributions of the functions ˆ () and ˆ () as
multivariate Gaussians, namely
ˆη ∼ N (¯η Γ)
ˆη ∼ N (¯η Γ)
• It turns out that these results are very good approximations of the finite dimensional reality, even
for relatively low values of  .
• This provides a tool to evaluate the resolution probability of the ML method according to
 = P
" 
=1
©
ˆ ()  ˆ
¡
¯
¢ª
#
= P
⎡
⎣
⎡
⎣
−1 1
... ...
−1 1
⎤
⎦ ˆη  0
⎤
⎦
which can be evaluated by computing the cdf of the Gaussian law.

Simulation results
ULA of  = 5 elements, two sources coming from 16 and 18 degrees with respect to the broadside.
−15 −10 −5 0 5 10 15 20 25
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
Prob.ofRes.
Prob. of res., M=5, Theta=[16,18] deg, corr=0
UML (Predicted)
UML (Simulated)
CML (Predicted)
CML (Simulated)
N=100
N=10
Uncorrelated sources
−15 −10 −5 0 5 10 15 20 25
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)Prob.ofRes.
Prob. of res., M=5, Theta=[16,18] deg, corr=0.95
UML (Predicted)
UML (Simulated)
CML (Predicted)
CML (Simulated)
N=100
N=10
Highly correlated sources

Concluding remarks

Conclusions
Random Matrix Theory offers the possibility of analyzing the behavior of different quantities depending
on ˆR when the sample size and the number of sensors/antennas have the same order of magnitude.
The objective is to describe the asymptotic behavior of a certain scalar function of ˆR, namely

³
ˆR
´
.
• Traditional Approach: Assuming that the number of samples is high, we might establish that

³
ˆR
´
→  (R) in some stochastic sense as  → ∞ while  remains fixed.
• New Approach: In order to characterize the situation where   have the same order of
magnitude, one might consider the limit   → ∞,  → , 0    ∞.
Results obtained under this asymptotic limit turn out to be extremely accurate, even for reasonably
low  .

Thank you for your attention!!!

Random Matrix Theory in Array Signal Processing: Application Examples

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Random Matrix Theory in Array Signal Processing: Application Examples