# On Blind MIMO System Identification Based on  Second-Order Cyclic Statistics

Authors: K. Sabri; M. El Badaoui; F. Guillet; A. Adib; D. Aboutajdine
Journal: Research Letters in Signal Processing
 (2008)

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## Abstract

This letter introduces a new frequency domain approach for either MIMO System Identification or Source Separation of convolutive mixtures in cyclostationary context. We apply the joint diagonalization algorithm to a set of cyclic spectral density matrices of the measurements to identify the mixing system at each frequency up to permutation and phase ambiguity matrices. An efficient algorithm to overcome the frequency dependent permutations and to recover the phase, even for non-minimum-phase channels, based on cyclostationarity is also presented. The new approach exploits the fact that each input has a different and specific cyclic
frequency. A comparison with an existing MIMO method is proposed.


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## Body

## 1. Introduction

In many real-world situations, man-made
signals such as those encountered in rotating machines, communications,
telemetry, radar, and sonar systems are nonstationary and very often
cyclostationary [1, 2]. The mixtures of real-world signals are usually
convolutive mixtures, so the conventional methods for the standard blind source
separation (BSS) problem which assume instantaneous mixtures of stationary
sources [3, 4] are no longer appropriate. In a cyclostationary context, BSS has
motivated much research which we summarize, for sake of space, as follows.
Liang et al. in [5] were the first, to our knowledge, to make use of the information
offered by the cyclic frequencies in order to separate signals. Abed-Meraim et
al. [1] introduce two second-order statistics- (SOS-) based criteria to blindly
separate sources. Ferreol and Chevalier [6] show
that, in a cyclostationary
context, the fourth-order (FO) empirical estimators generate asymptotically
biased estimates of the temporal mean of the quadricovariance. Jallon et al. in
[7] introduce an FO-based contrast function whose estimation does not require
the knowledge of the cyclic frequencies. Antoni et al. make use of the spectral
correlation density (SCD) matrices for the blind identification [8] and signal
separation [9] of sources with the same cyclic frequency.The method offered in this letter is a frequency
domain technique that exploits the cyclic spectral density matrices of the
measurements to identify the mixing channel and separate convolved
cyclostationary sources with different cyclic frequencies. The organization of
this letter is as follows. The problem statement including the set of required
assumptions is presented in Section2. Section 3 presents the algorithm to
estimate the magnitude of multiple-input multiple-output (MIMO) system as well
as the methods to correct the permutation and phase. We report some encouraging
results on simulated signals to demonstrate the robustness of the proposed
approach in Section 4. Ultimately, conclusions are presented in Section 5.


## 2. Problem Formulation

### 2.1. System Model

We considern-source m-sensor MIMO
linear model for the received signal for the convolutive mixing problem
(1)x(t)=∑k=0L−1H(k)a(t−k)+b(t),  where x(t)=[x1(t),…,xm(t)]T   is the vector
of measurements, a(t)=[a1(t),…,an(t)]T   is the source
vector were the components are real-valued, cyclostationary, i.i.d., mutually
statistically uncorrelated, centered (in the ensemble-average sense), and with
different cyclic frequencies {αi,i=1,…,n}. H(t) is the m×n impulse
response matrix whose elements are {hji(t)}, and b(t)=[b1(t),…,bm(t)]T   is the additive
noise vector which is assumed to be stationary, temporally, and spatially
white, zero-mean, and independent of the source signals. The superscript T denotes the
transpose and L is the length
of the real-valued channels. By taking the discrete Fourier transforms (DFTs)
(1) over N lines, we
obtain that (2)X(fk)=H(fk)A(fk)+B(fk),  where X(fk), H(fk), A(fk), and B(fk) are the DFT of x(t), H(t), a(t), and b(t), respectively. From (2), it is obvious that at each
frequency bin, the estimation of A(fk) can be seen as
an instantaneous complex BSS problem.


### 2.2. Cyclic Spectral Analysis

Let us introduce the SCD of the source
vector [10] (3)𝔼{A(fk)AH(fk−qαi)}=Saqαi(fk)=σi2(qαi)Ii,  where the
superscript H denotes the
complex conjugate transpose of a matrix and σi2(qαi) stands for the
SCD of the signal ai(t), at the qth multiple of αi, which is frequency-independent since ai(t) is i.i.d.. Ii is a matrix
with only one nonzero element which is set to 1 and i is the index of
the unique nonzero diagonal element of Saqαi(fk). This means that the SCD matrices of the inputs at
different harmonics are diagonal, with only one nonzero element, under the
independency hypothesis.We conclude from (3) that to blindly restore the
sources, we need to submit the measurements to some linear transformations so
that the estimated sources will have the same algebraic structure as the
primary sources.


## 2.1. System Model

We considern-source m-sensor MIMO
linear model for the received signal for the convolutive mixing problem
(1)x(t)=∑k=0L−1H(k)a(t−k)+b(t),  where x(t)=[x1(t),…,xm(t)]T   is the vector
of measurements, a(t)=[a1(t),…,an(t)]T   is the source
vector were the components are real-valued, cyclostationary, i.i.d., mutually
statistically uncorrelated, centered (in the ensemble-average sense), and with
different cyclic frequencies {αi,i=1,…,n}. H(t) is the m×n impulse
response matrix whose elements are {hji(t)}, and b(t)=[b1(t),…,bm(t)]T   is the additive
noise vector which is assumed to be stationary, temporally, and spatially
white, zero-mean, and independent of the source signals. The superscript T denotes the
transpose and L is the length
of the real-valued channels. By taking the discrete Fourier transforms (DFTs)
(1) over N lines, we
obtain that (2)X(fk)=H(fk)A(fk)+B(fk),  where X(fk), H(fk), A(fk), and B(fk) are the DFT of x(t), H(t), a(t), and b(t), respectively. From (2), it is obvious that at each
frequency bin, the estimation of A(fk) can be seen as
an instantaneous complex BSS problem.


## 2.2. Cyclic Spectral Analysis

Let us introduce the SCD of the source
vector [10] (3)𝔼{A(fk)AH(fk−qαi)}=Saqαi(fk)=σi2(qαi)Ii,  where the
superscript H denotes the
complex conjugate transpose of a matrix and σi2(qαi) stands for the
SCD of the signal ai(t), at the qth multiple of αi, which is frequency-independent since ai(t) is i.i.d.. Ii is a matrix
with only one nonzero element which is set to 1 and i is the index of
the unique nonzero diagonal element of Saqαi(fk). This means that the SCD matrices of the inputs at
different harmonics are diagonal, with only one nonzero element, under the
independency hypothesis.We conclude from (3) that to blindly restore the
sources, we need to submit the measurements to some linear transformations so
that the estimated sources will have the same algebraic structure as the
primary sources.


## 3. MIMO Identification

The singular value decomposition (SVD) ofH(fk) is(4)H(fk)=U(fk)Σ(fk)V(fk),  where U(fk) and V(fk) are m×n and n×n unitary
matrices, respectively, and Σ(fk) is n×n diagonal
matrix.Let us introduce the SCD matrix of the measurements
[10]:(5)Sxqαi(fk)=H(fk)Saqαi(fk)HH(fk−qαi)+Sbqαi=0(fk), where Sbqαi=0(fk) is the SCD
matrix of the noise. The eigen-value decomposition (EVD) of (5), evaluated when αi=0, allows the identification of U(fk) and Σ(fk) and hence the
whitening of the measurements.The SCD matrices of the whitened processes at
different cyclic frequenciesqαi are(6)Szqαi(fk)=V(fk)Saqαi(fk)VH(fk−qαi). Let
us consider the norm of the last relationship in
order to have a symmetric expression in the right-hand side(7)Szqαi(fk)=V(fk)Ssqαi(fk)VH(fk). The aim is to
find a unitary matrix that simultaneously and
jointly diagonalizes the set of Szqαi(fk) matrices. The
well-known approximate joint diagonalization (JAD) algorithm [4] will be used
to this end. Once V(fk) is identified,
one can estimate the mixing MIMO system H(fk) based on (4) as
follows:(8)H^(fk)=H(fk)ΔP(fk)D(fk), where H^(fk) is an
estimation of H(fk) with constant
diagonal matrix, Δ, which introduces a scaling ambiguity,
frequency-dependent permutation P(fk), and phase ambiguity D(fk) that introduces
circular shifts to the sources (the ith diagonal
element of D(fk) is of the form ejΦDi(fk)). Certainly,
the estimation offered by (8), which is inherent to SVD approaches, can no
longer provide a good estimation of the actual system H(fk) neither in the
frequency domain nor in the time one by involving the inverse DFT (IDFT).
Hereafter, we propose two techniques to cancel the permutation and the phase
ambiguities, so that the mixing MIMO system will be estimated up to a general
constant diagonal matrix and a uniform permutation
matrix, thus giving(9)H^opt(fk)=H(fk)ΔP which solves the blind identification issue.


### 3.1. Resolving the Frequency-Dependent Permutation Ambiguity

An interesting property of cyclostationary signals is
their spectral redundancy. In our case, the cyclic spectra of signals will not
overlap because they have distinct cyclic frequencies. This is the property
that we seek to exploit in our effort to derive a new algorithm for the
correction of permutations across all the frequency bins. More precisely, our
approach makes use of the cyclic cepstrum of the estimated signals which are
permuted.Let us first consider the estimating sources:(10)A^(fk)=H^#(fk)X(fk)=D*(fk)PT(fk)Δ−1A(fk), where the
superscripts # and * denote the
pseudoinverse and the complex conjugate of a matrix, respectively. The SCD
matrix of the estimated sources is given by(11)|Sa^αi(fk)|≃ϱi−2PT(fk)|Saαi(fk)|P(fk−αi), where ϱi is the ith diagonal
element of Δ. The following relationship can be easily deduced
from (11) for the elements in the main diagonal:(12)γjj2(fk)≃ϱi−2|σi2(αi)|pij(fk)pij(fk−αi), where γjj2(fk) is the SCD of a^j(t). Under the assumption that log(|Sa^αi(fk)|) is finite, the
real cepstrum of (12) exists and thus leads to(13)Cpij(τ)≃Cγjj2(τ)−log(ϱi−2|σi2(αi)|)δ(τ)1+exp(j2παiτ), where Cγjj2(τ)=Ca^αi(τ). Thus, the permutation pij(fk) can be computed from(14)pij(fk)=exp(∑τ=−∞∞Cpij(τ)e−j2πfkτ). For a given αi, (14) allows us to estimate the ith column of the
matrix P(fk). As we do not have any of these quantities ϱi−2|σi2(αi)|, it means that there is
no link between the actual and the estimated sources and therefore no
information is available about the actual order of the sources. A way to
estimate (13) is(15)Cpij(τ)≃Cγjj2(τ)−log(γ0i2(f1))δ(τ)1+exp(j2παiτ), where γ0i2(f1) is the
biggest value of the matrix Sa^αi(f1). Furthermore, there is no available solution whenever 2ταi=2l+1 or τ=(l+1/2)(1/αi). In fact, (15) assumes the permutation at f1 as the true one
and try to extend it to all frequency bins. Moreover, (8) will be simplified to(16)H^(fk)=H(fk)ΔPD(fk).


### 3.2. Resolving the Phase Ambiguity

The phase retrieval of single-input single-output (SISO) systems, excited by
cyclostationary inputs, has been studied in several papers [11–13]. Note
that, by exploiting second-order cyclic statistics, the identification of
nonminimum-phase channels becomes possible. Let us recall the phase retrieval
in the case of a SISO system:(17)ψ(fk)−θ(αi)=ϕ(fk)−ϕ(fk−αi), where ψ(fk)=arg{Sxαi(fk)}, θ(αi)=arg{Saαi(fk)}, ϕ(fk)=arg{H(fk)}, Sxαi(fk) and Saαi(fk) are the SCD of
the single measurement x(t) and the single
source a(t), with the cyclic frequency αi, respectively, and H(fk) is the transfer
function of the filter. We note that q has been set to 1 with θ(αi) being constant
since the source is i.i.d.. Li and Ding's solution for the phase retrieval in
the case of PAM signals is as follows:(18)ϕ(t)=ψ(t)−θαi(t)(1−exp(j2παit)),t≠pαi, where ϕ(t), ψ(t), and θαi(t)=θ(αi)δ(t) are the IDFT of ϕ(f), ψ(f), and θ(αi), respectively. One can interpolate the missing
information whenever t=p/αi, assuming the continuity of ϕ(t). Actually, θ(αi) can be computed
by summing up (17) over all discrete frequencies which
yield a
zero right-hand side; thus we get(19)θ(αi)=1N∑k=1Nψ(fk). Unfortunately,
the previous solution (18) is not appropriate for MIMO systems. The idea is to
transform the MIMO system in m×n SISO systems,
thanks to the cyclostationarity and hence, (18) will be appropriate. Let us
involve Sxαi(fk) for a given αi. Putting (3) into (5), the latter expression takes
the following forms:(20)Sxαi(fk)=σi2(αi)H(fk)IiHH(fk−αi) and, moreover,(21)Sxiiαi(fk)=σi2(αi)Hii(fk)Hii*(fk−αi). The relationship (21) is important because it transforms the m×n MIMO system to n, m×1, single-input multiple-output (SIMO) systems, according to each αi in order to
have the contribution of only one source. Then, we successively apply (18) over
only the diagonal elements of (21) since it comprises m SISO systems.
This is the key relationship that can lead to the estimation of the ith column of Φ(fk). Finally, by sweeping the other cyclic frequencies,
the remaining columns can be estimated in the same way.Actually, we have no knowledge on the true order ofΦ(fk) columns. The
estimated columns of Φ(fk) are affected by
a permutation matrix P′ which is
different from P:(22)Φ^(fk)=Φ(fk)P′.


### 3.3. Connection between the Magnitudes and the Phases

As the processes
of estimating the MIMO system's magnitude and phase are disconnected, the
correspondence between channel magnitude and phase is missed. This can be seen
by taking the absolute value of the relationship (16):(23)|H^(fk)|=|H(fk)Δ|P. It is clear
therefore that the columns of the MIMO system's magnitudes are permuted by the
matrix P, which was taken as reference of permutations over
all frequencies, and the MIMO system's phases are permuted by P′. However, to correctly identify H(fk), we need to cancel the effect of P′ as follows:(24)H^(fk)=|H(fk)Δ|P⊙exp(jΦ^(fk)P′TP), where the
symbol ⊙ denotes the
Hadamard product. In other words, the knowledge of P′ is
indispensable in order to make the true connection between the magnitude and
the phase. Let us take the argument of (16):(25)ΦH^(fk)=Φ(fk)P+ΦD(fk), where ΦH^(fk) is the phase of H^(fk) and ΦD(fk) is an m×n matrix whose
column i, i=1,…,n, contains the
phase of the ith diagonal
element of D(fk). The jth line of ΦD(fk) is as follows:(26)ΦDj(fk)=[ΦD1(fk),…,ΦDi(fk),…,ΦDn(fk)]. Of course, (25)
shows the existence of correlation between ΦH^(fk) and Φ^(fk). However, the matrix ΦD(fk) masks this
correlation and hence, it must be removed to well estimate P′. Let us subtract the jth line of ΦH^(fk) from its ith line:(27)ΦH^i(fk)−ΦH^j(fk)=[∑l=1n(Φil(fk)−Φjl(fk))Pl1,…,∑l=1n(Φil(fk)−Φjl(fk))Pln]. The way to
cancel P′ is to evaluate
the correlation between the elements of (27) and the following equation:(28)Φ^i(fk)−Φ^j(fk)=[∑l=1n(Φil(fk)−Φjl(fk))Pl1′,…,∑l=1n(Φil(fk)−Φjl(fk))Pln′], where Φ^i(fk) and Φ^j(fk) are,
respectively, the ith and the jth lines of Φ^(fk). The correlation will be made element by element and
over all frequency bins. Actually, P′ is canceled and
replaced by P, therefore, (22) becomes(29)Φ^(fk)=Φ(fk)P. The ambiguity ΦD(fk) can easily be
deducted from (25):(30)ΦD(fk)=ΦH^(fk)−Φ(fk)P. Finally, the
MIMO system is identified as in (9).


## 3.1. Resolving the Frequency-Dependent Permutation Ambiguity

An interesting property of cyclostationary signals is
their spectral redundancy. In our case, the cyclic spectra of signals will not
overlap because they have distinct cyclic frequencies. This is the property
that we seek to exploit in our effort to derive a new algorithm for the
correction of permutations across all the frequency bins. More precisely, our
approach makes use of the cyclic cepstrum of the estimated signals which are
permuted.Let us first consider the estimating sources:(10)A^(fk)=H^#(fk)X(fk)=D*(fk)PT(fk)Δ−1A(fk), where the
superscripts # and * denote the
pseudoinverse and the complex conjugate of a matrix, respectively. The SCD
matrix of the estimated sources is given by(11)|Sa^αi(fk)|≃ϱi−2PT(fk)|Saαi(fk)|P(fk−αi), where ϱi is the ith diagonal
element of Δ. The following relationship can be easily deduced
from (11) for the elements in the main diagonal:(12)γjj2(fk)≃ϱi−2|σi2(αi)|pij(fk)pij(fk−αi), where γjj2(fk) is the SCD of a^j(t). Under the assumption that log(|Sa^αi(fk)|) is finite, the
real cepstrum of (12) exists and thus leads to(13)Cpij(τ)≃Cγjj2(τ)−log(ϱi−2|σi2(αi)|)δ(τ)1+exp(j2παiτ), where Cγjj2(τ)=Ca^αi(τ). Thus, the permutation pij(fk) can be computed from(14)pij(fk)=exp(∑τ=−∞∞Cpij(τ)e−j2πfkτ). For a given αi, (14) allows us to estimate the ith column of the
matrix P(fk). As we do not have any of these quantities ϱi−2|σi2(αi)|, it means that there is
no link between the actual and the estimated sources and therefore no
information is available about the actual order of the sources. A way to
estimate (13) is(15)Cpij(τ)≃Cγjj2(τ)−log(γ0i2(f1))δ(τ)1+exp(j2παiτ), where γ0i2(f1) is the
biggest value of the matrix Sa^αi(f1). Furthermore, there is no available solution whenever 2ταi=2l+1 or τ=(l+1/2)(1/αi). In fact, (15) assumes the permutation at f1 as the true one
and try to extend it to all frequency bins. Moreover, (8) will be simplified to(16)H^(fk)=H(fk)ΔPD(fk).


## 3.2. Resolving the Phase Ambiguity

The phase retrieval of single-input single-output (SISO) systems, excited by
cyclostationary inputs, has been studied in several papers [11–13]. Note
that, by exploiting second-order cyclic statistics, the identification of
nonminimum-phase channels becomes possible. Let us recall the phase retrieval
in the case of a SISO system:(17)ψ(fk)−θ(αi)=ϕ(fk)−ϕ(fk−αi), where ψ(fk)=arg{Sxαi(fk)}, θ(αi)=arg{Saαi(fk)}, ϕ(fk)=arg{H(fk)}, Sxαi(fk) and Saαi(fk) are the SCD of
the single measurement x(t) and the single
source a(t), with the cyclic frequency αi, respectively, and H(fk) is the transfer
function of the filter. We note that q has been set to 1 with θ(αi) being constant
since the source is i.i.d.. Li and Ding's solution for the phase retrieval in
the case of PAM signals is as follows:(18)ϕ(t)=ψ(t)−θαi(t)(1−exp(j2παit)),t≠pαi, where ϕ(t), ψ(t), and θαi(t)=θ(αi)δ(t) are the IDFT of ϕ(f), ψ(f), and θ(αi), respectively. One can interpolate the missing
information whenever t=p/αi, assuming the continuity of ϕ(t). Actually, θ(αi) can be computed
by summing up (17) over all discrete frequencies which
yield a
zero right-hand side; thus we get(19)θ(αi)=1N∑k=1Nψ(fk). Unfortunately,
the previous solution (18) is not appropriate for MIMO systems. The idea is to
transform the MIMO system in m×n SISO systems,
thanks to the cyclostationarity and hence, (18) will be appropriate. Let us
involve Sxαi(fk) for a given αi. Putting (3) into (5), the latter expression takes
the following forms:(20)Sxαi(fk)=σi2(αi)H(fk)IiHH(fk−αi) and, moreover,(21)Sxiiαi(fk)=σi2(αi)Hii(fk)Hii*(fk−αi). The relationship (21) is important because it transforms the m×n MIMO system to n, m×1, single-input multiple-output (SIMO) systems, according to each αi in order to
have the contribution of only one source. Then, we successively apply (18) over
only the diagonal elements of (21) since it comprises m SISO systems.
This is the key relationship that can lead to the estimation of the ith column of Φ(fk). Finally, by sweeping the other cyclic frequencies,
the remaining columns can be estimated in the same way.Actually, we have no knowledge on the true order ofΦ(fk) columns. The
estimated columns of Φ(fk) are affected by
a permutation matrix P′ which is
different from P:(22)Φ^(fk)=Φ(fk)P′.


## 3.3. Connection between the Magnitudes and the Phases

As the processes
of estimating the MIMO system's magnitude and phase are disconnected, the
correspondence between channel magnitude and phase is missed. This can be seen
by taking the absolute value of the relationship (16):(23)|H^(fk)|=|H(fk)Δ|P. It is clear
therefore that the columns of the MIMO system's magnitudes are permuted by the
matrix P, which was taken as reference of permutations over
all frequencies, and the MIMO system's phases are permuted by P′. However, to correctly identify H(fk), we need to cancel the effect of P′ as follows:(24)H^(fk)=|H(fk)Δ|P⊙exp(jΦ^(fk)P′TP), where the
symbol ⊙ denotes the
Hadamard product. In other words, the knowledge of P′ is
indispensable in order to make the true connection between the magnitude and
the phase. Let us take the argument of (16):(25)ΦH^(fk)=Φ(fk)P+ΦD(fk), where ΦH^(fk) is the phase of H^(fk) and ΦD(fk) is an m×n matrix whose
column i, i=1,…,n, contains the
phase of the ith diagonal
element of D(fk). The jth line of ΦD(fk) is as follows:(26)ΦDj(fk)=[ΦD1(fk),…,ΦDi(fk),…,ΦDn(fk)]. Of course, (25)
shows the existence of correlation between ΦH^(fk) and Φ^(fk). However, the matrix ΦD(fk) masks this
correlation and hence, it must be removed to well estimate P′. Let us subtract the jth line of ΦH^(fk) from its ith line:(27)ΦH^i(fk)−ΦH^j(fk)=[∑l=1n(Φil(fk)−Φjl(fk))Pl1,…,∑l=1n(Φil(fk)−Φjl(fk))Pln]. The way to
cancel P′ is to evaluate
the correlation between the elements of (27) and the following equation:(28)Φ^i(fk)−Φ^j(fk)=[∑l=1n(Φil(fk)−Φjl(fk))Pl1′,…,∑l=1n(Φil(fk)−Φjl(fk))Pln′], where Φ^i(fk) and Φ^j(fk) are,
respectively, the ith and the jth lines of Φ^(fk). The correlation will be made element by element and
over all frequency bins. Actually, P′ is canceled and
replaced by P, therefore, (22) becomes(29)Φ^(fk)=Φ(fk)P. The ambiguity ΦD(fk) can easily be
deducted from (25):(30)ΦD(fk)=ΦH^(fk)−Φ(fk)P. Finally, the
MIMO system is identified as in (9).


## 4. Comparison with Existing MIMO Methods

We provide here
a comparison between the proposed approach against the method of [14]. For each
value of the signal-to-noise ratio (SNR), 20 Monte Carlo
runs were implemented. For the sources, we use two independent white gaussian
processes multiplied by cosine signals. The sources are filtered by a 3×2 finite-impulse
response MIMO system. N is set to 128. For the algorithm of [14], the number of samples in
each epoch (a short time interval) is 1000 and the number
of cross-spectral density matrices is 100.LetC(fk) represent the
composite mixing-unmixing system:(31)C(fk)=H^#(fk)H(fk), where its (ij)th element is cij(fk). As a performance index to measure the separation
performance and to make comparison as well, we here used the same index as the
one used in [14], namely, the signal-to-interference ratio (SIR) given by the
formula(32)SIR(i)=∑l=1Mcmaxj(Sijl)∑l=1Mc[∑j=1nSijl]−maxj(Sijl), where Sijl=∑k=1M|cijl(fk)|2, l is an index to
the lth Monte Carlo
run, and Mc is the total
number of Monte Carlo runs. SIR(1) and SIR(2) stand for the
SIR of the 1st and 2nd
estimated sources.Figure1 shows the variation of each output's SIR,
for the proposed method and the one of [14] as well, with the SNR. As can be
seen from this figure, by increasing the SNR, the output SIR improves
(increases). However, it shows that the proposed approach performs better than that
of [14] and provides better SIRs even for low SNRs as well. Actually, Karman's
method constraints the source signals to be second-order quasistationary over
an epoch. This constraint is not anymore appropriate to cyclostationary
signals. Based on the previous comparison results, it is clear that the
exploitation of cyclostationarity makes our algorithm more robust.Figure 1
Comparison of the proposed method with the method of [14].


## 5. Conclusion

The spectral redundancy allowed us to apply the AJD
algorithm to a set of SCD matrices for every frequency. A robust algorithm to
overcome the frequency-dependent permutations and to remove the phase ambiguity,
based on cyclostationarity, was presented. The performance of the new algorithm
was demonstrated, it is apparent, therefore, that the proposed method performs
better than [14].



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*Source: 539139-2008-02-25.xml*