Blind Extraction of Cyclostationary Signal fromConvolutional Mixtures
Yong Xiang, Indivarie Ubhayaratne, Zuyuan Yang
School of Information TechnologyDeakin UniversityBurwood CampusBurwood, VIC 3125, AustraliaEmail:
{
yxiang, kubhayar, zuyuan.yang
}
@deakin.edu.au
Bernard Rolfe
School of EngineeringDeakin UniversityWaurn Ponds CampusGeelong, VIC 3220, AustraliaEmail: bernard.rolfe@deakin.edu.au
Abstract
—Extracting a signal of interest from available measurements is a challenging problem. One property which canbe utilized to extract the signal is cyclostationarity, which existsin many signals. Various blind source separation methods basedon cyclostationarity have been reported in the literature but theyassume that the mixing system is instantaneous. In this paper, wepropose a method for blind extraction of cyclostationary signalfrom convolutional mixtures. Given that the signal of interest hasa unique cyclostationary frequency and the sensors are placedclose to the concerned signal, we show that the signal of interestcan be estimated from the measured data. Simulations resultsshow the effectiveness of our method.
I. I
NTRODUCTION
Blind source separation (BSS) is a fundamental problemencountered in many applications [1][4]. It considers practicalmultipleinput multipleoutput (MIMO) scenarios and aims toestimate the unknown source signals from the measured databy sensors. To achieve BSS, the sources should have some sortof differences, e.g., they are mutually independent/uncorrelated[1], cyclostationary [5], or have constant modulus [3], [6].In some manmade systems such as wireless communicationsystems, the source signals can be preprocessed such that theypossess certain diversity [4], [7]. Among these diversity properties, cyclostationarity exists in many practical applicationssuch as communication, telemetry, radar, sonar and mechanicalsystems. For example, cyclostationarity is shared by mostcommunication signals as a result of periodic switching, gating, or mixing operations at the transmitter [8]. Many signalsgenerated by mechanical systems are also cyclostationary dueto the periodical rotation or movement of some parts [2].While the higherorder cyclostationary statistics of signals can be employed for blind source separation [9], it isknown that the higherorder statistics based methods normallyrequire a larger number of data samples for accurate timeaverage approximations of higherorder statistics to achievegood statistical performance. This would signiﬁcantly increasecomputation cost. In contrast, the methods based on secondorder statistics are often more efﬁcient in computation. So, thispaper restricts its attention to methods based on secondordercyclostationary statistics. In [5], AbedMeraim
et al.
showsthat blind source separation can be achieved using the outputcyclic correlation matrices if there do not exist two distinctsource signals whose cycle frequencies are the same and whosecyclic autocorrelation vectors are linearly dependent. A set of algorithms are proposed to perform BSS. The method in [10]exploits the phase and frequency redundancy of cyclostationarysignals in a complementary way. However, both the methodsin [5] and [10] assume that the mixing system is instantaneous.In this paper, we relax the assumption on the mixingsystem to allow it to be convolutional, which can result fromapplications where multipath propagation exists. Speciﬁcally,we assume that the MIMO mixing system is of ﬁnite impulseresponse (FIR). Our aim is to extract a cyclostationary signalof interest from the outputs of the MIMO FIR mixing system.Extracting a cyclostationary signal from convolutional mixtures is very challenging if not impossible. We show that thiscan be achieved if the signal of interest has a distinct cyclicfrequency and the sensors are placed close to the concernedsignal. A secondorder statistics based algorithm is proposedto perform signal extraction. The effectiveness of the proposedmethod is veriﬁed by simulations.The remainder of the paper is as follows. Section IIintroduces the problem of blind signal extraction together withrelevant assumptions. The new signal extraction method isproposed in Section III. Simulation results are provided inSection IV to illustrate the performance of our method. SectionV concludes the paper.II.
PROBLEM
F
ORMULATION
The mixing model we consider is as follows:
y
(
n
) =
H
(
z
)
x
(
n
) +
w
(
n
)
(1)where
x
(
n
) = [
x
1
(
n
)
,x
2
(
n
)
,...,x
I
(
n
)]
T
is the source signal vector,
y
(
n
) = [
y
1
(
n
)
,y
2
(
n
)
,...,y
J
(
n
)]
T
is the systemoutput vector,
w
(
n
) = [
w
1
(
n
)
,w
2
(
n
)
,...,w
J
(
n
)]
T
is theadditive noise vector,
H
(
z
)
is the
J
×
I
FIR mixing matrix,and the superscript
T
denotes transpose. Assume thatA1) The source signals
x
1
(
n
)
,x
2
(
n
)
,...,x
I
(
n
)
are mutuallyindependent with zero mean and temporally white, and
x
1
(
n
)
is cyclostationary with a unique nonzero cyclic frequency
β
1
.A2) The entries in the ﬁrst column of
H
(
z
)
are constants andthe rest entries are polynomials, and
J > I
. Besides,
H
(
z
)
isirreducible and columnreduced.A3) The noise signals
w
1
(
n
)
,w
2
(
n
)
,...,w
I
(
n
)
are mutually
independent with zero mean and equal variance
σ
2
w
. They arealso independent of the source signals.From the assumptions A1) and A3), it follows:
x
1
(
n
)
x
∗
1
(
n
)
e
β
1
n
>
0
(2)
x
1
(
n
)
x
∗
1
(
n
−
τ
)
e
β
1
n
= 0
,
if
τ
= 0
(3)
x
i
(
n
)
x
∗
i
(
n
)
e
β
1
n
= 0
,
if
i
= 1
(4)
x
i
(
n
)
x
∗
j
(
n
)
e
β
1
n
= 0
,
if
i
=
j
(5)
x
i
(
n
)
w
∗
j
(
n
)
e
β
1
n
= 0
,
∀
i,j
(6)
w
i
(
n
)
w
∗
j
(
n
)
e
β
1
n
= 0
,
∀
i,j
(7)where
=
√ −
1
, the superscript
∗
is the complex conjugateoperator, and
<
·
>
denotes the time averaging operatordeﬁned as follows:
x
i
(
n
)
x
∗
i
(
n
−
τ
)
e
β
1
n
= lim
N
→∞
1
N
N
−
1
n
=0
x
i
(
n
)
x
∗
i
(
n
−
τ
)
e
β
1
n
(8)where
N
is the number of samples. Here, the objective isto extract the signal of interest,
x
1
(
n
)
, from the mixtures
y
1
(
n
)
,y
2
(
n
)
,...,y
J
(
n
)
.Assumption A1) and A3) are common in the context of BSS. Assumption A2) holds if the sensors are placed closeto the source generating
x
1
(
n
)
, such that we can neglectthe multipath contributions from
x
1
(
n
)
to all sensor measurements. In some applications such as certain mechanicalsystems, placing sensors close to a source is achievable.III. P
ROPOSED METHOD
Let
H
(
z
) =
L
l
=0
H
(
l
)
z
−
l
(9)where
L
is the order of the MIMO FIR channel matrix
H
(
z
)
,and denote the order of the
j
th column of
H
(
z
)
by
L
j
. Clearly,
L
1
= 0
and
L
= max(
L
1
,L
2
,...,L
I
)
. Based on the
i
thsystem output
y
i
(
n
)
, we deﬁne
˜
y
i
(
n
) = [
y
i
(
n
)
,y
i
(
n
−
1)
,...,y
i
(
n
−
W
+ 1)]
T
(10)where the slidewindow width
W
is chosen to satisfy
W >
¯
L
=
I
i
=1
L
i
.
(11)Also, denote the
(
i,j
)
th entry of
H
(
l
)
by
H
i,j
(
l
)
. Then, from(1), (9) and (10), it follows
˜
y
i
(
n
) =
I
j
=1
H
i,j
˜
x
j
(
n
) + ˜
w
i
(
n
)
(12)where
˜
x
j
(
n
) = [
x
j
(
n
)
,x
j
(
n
−
1)
,...,x
j
(
n
−
W
−
L
j
+ 1)]
T
(13)
˜
w
i
(
n
) = [
w
i
(
n
)
,w
i
(
n
−
1)
,...,w
i
(
n
−
W
+ 1)]
T
(14)and
H
i,j
is shown at the top of the next page. Since
L
1
= 0
,it is obvious that
H
i,
1
is a
W
×
W
matrix with the followingform:
H
i,
1
=
H
i,
1
(0) 0 0
···
0
···
00
H
i,
1
(0) 0
···
0
···
0
... ...
0
···
0
H
i,
1
(0) 0
···
0
(16)where
i
= 1
,
2
,
···
,J
.Furthermore, deﬁne the cyclic autocorrelation function
ρ
β
1
ii
(
τ
)
as
ρ
β
1
ii
(
τ
) =
x
i
(
n
)
x
∗
i
(
n
−
τ
)
e
β
1
n
(17)and the cyclic autocorrelation matrix
R
β
1
˜
x
j
˜
x
j
(
τ
)
as
R
β
1
˜
x
j
˜
x
j
(
τ
) =
˜
x
j
(
n
)˜
x
H j
(
n
−
τ
)
e
β
1
n
(18)where the superscript
H
stands for complex conjugate transpose. From (2)(7), (13), (14), (17) and (18), we obtain
R
β
1
˜
x
1
˜
x
1
(0) =
ρ
β
1
11
(0) 0 0
···
0 00
ρ
β
1
11
(0) 0
···
0 0
............
0 0 0
···
0
ρ
β
1
11
(0)
=
ρ
β
1
11
(0)
·
I
W
+
L
1
(19)
R
β
1
˜
x
i
˜
x
i
(
τ
) =
0
,
if
i
= 1
(20)
R
β
1
˜
x
i
˜
x
j
(
τ
) =
0
,
if
i
=
j
(21)
R
β
1
˜
x
i
˜
w
j
(
τ
) =
0
,
∀
i,j
(22)
R
β
1
˜
w
i
˜
w
j
(
τ
) =
0
,
∀
i,j
(23)where
I
i
stands for the
i
×
i
identity matrix. Also, deﬁne
ρ
ii
(0) =
x
i
(
n
)
x
∗
i
(
n
)
.
(24)It follows
R
˜
x
i
˜
x
i
(0) =
ρ
ii
(0) 0 0
···
0 00
ρ
ii
(0) 0
···
0 0
..................
0 0 0
···
0
ρ
ii
(0)
=
ρ
ii
(0)
·
I
W
+
L
i
.
(25)Denoting
˜
y
(
n
) = [˜
y
1
(
n
)
,
˜
y
2
(
n
)
,...,
˜
y
J
(
n
)]
(26)we have
˜
y
(
n
) =
H
˜
x
(
n
) + ˜
w
(
n
)
(27)where
˜
x
(
n
) = [˜
x
1
(
n
)
,
˜
x
2
(
n
)
,...,
˜
x
I
(
n
)]
(28)
˜
w
(
n
) = [˜
w
1
(
n
)
,
˜
w
2
(
n
)
,...,
˜
w
J
(
n
)]
(29)
H
i,j
=
H
i,j
(0)
··· ···
H
i,j
(
L
j
) 0
···
00
H
i,j
(0)
··· ···
H
i,j
(
L
j
)
···
0
... ...
0
···
0
H
i,j
(0)
··· ···
H
i,j
(
L
j
)
W
×
(
W
+
L
j
)
(15)
H
=
H
1
,
1
H
1
,
2
···
H
1
,I
H
2
,
1
H
2
,
2
···
H
2
,I
............
H
J,
1
H
J,
2
···
H
J,I
JW
×
(
IW
+¯
L
)
.
(30)Since
J > I
, the matrix
H
is a tall matrix, i.e., it has morerows than columns. Also, under the assumption A1) and theinequality in (11), it is shown in [11] that the matrix
H
hasthe full column rank
IW
+ ¯
L
. Let
u
1
be a
JW
×
1
vector anddeﬁne the compound vector
c
1
=
u
H
1
H
H
.
(31)Clearly, since
L
1
= 0
, the ﬁrst
W
elements of
c
1
are related tothe ﬁrst source signal
x
1
(
n
)
and its delayed versions
x
1
(
n
−
1)
,x
1
(
n
−
2)
,
···
,x
1
(
n
−
W
+1)
, respectively. So, to estimate
x
1
(
n
−
τ
)
,
0
≤
τ
≤
W
−
1
from
˜
y
(
n
)
, we need to ﬁnd such
u
1
that one of the ﬁrst
W
elements of
c
1
is nonzero and therest elements of
c
1
are zero.From (19)(30), we have
R
β
1
˜
y
˜
y
(0) =
H
R
β
1
˜
x
˜
x
(0)
H
H
+
H
R
β
1
˜
x
˜
w
(0)+(
H
R
β
1
˜
x
˜
w
(0))
H
+
R
β
1
˜
w
˜
w
(0)=
H
R
β
1
˜
x
˜
x
(0)
H
H
=
H
ρ
β
1
11
(0)
I
W
+
L
1
0
···
00 0
···
0
............
0 0
···
0
H
H
.
(32)From (16) and (30), we can see that the ﬁrst
W
columns of the
JW
×
(
IW
+ ¯
L
)
matrix
H
are orthogonal. Thus, from(32), the singular vectors corresponding to the
W
−
1
largestsingular values of
R
β
1
˜
y
˜
y
(0)
are orthogonal to one of the ﬁrst
W
columns of
H
. Denoting these singular vectors as
v
1
,j
,
j
= 1
,
2
,...,W
−
1
, we deﬁne
V
1
= [
v
1
,
1
,
v
1
,
2
,...,
v
1
,W
−
1
]
.
(33)Furthermore, we can obtain the autocorrelation matrix of
˜
y
(
n
)
by
R
˜
y
˜
y
(0) =
H
R
˜
x
˜
x
(0)
H
H
+
H
R
˜
x
˜
w
(0)+(
H
R
˜
x
˜
w
(0))
H
+
R
˜
w
˜
w
(0)=
H
R
˜
x
˜
x
(0)
H
H
+
R
˜
w
˜
w
(0)=
H
R
˜
x
˜
x
(0)
H
H
+
σ
2
w
I
JW
.
(34)Since
H
is a tall matrix, (34) implies that
σ
2
w
is the smallesteigenvalue of
R
˜
y
˜
y
(0)
and thus can be estimated from
R
˜
y
˜
y
(0)
.Consequently, we can remove
σ
2
w
I
JW
from
R
˜
y
˜
y
(0)
to obtain
¯
R
˜
y
˜
y
(0)=
R
˜
y
˜
y
(0)
−
σ
2
w
I
JW
=
H
R
˜
x
˜
x
(0)
H
H
=
H
R
˜
x
1
˜
x
1
(0)
0
···
00 R
˜
x
2
˜
x
2
(0)
···
0
............
0 0
···
R
˜
x
J
˜
x
J
(0)
H
H
(35)where
R
˜
x
i
˜
x
i
(0) =
ρ
ii
(0)
·
I
W
+
L
i
,
i
= 1
,
2
,...,J
. Basedon the mechanism from which the cyclostationary signal isgenerated, there exists certain relationship between
ρ
β
1
11
(0)
and
ρ
11
(0)
, i.e.,
ρ
11
(0) =
ξ
·
ρ
β
1
11
(0)
, where
ξ
is known. Thus, itfollows from (32) and (35) that
¯
R
˜
y
˜
y
(0)
−
ξ
·
R
β
1
˜
y
˜
y
(0)=
H
0 0
···
00 R
˜
x
2
˜
x
2
(0)
···
0
............
0 0
···
R
˜
x
J
˜
x
J
(0)
H
H
.
(36)We can see from (36) that the rank ok of the matrix
¯
R
˜
y
˜
y
(0)
−
ξ
·
R
β
1
˜
y
˜
y
(0)
is
(
I
−
1)
W
+ ¯
L
−
L
1
. Denoting thesingular vectors corresponding to the
(
I
−
1)
W
+ ¯
L
−
L
1
largest singular values of the matrix
¯
R
˜
y
˜
y
(0)
−
ξ
·
R
β
1
˜
y
˜
y
(0)
as
v
2
,j
,
j
= 1
,
2
,...,
(
I
−
1)
W
+ ¯
L
−
L
1
, we deﬁne
V
2
=
v
2
,
1
,
v
2
,
2
,...,
v
2
,
(
I
−
1)
W
+¯
L
−
L
1
.
(37)Based on (33) and (37), we deﬁne
V
= [
V
1
,
V
2
]
.
Then, the singular vector associated with the largest singularvalue of
V
is proportional to one of the ﬁrst
W
columns of
H
but orthogonal to the other columns of
H
. Thus, this singularvector can be chosen as the desired extraction vector
u
1
.IV. S
IMULATION
R
ESULTS
In this section, we present simulation examples to illustratethe performance of the proposed method. In the simulations,we consider a mixing system with three inputs, which aregenerated as follows:
x
1
(
n
) =
s
1
(
n
)
∗
cos(
α
1
·
n
)
x
2
(
n
) =
s
2
(
n
)
x
3
(
n
) =
s
3
(
n
)
(38)
0 1000 3000 5000 7000 9000 11000 13000 15,000−33−32.5−32−31.5−31−30.5−30−29.5−29Samples
M I R L ( d B )
Figure 1. MIRL versus sample size, where
I
= 3
,
J
= 4
,
L
= 3
andSNR
= 20
dB.
where
s
1
(
n
)
,
s
2
(
n
)
and
s
3
(
n
)
are randomly generated temporarily white sequences. The signal of interest,
x
1
(
n
)
, iscyclostationary with cyclic frequency
β
1
= 2
α
1
and
ρ
11
(0) =2
ρ
β
1
11
(0)
. The MIMO FIR mixing matrix
H
(
z
)
is randomlygenerated in each simulation run. Randomly generated whiteGaussian noise is added to the signal mixtures and the signaltonoise ratio (SNR) is deﬁned as
SNR =
−
10log
10
σ
2
w
.Ideally, for the compound vector
c
1
deﬁned in (31), onlyone of its ﬁrst
W
elements is nonzero and the rest elementsshould be zero. However, this is not practically achievable dueto ﬁnite sample size and computational inaccuracy. Therefore,the performance of our method is measured by means of theMean Interference Rejection Level (MIRL) of the extractionvector. Let
c
H
1
= [
c
H
1
,
1
,c
H
2
,
1
,...,c
H k,
1
,...,c
H IW
+¯
L,
1
]
and assume
c
k,
1
is the maximum element out of the ﬁrst
W
elementsof
c
1
. The MIRL index is deﬁned as follows:
MIRL(dB) = 20log
10
1
IW
+ ¯
L
−
1
· 
c
k,
1

IW
+¯
L
i
=1
,i
=
k

c
i,
1

.
We compute MIRL by averaging 200 independent runs.Clearly, the smaller MIRL, the better signal extraction performance.Given a mixing system of 3 inputs, 4 outputs and order3, Fig. 1 shows MIRL versus sample size where SNR
= 20
dBand Fig. 2 shows MIRL versus SNR where the sample size is
5000
. It can be seen that satisfactory performance is achievedfor all tested sample sizes and SNRs, including small samplesizes and low SNR levels. Also, with the increase of samplesize or SNR, MIRL decreases accordingly.Fig. 3 shows MIRL against different number of sensors,where
I
= 3
,
L
= 3
,
N
= 5000
and SNR
= 20
dB. As we cansee, when 3 sensors are used, the performance is not satisfactory. This is expected as the assumption A2) is not satisﬁedin this case. In contract, when the number of sensors used is4 or above, the MIRL is lower than
−
30
dB, indicating good
0 5 10 15 20 25 30−32−30−28−26−24−22−20SNR (dB)
M I R L ( d B )
Figure 2. MIRL versus SNR, where
I
= 3
,
J
= 4
,
L
= 3
and
N
= 5000
.
3 4 5 6 7 8−34−32−30−28−26−24−22−20−18−16−14Number of Sensors
M I R L ( d B )
Figure 3. MIRL versus the number of sensors, where
I
= 3
,
L
= 3
,
N
= 5000
and SNR
= 20
dB.
extraction performance. Finally, the MIRL is evaluated underdifferent system orders, where the other simulation parametersare
I
= 3
,
J
= 4
,
N
= 5000
and SNR
= 20
dB. It is shown inFig. 4 that the MIRL increases (i.e., performance decreases)with the rise of system order. For the considered system orders,the proposed method demonstrate good performance.V. C
ONCLUSION
This paper deals with the problem of extracting a cyclostationary signal from the outputs of a MIMO FIR system. Underthe assumption that the sensors are placed close to the cyclostationary signal of interest, a secondorder statistics basedmethod is proposed to extract the cyclostationary signal byusing its unique cyclic frequency. Several simulation examplesare provided to illustrate the validity of our method.
1 2 3 4 5−35.5−35−34.5−34−33.5−33−32.5−32−31.5−31Order of mixing system
M I R L ( d B )
Figure 4. MIRL versus channel order, where
I
= 3
,
J
= 4
,
N
= 5000
andSNR
= 20
dB.
R
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