Vol.29 No.3/4 JOURNAL OF ELECTRONICS (CHINA) July 2012

GEOMETRIC METHOD FOR BLIND SIGNAL SEPARATION OF MULTICHANNEL MPSK SIGNALS1

Fang Yong Ji Shuai Wang Min (School of Communication and Information Engineering, Key Laboratory of Specialty Fiber Optics and Optical

Access Networks, Shanghai 200072, China)

Abstract In this letter, the problem of blind source separation of Multiple-Phase-Shift-Keying (MPSK) digital signal is considered. The geometry of received MPSK signals constellation is exploited. The column vectors of received signals can be regarded as the points of hyper-cube. All the possible distinct vectors of received signals are found by clustering, and mixing matrix and sources are estimated by searching out the pairing vectors and eliminating redundant information in all possible distinct vectors. Simulation results give the polar diagram of estimated original signals. They show that the proposed algorithm is effective when the original signals is Quadrature-Phase-Shift-Keying (QPSK) or 8-Phase-Shift-Keying (8PSK).

Key words Multiple-Phase-Shift-Keying (MPSK); Blind source separation; Higher order modulation

CLC index TN911.7

DOI 10.1007/s11767-012-0877-9

I. Introduction Blind source separation is the separation of a

set of signals from a set of mixed signals, without the aid of information (or with very little infor-mation) about the source signals or the mixing process. It is of considerable interest in many kinds of fields such as speech recognition, telecommuni-cation, and medical signal processing. With the increasing use of computers, the needs for digital signal processing has increased, therefore blind signal separation of digital signals such as Multi-ple-Quadrature-Amplitude-Modulation (MQAM), Multiple-Phase-Shift-Keying (MPSK) and Multi-ple-Amplitude-Shift-Keying (MASK) has become more and more popular at present[1] . Nevertheless, the proposed most methods[2–5] mainly deal with the blind separation problems for multichannel Bi-nary-Phase-Shift-Keying (BPSK) signals, which is

1 Manuscript received date: April 27, 2012; revised date:

May 30, 2012. Supported by the National Natural Science Foundation of China (No. 60872114, 60972056, 61132004), Shanghai Leading Academic Discipline Project and STCSM (S30108 and 08DZ2231100). Communication author: FangYong, born in 1964, male, Professor. Box 160, School of Communication and In-formation Engineering, Key Laboratory of Specialty Fiber Optics and Optical Access Networks, Shanghai University, No. 149 Yanchang Road, Shanghai 200072, China. Email: yfang@staff.shu.edu.cn.

also called binary sources in some papers. When transmitted data are high-order MPSK signals, those methods which are proposed for solving BPSK signals is invalid. Several methods[6–8] for separating mixed MPSK signals had been devel-oped as well. However, these methods have high complexity when MPSK signal’s order M is larger than four or there are more than two received signals.

In this letter, a novel method of blind source separation for multichannel MPSK digital signals is proposed. Firstly, all the possible distinct vectors of received signals are found by clustering. Secondly, the received data is whitened by whitening matrix. And finally, through searching out those paring vectors which have the closed distance and elimi-nating redundant information, the sources and mixing matrix can be estimated. Simulation results demonstrate that the proposed algorithm is effec-tive.

II. Data Modeling Let m MPSK signals impinge at an array of

d sensors with arbitrary characteristics. The wire-less communication systems model is described as follow[2]

( ) ( )1 1

( ) ( ) ( ),m

l lk lk k k kk j

x t p a b j s t jT w tθ τ∞

= =

= − − +∑ ∑

FANG et al. Geometric Method for Blind Signal Separation of Multichannel MPSK Signals 335

1,2, ,l d= (1)

where T is period, kp is the amplitude of the k-th signal, ( )lk ka θ is the array response of the l-th sensor to the k-th user’s signals from direction ,kθ

()kb ⋅ represents the k-th user’s bit stream, ()s ⋅ is the k-th signal waveform, kτ represents the time delay of the k-th signal to the array, and ( )lw t is additive white noise with covariance 2 .σ I

Assuming that all kτ is equal, and the period T is eliminated by matched filter, the following equivalent can be obtained:

( )1

( ) ( ) ( )m

l k lk k k lk

x n p a s n w nθ=

= +∑ (2)

It can be rewritten in matrix form:

( ) ( ) ( ), 1 , 2 , ,n n n n N= + =x As w (3)

where T1 2( ) [ ( ) ( ) ( )]d d Nn x n x n x n ×=x represents

the received signals from antenna array, ( )n =w T

1 2[ ( ) ( ) ( )]d d Nw n w n w n × is the additive Gaussian white noise, T

1 2( ) [ ( ) ( ) ( )]m m Nn s n s n s n ×=s is the original signal, and A is a d m× unknown mixture matrix.

MPSK signals use a finite number of phases to represent digital data. So we can obtain that the elements of sources belong to finite alphabet

{ }MPSKˆ ; (2 1) / ] ,

1,2, , /2

jS e n M

n M

ϕ ϕ π= ± = −

= (4)

III. Geometrical Property of MPSK Signals

First note that ( )ns contains exactly all L = mM distinct columns. 1 2[ ]L m L×=S s s s repre-

sents the matrix of all L distinct vectors. With the channel white noise ( ),nw the received data is a union of clusters centered around the points

,i i=x As 1,2, , .i L= Since any extra columns in ( )nx would yield redundant information, we should get rid of these redundant columns by clustering firstly. The hybrid clustering algorithm[2]

can be used to obtain the cluster vectors. Assuming that these cluster vectors compose the ma-trix 1 2[ ] .L d L×=X x x x Apparently, the matrix

= +X AS W (5)

whereW is the white noise. Without the white noise or the noise is small, the formula can be written as

=X AS (6)

1. Problem simplification using symmetric prop-erty

Though the received signals have been simpli-fied through clustering, its cluster vectors still have part of redundant information.

Here, we simplify X by exploiting the sym-metric property. Consider those columns in .S They are composed by those elements in the set MPSK,S so both is and i−s exist whenever one of them is generated in the columns of .S Hence, if the matrix S contains all L possible distinct vec-tors, S can be partitioned as follows

⎡ ⎤= −⎢ ⎥⎣ ⎦S S S (7)

where S represents the first half part of columns in matrix S . So we transform Eq. (6) into Eq. (8)

=X AS (8)

2. Whitening

We know that S there is a verified property T

,K=SS I where /2.mK M= Thus we have[9] T

T T

K= =

XXAA U UΣ (9)

whereU is a unitary matrix, 2 21=diag{ , , },dσ σΣ

iσ denotes -thi singular value of .A We define whitening matrix as 1/2 T.−=W UΣ Then the whitened data set is formed as

= = =Z WX WAS QS (10)

whereQ is a d m× real unitary channel matrix to be determined, [ (1) (2) ( )] ,d KK ×= =Z z z z S [ (1) (2) ( )] .m KK ×s s s

3. Geometry of S

Here, we consider the geometric description of .S We can take the K columns of S as the verti-

ces of a d-cube. Instead of viewing the S as vertices, we can also view them as the intersection of 2d hyper planes. A hyper plane[10] can be described using the formula:

336 JOURNAL OF ELECTRONICS (CHINA), Vol.29 No.3/4, July 2012

H ,( , ) :

0,

d

dC

H CC

ββ

β

⎛ ⎞= ∈ ⎟⎜ ⎟⎜≡ ∈ ⎟⎜ ⎟⎜ ≠ ∈ ⎟⎟⎜⎝ ⎠

a sa x

α

α (11)

where the vector a is the normal vector. Letting

ie denotes the i-th unit vector, we have that

{ }MPSK

T

1:

d

ii Sγ

γ= ∈

= =S s e s∩ ∪ (12)

where

{ }MPSK ; (2 1) / , 1,2, , /2jS e n M n Mϕ ϕ π= = − =

(13)

here, ϕ only have /2M angles for we have elimi-nated the symmetric part of .S

From Eqs. (11) and (12), we can get

( )MPSK1

,h

d

ii S

Hγ

γ= ∈

=S e∩ ∪ (14)

In other words, Eq. (14) tells us that the points in the signal space S are just the vertices of the hypercube enclosed by the hyper planes. S can be expressed in an iterative form. Let

1 2{ , , , }Lr r rγ = be the element of S and define

1 1 2 Kγ γ γ⎡ ⎤= ⎢ ⎥⎣ ⎦S (15)

1 1 1

T T T1 2

m m m

m

Kγ γ γ

− − −⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

S S SS

1 1 1 (16)

where the all-ones vector 1 is of appropriate di-mension, {1,2, , }.m m∈ Obviously, when m =

,m .m=S S

4. Properties of sources

Let us defineij

∇S be the distance vector be-tween two columns

( ) ( ), 1,2, ,

1,2, , ,

iji j i K

j K i j

∇ = − =

= ≠

S S S

(17)

and we assume that12 13

[ , , ,ij

∇ = ∇ ∇S S S 1,

].K K−

∇S Let’s define the element in the p-th row

and the q-th column of matrix ∇S is written as

,pq∇S and2

1( ) .mpq pqp=⏐⏐∇ ⏐⏐= ∇∑S S It is obviously

that pq⏐⏐∇ ⏐⏐S is the Euclidean distance between two

columns of .S

Property 1 The minimum Euclidean distance between each columns of S is 1 ,i id γ γ−=⏐ − ⏐

2, 3, , .i K= Proof When the elements of pq∇S are all equal to zero except one element. We can express

,k

pq pq⏐⏐∇ ⏐⏐=⏐∇ ⏐S S where k represents that the -thk element is not equal to zero. Because

kpq⏐∇ ⏐=S

, , 1,2, , ,i j i j Kγ γ⏐ − ⏐ = the minimum distance between iγ and jγ is 1 , 2, , .i i i Kγ γ−⏐ − ⏐ = Therefore, the minimum Euclidean distance be-tween each columns of S is 1 ,i id γ γ−=⏐ − ⏐ 2,i = 3, , .K Q.E.D. Property 2 If ,pq d⏐⏐∇ ⏐⏐=S pq∇S only has one element which is not equal to zero. Proof Consider that there is more than one ele-ment in .pq∇S Only when those elements are all equal to d, pq⏐⏐∇ ⏐⏐S is the minimum value. However even if those elements are all equal to minimum value, pq⏐⏐∇ ⏐⏐S still is larger than d.

It is know that the columns S can be regarded as the vertices of d-cube, so the distance between two vertices is also equal to d. However, the mixing matrix defines an invertible transformation from the d-cube to the d-parallel tope. Therefore, the distance between two vertices has been changed by mixing matrix[10]. Q.E.D.

IV. Geometric Approach for Channel Estimation and Source Recovery

Let dis( ( ), ( )) ( ) ( )i j i j=⏐⏐ − ⏐⏐z z z z denotes the Euclidean distance between the i-th column and j-th column of .Z It is clear that we have the fol-lowing formulation:

( ) ( )

( )

dis ( ), ( ) ( ) ( ) ( ) ( )

( ) ( ) dis ( ), ( )

i j i j i j

i j i j

= − = −

= − =

z z z z Q s s

s s s s (18)

Notice that for any .i j≠ According to Prop-erty 1, the minimum of dis( ( ), ( ))i js s is d. So the minimum of dis( ( ), ( ))i jz z is also equal to d.

If ( ) ( ) , ( ), ( ),i j d i j′ ′ ′ ′⏐⏐ − ⏐⏐=z z z z satisfy Eq. (19)

( ) ( )i j d′ ′− = ±z z Qek (19)

where ke denotes the unit vector with its -thk entry equal to one, and its other entries equal to zero. Thus some column of the unitary matrixQ can be determined up to a sign.

The mixing matrix can be estimated by formula

FANG et al. Geometric Method for Blind Signal Separation of Multichannel MPSK Signals 337

1 .−=A W Q And input signals are estimated as 1

e( ) ( )n n−=s Q Wx (20)

V. Simulation Results QPSK signals and 8PSK signals were used to

demonstrate the effectiveness of the proposed al-gorithm. Assume that there were three QPSK signals and three receivers. The simulations were taken when the power of white noise was 30 dB. The mixing matrix was

0.0788 0.2570 0.7419

0.3171 0.5991 1.7725

0.6494 0.2205 0.6369

⎡ ⎤−⎢ ⎥⎢ ⎥

= − −⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥⎣ ⎦

A

And by Eq. (20), the estimated mixing matrix was

0.7547 0.2557 0.0772

1.7587 0.5872 0.3098

0.6430 0.2579 0.6451

⎡ ⎤− − −⎢ ⎥⎢ ⎥

= − −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Aes

The sources e( )ns were estimated. The polar diagram of sources was as Fig. 1.

Fig. 1 QPSK polar diagram of sources with 30 dB noise (3 re-ceivers, 3 sources)

Fig. 1 shows that the proposed is efficient for the separation of QPSK sources with four phases. We can transform the information of phases to bit signals by setting four threshold values. The simulation results show that the Bit Error Rate (BER) is 0.

Assume that there were three 8PSK signals and three receivers with 30 dB noise. The original

mixing matrix was

0.1296 0.8435 0.1163

1.3927 2.8334 1.1891

1.1004 0.1224 0.9708

⎡ ⎤⎢ ⎥⎢ ⎥

= −⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥⎣ ⎦

A

By Eq. (19), the mixing matrix was estimated as

0.0968 0.8956 0.1184

1.3978 2.8987 1.1896

1.1292 0.1003 0.9388

⎡ ⎤− −⎢ ⎥⎢ ⎥

= − − −⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥⎣ ⎦

Aes

Then, the original signals were estimated. Fig. 2 is the polar diagram of estimated original signals.

Fig. 2 8PSK polar diagram of sources with 30 dB noise (3 re-ceivers, 3 sources)

Fig. 2 shows that the proposed algorithm also can separate the 8PSK sources.

VI. Conclusion This letter proposed an algorithm which ex-

ploits the geometric property of signals to separate the mixing MPSK signals. As long as received signals can be clustered out, original signals can be estimated according this geometric method. Simulation results show us that even if M is 8 and there are three channels, this method is still effec-tive.

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