Open Access

Blind recognition of space-time block code in MISO system

  • Guobing Qian1, 2Email author,
  • Liping Li1,
  • Minggang Luo1,
  • Hongshu Liao1, 2 and
  • Huaguo Zhang1
EURASIP Journal on Wireless Communications and Networking20132013:164

https://doi.org/10.1186/1687-1499-2013-164

Received: 25 March 2013

Accepted: 30 May 2013

Published: 13 June 2013

Abstract

Blind recognition of space-time block codes (STBCs) used in multiple transmitter communication is an important research topic in the non-cooperative scenario, which has attracted more and more attention recently. However, all of the current recognition algorithms can only work well in multiple-input multiple-output system, i.e., the system employs multiple receive antennas. To our knowledge, there is no report in the literature on blind recognition of STBCs in multiple-input single-output (MISO) system, i.e., the system employs only one receive antenna. In this paper, this matter is addressed. An original method of feature extraction for STBCs in the MISO system is proposed using the second-order and higher-order statistics of the reconstructed receiver. After feature extraction, the classification of space-time code can be considered as a pattern recognition problem. A classifier based on a support vector machine is proposed for the recognition of STBCs by mapping these features into a high dimensional space. Simulations show that the proposed classifier can recognize STBCs with high performance and be robust to modulation.

Keywords

Blind recognition; Space-time block codes; Multiple-input single-output; SVM

1 Introduction

Over the past decade, space-time block codes (STBCs) have been firmly established as an effective technique for achieving reliable transmission in multiple transmitter wireless communication systems [13]. The blind recognition of communication parameters of these systems is an important research topic in the non-cooperative scenario, which has attracted more and more attention recently. Most researches are focused on blind channel estimation [48]. These blind channel estimation algorithms are based on the assumption that the space-time code is known at the receiver side. However, this is not practical since the space-time code is not known at the receiver side in the non-cooperative scenario. Therefore, one essential step in the signal interception process is to blindly recognize the STBCs used in multiple transmitter communications.

The blind recognition of STBCs is a challenging topic, which is a key research issue in non-cooperative communication systems with military and commercial applications. Recently, it has been presented in [913]. These methods can be divided into three categories: feature-based methods [911, 13], maximum likelihood (ML) approach [12], and methods derived from ML [12]. These methods can only work well in multiple-input multiple-output (MIMO) system. However, STBCs are generally designed for transmit diversity in the downlink. One of the most appealing researches is to recognize the space-time block codes with a single receive antenna.

To our knowledge, none of the previous algorithms is able to blindly recognize the STBCs with a single receive antenna. In this paper, an original support vector machine (SVM)-based classifier is proposed which is well suited for this case. The received one-dimensional signal is reconstructed as a multidimensional signal, and then the SVM-based classifier is proposed based on second-order and higher-order statistics of the reconstructed receiver. The method proposed in this paper can be easily implemented without a priori knowledge about the number of transmitters, the modulation of the transmitted symbols, and the channel state information (CSI). The only assumption lies on the perfect estimation of timing synchronization (one sample per symbol, optimum sampling time). The candidate STBCs for recognition, which are the same as those in [10], include spatial multiplexing (SM), Alamouti code, orthogonal STBC using three transmit antennas (OSTBC3)with rate 3/4 (first and second codes), and OSTBC3 with rate 1/2. These codes belong to the class of linear space-time block codes, and transmitted matrices of these codes are presented in Section 2.

The remainder of this paper is organized as follows. The system model of communication is introduced in Section 2. Feature extractions based on the second-order and higher-order statistics of the reconstructed receiver of different STBCs are presented in Section 3. It is shown that the second-order or higher-order statistics of the reconstructed receiver presents distinguished features. Based on these features, Section 4 proposes an SVM-based classifier for the blind recognition of five linear STBCs. Finally, simulation results are presented in Section 5, and conclusions are drawn in Section 6.

Notations adopted in this paper are mostly standard. Uppercase and bold letters denote matrices, lower case and bold letters denote vectors, and lower case and non-bold letters denote scalars. Superscript (·) T , (·), (·) H , and (·) stand for transpose, complex conjugate, Hermitian transpose, and pseudo-inverse, respectively. The operator Re(·) denotes the real parts. The operator E[·] denotes the expectation, and · F represents the Frobenius norm.

2 System models

2.1 Signal models of linear space-time block codes

In a linear STBC communications system of multiple transmitters, a signal vector s v =[s1,,s n ] T (v=1,,N b ) composed of n symbols is encoded into a n t ×l transmitted matrix C(s v ), where N b corresponds to the number of transmitted blocks, n t and l denote the number of transmitting antennas and the length of a block, respectively. Furthermore, the transmitted symbols s are assumed to be zero mean, non-Gaussian, independent, and identically distributed (i.i.d.), and they belong to the same linear modulation scheme (pluggable authentication module (PAM), quadrature amplitude modulation (QAM), or phase-shift keying (PSK)). Parameters n, n t , and l depend only on the STBC employed at the transmitter side.

The candidate STBCs for recognition include SM, Alamouti code, OSTBC3 with rate 3/4 (first and second codes), and OSTBC3 with rate 1/2. The transmitted matrices of these STBCs are as follows:
  1. 1.
    Spatial multiplexing
    C ( s v ) = [ s 1 s n t ] T
    (1)
     
  2. 2.
    Alamouti
    C ( s v ) = s 1 s 2 s 2 s 1
    (2)
     
  3. 3.
    First code of OSTBC3 with rate 3/4
    C ( s v ) = s 1 0 s 2 s 3 0 s 1 s 3 s 2 s 2 s 3 s 1 0
    (3)
     
  4. 4.
    Second code of OSTBC3 with rate 3/4
    C ( s v ) = s 1 s 2 s 3 0 s 2 s 1 0 s 3 s 3 0 s 1 s 2
    (4)
     
  5. 5.
    OSTBC3 with rate 1/2
    C ( s v ) = s 1 s 2 s 3 s 4 s 1 s 2 s 3 s 4 s 2 s 1 s 4 s 3 s 2 s 1 s 4 s 3 s 3 s 4 s 1 s 2 s 3 s 4 s 1 s 2
    (5)
     

2.2 System models of the multiple-input single-output communication

In this paper, we assume that transmitted signals are narrow band, and the channel is quasi-static and frequency flat. Moreover, the receiver is perfectly synchronized with the transmitter, i.e., one sample per symbol and an optimum sampling time. In this case, the multiple-input single-output (MISO) communication model of the v th block is expressed as follows [1]:
y v = hC ( s v ) + w v
(6)

where y v = [y(v−1)l+1,y(v−1)l+2,y v l ] denotes the v th block of received samples, h = [ h 1 , h 2 h n t ] represents the channel between the transmit antennas and receive antenna, and w v = [w(v−1)l+1,w(v−1)l+2,…w v l ] refers to the v th block of additive receiver noise, which is assumed to be white and Gaussian.

Our aim is to blindly recognize the space-time block code of a communication with a single receive antenna from N received samples. Hereafter in this paper, N is assumed to be an integer multiple of 4 to allow simplifications of the following mathematical expressions. Moreover, if this assumption does not hold, the excess samples can be discarded to meet this assumption easily.

3 Feature extraction

In this section, we reconstruct the receiver and analyze the second- or higher-order statistics of the reconstructed receiver. Feature extractions of different STBCs are based on the second- and higher-order statistics of the reconstructed receiver.

3.1 Feature extraction of SM’s coded signal

Based on the assumptions of Section 2, it can be easily obtained that received samples of SM’s coded signal can be expressed as follows:
y = y 1 y 2 y N = hC ( s ) + w = h 1 h 2 h n t s 1 s n t + 1 s n t ( N 1 ) + 1 s 2 s n t + 2 s n t ( N 1 ) + 2 s n t s 2 n t s n t N + w 1 w 2 w N .
(7)
  1. 1.
    If the received samples are reshaped as follows:
    Y 1 = y 1 y 5 y N 3 y 2 y 6 y N 2 y 3 y 7 y N 1 y 4 y 8 y N ,
    (8)
     
then the reshaped samples can be expressed as:
Y 1 = h 1 h n t 0 0 0 0 0 0 0 0 h 1 h n t 0 0 0 0 0 0 0 0 h 1 h n t 0 0 0 0 0 0 0 0 h 1 h n t × s 1 s ( n t 4 ) N + 1 s n t s ( n t 3 ) N s n t + 1 s ( n t 3 ) N + 1 s 2 n t s ( n t 2 ) N s 2 n t + 1 s ( n t 2 ) N + 1 s 3 n t s ( n t 1 ) N s 3 n t + 1 s ( n t 1 ) N + 1 s 4 n t s n t N + w 1 w 5 w N 3 w 2 w 6 w N 2 w 3 w 7 w N 1 w 4 w 8 w N .
(9)
Based on the independence of s i , we can easily get that
R 1 = lim N 1 N / 4 Y 1 Y 1 H = ( h 1 2 + h n t 2 ) E ( s 2 ) + σ 2 I 4 .
(10)
In practice, R1 can only be estimated by limited received samples N. Therefore, we define performance index I 1 ( R 1 ) = 1 2 M ( M 1 ) i = 1 M j = 1 M r ij r ii 1 + j = 1 M i = 1 M r ij r jj 1 to measure the feature of R1, where M=4 denotes the order of matrix R1, ri,j denotes the element on the i th row and the j th column of matrix R1. The lower the I1(R1) value, the better of the performance. This performance index is similar to the Amari index defined in [14].
  1. 2.
    If the received samples are reconstructed as follows:
    Y 2 = y 1 y 3 y N 1 y 2 y 4 y N ,
    (11)
     
then the reconstructed samples can be expressed as:
Y 2 = h 1 h n t 0 0 0 0 h 1 h n t H s 1 s ( n t −2)N+1 s n t s ( n t −1)N s n t +1 s ( n t −1)N+1 s 2 n t s n t N S + w 1 w 3 w N 1 w 2 w 4 w N .
(12)
It can be easily obtained that
C 4 [ 1 , 2 ] = c 4 Y 2 ( : , i ) , Y 2 ( : , i ) H , Y 2 ( 1 , i ) , Y 2 ( 2 , i ) H = c 4 Y 2 ( 1 , i ) , Y 2 ( 1 , i ) H , Y 2 ( 1 , i ) , Y 2 ( 2 , i ) H c 4 Y 2 ( 2 , i ) , Y 2 ( 1 , i ) H , Y 2 ( 1 , i ) , Y 2 ( 2 , i ) H c 4 Y 2 ( 1 , i ) , Y 2 ( 2 , i ) H , Y 2 ( 1 , i ) , Y 2 ( 2 , i ) H c 4 Y 2 ( 2 , i ) , Y 2 ( 2 , i ) H , Y 2 ( 1 , i ) , Y 2 ( 2 , i ) H = 0 0 0 0 ,
(13)

where Y2(i,j) denotes the element on the i th row and the j th column of Y2, Y2(:,i) denotes the elements on the i th column of Y2, c4(x1,x2,x3,x4) is defined as c 4 ( x 1 , x 2 , x 3 , x 4 ) = E ( x 1 x 2 x 3 x 4 ) E ( x 1 x 2 ) E ( x 3 x 4 ) E ( x 1 x 3 ) E ( x 2 x 4 ) E ( x 1 x 4 ) E ( x 2 x 3 ) .

For proof of (13), see Appendix 1.

3.2 Feature extraction of an Alamouti’s coded signal

The received samples of Alamouti’s coded signal can be expressed as follows:
y = y 1 y 2 y N b = [ y 1 y 2 y N ] = hC ( s ) + w = [ h 1 h 2 ] s 1 s 2 s 3 s 4 s N 1 s N s 2 s 1 s 4 s 3 s N s N 1 + [ w 1 w 2 w N ] .
(14)
  1. 1.
    If the received samples are reshaped as (8), then the reshaped samples can be expressed as follows:
    Y 1 = h 1 h 2 0 0 0 0 0 0 0 0 h 1 h 2 0 0 0 0 0 0 0 0 h 1 h 2 0 0 0 0 0 0 0 0 h 1 h 2 s 1 s N 3 s 2 s N 2 s 2 s N 2 s 1 s N 3 s 3 s N 1 s 4 s N s 4 s N s 3 s N 1 + w 1 w 5 w N 3 w 2 w 6 w N 2 w 3 w 7 w N 1 w 4 w 8 w N .
    (15)
     
  1. (i)
    If s i belongs to modulation scheme I (≥4 PSK or 4 QAM or 16 QAM or 32 QAM), then E ( s i s i ) = E ( s i s i ) = 0 . It can be easily obtained that
    R 1 = lim N 1 N / 4 Y 1 Y 1 H = ( h 1 2 + h 2 2 ) E ( s 2 ) + σ 2 I 4 .
    (16)
     
  2. (i)
    If s i belongs to modulation scheme II (BPSK or 8QAM or PAM), then E ( s i s i ) 0 , E ( s i s i ) 0 . It can be easily obtained that
    R 1 = lim N 1 N / 4 Y 1 Y 1 H = h 1 2 + h 2 2 E ( s 2 ) + σ 2 h 1 h 2 h 1 h 2 E ( s 2 ) 0 0 h 1 h 2 h 1 h 2 E ( s 2 ) h 1 2 + h 2 2 E ( s 2 ) + σ 2 0 0 0 0 h 1 2 + h 2 2 E ( s 2 ) + σ 2 h 1 h 2 h 1 h 2 E ( s 2 ) 0 0 h 1 h 2 h 1 h 2 E ( s 2 ) h 1 2 + h 2 2 E ( s 2 ) + σ 2 .
    (17)
     
In order to measure the feature of R1, we partition it as R 1 = P 1 P 2 P 3 P 4 , where P1, P2, P3, and P4 are 2×2 dimensional matrix. The feature of R1 can be measured by performance index I 2 ( R 1 ) = 1 4 P 3 F + P 2 F P 1 F + P 3 F + P 2 F P 4 F . The lower the I2(R1)value, the better the performance.
  1. 2.
    If the received samples are reconstructed as (11), then the reconstructed samples can be expressed as follows:
    Y 2 = h 1 h 2 h 2 h 1 H 1 s 1 s 3 s N 1 s 2 s 4 s N + w 1 w 3 w N 1 w 2 w 4 w N .
    (18)
     
It can be easily obtained that
C 4 [ 1 , 2 ] = c 4 Y 2 ( : , i ) , Y 2 ( : , i ) H , Y 2 ( 1 , i ) , Y 2 ( 2 , i ) H = k 4 h 1 h 2 H 1 1 0 0 1 H 1 H , where k 4 = c 4 ( s 1 , s 1 , s 1 , s 1 ) = c 4 ( s 2 , s 2 , s 2 , s 2 )
(19)

For the proof of (19), see Appendix 2.

Therefore, if C 4 [ k , j ] is decomposed as C 4 [ k , j ] = U Λ U H , then the relationship between U and H1 can be expressed as H1=U P1Λ1, where P1 and Λ1 denote permutation matrix and diagonal matrix, respectively. Moreover, matrix H1 can be identified from Y2 by independent component analysis (ICA)-based method (for example, joint approximate diagonalization of eigenmatrices (JADE) [15]). It should be noted that the relationship between H1 and H ̂ 1 (which denotes the identification of matrix H1 by ICA-based method) can be expressed as H 1 = H ̂ 1 P 2 Λ 2 , where P2 and Λ2 denote permutation matrix and diagonal matrix, respectively. Therefore, the feature of T = U H ̂ 1 = P 1 Λ 1 Λ 2 1 P 2 1 can be measured by Amari index I ( T ) = 1 2 M ( M 1 ) i = 1 M j = 1 M t ij max k t ik 1 + j = 1 M i = 1 M t ij max k t kj 1 defined in [14], where M=4 denotes the order of matrix T, ti,j denotes the element on the i th row and the j th column of matrix T.

3.3 Feature extraction of code of OSTBC3 with rate 3/4’s coded signal

The received samples of the first code of OSTBC3 with rate 3/4’s coded signal can be expressed as:
y = h 1 h 2 h 3 s 1 0 s 2 s 3 s 3 N b 2 0 s 3 N b 1 s 3 N b 0 s 1 s 3 s 2 0 s 3 N b 2 s 3 N b s 3 N b 1 s 2 s 3 s 1 0 s 3 N b 1 s N b s 3 N b 2 0 + w 1 w 2 w N .
(20)
Here, N b = N 4 denotes the number of transmission blocks.
  1. 1.
    If the received samples are reshaped as (8), then the reshaped samples can be expressed as:
    Y 1 = h 1 0 0 0 h 3 0 h 2 0 h 3 0 0 0 h 3 h 1 0 0 0 h 2 0 0 h 1 0 h 2 0 s 1 s 4 s 3 N b 2 s 2 s 5 s 3 N b 1 s 3 s 6 s 3 N b s 1 s 4 s 3 N b 2 s 2 s 5 s 3 N b 1 s 3 s 6 s 3 N b + w 1 w 5 w N 3 w 2 w 6 w N 2 w 3 w 7 w N 1 w 4 w 8 w N .
    (21)
     
  1. (i)
    If s i belongs to modulation scheme I, then it can be easily obtained that
    R 1 = lim N 1 N / 4 Y 1 Y 1 H = E ( s 2 ) h 1 2 + h 3 2 h 1 h 2 h 1 h 3 h 2 h 3 h 1 h 2 h 2 2 + h 3 2 h 2 h 3 h 1 h 3 h 1 h 3 h 2 h 3 h 1 2 + h 2 2 + h 3 2 0 h 2 h 3 h 1 h 3 0 h 1 2 + h 2 2 + σ 2 I 4 .
    (22)
     
  2. (i)
    If s i belongs to modulation scheme II, then it can be easily obtained that
    R 1 = lim N 1 N / 4 Y 1 Y 1 H = h 1 2 + h 3 2 E ( s 2 ) + σ 2 h 1 h 2 E ( s 2 ) h 1 h 3 E ( s 2 ) h 1 h 3 E ( s 2 ) h 2 h 3 E ( s 2 ) h 1 h 2 E ( s 2 ) h 2 2 + h 3 2 E ( s 2 ) + σ 2 h 2 h 3 E ( s 2 ) h 2 h 3 E ( s 2 ) h 1 h 3 E ( s 2 ) h 1 h 3 E ( s 2 ) h 1 h 3 E ( s 2 ) h 2 h 3 E ( s 2 ) + h 2 h 3 E ( s 2 ) h 1 2 + h 2 2 + h 3 2 E ( s 2 ) + σ 2 h 1 h 2 E ( s 2 ) h 1 h 2 E ( s 2 ) h 2 h 3 E ( s 2 ) h 1 h 3 E ( s 2 ) h 1 h 2 E ( s 2 ) h 1 h 2 E ( s 2 ) h 1 2 + h 2 2 E ( s 2 ) + σ 2 .
    (23)
     
  3. 2.
    If the received samples are reconstructed as follows:
    Y 3 = y 1 y 5 y N 3 y 2 y 6 y N 2 y 3 y 7 y N 1 y 4 y 8 y N .
    (24)
     
then the reconstructed samples can be expressed as:
Y 3 = h 1 h 3 0 h 2 0 h 3 h 3 h 1 h 2 0 h 2 h 1 s 1 s 4 s 3 N b 2 s 2 s 5 s 3 N b 1 s 3 s 6 s 3 N b + w 1 w 5 w N 3 w 2 w 6 w N 2 w 3 w 7 w N 1 w 4 w 8 w N .
(25)
It can be easily obtained that
R 3 = lim N 1 N / 4 Y 3 Y 3 H = E ( s 2 ) h 1 2 + h 3 2 h 1 h 2 0 h 2 h 3 h 1 h 2 h 2 2 + h 3 2 0 h 1 h 3 0 0 h 1 2 + h 2 2 + h 3 2 0 h 2 h 3 h 1 h 3 0 h 1 2 + h 2 2 + σ 2 I 4
(26)

The feature of R3 can be measured by I 3 ( R 3 ) = 1 2 ( M 1 ) j = 1 M r 3 , j r 3 , 3 1 + i = 1 M r i , 3 r 3 , 3 1 , where M=4 denotes the order of matrix R3, ri,j denotes the element on the i th row and the j th column of matrix R3. The lower the I3(R 3) value, the better the performance.

3.4 Feature extraction of the second code of OSTBC3 with rate 3/4’s coded signal

The received samples of the second code of OSTBC3 with rate 3/4’s coded signal can be expressed as:
y = y 1 y 2 y N = hC ( s ) + w = h 1 h 2 h 3 s 1 s 2 s 3 0 s 3 N b 2 s 3 N b 1 s 3 N b 0 s 2 s 1 0 s 3 s 3 N b 1 s 3 N b 2 0 s 3 N b s 3 0 s 1 s 2 s 3 N b 0 s 3 N b 2 s 3 N b 1 + w 1 w 2 w N
(27)
Here, N b = N 4 denotes the number of transmission blocks.
  1. 1.
    If the received samples are reshaped as (8), then the reshaped samples can be expressed as:
    Y 1 = h 1 h 2 h 3 0 0 0 0 0 0 h 2 h 1 0 0 0 0 h 3 h 1 0 0 0 0 h 3 h 2 0 s 1 s 4 s 3 N b 2 s 2 s 5 s 3 N b 1 s 3 s 6 s 3 N b s 1 s 4 s 3 N b 2 s 2 s 5 s 3 N b 1 s 3 s 6 s 3 N b + w 1 w 5 w N 3 w 2 w 6 w N 2 w 3 w 7 w N 1 w 4 w 8 w N
    (28)
     
  1. (i)
    If s i belongs to modulation scheme I, then it can be easily obtained that
    R 1 = lim N 1 N / 4 Y 1 Y 1 H = E ( s 2 ) h 1 2 + h 2 2 + h 3 2 0 0 0 0 h 1 2 + h 2 2 h 2 h 3 h 1 2 h 2 h 3 + h 1 h 2 0 h 2 h 3 h 1 2 h 1 2 + h 3 2 h 1 h 2 h 3 2 0 h 2 h 3 + h 1 h 2 h 1 h 2 h 3 2 h 2 2 + h 3 2 + σ 2 I 4
    (29)
     
The performance index of R1 is defined by I 4 ( R 1 ) = 1 2 ( M 1 ) j = 1 M r 1 , j r 1 , 1 1 + i = 1 M r i , 1 r 1 , 1 1 , where M=4 denotes the order of matrix R1, ri,j denotes the element on the i th row and the j th column of matrix R1. The lower the I4(R1) value, the better the performance.
  1. (i)
    If s i belongs to modulation scheme II, then it can be easily obtained that
    R 1 = lim N 1 N / 4 Y 1 Y 1 H = h 1 2 + h 2 2 + h 3 2 E ( s 2 ) + σ 2 ( h 1 h 2 h 1 h 2 ) E ( s 2 ) ( h 1 h 3 + h 1 h 2 ) E ( s 2 ) ( h 1 h 3 h 2 2 ) E ( s 2 ) ( h 1 h 2 + h 1 h 2 ) E ( s 2 ) h 1 2 + h 2 2 E ( s 2 ) + σ 2 ( h 2 h 3 h 1 2 ) E ( s 2 ) ( h 2 h 3 + h 1 h 2 ) E ( s 2 ) ( h 1 h 3 + h 1 h 2 ) E ( s 2 ) ( h 2 h 3 h 1 2 ) E ( s 2 ) h 1 2 + h 3 2 E ( s 2 ) + σ 2 ( h 1 h 2 h 3 2 ) E ( s 2 ) ( h 1 h 3 h 2 2 ) E ( s 2 ) ( h 2 h 3 + h 1 h 2 ) E ( s 2 ) ( h 1 h 2 h 3 2 ) E ( s 2 ) h 2 2 + h 3 2 E ( s 2 ) + σ 2
    (30)
     
  2. 2.
    If the received samples are reconstructed as (24), then the reshaped samples can be expressed as:
    Y 3 = h 1 h 2 h 3 0 0 0 0 0 0 h 2 h 1 0 h 3 0 h 1 0 0 0 0 0 0 h 3 h 2 0 s 1 s 4 s 3 N b 2 s 2 s 5 s 3 N b 1 s 3 s 6 s 3 N b s 1 s 4 s 3 N b 2 s 2 s 5 s 3 N b 1 s 3 s 6 s 3 N b + w 1 w 5 w N 3 w 2 w 6 w N 2 w 3 w 7 w N 1 w 4 w 8 w N
    (31)
     
  1. (i)
    If s i belongs to modulation scheme I, then it can be easily obtained that
    R 3 = lim N 1 N / 4 Y 3 Y 3 H = E ( s 2 ) h 1 2 + h 2 2 + h 3 2 0 0 0 0 h 1 2 + h 2 2 0 h 2 h 3 + h 1 h 2 0 0 h 1 2 + h 3 2 0 0 h 2 h 3 + h 1 h 2 0 h 2 2 + h 3 2 + σ 2 I 4
    (32)
     
  2. (i)
    If s i belongs to modulation scheme II, then it can be easily obtained that
    R 3 = lim N 1 N / 4 Y 3 Y 3 H = h 1 2 + h 2 2 + h 3 2 E ( s 2 ) + σ 2 h 1 h 2 h 1 h 2 E ( s 2 ) 0 h 1 h 3 h 2 2 E ( s 2 ) h 1 h 2 h 1 h 2 E ( s 2 ) h 1 2 + h 2 2 E ( s 2 ) + σ 2 h 2 h 3 E ( s 2 ) h 2 h 3 + h 1 h 2 E ( s 2 ) 0 h 2 h 3 E ( s 2 ) h 1 2 + h 3 2 E ( s 2 ) + σ 2 h 3 2 E ( s 2 ) h 3 h 1 h 2 2 E ( s 2 ) h 2 h 3 + h 1 h 2 E ( s 2 ) h 3 2 E ( s 2 ) h 2 2 + h 3 2 E ( s 2 ) + σ 2
    (33)
     

3.5 Feature extraction of OSTBC3 with rate 1/2’s coded signal

The received samples of OSTBC3 with rate 1/2’s coded signal can be expressed as:
y = y 1 y 2 y N b = y 1 y 2 y N = h 1 h 2 h 3 T s 1 s 2 s 3 s 4 s 1 s 2 s 3 s 4 s 2 s 1 s 4 s 3 s 2 s 1 s 4 s 3 s 3 s 4 s 1 s 2 s 3 s 4 s 1 s 2 s 4 N b 3 s 4 N b 2 s 4 N b 1 s 4 N b s 4 N b 3 s 4 N b 2 s 4 N b 1 s 4 N b s 4 N b 2 s 4 N b 3 s 4 N b s 4 N b 1 s 4 N b 2 s 4 N b 3 s 4 N b s 4 N b 1 s 4 N b 1 s 4 N b s 4 N b 3 s 4 N b 2 s 4 N b 1 s 4 N b s 4 N b 3 s 4 N b 2 + w 1 w 2 w N
(34)
If the received samples are reshaped as (8), then the reshaped samples can be expressed as:
Y 1 = h 1 h 2 h 3 0 h 2 h 1 0 h 3 h 3 0 h 1 h 2 0 h 3 h 2 h 1 s 1 s 1 s 4 N b 3 s 4 N b 3 s 2 s 2 s 4 N b 2 s 4 N b 2 s 3 s 3 s 4 N b 1 s 4 N b 1 s 4 s 4 s 4 N b s 4 N b + w 1 w 5 w N 3 w 2 w 6 w N 2 w 3 w 7 w N 1 w 4 w 8 w N
(35)
It can be easily obtained that
R 1 = lim N 1 N / 4 Y 1 Y 1 H = E ( s 2 ) h 1 2 + h 2 2 + h 3 2 h 1 h 2 h 1 h 2 h 1 h 3 h 1 h 3 h 2 h 3 h 2 h 3 h 1 h 2 h 1 h 2 h 1 2 + h 2 2 + h 3 2 h 2 h 3 h 2 h 3 h 1 h 3 h 1 h 3 h 1 h 3 h 1 h 3 h 2 h 3 h 2 h 3 h 1 2 + h 2 2 + h 3 2 h 1 h 2 h 1 h 2 h 2 h 3 h 2 h 3 h 1 h 3 h 1 h 3 h 1 h 2 h 1 h 2 h 1 2 + h 2 2 + h 3 2 + σ 2 I 4
(36)
Furthermore,
Re ( R 1 ) = E ( s 2 ) h 1 2 + h 2 2 + h 3 2 + σ 2 I 4
(37)

The performance can be measured by I1[Re(R1)]. The lower the I1[Re(R1)] value, the better the performance.

4 Classification of space-time code

4.1 Analysis of the features of different STBCs

In this subsection, we analyze the features of different STBCs mentioned in Section 3. The features of different STBCs are summarized in Table 1. It should be noted that the features extracted from the reconstructed receiver are independent of the channel matrix. Figure 1 illustrates these features in the form of decision tree. First, SM and Alamouti codes are recognized by the values of Amari 1=10 log[I1(R1)], Amari 2=10 log[I2(R1)], and Amari 5=10 log[I(T)]. Then the OSTBC3 with rate 1/2 is recognized by the values of Amari3=10 log[I1Re(R1)]. Finally, Amari 4=10 log[I4(R1)]and Amari6=10 log[I3(R3)] are used to distinguish the first code of OSTBC3 with rate 3/4 from the second code of OSTBC3 with rate 3/4. Unfortunately, it appears to be difficult to find theoretical values of these thresholds. Therefore, we propose SVM-based classifier for the blind recognition of five linear STBCs in the next subsection.
Table 1

The features of different STBCs

Code

Features

SM

Amari 1<ε1; Amari 5>ε5

Alamouti

If s i belongs to modulation scheme I, Amari 1<ε1 and Amari 5<ε5;

 

If s i belongs to modulation scheme II, Amari 1>ε1, Amari 5<ε5, Amari 2<ε2,

 

Amari 3<ε3

3 4 OSTBC3

Amari 1>ε1, Amari 2>ε2, Amari 3>ε3;

Code 1

If s i belongs to modulation scheme I, Amari 4>ε4, Amari 6<ε6

 

If s i belongs to modulation scheme II, Amari 4>ε4, Amari 6<ε6

3 4 OSTBC3

Amari 1>ε1, Amari 2>ε2, Amari 3>ε3;

Code 2

If s i belongs to modulation scheme I, Amari 4<ε4, Amari 6<ε6

 

If s i belongs to modulation scheme II, Amari 4>ε4, Amari 6>ε6

1 2 OSTBC3

 

Code 3

Amari 1>ε1, Amari 2>ε2, Amari 3<ε3

Amari 1=10 log[I1(R1)], Amari 2=10 log[I2(R1)], Amari 3=10 log[I1Re(R1)], Amari 4=10 log[I4(R1)], Amari 5=10 log[I(T)], Amari 6=10 log[I3(R3)]. ε1, ε2, ε3, ε4, ε5, and ε6 are thresholds to discriminate the features of different STBCs.

Figure 1

The features of 5 STBCs. This illustrates the features in the form of a decision tree. First, SM and Alamouti codes are recognized by the values of Amari 1=10 log[I1(R1)], Amari 2=10 log[I2(R1)], and Amari 5=10 log[I(T)]. Then the OSTBC3 with the rate 1/2 is recognized by the values of Amari3=10 log[I1Re(R1)]. Finally, Amari 4=10 log[I4(R1)]and Amari6=10 log[I3(R3)] are used to distinguish the first code of OSTBC3 with the rate 3/4 from the second code of OSTBC3 with the rate 3/4.

4.2 Classification of space-time code by support vector machine

SVM has demonstrated good performance in numerous pattern recognition problems [1621]. It can discriminate non-linear separable data by mapping original feature space into a higher dimensional space via kernel functions. The commonly used kernel functions are as follows [16]:
K ( x , y ) = x , y + 1 p ( Polynomial )
(38)
K ( x , y ) = e x y 2 / 2 σ 2 ( Gaussian )
(39)
K ( x , y ) = tanh ( κ x · y δ ) ( Sigmoidal )
(40)
In this paper, we select a polynomial function as the kernel function of SVM classifier. Figures 2 and 3 illustrate the feature distribution of five STBCs in two different three-dimensional spaces projected by the six-dimensional feature space of Amari1, Amari2, Amari3, Amari4, Amari5, and Amari6. The number of received samples, the type of modulation, the number of Monte Carlo trials, and the signal-to-noise ratio (SNR) are N=2048, QPSK, 50 times, and 20 dB, respectively. It shows that SM and Alamouti cluster at different places in the three-dimensional spaces mapped by Amari1, Amari2, and Amari5; OSTBC3 with rate 1/2, first code of OSTBC3 with rate 3/4 and second code of OSTBC3 with rate 3/4 cluster at different place in the three-dimensional spaces mapped by Amari3, Amari4, and Amari6. Therefore, five STBCs can be separated easily by a hyperplane in the six-dimensional feature space. The optimal hyperplane for classification is obtained from the training set based on the criteria of maximizing the margin and minimizing the empirical risk.
Figure 2

The feature distribution of five STBCs in the first three-dimensional spaces. It illustrates the feature distribution of five STBCs in the three-dimensional spaces mapped by Amari1, Amari2, and Amari5. The number of received samples, the type of modulation, the number of Monte Carlo trials and the signal-to-noise ratio are N=2048, QPSK, 50 times, and 20 dB, respectively. It shows that SM and Alamouti cluster at different places in the three-dimensional spaces mapped by Amari1, Amari2, and Amari5.