Globecom 2014 - Wireless Communications Symposium

OMP-Based Detector Design for Space Shift Keying in Large MIMO Systems Chien-Hsien Wu∗ , Wei-Ho Chung∗† , and Han-Wen Liang∗ ∗ Research

Center for Information Technology Innovation, Academia Sinica, Taipei, Taiwan † Email:

[email protected]

Abstract—We investigate the detector design in generalized space shift keying (GSSK) modulation for large MIMO systems. An orthogonal matching pursuit (OMP) based detector design is adopted due to its low complexity compared with the maximum likelihood (ML) detector. To improve the performance of the OMP algorithm, our first design is to propose an equalizer at the receiver in order to orthogonalize the equivalent channel matrix; our second design is to propose an equalizer whose columns pursue orthonormality to the columns of the channel matrix. We obtain closed-form expressions of the equalizers under the two design objectives. Simulation results demonstrate the performance superiority compared with the standard OMP algorithm. Index Terms—Large MIMO, space shift keying, detection, orthogonal matching pursuit.

I. I NTRODUCTION

L

ARGE multiple-input multiple-output (MIMO) systems have attracted tremendous attention in recent years due to its potential to provide improvements in data rates and link reliability [1], [2]. In large MIMO systems, the transmitter and receiver are equipped with number of antennas in the order of tens to hundreds and thus high spectral efficiency can be achieved by exploiting degrees of freedom in spatial domain. Despite the advantages, the increase in the number of antennas is costly in terms of the hardware (e.g. the number of RF chains) and the signal processing complexity at the transmitter/receiver sides. One approach to mitigate these issues is to use spatial modulation (SM) [3]–[5]. In SM MIMO systems, the information is carried by the transmit antenna index, where the constellation mapping scheme stipulates the correspondence between the information bits and the activated antenna index. A special case of SM is the space shift keying (SSK) that carries information only on the transmit antenna index to avoid any form of conventional modulations and provides trade-off on receiver complexity [6]. By allowing multiple transmitting antennas to be activated, the generalized SSK (GSSK) modulation improves the transmission rate [7]. If the detection of GSSK is to be carried out in the optimal maximum likelihood (ML) sense, This work was supported in part by Ministry of Science and Technology, Taiwan, under grants NSC 102-2221-E-001-006-MY2 and MOST 103-3113E-110-002.

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the complexity grows exponentially in the number of transmit antennas, which is computationally prohibitive in large MIMO systems. Recently, the design of low complexity detectors for GSSK-modulated MIMO systems has attracted research attention [8], [9]. In this paper, we consider large MIMO systems with GSSK modulation. Motivated by the work in [9], we adopt an orthogonal matching pursuit (OMP) based rationale due to the low computational complexity [10]. The OMP algorithm, a sequential selection method, can be potentially regarded as a successive interference cancellation (SIC) scheme and the performance limit of OMP based on the SIC criterion was developed in [11]. It is shown in [12] that the signal recovery performance based on the OMP algorithm is good when the correlations between any two columns of the channel matrix are small. Many works [13], [14] consider to improve the performance of the OMP algorithm by minimizing the sum of correlations between all pairs of columns in the equivalent matrix, which is related to an equalizer to be designed and the channel matrix. In this work, we propose two methods that design matrices, i.e., equalizers, at the receiver. The objective of the first method is to design a matrix that orthogonalizes an equivalent channel matrix; the objective of the second method is to design a matrix such that its columns are as orthonormal as possible to the columns of the channel matrix. Both methods consider the problem of achieving objectives mentioned above as well as minimizing the total noise power. In achieving the aforementioned designs, we obtain closed-form solutions of the two matrices. Simulation results demonstrate that the proposed methods indeed improve the performance particularly at high signal-to-noise ratio (SNR). This paper is organized as follows. Section II describes the system model including the GSSK modulation and the OMP algorithm. Section III gives our proposed two methods by which the performance of the OMP algorithm can be further improved. Simulation results are presented to validate our proposed methods in Section IV. Finally, Section V is a brief conclusion.

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II. S YSTEM M ODEL A. GSSK Modulation We consider a MIMO system with N transmit antennas and m receive antennas using GSSK modulation, where m < N . The GSSK modulated symbol x is selected equiprobably from PN the modulation alphabet A = {x = [x1 · · · , xN ]T | i=1 xi = k, xi ∈ {0, 1}}. The nonzero entries of x correspond to activated antennas whereas the zero entries correspond to idle antennas. By allocating transmit power Es to each activated antenna, the modulated symbol is then transmitted to the receiver through the flat-fading channel H ∈ Cm×N . The baseband expression of the received signal is given by p y = Es Hx + n, (1)

where n ∈ Cm is additive noise vector with zero mean and covariance matrix E[nnH ] = σn2 Im . It is noted that the amount of information that can be transmitted by GSSK

modulation is given by b = ⌊log2 N ⌋ bits, where ⌊·⌋ k denotes the floor operator. Assuming the channel matrix H is perfectly known at the receiver, and given the received signal y in (1), the optimal detector is the maximum likelihood (ML) detector, which is written as p xML = arg min ky − Es Hxk22 . (2) x∈A

B. OMP-Based Detector

The optimal ML detector in (2) requires O(mN k ) complexity, which becomes computationally prohibitive for large MIMO systems. To reduce complexity, Yu et al. [9] proposed to use the orthogonal matching pursuit (OMP) that requires only O(kmN ) complexity for GSSK symbol detection. In the following, we briefly review the OMP and its application to GSSK symbol detection. ˆ We express the channel matrix as H = HD, where ith ˆ ˆ ˆ i k2 = column of H is denoted as hi and it is normalized as kh 1 for i = 1, · · · , N ; the D = diag{kh1 k, · · · , khN k} is a ˆ Λ denotes diagonal matrix. For any subset Λ ⊆ {1, · · · , N }, H ˆ ˆ the submatrix of H consisting of the columns hi with i ∈ Λ. ˆ where x has k nonzero entries, For the measurement y = Hx, the procedure of OMP algorithm [10] is stated in Algorithm 1. After k iterations, the OMP algorithm can identify a set of ˆ as column indices Λk . Define the mutual coherence of H µ=

max

1≤i6=j≤N

ˆH h ˆ j |, |h i

(3)

ˆ H H. ˆ which is the maximum value of off-diagonal entries of H It has been shown that, when µ is sufficiently small, the set of column indices is correct and x can be recovered exactly with high probabilities [12]. To apply the OMP algorithm to GSSK symbol detection, the columns of the channel matrix H must be normalized. The received signal in (1) is rewritten as p ˆ x + n, y = Es Hˆ (4)

Algorithm 1 Orthogonal matching pursuit algorithm. Input: the received signal y, the normalized channel matrix ˆ and the number of activated antennas, k. H, Output: the set of column indices Λk . 1: Initialization: Set the residual r0 = y, the index set Λ0 = ∅, and the iteration counter t = 1; 2: Identification: ˆ j , rt−1 i|; it = arg max |hh 1≤j≤N

3:

Reconstruction: Λt = Λt−1 ∪ it ; ˆ Λt xk2 ; xt = arg min ky − H x

4:

Update: ˆ Λt xt ; rt = y − H

5: 6:

If t = k, stop; otherwise, set t = t + 1 and go to step 2; return Λk .

ˆ = Dx. Since D is a diagonal matrix, the nonzero where x ˆ are unchanged. Using the OMP algorithm on entries in x and x the received signal in (4), we can obtain a set of column indices Λk which correspond to the index set of activated transmit antennas in the GSSK system. The recovered signal can be expressed as x ˜i = 1 if i ∈ Λk ; otherwise x ˜i = 0. III. P ROPOSED M ETHODS To detect the GSSK symbol by the OMP algorithm, a ˆ is crucial for correct indices small mutual coherence µ of H identification and exact signal recovery provided that the noise ˆ the mutual coherence is is small. However, for a given H, fixed. In this section, we propose two methods to design equalizers at the receiver to obtain a small µ. In the proposed methods, instead of minimizing µ directly, we consider to use a different design metric to simplify the problem. Specifically, we propose two methods to design matrices, i.e., equalizers, at the receiver so that i) the equivalent channel matrix, which is composed of the design matrix and the channel matrix, is close to orthogonal; and ii) the columns of the design matrix are approximately orthonormal to the columns of the channel matrix. Based on the proposed methods, the performance of the OMP algorithm can be further improved. A. Method I In the first method, we multiply y in (4) by an equalizer W ∈ Cm×m to obtain z =Wy p ˜x+n ˜, = Es Hˆ

(5)

˜ = WH ˆ is the equivalent channel matrix and where H ˜ = Wn is the equivalent noise vector with zero mean and n

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˜ H ] = σn2 WWH . It is noted that the covariance matrix E[˜ nn noise power may be amplified by W. Hence, the objectives of the design matrix W are that i) the equivalent channel matrix ˜ is close to orthogonal, which implies that µ is close to 0; and H ii) the total noise power is minimized. To achieve a balance between the two objectives, we define the normalized noise ˜ as Pn = σn2 /Es and consider power of n to  minimize the total normalized noise power Pn tr WWH . The problem can be reformulated as ˆ H WH WHk ˆ 2F + Pn tr{WWH }, min kIN − H W

(6)

where k · kF denotes the Frobenius norm and tr{·} is the trace operator. Let Q = WH W, where Q is a symmetric positive-semidefinite matrix, denoted as Q  0, with its diagonal entries qii ≥ 0 for i = 1, · · · , m. Using the property tr{AB} = tr{BA} in the second term of the objective function, the equivalent problem of (6) is   ˆH ˆ 2   minQ kIN − H QHkF + Pn tr {Q} . (7) s.t. Q0    q ≥ 0, i = 1, · · · , m i,i

Remark 1: Pn is regarded as a weighting factor. As the noise power is small, W makes an effort to orthogonalize the equivalent channel matrix, otherwise, W tries to minimize the total normalized noise power. ˆ as the singular value decomposition (SVD) Express H ˆ = U ˆ Σ ˆ VHˆ , H H H H

(8)

H N ×N where UHˆ ∈ Cm×m and VH are unitary matrices, ˆ ∈ C m×N ˜ = UH QU ˆ and ΣHˆ ∈ R is a diagonal matrix. Let Q ˆ H H and ΛHˆ = ΣHˆ ΣH . The objective function in (7) can be ˆ H expressed as

ˆ H QHk ˆ 2 + Pn tr {Q} kIN − H F n o (a) H H ˜ H ˜ ˜ = tr IN − 2VHˆ ΣH QΣ V + V Σ QΛ QΣ V ˆ H ˆ H ˆ ˆ H ˆ ˆ ˆ ˆ H H H H H n o ˜ + Pn tr Q n o n o (b) ˜ ˆ QΛ ˜ ˆ − 2QΛ ˜ ˆ + Pn tr Q ˜ = N + tr QΛ H H H   Pn ˜ ˆ QΛ ˜ ˆ − 2Q(Λ ˜ Im ) , (9) =N + tr QΛ ˆ − H H H 2 where (a) follows from kAk2F = tr{AH A}, and (b) follows from tr{A + B} = tr{A} + tr{B} and tr{AB} = tr{BA}. From (9), the problem in (7) can be rewritten as n o  Pn ˜ ˆ QΛ ˜ ˆ − 2Q(Λ ˜  min tr QΛ − I )  ˜ ˆ m Q H H H 2  ˜ 0 . (10) s.t. Q    q˜ ≥ 0, i = 1, · · · , m i,i

The solution of (10) is given in the following proposition.

ˆ H Pn

Compute W W by (13)

y ˆ H

z Compute ˜ in (5) H ˜ z and H k

OMP algorithm

Λk

Fig. 1. Functional diagram of the OMP-based detector for the proposed method I.

given by    Pn  /λ2H,i λ −  ˆ ˆ H,i 2  q˜ij = 0    0

, if i = j and λH,i ˆ ≥ , if i = j and λH,i ˆ
i

m X

q˜ii (λH,i ˆ −

i=1

Pn ), 2

(12)

∗ where we use the fact that q˜ii is real and q˜ij = q˜ji . From (12), we see that for fixed q˜ii , any nonzero q˜ij leads to the increase ˜ Thus, we conclude that q˜ji = 0. Furthermore, to of J(Q). ˜ solve q˜ii , we use ∂J(Q)/∂ q˜ii = 0 and the constraint q˜ii ≥ 0, and the solution in (11) can be established.

˜ is a diagonal matrix From Proposition 1, the optimal Q with nonnegative entries and thus it can be expressed as ˜ = Q ˜ 1/2 Q ˜ 1/2 . Since Q ˜ = UH QU ˆ and Q = WH W, Q ˆ H H the optimal W to the problem (6) is ˜ 1/2 UHˆ . W=Q H

(13)

The functional diagram of the method is illustrated in Fig. 1. When obtaining W in (13), we have the received signal given ˆ replaced in (5). Applying Algorithm 1 with its input y and H ˜ in (5), we then obtain the indices of k activated by z and H antennas. Remark 2: From (6), we see that the basic idea of choosing ˜ in (5) be better W is to make the equivalent channel matrix H conditioned than the original one. This is similar to the lattice reduction detection scheme [15], which achieves this goal by choosing a precoding matrix with unit magnitude determinant at the transmitter. B. Method II In GSSK-modulated system, the identification of indices of activated transmit antennas is the step 2 of Algorithm 1, which is written as

˜ and Λ ˆ = Proposition 1. Let q˜ij be the (i, j)th entry of Q H diag{λH,1 , · · · , λ }. The solution to the problem (10) is ˆ ˆ H,m

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ˆ j , rt−1 i|. it = arg max |hh 1≤j≤N

(14)

Globecom 2014 - Wireless Communications Symposium

ˆ H Pn

m = 24, N = 64, k = 3

0

Compute Φ Φ by (18) y ˆ H k

10

Modified OPM algorithm with (14) replaced by (15)

MMSE OMP in [8] Proposed I Proposed II ML

Λk −1

10

Fig. 2. Functional diagram of the OMP-based detector for the proposed method II.

−2

SER

10

−3

10

ˆ the mutual coherence in (3) is fixed. The OMP For a given H, algorithm may fail to identify the correct index by (14) under a large µ. Hence, instead of using (14), Schnass et al. [13] proposed a modified OMP algorithm that uses a designed matrix Φ ∈ Cm×N to identify the index set, which can be written as

−4

10

−5

10

0

5

10

15

20

25

SNR (dB)

it = arg max |hφj , rt−1 i|, 1≤j≤N

(15)

m

where φi ∈ C denotes the ith column of Φ. The goal is to choose φi to satisfy ( 1, if i = j Hˆ φi hj = . 0, if i 6= j Consider the received signal in (4). The identification in (15) with t = 0 is to find the index of the maximum absolute value from z¯ ∈ CN , which is written as ˆ + ΦH n, z¯ = ΦH y = ΦH Hx where the equivalent noise is zero mean with covariance matrix E[ΦH nnH Φ] = σn2 ΦH Φ. By the similar design objectives described in Section III-A, the problem can be formulated as ˆ 2 + Pn tr{ΦH Φ}. min kIN − ΦH Hk F Φ

(16)

Since kAk2F = tr{AH A}, the objective function in (16) can be written as ˆ 2 + Pn tr{ΦH Φ} kIN − Φ Hk F H ˆ ˆ HΦ + H ˆ H ΦΦH H ˆ + Pn ΦH Φ}. (17) =tr{IN − Φ H − H H

It can be shown that the solution that minimizes (17) is  −1 ˆH ˆ H + Pn Im ˆ Φ= H H. (18) As Φ in (18) is obtained, instead of using (14) at the step 2 of Algorithm 1, we use (15) to identify the indices of the k activated antennas, which is regarded as a modified OMP algorithm. The functional diagram of the method is illustrated in Fig. 2. Remark 3: Although the proposed method II is very similar to the MMSE-SIC detection [16], they estimate the symbols in a different way. In the proposed method, the symbols are estimated based on the selected channel columns while they are estimated based on full channel matrix in the MMSE-SIC technique.

Fig. 3. SER versus SNR with different methods: MMSE, standard OMP, proposed OMP-based methods, and ML.

C. Complexity The computational complexity of the Algorithm 1 is dominated by Step 2, which requires O(kmN ) operations [10]. In the proposed method I, we only need UH ˆ and ΣH ˆ from the ˆ SVD of H, which dominates the computation cost and requires O(m2 N ) operations [17, p.254]. In the proposed method II, the computation cost of the matrix inverse in (18) requires O(m3 ) operations. Since we consider the case m < N in our MIMO system, the cost of method II is dominated by evaluating the matrix multiplication in (18) which requires O(m2 N ) operations. Note that the complexity of the proposed methods is smaller than that of the ML detector, which needs O(mN k ) operations. IV. S IMULATION R ESULTS In this section, we use numerical simulations to illustrate the results established in the previous section. The entries of noise n are independent and identically complex Gaussian with zero mean and variance σn2 . Each entry of channel matrix H is assumed to have complex Gaussian distribution with zeromean and unit variance. The SNR is defined as kEs /σn2 in our simulations. For comparison, we also simulate the minimummean-square-error (MMSE) linear detector. Fig. 3 shows the comparison of the symbol error rate (SER) for the proposed two OMP-based methods, the standard OMP, the MMSE detector, and the ML detector with N = 64, m = 24, k = 3. It shows that the MMSE detector performs poorly while the ML detector exhibits the best performance among all the detectors. By using the matrix design methods proposed in Section III, we see that the performances of the proposed OMP-based methods are improved compared to the standard OMP algorithm. Moreover, for SNR> 15 dB, the SER of the proposed methods performs ten times better compared with that of the standard OMP algorithm.

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N = 64, k = 3, SNR = 10dB

0

10

MMSE OMP in [8] Proposed I Proposed II ML

−1

10

−2

SER

10

−3

10

−4

10

−5

10

5

10

15

20 25 30 Number of received antennas (m)

35

40

Fig. 4. SER versus number of receive antennas with different methods: MMSE, standard OMP, proposed OMP-based methods, and ML.

The comparisons among the OMP-based methods, the MMSE detector, and the ML detector are shown in Fig. 4 with SER as a function of m. Again the figure shows that the performance of the ML detector is better than other methods; also the performances of the proposed OMP-based methods are very close. As the number of receive antennas m increases, the performance gap between the proposed OMP-based methods and the standard OMP algorithm becomes large. Moreover, all OMP-based methods outperform the MMSE detector.

[5] M. Di Renzo, H. Haas, A. Ghrayeb, S. Sugiura, and L. Hanzo, “Spatial modulation for generalized MIMO: Challenges, opportunities, and implementation,” Proc. IEEE, vol. 102, no. 1, pp. 56–103, 2014. [6] J. Jeganathan, A. Ghrayeb, L. Szczecinski, and A. Ceron, “Space shift keying modulation for MIMO channels,” IEEE Trans. Wireless Commun., vol. 8, no. 7, pp. 3692–3703, Jul. 2009. [7] J. Jeganathan, A. Ghrayeb, and L. Szczecinski, “Generalized space shift keying modulation for MIMO channels,” in Proc. IEEE PIMRC’08, Cannes, France, Sep. 2008, pp. 1–5. [8] R. Y. Chang, W.-H. Chung, and S.-J. Lin, “Detection of space shift keying signaling in large MIMO systems,” in IEEE IWCMC’12, Limassol, Cyprus, Aug. 2012, pp. 1–6. [9] C.-M. Yu, S.-H. Hsieh, H.-W. Liang, C.-S. Lu, W.-H. Chung, S.-Y. Kuo, and S.-C. Pei, “Compressed sensing detector design for space shift keying in MIMO systems,” IEEE Commun. Lett., vol. 16, no. 10, pp. 1556–1559, Oct. 2012. [10] J. Tropp and A. Gilbert, “Signal recovery from random measurements via ortogonal matching pursuit,” IEEE Trans. Inf. Theory, vol. 53, no. 12, pp. 4655–4666, Dec. 2007. [11] Y. Jin and B. Rao, “Performance limits of matching pursuit algorithms,” in Proc. IEEE ISIT’08, Toronto, Canada, Jun. 2008, pp. 2444–2448. [12] J. Tropp, “Greed is good: Algorithmic results for sparse approximation,” IEEE Trans. Inf. Theory, vol. 50, no. 10, pp. 2231–2242, Oct. 2004. [13] K. Schnass and P. Vandergheynst, “Dictionary preconditioning for greedy algorithms,” IEEE Trans. Signal Process., vol. 56, no. 5, pp. 1994–2002, May 2008. [14] L. Zelnik-Manor, K. Rosenblum, and Y. C. Eldar, “Sensing matrix optimization for block-sparse decoding,” IEEE Trans. Signal Process., vol. 59, no. 9, pp. 4300–4312, Sep. 2011. [15] E. Larsson, “MIMO detection methods: How they work,” IEEE Signal Process. Mag., vol. 26, no. 3, pp. 91–95, May 2009. [16] J.-H. Park, Y. Whang, and K. S. Kim, “Low complexity MMSE-SIC equalizer employing time-domain recursion for OFDM systems,” IEEE Signal Process. Lett., vol. 15, pp. 633–636, Oct. 2008. [17] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. MD: Johns Hopkins Univ. Press, 1996.

V. C ONCLUSION We consider large MIMO systems with GSSK modulation. We propose two OMP-based methods to improve the performances compared with the standard OMP algorithm by designing equalizers at the receiver. By jointly factoring the noise effects in the design objective, the first method is to design a matrix so that the equivalent channel matrix is close to orthogonal; the second method is to design a matrix whose columns are as orthonormal as possible to the columns of the channel matrix. Two closed form expressions of the design matrices are obtained. Simulations show that the performances of the proposed methods outperform previous works. R EFERENCES [1] K. V. Vardhan, S. K. Mohammed, A. Chockalingam, and B. S. Rajan, “A low-complexity detector for large MIMO systems and multicarrier CDMA systems,” IEEE J. Sel. Areas Commun., vol. 26, no. 3, pp. 473– 485, Apr. 2008. [2] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, “Scaling up MIMO: Opportunities and challenges with very large arrays,” IEEE Signal Process. Mag., vol. 30, no. 1, pp. 40–60, Jan. 2013. [3] R. Y. Mesleh, H. Haas, S. Sinanovi´c, C. W. Ahn, and S. Yun, “Space modulation,” IEEE Trans. Veh. Technol., vol. 57, no. 4, pp. 2228–2241, Jul. 2008. [4] M. D. Renzo, H. Haas, and P. M. Grant, “Spatial modulation for multiple-antenna wireless systems: a survey,” IEEE Commun. Mag., vol. 49, no. 12, pp. 182–191, Dec. 2011.

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