Effect of imperfect channel estimation on high rate spatial modulation detectors

In this paper, a general channel model including the correlation effect at the transceiver and the I-CSI is proposed. Performance of the existing HRSM detectors is investigated under this model. Besides, in order to obtain comprehensive understanding about the HRSM system, the system capacity under two cases: P-CSI and I-CSI is studied.

EFFECT OF IMPERFECT CHANNEL ESTIMATION ON HIGH- RATE SPATIAL MODULATION DETECTORS

Nguyen Tien Dong*, Le Thi Thanh Huyen, Tran Xuan Nam

Abstract:Recently, anew Spatial Modulation (SM) system, called high rate

spatial modulation (HRSM), was proposed to achieve spectral efficiency linearly increased with the number of transmit antennas In this system, the HRSM receiver

is assumed to exactly know the channel state information (CSI) However, in practice, this assumption is not realistic due to channel estimation error Furthermore, the spatial correlation between antennas at the transceiver affects on the CSI As a result, the HRSM performance deteriorates due to the imperfect CSI (I-CSI) and the correlation effect In this paper, a general channel model including the correlation effect at the transceiver and the I-CSI is proposed Performance of the existing HRSM detectors is investigated under this model Besides, in order to obtain comprehensive understanding about the HRSM system, the system capacity under two cases: P-CSI and I-CSI is studied

Keywords: Spatial modulation, Detector, Channel state information, Correlation effect

1 INTRODUCTION

Recently, a promising multi-input multi-output (MIMO) scheme, called spatial modulation (SM) has been proposed [1] In this model, antenna indices are exploited as an additional source to convey information bits Since only one transmit antenna is activeat a time, the SM scheme totally avoids the Interchannel Interference (ICI) effect.The SM transmitter can also facilitate the antenna synchronization requirements as well as require less radio frequency (RF) chains compared with the conventional MIMO systems However, the SM spectral efficiency is limited tolog2 n T where n is the number of T

transmit antennas Therefore, many researchers have focused on methods to increase the

SM spectral efficiency In [2], by activating more than one transmit antenna at a time, a generalized SM (GSM) was proposed The spectral efficiency of this model is higher than that of the SM However, this technique requires multiple RF chains Furthermore, by

applying the Spatial Constellation (SC) concept and Almouti code matrix, Le et al

proposed a Spatially Modulated Orthogonal Space Time Block Coding (SM-OSTBC) scheme [3] This scheme obtains the maximum spectral efficiency of n T  2 log2M bpcu when the number of transmit antennas is equal the number of active antennas, i.e n T n A

where M is the signal constellation level Recently, the so-called High Rate Spatial

Modulation (HRSM) scheme which has the spectral efficiency linearly increased with the number of transmit antennas, i.e, 2(n T 1)log2M ,was proposed in [4] However, in this model the CSI was assumed exactly known or perfectly estimated at the receiver Related

to the imperfect channel problem, in [5] Renzo et al studied the SM-MIMO system under

generalized fading channels Furthermore, the SM scheme is surveyed in the presence of

channel estimation error in [6] Mesleh et al studied the SM systemperformance under the

Gaussian imperfect channel estimation [7] Besides, the SM system was analyzed over correlated fading channels in [8] Recently, the authors of [9] compared the performance

of the SM detectors under channel impairments where only correlation effectwas considered between neighboring antennas at the receiver

In this paper, basing on the exponential correlation model [10], a comprehensive model including the correlation effect at the transceiver and the I-CSI is proposed We then use this model to investigate the performance of existing HRSM detectors including MVBLAST (Modified Vertical Bell Laboratories Space-Time), MSQRD (Modified Sorted

QR Decomposition), ISQRD (Improved SQRD) [11] and Maximum likelihood (ML) detector [4] in flat Rayleigh fading channel under perfect CSI (P-CSI) Furthermore, the HRSM system capacity in I-CSI and P-CSI is studied

2 HRSM System model

S/P

M-QAM/PSK

Data

(l+m) bits

m bits

SC codewords

l bits

X

ML detector

Data

R n

T

n

1

n

R

n

n

+

S

x

n

+

+

Figure 1 High rate spatial modulation scheme

Fig 1 illustrates a general model of the HRSM scheme withn receive and R n T

transmit antennas At each transmission instant, a block ( lm) of data bits is separated in

two groups of l bits and m bits where the l bits are used to select an arbitrary SC matrix

s from the spatial constellation set  and the latter is modulated to generate a modulated S

symbol x A set of SC matrices is designed by fixing the first entry of s by 1 and

randomly allocating the others from a set  1, j where 2

1

j   Generally, with n T

transmit antennas, the total number of the SC matrices in  isS 1

 Therefore, the number of information bits carried by the SC matrices increase linearly with the number of transmit antennas, llog2K2n T 1 bit per channel use (bpcu) Finally, the HRSM

codeword c is the product of the chosen SC matrix s and the modulated symbol x, i.e

x



At the receiver, the n  receive signal vector y is given by R 1

x

where H is ann Rn T channel matrix and nis noise vector H and nare assumed to have

independent and identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance E is the average symbol energy of x and s  is the average Signal-to-Noise Ratio (SNR) at each receive antenna

3 HRSM DETECTORS

An optimal detector [4] and suboptimal detectors [11] in the HRSM scheme are briefly introduced in this section

A ML Detector

Corresponding to SC codeword sk S,k 1, ,K, ann  equivalent channel matrix R 1

k  k

h Hs is obtained Then, the pair signals ˆ, xs  is recovered as follows

Detailed detection algorithm is summarized in Table I

Table I:An optimal ML decoding algorithm of HR-SM scheme

1 Compute the equivalent channel matrix hk Hsksk S

2 For each matrix k

h and for modulated symbolx   compute the Euclidean x

distances based on (2), m   1, ,

d d x  m M

3 Find d kmin among M values of m

k

d and define ˆ x k

4 Find index ˆmin k corresponding to the minimum distance dmin among K value of

k

5 The estimated SC codeword and modulated symbol is given sˆskˆ,xx kˆ

6 Data bits are recovered from sˆ ˆ, x

B MVBLAST Detector

Considering the HRSM as a MIMO system, the MMSE (Minimum Mean Square Error) filter matrix is given by

1

1

R

s

E



Where:

T s

n E





H H TheMVBLAST algorithm is summarized in Table II

Table II.MVBLAST detection algorithm

1 Compute

1

1

T

H

n s

E



  

2

Find the strongest signal index based on arg min j j,

j

k  P where Pj j, is the -thj

diagonal entry of P , and reorder the entries of c so that the smallest diagonal

entry is the first one

3 Compute the MMSE filter matrix GMMSE and form the LMS estimate

MMSE ,

c g y where gMMSE ,k is the kth row of GMMSE

4 Obtain ˆc by slicing k c k

5 Cancel effect of ˆc from y and re-organize the channel matrix k H by deleting its

kth column

6 Repeat Steps 2 to 6 until all element of vector c are detected

7 Re-arrange the vector c following the transmitted order

8 Recover the pair signals ˆx  c and ˆ1 ˆ ˆ

ˆx

c

To overcome the matrix inversion step in the MVBLAST, we apply MMSE-SQRD [12] to recover the received signal

Ann T n Rn T extended channel matrix D and a received vector z is defined as

, 1

0

T

n s

E

 

    

 

H

y

I



(4)

Table III.MSQRD detection algorithm

1 Decompose D by MMSE-SQRD [12]toget Q,R and the permutation vector p

2 Compute vQ z H

3

Detection and cancellation:

for n T: 1:1

if kn T obtain ˆc by slicing k v k k, r k k,

elsefor lk1:n T

, ˆ

k k k l k

end

Obtain ˆc by slicing k v k k, r k k,

end

end

4 Re-arrange the recovered vector ˆc by the permutation vector p

5 The pair of signals is given ˆ ˆ ,1 ˆ ˆ

ˆ

x

 sc

D ISQRD Detector

The equivalent of the system is rewritten as

 

where tyh1x,  1 2 3

T

n  n  H h h  h  with , 1, ,

k k  n T

h being the

k-thcolumn of the equivalent channel matrix H and c is the n  T 11 new transmitted codeword consisting the last n  T 1 entries of c The remaining vector c is applied in

MSQRD The ISQRD algorithm is summarized in Table IV

Table IV.ISQRD detection algorithm

1 Decompose H by MMSE-SQRD [12] to getQ, R andthe permutation vector p

2

Detection and cancellation:

for m1:M

Compute tmyh1x m and

0

m

H  

  

 

t

for kn T 1: 1:1

if kn T  obtain 1 cˆm k, by slicing v k k, r k k,

else

for lk1:n T  1

, ˆ ,

k k k l m l

end

obtain cˆm k, by slicing v k k, r k k,

end

end

Compute d  m tmHcˆm,p 2

End

3 Find mˆ : ˆmarg min m d m

4 Find index ˆmin k corresponding to the minimum distance dmin among K values of

k

5 The estimated SC codeword and modulated symbol are given by:sˆskˆ,xx kˆ

6 The pair of signals is given ˆ ˆ,ˆ 1 ˆ ˆˆ ,

ˆ

T

 s  c p

4 IMPERFECT CHANNEL MODEL

In this section, we introduce a general imperfect channel model including the correlation effect (CE) and the channel estimation error (CEE)

A Channel estimation error

In [9], a measurement metric is proposed to denote the estimation error value,

0 and P-CSI is 1  i.e., 11  The HRSM channel matrix is given by 0

1 

(6) Where: is a complex Gaussian variable

B Correlation effect

Based onexponential correlation model [10] and Kronecker model [13], a comprehensive correlation model is proposed as

T

 

Each element of R is given by:

, , 1 ,

j i

i j ji









(8)

where r is the complex correlation coefficient of the neighboring branches

Generally, a comprehensive HRSM channel matrix for both effects are given by

T

5 SIMULATION RESULTS

In this section, the performance of the four HRSM detectors is compapred in I-CSI case where the HRSM is equipped with 4 transmit and receive antennas, 4-QAM is used for modulation technique The channel is assumed to be Rayleigh fading and varies after each transmit data block

Figure 2 Performance comparison of the HRSM detectors under I-CSI

Figure 2 compares the performance of various HRSM detectors under the I-CSI with two estimation error rates: 0.85and0.7 Simulation results show that all

algorithms robust with the channel estimation error (CEE) In these outcomes, the

MSQRD detector is affected by estimation error more heavily than the others Although the ISQRD detector achieves lower performance than the ML at the high SNR region, the ISQRD detector is still a potential candidate to substitute the ML in practice

Figure 3 HRSM detector preformances under proposed model

Figure 3 shows the HRSM detectors’ performance under the proposed model with correlation factor r 0.3 and channel error factor 0.8 It is seen that all detectors perform relatively well under correlation effect (CE) and CEE Specifially, the ISQRD surpasses the ML It can be explained by the fact that due to CE and CEE the ML

10-4

10-3

10-2

10-1

100

SNR (dB)

MSQRD,=0.7 MSQRD,=0.85 MVBLAST,=0.7 MVBLAST,=0.85 ISQRD,=0.7 ISQRD,=0.85 ML,=0.7 ML,=0.85

10-5

10-4

10-3

10-2

10-1

100

SNR (dB)

MSQRD,r=0.3,=0.8 MBLAST,r=0.3,=0.8 ISQRD,r=0.3,=0.8 ML,r=0.3,=0.8

incorrectly indentifies the transmited SC codeword while the ISQRD first chooses the best symbol to detect As a result, the ISQRD removes the chain detection error effect

In [14], the SM system capacity is given by

SM 1 elog2 e 1 e log 12 e

C m  p p  p p  (10)

Where: m is the number of transmit bits in a transmission period and p is the HRSM e

system probability error,p e  1 1p S1p xwherep and S p are the SC codeword x

probability error and the modulated signal probability error, respectively

Figure 4 Capacity comparison of HRSM, SM-OSTBC, STBC-SM, and SM

at 8 bpcu under P-CSI

Figure 5 Capacity comparison of HRSM, SM-OSTBC, STBC-SM, and SM

at 8 bpcu under I-CSI,0.7

0 1 2 3 4 5 6 7 8

SNRdB

HRSM(4,4,4),4QAM SM-OSTBC(4,4,4),64QAM STBC-SM(4,4,2),64QAM SM(4,4,1),64QAM

0 1 2 3 4 5 6 7 8

SNRdB

HRSM(4,4,4),  =0.7 SM-OSTBC(4,4,4),  =0.7 STBC-SM(4,4,2),  =0.7 SM(4,4,1),  =0.7

The HRSM capacity is compared with the SM [1], the STBC-SM [14], and the SM-OSTBC [3] ones under two cases: P-CSI and I-CSI Each system is equipped with 4 transmit, 4 receive antennas, and suitable active antennas and chooses an appropriate modulation technique at the same spectral efficiency 8 bits per channel use (bpcu) Figure

4 shows that at the low range of capacity, the HR-SM is less than the STBC-SM, but it is better than the others.At the high range of capacity, the HR-SM surpasses the others Particularly, at 6 bits per second per Hz (bits/s/Hz), the HR-SM obtains at approximately

11 dB Signal-to-Noise (SNR) gain while the STBC-SM and the SM-OSTBC works at 13

dB and the SM is 17 dB Figure 5 compares the capacity of the HR-SM, the STBC-SM, SM-OSTBC, and the SM under I-CSI with the estimation error rate 0.7 All systems’ performances moves 2 SNRdB gain in the left compared with these systems performance under P-CSI

6 CONCLUSION

In this paper, a general channel model includingboth channel estimation error and correlation effect is proposed Performance of variousexisting HRSM detectors was investigated under the proposed model Simulation results show that all HRSM detectors are robust in this case Specifically, the ISQRD detector can be a potential candidate in the HRSM system Finally, it is also shown that the HRSM system outperforms the existing

SM based MIMO ones in two cases: P-CSI and I-CSI

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TÓM TẮT

PHẨM CHẤT CÁC BỘ TÁCH TÍN HIỆU ĐIỀU CHẾ KHÔNG GIAN TỐC ĐỘ CAO

TRONG ĐIỀU KHIỂN KÊNH TRUYỀN KHÔNG HOÀN HẢO

Hệ thống điều chế không gian tốc độ cao (HRSM) đã được đề xuất với các ưu điểm nổi trội về hiệu suất sử dụng phổ tần Tuy nhiên, việc đánh giá phẩm chất HRSM mới chỉ được thực hiện trong điều kiện kênh truyền được ước lượng hoàn hảo Trong bài báo này chúng tôi đề xuất một mô hình kênh truyền không hoàn hảo tổng quát bao gồm cả lỗi ước lượng kênh truyền và hiệu ứng tương quan giữa các ăng-ten Các bộ tách sóng đã đề xuất trong hệ thống HRSM sẽ được khảo sát và đánh giá sử dụng mô hình này Ngoài ra, dung lượng của hệ thống HRSM cũng được phân tích trong trường hợp kênh truyền hoàn hảo (P-ISI) và kênh truyền không hoàn hảo (I-ISI)

Từ khóa: Điều chế không gian, Bộ tách sóng, Thông tin trạng thái kênh truyền, Hiệu ứng tương quan

Received date, 24 th February 2017

Revised manuscript, 4 th April 2017 Published on 26 th April 2017

Author affiliations:

Military Technical Academy;

*Corresponding author: qttdong@gmail.com