Low Complexity Turbo Equalization for LTE Uplink with ... - IEEE Xplore

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Abstract—In this paper, we propose two improved turbo equalizers for Long Term Evolution (LTE) uplink in presence of transmitter (Tx) in-phase and ...
Low Complexity Turbo Equalization for LTE Uplink with Transmitter IQ Imbalance Rui Liu, Xiao Li, Lv Ding and Xiqi Gao National Mobile Communication Research Laboratory, Southeast University Nanjing 210096, China Email:{ruiliu, li xiao, lvding, xqgao}@seu.edu.cn

Abstract—In this paper, we propose two improved turbo equalizers for Long Term Evolution (LTE) uplink in presence of transmitter (Tx) in-phase and quadrature-phase imbalance (IQI) which are known at the receiver. For multi-user multipleinput multiple-output (MU-MIMO) single-carrier frequency division multiple access (SC-FDMA) systems, we derive a optimal joint linear turbo equalizer under minimum mean-square error (MMSE) criterion. For the special scenario where there is only one user equipment (UE) and it occupies all sub-carriers, we derive an optimal widely linear MMSE (WLMMSE) turbo equalizer. Both of two equalizers are implemented in discrete frequency domain and only cause slightly increase in computational complexity compared to the conventional turbo equalizers. Simulation results show that both the proposed equalizers outperform the conventional turbo equalizers under corresponding scenarios. Index Terms—LTE, SC-FDMA, Tx IQI, MMSE, turbo equalization

I. I NTRODUCTION In modern communication systems, radio frequency (RF) transceivers contain various architectures such as heterodyne, homodyne, image-reject and so on. Among them, the homodyne (also called zeros intermediate-frequency, ZIF) architecture offers several important advantages such as low cost, amenable to monolithic integration, low power consumption and circumvented image problem. However, this architecture suffers from a number of inferior issues such as DC offset, in-phase and quadrature-phase imbalance (IQI) [1]. IQI is the mismatch between I and Q balances both in the RF transmitter (Tx) and receiver (Rx) and deteriorate the system performance. Because IQI is difficult to be efficiently and entirely eliminated in the analog domain due to power consumption, size and cost of the devices [8], many compensation techniques in the digital baseband domain are proposed. In [2]–[4], several methods are proposed to compensate Tx and Rx IQI in the receiver using frequency domain symbol processing for downlink. In [5], Rx IQI is estimated and cancelled to improve the performance of linear turbo receiver for uplink. Recently, in uplink, several methods are proposed to compensate the Tx IQI in the receiver, and they have an advantage that the cost of user equipment (UE) is able to be reduced by using the computing resources of the base station (BS). In [6], [7], with reference to SCBT systems, widely linear minimum mean-square error (WLMMSE) equalizers are proposed to improve the anti-Tx-IQI ability of the receiver. In [8], with reference to OFDMA and SC-FDMA systems with one BS

antenna, the effects of inter-user interference (IUI) caused by Tx IQI are analyzed, and a WLMMSE equalizer is used to reduce the interference. A sub-carrier allocation scheme is also proposed to avoid the IUI due to the Tx IQI. In [9], with reference to OFDMA systems, a pair of frequency tiles are allocated symmetrically beside the central subcarrier, and a tile pair is jointly detected. Those methods improve the anti-Tx-IQI abilities of the relative systems, but to the best of our knowledge, few works has been done focusing on both Tx IQI and inter-symbol interference (ISI) problem in uplink. Further more, when multiple users are sharing the same sub-carriers, the IUI can not be avoid by the subcarrier allocation in [8]. In this paper, when the Tx IQI is known at the BS receiver, we derive a low complexity joint linear soft input soft output (SISO) equalizer under MMSE criterion for multi-user multiple input multiple output (MU-MIMO) single carrier frequency domain division multiple access (SC-FDMA) systems in LTE, and then a low complexity WLMMSE SISO equalizer is also proposed for the special scenario when there is only one UE occupying all sub-carriers. Both the computational complexities of the two equalizers are distinctly lowered by matrix transformation. Simulation results show that both the proposed equalizers outperform the conventional turbo equalizers under corresponding scenarios. The paper is organized as follows. In section II, the equivalent baseband signal model with Tx IQI and the joint linear system model are described. In section III, a low complexity joint LMMSE SISO equalizer is derived for MU-MIMO SCFDMA systems. In section IV, a low complexity WLMMSE SISO equalizer is derived for the special single user scenario. In section V, the simulation environment and results are showed. Section VI gives the conclusions. Notation: matrix is represented by bold capital letter. 𝑁 ×𝑁 identify matrix is represented by I𝑁 , and its 𝑘-th column is represented by e𝑘 . Diagonal matrix is represented by Λ, and the 𝑘-th element of the diagonal is represented by (Λ)𝑘 . 𝐾 ×𝐿 zero matrix is represented by O𝐾×𝐿 . Vector is represented by bold lowercase letter. Scalar is represented by ordinary lower∗ 𝑇 𝐻 † case letter. (⋅) , (⋅) , (⋅) , (⋅) , ∣⋅∣, 𝔼 {⋅}, cov {⋅}, ℜ {⋅}, ℑ {⋅}, diag {⋅}, bdiag {⋅}, badiag {⋅}, vec {⋅} and ≜ respectively denote the conjugate, transpose, conjugate transpose, inverse, norm, expectation, covariance, real part, image part, diagonal matrix, block diagonal matrix, block anti-diagonal matrix, stacking and definition. The Kronecker product is represented

978-1-4244-7555-1/10/$26.00 ©2010 IEEE

d1,1

FM

s1,1

Q1 F

BICM

d L ,1

s L ,1 #

BICM

d L , NT Fig. 1.

FM

Q L FNH

DFT-spread OFDM

FM

s L , NT

Q L FNH

I s1,1

+CP &

I P/S sL ,1

+CP & P/S

I s L , NT

IQI

# IQI

IQI

I x1,1 #

Cluster l &l

FN

-CP &

1

S/P

S/P

#

y

II. SYSTEM MODEL A. Equivalent baseband signal model with Tx IQI The issues of Tx IQI arise in the radio frequency (RF) modulation stage in the low cost ZIF transceivers and cause the distortion of transmit RF signal. According to [2], the equivalent baseband transmit signal of the distorted transmit RF signal is (1) 𝑥 (𝑡) = 𝛼𝑠 (𝑡) + 𝛽𝑠∗ (𝑡) , where 𝛼 = cos 𝜑 + 𝑗𝜀 sin 𝜑 and 𝛽 = 𝜀 cos 𝜑 + 𝑗 sin 𝜑. 𝜀 and 𝜑 denote the amplitude and phase imbalances at the transmitter respectively. In block transmission systems, (1) can be rewritten in vectors by sampling.

s 𝑙,𝑛𝑡 = F𝐻 𝑁 Q𝑙 F𝑀 s𝑙,𝑛𝑡 ,



(2)

where Q𝑙 is the sub-carrier mapping matrix of cluster 𝑙. In this paper, localized mapping matrix according to LTE release 8 is adopted [15], and it can be represented by ] [ (3) Q𝑙 = O𝑀 ×𝑣𝑙 I𝑀 O𝑀 ×(𝑁 −𝑀 −𝑣𝑙 ) ,

L

˜

P/S

˜

*

#

f l1 , N R *

  V

#

where 𝑣𝑙 is the starting index of the contiguous sub-carriers of the 𝑙-th cluster. After inserting cyclic prefixes (CP), the signals are disturbed by the Tx IQI before they are send to the antenna. The relations between the transmit signals and corresponding DFT-spread OFDM signals are ↔



↔∗

x 𝑙,𝑛𝑡 = 𝛼𝑙,𝑛𝑡 s 𝑙,𝑛𝑡 + 𝛽𝑙,𝑛𝑡 s 𝑙,𝑛𝑡 ,

(4)

where 𝛼𝑙,𝑛𝑡 and 𝛽𝑙,𝑛𝑡 are known at the receiver [8]. Finally the disturbed transmit signals are send over triple selective fading channels. At the BS, the signals are received by 𝑁𝑅 antennas. In the 𝑛𝑟 -th antenna, after removing the CP, the signal is fed to an 𝑁 -point DFT block. The output discrete frequency domain receive signals can be expressed as 𝑓

y 𝑛𝑟 =

In accordance with [11], the transmission structure of MUMIMO SC-FDMA systems in LTE uplink in presence of Tx IQI is shown in Fig.1. A cluster described in the figure is a UE group in which 𝑁𝑇 users share the same time-slots and sub-carriers allocated to that cluster. Each UE is equipped with one antenna. In the following, we describe a transmitter IQ-imbalanced MU-MIMO SC-FDMA transmission with 𝐿 clusters and derive its joint linear system model. For the 𝑛𝑡 -th UE in the 𝑙-th cluster, a frame of information bits d𝑙,𝑛𝑡 are converted into symbols after bit-interleaved coded modulation (BICM). The symbols are divided into several 𝑀 -length packets represented by s𝑙,𝑛𝑡 ,𝑝 where 𝑝 is the packet index and is ignored in the following for conveniens. The symbol packets are then spread by DFT-spread OFDM modules yielding outputs

P/S

The proposed BS structure of MU-MIMO SC-FDMA systems



B. MU-MIMO SC-FDMA transmission

H l

P/S

f l ,1

H l

y

Q

Transmission structure of MU-MIMO SC-FDMA systems

#

y lf, N R

#

Fig. 2.

P/S

QlH

Q

by ⊗. The normalized discrete Fourier transform (DFT) matrix 𝐻 is represented by F𝑁 , and F𝐻 𝑁 F𝑁 = F 𝑁 F𝑁 = I𝑁 .

H l

#

FN I y Nf R

-CP &

NR

y mf

y lf,1

Q

I y1f

#

I x L ,1 I x1, NT

I x L , NT

Other cluster couple

LMMSE SISO Equalizer

Cluster L

+CP & P/S

2 NT User's LLR

BICM

#

H N

2

#

Cluster 1

𝑁𝑇 𝐿 ∑ ∑ 𝑙=1 𝑛𝑡 =1

+

Λ𝑓𝑙,𝑛𝑟 ,𝑛𝑡 (𝛼𝑙,𝑛𝑡 Q𝑙 F𝑀 s𝑙,𝑛𝑡

𝛽𝑙,𝑛𝑡 P𝑁 Q𝑙 F∗𝑀 s∗𝑙,𝑛𝑡 )

+

(5)

z𝑓𝑛𝑟 ,

where Λ𝑓𝑙,𝑛𝑟 ,𝑛𝑡 represent the frequency domain channel gains from the 𝑛𝑡 -th UE in the 𝑙-cluster to the 𝑛𝑟 -th receive antenna in the BS. And it is a diagonal matrix with the 𝑛-th diagonal elements represent as 𝑁 ) ( ∑ Λ𝑓𝑙,𝑛𝑟 ,𝑛𝑡 = h𝑙,𝑛𝑟 ,𝑛𝑡 (𝑘) 𝑒−𝑗2𝜋(𝑘−1)(𝑛−1)/𝑁 , 𝑛

(6)

𝑘=1

where h𝑙,𝑛𝑟 ,𝑛𝑡 are the corresponding time domain channel gains which are assumed to be invariant during a block. In (5), P𝑁 is a mirror permutation matrix defined as P𝑁 = F𝑁 F𝑇𝑁 , and it has the property (P𝑁 x)𝑘 = x ((𝑁 − 𝑘 + 1) mod 𝑁 + 1) ,

(7)

where (⋅)𝑘 represent the 𝑘-th element of a vector. From (5) we can see that for each cluster 𝑙 located from subcarrier 𝑣𝑙 to 𝑣𝑙 + 𝑀 , its frequency interference caused by Tx IQI is located from [(𝑁 − 𝑣𝑙 − 𝑀 + 1) mod 𝑁 + 1] to [(𝑁 − 𝑣𝑙 + 1) mod 𝑁 + 1] which means the location of the interference is mirrored over the (𝑁/2 + 1)-th subcarrier (also

called DC subcarrier [9]). If cluster ¯𝑙 is allocated with the mirror location of cluster 𝑙, then both clusters are called a cluster couple in which the interference of one cluster caused by the Tx IQI is mirrored to the opposite cluster and will not affect other clusters in the system. In this paper, all the clusters are arranged into couples, and a cluster couple is jointly equalized. The structure of MU-MIMO SC-FDMA receiver with joint equalizer is illustrated in Fig.2. For a couple contains cluster 𝑙 and ¯𝑙, the frequency receive signals of cluster 𝑙 on the 𝑛𝑟 -th antenna are obtained by ↔𝑓 applying Q𝐻 𝑙 to the receive frequency signal y 𝑛𝑟 , yielding 𝑓 y𝑙,𝑛 = 𝑟

𝑁𝑇 ∑ 𝑛𝑡 =1

+

𝑓 (𝛼𝑙,𝑛𝑡 Q𝐻 𝑙 Λ𝑙,𝑛𝑟 ,𝑛𝑡 Q𝑙 F𝑀 s𝑙,𝑛𝑡

The indicator badiag {⋅} in (12) means block anti-diagonal matrix which equals to the left-right flipping of bdiag {⋅}. ˜ 𝑓 is a block cross-diagonal matrix. From (12) we can see H III. L INEAR T URBO E QUALIZATION FOR MU-MIMO SC-FDMA SYSTEMS WITH T X IQI The turbo receiver is an iterative receiver which performs turbo detection by passing soft information of code bits between a SISO equalizer and a SISO decoder. In this section, we review general linear system of turbo receiver and propose a MMSE-based SISO linear equalizer for MU-MIMO SCFDMA systems with Tx IQI. The LMMSE SISO equalizer contains a linear equalizer and a soft demodulator [10], [11]. The equalizer uses the channel information and a priori information of transmit symbols to equalize the receive symbols and deliver the MMSE estimation of transmit symbols and their conditional variances to the soft demodulator where the inputs are translated into bit log likelihood ratios (LLR) and delivered to the soft decoder. According to [10] the linear MMSE equalization of the combined packet symbols ˜ s in (10) can be represented as ( )† ˆ ˜𝑓F ˜ 𝑓𝐻 H ˜V ˜F ˜𝐻 H ˜ 𝑓 𝐻 + Φ𝑧˜ ˜𝐻 H ˜ s =F (14) ) ( ˜ 𝝁 + diag {𝝆} ˜ 𝑓 F˜ ˜ 𝝁 ˜, ⋅ y ˜𝑓 − H

(8)

𝑓 𝛽¯𝑙,𝑛𝑡 Q𝐻 P Q𝑙 F∗𝑀 s¯∗𝑙,𝑛𝑡 ) 𝑙 Λ¯ 𝑙,𝑛𝑟 ,𝑛𝑡 𝑁 ¯

+

𝑓 Q𝐻 𝑙 z𝑛𝑟 .

For the opposite cluster ¯𝑙 on the 𝑛𝑟 -th antenna , the frequency receive signals are conjugated to make the combined receive signal being a linear transformation of the combined packet symbols of two clusters, yielding 𝑓∗ = y¯𝑙,𝑛 𝑟

𝑁𝑇 ∑ 𝑛𝑡 =1

𝑓∗ ∗ (𝛽𝑙,𝑛 Q¯𝐻 𝑙 Λ𝑙,𝑛𝑟 ,𝑛𝑡 P𝑁 Q𝑙 F𝑀 s𝑙,𝑛𝑡 𝑡

𝑓∗ ∗ + 𝛼¯𝑙,𝑛 Q¯𝐻 𝑙 Λ¯ 𝑙,𝑛 𝑡

𝑟 ,𝑛𝑡

(9)

𝑓∗ Q¯𝑙 F∗𝑀 s¯∗𝑙,𝑛𝑡 ) + Q¯𝐻 𝑙 z𝑛𝑟 .

Combining (8) and (9) and rewriting them into the matrix form, we get the joint linear system model of a couple as ˜s + ˜ ˜ 𝑓 F˜ z𝑓 , y ˜𝑓 = H where 𝑓

y ˜ =

[

𝑓 vec([y𝑙,1 ,⋅⋅⋅

[

𝝁 ˜ =𝔼 {˜ s} [ = 𝜇𝑙,1,1 , 𝜇𝑙,2,1 , ⋅ ⋅ ⋅ , 𝜇𝑙,𝑁𝑇 ,1 , ⋅ ⋅ ⋅ , 𝜇𝑙,𝑁𝑇 ,𝑀 ,

(10) ]𝑇 𝑓∗ , y¯𝑙,𝑁 ]𝑇 ) 𝑇 𝑅 ]𝑇 , s¯∗𝑙,𝑁𝑇 ]𝑇 )𝑇 ,

𝑓 𝑓∗ , y𝑙,𝑁 ]𝑇 )𝑇 , vec([y¯𝑙,1 ,⋅⋅⋅ 𝑅

˜ s = vec([s𝑙,1 , ⋅ ⋅ ⋅ , s𝑙,𝑁𝑇 ]𝑇 )𝑇 , vec([s¯∗𝑙,1 , ⋅ ⋅ ⋅ ]𝑇 [ 𝑓∗ 𝑓∗ , ⋅ ⋅ ⋅ , z¯𝑙,𝑁 ]𝑇 ) 𝑇 , ˜ z𝑓 = vec([z𝑓𝑙,1 , ⋅ ⋅ ⋅ , z𝑓𝑙,𝑁𝑅 ]𝑇 )𝑇 , vec([z¯𝑙,1 𝑅 [ ] F ⊗ I 𝑀 𝑁 𝑇 ˜= F . (11) F∗𝑀 ⊗ I𝑁𝑇 The frequency noises of each UE in each cluster are z𝑓𝑙,𝑛𝑟 = 𝑓∗ 𝑓 = Q¯𝐻 z𝑓 ∗ . The reshaped frequency channel Q𝐻 𝑙 z𝑛𝑟 and z¯ 𝑙 𝑛𝑟 𝑙,𝑛𝑟 gains matrix is 𝑓∗ ˜ 𝑓 = bdiag{H𝑓 , ⋅ ⋅ ⋅ , H𝑓 H , ⋅ ⋅ ⋅ , H𝑓𝛼,∗¯𝑙,𝑀 } 𝛼,𝑙,1 𝛼,𝑙,𝑀 , H𝛼,¯ 𝑙,1 ∗ ∗ , ⋅ ⋅ ⋅ , H𝑓𝛽,𝑙,𝑀 }, + badiag{H𝑓𝛽,¯𝑙,1 , ⋅ ⋅ ⋅ , H𝑓𝛽,¯𝑙,𝑀 , H𝑓𝛽,𝑙,1 (12)

where each element in (12) is a 𝑁𝑅 × 𝑁𝑇 matrix with subelements defined as ) ( ) ( = 𝛼𝑙,𝑛𝑡 Λ𝑓𝑙,𝑛𝑟 ,𝑛𝑡 , H𝑓𝛼,𝑙,𝑚 ( )𝑛𝑟 ,𝑛𝑡 ( )𝑣𝑙 +𝑚 H𝑓𝛽,¯𝑙,𝑚 = 𝛽¯𝑙,𝑛𝑡 Λ¯𝑓𝑙,𝑛 ,𝑛 , 𝑟 𝑡 𝑣 +𝑚 𝑛𝑟 ,𝑛𝑡 ( ) ( ) 𝑙 (13) 𝑓∗ ∗ H𝑓𝛼,∗¯𝑙,𝑚 Λ¯𝑙,𝑛 = 𝛼¯𝑙,𝑛 , ,𝑛 𝑡 𝑟 𝑡 𝑣¯+𝑚 𝑛 ,𝑛 ( ) 𝑟 𝑡 ( ) 𝑙 𝑓∗ 𝑓∗ ∗ H𝛽,𝑙,𝑚 = 𝛽𝑙,𝑛𝑡 Λ𝑙,𝑛𝑟 ,𝑛𝑡 . 𝑛𝑟 ,𝑛𝑡

where

𝑣𝑙¯+𝑚

,

]𝑇 , 𝜇¯∗𝑙,1,1 , 𝜇¯∗𝑙,2,1 , ⋅ ⋅ ⋅ , 𝜇¯∗𝑙,𝑁𝑇 ,1 , ⋅ ⋅ ⋅ , 𝜇¯∗𝑙,𝑁𝑇 ,𝑀 { } 𝐻 ˜ =𝔼 (˜ V s−𝝁 ˜ )(˜ s−𝝁 ˜) { =diag 𝑣𝑙,1,1 , 𝑣𝑙,2,1 , ⋅ ⋅ ⋅ , 𝑣𝑙,𝑁𝑇 ,1 , ⋅ ⋅ ⋅ 𝑣𝑙,𝑁𝑇 ,𝑀 , } 𝑣¯𝑙,1,1 , 𝑣¯𝑙,2,1 , ⋅ ⋅ ⋅ , 𝑣¯𝑙,𝑁𝑇 ,1 , ⋅ ⋅ ⋅ 𝑣¯𝑙,𝑁𝑇 ,𝑀 , { 𝐻} Φ𝑧˜ =𝔼 ˜ = 𝜎𝑧2˜ I2𝑀 𝑁𝑇 , z˜ z 𝝆˜ =[𝜌1 , 𝜌2 , ⋅ ⋅ ⋅ , 𝜌2𝑀 𝑁𝑇 ]𝑇 ,

(15)

in which

( )† ˜𝑓F ˜𝐻 ˜ 𝑓𝐻 H ˜V ˜F ˜𝐻 H ˜ 𝑓 𝐻 + Φ𝑧˜ H ˜ 𝑓 Fe ˜ 𝑘. 𝜌𝑘 = e𝐻 𝑘 F H

(16)

In (14), The symbol 𝜇𝑙,𝑛𝑡 ,𝑚 products 𝜌𝑚𝑁𝑇 +𝑛𝑡 and the symbol 𝜇¯∗𝑙,𝑛𝑡 ,𝑚 products 𝜌(𝑀 +𝑚)𝑁𝑇 +𝑛𝑡 . To simplify (14), the reconstructed variance of the transmit packet symbols are assumed to be identical within a packet [10], so the covariance matrix of the packet symbols of the 𝑛𝑡 -th UE in the 𝑙-th cluster can be represented as cov (s𝑙,𝑛𝑡 , s𝑙,𝑛𝑡 ) =diag {𝑣𝑙,𝑛𝑡 ,1 , ⋅ ⋅ ⋅ , 𝑣𝑙,𝑛𝑡 ,𝑀 } ≜¯ 𝑣𝑙,𝑛𝑡 I𝑀 .

(17)

where 𝑣¯𝑙,𝑛𝑡 are corresponding average variance. The covariance matrix of the combined transmit symbols ˜ s can be simplified as ] [ ¯𝑙 ˜ = I𝑀 ⊗ v , (18) V I𝑀 ⊗ v ¯¯𝑙

where v ¯𝑙 = [¯ 𝑣𝑙,1 , ⋅ ⋅ ⋅ , 𝑣¯𝑙,𝑁𝑇 ]𝑇 and v ¯¯𝑙 = [¯ 𝑣¯𝑙,1 , ⋅ ⋅ ⋅ , 𝑣¯¯𝑙,𝑁𝑇 ]𝑇 . Applying (18) to (14), we get the simplified equalizer in the frequency domain represented as ( )† ˆ ˜ 𝑓𝐻 H ˜𝑓V ˜ + 𝜎𝑧2˜ I2𝑀 𝑁𝑇 H ˜ 𝑓𝐻 ˜𝐻 H ˜ s =F (19) ) ( ˜ 𝝁 + diag {𝝆} ˜ 𝑓 F˜ ˜ 𝝁 ˜. ⋅ y ˜𝑓 − H For conveniens of discussion, we define the sum matrix as ˜𝑓V ˜ + 𝜎𝑧2˜ I2𝑀 𝑁𝑇 . ˜ 𝑓𝐻 H ˜𝑓 ≜ H Ω

(20)

˜ 𝑓 is a 2𝑀 𝑁𝑅 × 2𝑀 𝑁𝑇 block cross-diagonal maBecause H ˜ 𝑓 is also a 2𝑀 𝑁𝑇 ×2𝑀 𝑁𝑇 block crosstrix, the sum matrix Ω diagonal matrix which all sub-matrixes are 𝑁𝑇 × 𝑁𝑇 . The ˜ 𝑓 † can be fast achieved using ˜ 𝑓 represented by Ω inverse of Ω the property of block matrix and the calculation ) ( cross-diagonal to the inverse of complexity is 𝒪 𝑀 (2𝑁𝑇()3 comparing ) block diagonal matrix 𝒪 𝑀 𝑁𝑇3 of the conventional turbo equalizer. ˜ 𝑓 † is achieved, 𝜌𝑘 in (16) can be simplified as When Ω ˜𝐻 ˜ 𝑓† ˜ 𝑓𝐻 H ˜ 𝑓 Fe ˜ 𝑘. 𝜌𝑘 = e𝐻 𝑘 F Ω H

(21)

To fast achieve 𝜌𝑘 , we define its frequency products as ˜ 𝑓𝐻 H ˜𝑓. ˜ 𝑓 †H Υ𝑓 ≜ Ω

(22)

where Υ𝑓 is also a block cross-diagonal matrix and can be written in the expanding form as Υ𝑓 =bdiag{Υ1,1 , ⋅ ⋅ ⋅ , Υ1,𝑀 , Υ2,1 , ⋅ ⋅ ⋅ , Υ2,𝑀 } +badiag{Υ3,1 , ⋅ ⋅ ⋅ , Υ3,𝑀 , Υ4,1 , ⋅ ⋅ ⋅ , Υ4,𝑀 },

(23)

where all elements Υ𝑘,𝑚 are 𝑁𝑇 × 𝑁𝑇 matrixes. And 𝜌𝑘 can be fast achieved in its vector form as [ ∑𝑀 Υ1,𝑚 /𝑀 𝑁𝑇 }𝑇 , 𝝆˜ = 1𝑇𝑀 ⊗ diag{ 𝑚=1 (24) ]𝑇 ∑𝑀 Υ2,𝑚 /𝑀 𝑁𝑇 }𝑇 , 1𝑇𝑀 ⊗ diag{ 𝑚=1

where 1𝑀 is a 𝑀 -length all-ones vector. We can see that 𝝆˜ is irrelevant with the anti-diagonal blocks. So far the equalization (19) is fast achieved in the frequency domain. Remind the equalized symbols of the couple cluster are in conjugate form and another conjugate operation is needed for correction. The equalizer delivers the MMSE linear equalized packet symbols 𝑠ˆ𝑙,𝑛𝑡 ,𝑚 with their variances 𝑣𝑙,𝑛𝑡 ,𝑚 and corresponding 𝜌𝑘 to the soft demodulator to generate bit LLR of the coded bits of the 𝑛𝑡 -th UE of the 𝑙-th cluster. The proceeding is the same as in [10], [11]. IV. W IDELY L INEAR T URBO E QUALIZATION FOR S INGLE U SER U PLINK WITH T X IQI In last section, a joint liner equalizer working with a couple of cluster users is proposed. However, when there is a special user occupying all sub-carriers which means no couple cluster can be used, the joint linear equalizer is unfit and a new equalizer is proposed for this scenario in this section.

A. Widely linear Equalizer with a priori information A complex random variable is called proper when its real part and its image part is independent with each other [14]. When the variable 𝑥 is complex Gaussian, it is proper if and only if (25) 𝑐𝑥 = cov(𝑥, 𝑥∗ ) = 0 where 𝑐𝑥 is also called the pseudo variance of 𝑥. In last section, the estimate symbol 𝑠ˆ˜𝑘 in (19) is a Gaussian complex random variable according to central limit theorem. And it is also s proper because 𝑠ˆ˜𝑘 is a linear function of proper variables ˜ and n ˜ . So 𝑠ˆ˜𝑘 can be correctly translate into bit LLR by the conventional soft demodulator which assume the inputs are proper Gaussian. When there is only 1 UE occupying all sub-carriers in Fig.1, the system degrades to a single carrier system. We consider the most simplified scenario where there is 1 receive antenna at the BS, and the system can be represent as y = H (𝛼s + 𝛽s∗ ) + n ≜ Hx + n

(26)

where 𝛼 and 𝛽 are Tx IQI parameters of the UE which are assumed to be known at the receiver, and x are the Tx-IQIdisturbed transmit signals. We can see that y is not a linear function of the proper variables s, but is linear in the improper variables x. Such a system belongs to the widely linear (WL) systems [12], and the improper random variables x should be jointly estimated from the observations and their conjugates. In [6]–[8], y and y∗ are used as observations, and that leads to large residual ISI and deteriorates the receiver performance. In fact, (26) can be rewritten as x𝑘 + n, y = He𝑘 𝑥𝑘 + H˘

(27)

where x ˘𝑘 = [𝑥1 , ⋅ ⋅ ⋅ , 𝑥𝑘−1 , 0, 𝑥𝑘+1 , ⋅ ⋅ ⋅ 𝑥𝑁 ]𝑇 . The terms located orderly at the right hand side of (27) are called signal term, ISI term and noise term respectively. When 𝑥𝑘 is to be equalized, if the a priori information of the transmit signals defined as the expectation ¯ s and the covariance matrix V𝑠 is known from the SISO decoder, the expectation of ISI term in ˘ 𝑥1 , ⋅ ⋅ ⋅ , 𝑥 ¯𝑘−1 , 0, 𝑥 ¯𝑘+1 , ⋅ ⋅ ⋅ 𝑥 ¯𝑁 ]𝑇 . The (27) are known as x ¯𝑘 = [¯ WL equalization of 𝑥𝑘 with a priori information should use ´𝑘∗ both of which the interference terms observations y ´𝑘 and y are cancelled, where ˘ ¯𝑘 . y ´𝑘 ≜ y − Hx

(28)

According to [12], the WL equalization of 𝑥𝑘 with a priori information is [ 𝑇 𝐻 ]𝑇 ´𝑘 y ´𝑘 , (29) 𝑥 ˆ𝑘 = g𝑘𝐻 y where g𝑘𝐻 is a 1 × 2𝑁 row vector representing the WL equalizer coefficients. According to the MMSE orthogonality principle, they should meet the equation { [ 𝑇 𝐻 ]𝑇 } = 0. (30) ´𝑘 y ´𝑘 𝔼 (ˆ 𝑥𝑘 − 𝑥 𝑘 ) y Assuming V𝑠 = 𝑣¯𝑠 I𝑁 for simplifying [10], we solve (30) and obtain the WLMMSE coefficients ( )† ˜ +퓗 ˜ 𝑘 Γ퓗 ˜𝐻 ˜𝐻 A , (31) g𝑘𝐻 = [𝑟𝑥 𝑚𝑥 ] 퓗 𝑘 𝑘

TABLE I S IMULATION PARAMETERS

where

2

(32)

2

𝑟𝑥 = ∣𝛼∣ + ∣𝛽∣ , 𝑚𝑥 = 2𝛼𝛽, ( 2 2) 𝑣¯𝑥 = ∣𝛼∣ + ∣𝛽∣ 𝑣¯𝑠 , 𝑐¯𝑥 = 2𝛼𝛽¯ 𝑣𝑠 . (31) can be further simplified by using matrix transformation ( )† ( )† 𝐻 ˜𝐻 ˜ ˜ ˜† ˜ ˜ ˜† ˜ 𝐻 = I2 + 퓗 ˜𝐻 퓗 𝑘 A + 퓗 𝑘 Γ 퓗𝑘 𝑘 A 퓗𝑘 Γ 퓗𝑘 A , (33) †˜ ˜ ˜𝐻 where I2 + 퓗 A 퓗 Γ is a 2 × 2 matrix. Applying (33) to 𝑘 𝑘 (31) and (29), assuming the channel is time invariant within a block, we get the WLMMSE turbo equalizer for single user uplink with Tx IQI which is quickly achieved in the discrete frequency domain [ ] [ ] ) 𝐻 Λ∗ℎ Λ1 Λ∗ℎ Λ2 ( y − H¯ x ˜ ˜ F x ˆ = 𝝎 𝐻 ⊗ I𝑁 F ∗ (y − H¯ x) Λ′ℎ Λ∗2 Λ′ℎ Λ′1 (( 𝐻 ) ) [ 𝑇 𝐻 ]𝑇 + 𝝎 Ψ𝜌 ⊗ I 𝑁 x ¯ x ¯ , (34)

FFT size

2048

Modulation

Symbol rate

30.72M

Mapping

Gray

Carrier Frequency

2.0G Hz

Channel coding

Turbo

Cluster subcarriers

300

Coding rate

1/2

Channel Model

EVA

Generators

(11, 13)8

Vehicle Speed

120km/h

Memory length

3

Channel Estimation

Perfect

Bit interleaver length

8400

frame length

1ms

Cluster bandwidth

5 MHz

Equalizer Iterative

6 Times

Decoder Iterative

4 Times

0

∑𝑁 𝑛=1

h (𝑛) 𝑒−𝑗(𝑛−1)(𝑘−1)/𝑁

−3

10

−4

10

2 ∣(Λ′ℎ )𝑘 ∣ ,

− ∣¯ 𝑐𝑥 ∣ ∣(Λℎ )𝑘 ∣ ∑𝑁 ∗ 𝜌1 = 𝑁1 (Λℎ )𝑘 (Λℎ )𝑘 (Λ1 )𝑘 , 𝑘=1 ∑𝑁 ∗ ∗ (Λ′ℎ )𝑘 (Λℎ )𝑘 (Λ2 )𝑘 , 𝜌2 = 𝑁1 𝑘=1 ] [ 𝜌1 𝜌2 ˜ = bdiag {F𝑁 , F𝑁 } , , F Ψ𝜌 = 𝜌∗2 𝜌1

(35)

By using (33), the complexity of the equalizer is reduced from 𝒪(𝑁 4 ) which is obtained by applying (31) to each 𝑥𝑘 to 𝒪(𝑁 log(𝑁 )). B. Soft demodulator for WLMMSE turbo equalizer ´𝑘 , y ´𝑘∗ and 𝑥 ˆ𝑘 are also improper. Because 𝑥𝑘 is improper, y According to the central limit theorem 𝑥 ˆ𝑘 is a improper Gaussian random variable. According to [13], its condition probability density function (PDF) is 2

∣ˆ 𝑥𝑘 − 𝜇𝑎 ∣ 𝑣𝑎 − ℜ{(ˆ 𝑥𝑘 − 𝜇𝑎 ) 𝑐𝑎 } 𝑣𝑎2 − ∣𝑐𝑎 ∣

2

2

3 SNR (dB)

4

5

6

𝑥𝑘 , 𝑥 ˆ𝑘 ∣𝑠𝑎 } = 𝝎 𝐻 Ψ𝜌 (I2 − Φ𝑥 Ψ𝜌 ) 𝝎, 𝑣𝑎 = cov {ˆ 𝑐𝑎 = cov {ˆ 𝑥𝑘 , 𝑥 ˆ∗𝑘 ∣𝑠𝑎 } = 𝝎 𝐻 Ψ𝜌 (I2 − Φ𝑥 Ψ𝜌 ) J2 𝝎 ∗ , (37)

𝑃 (𝑠𝑎 ∣ˆ 𝑥𝑘 ) = ∑

𝝎 𝐻 = [𝑟𝑥 𝑚𝑥 ] (I2 + Ψ𝜌 Γ) .

∝ exp −

1

in which J2 is the left-right flip matrix of I2 , and 𝑠𝑎 comes from the transmit modulation constellation 𝔖. The a posterior probability (APP) of 𝑠𝑎 is obtained by



2

0

[ ]𝑇 𝑥𝑘 ∣𝑠𝑎 } = 𝝎 𝐻 Ψ𝜌 (𝛼𝑠𝑎 + 𝛽𝑠∗𝑎 ), (𝛼∗ 𝑠∗𝑎 + 𝛽 ∗ 𝑠𝑎 ) , 𝜇𝑎 = 𝔼 {ˆ

(Λ2 )𝑘 = − )( ) ( 2 2 (Λ)𝑘 = 𝑣¯𝑥 ∣(Λℎ )𝑘 ∣ + 𝜎𝑛2 𝑣¯𝑥 ∣(Λ′ℎ )𝑘 ∣ + 𝜎𝑛2

𝑃 (ˆ 𝑥𝑘 ∣𝑠𝑎 ) {

EVA Convent. It1 EVA Convent. It6 EVA Proposed It1 EVA Proposed It6 EVA Convent. It6 No IQI AWGN Proposed It6 AWGN Convent. It6 No IQI

where

𝑐¯𝑥 (Λℎ )𝑘 (Λ′ℎ )𝑘 /(Λ)𝑘 , 2

−2

10

Fig. 3. BER performance. 4 Tx 4 Rx, only 1 bad UE with Tx IQI 𝜖 = 0.3, 𝜙 = 5 in each cluster

(Λ′ℎ )𝑘 ≜(Λℎ )(𝑁 −𝑘+1) mod 𝑁 +1 , ) ( 2 (Λ1 )𝑘 = 𝑣¯𝑥 ∣(Λℎ )𝑘 ∣ + 𝜎𝑛2 /(Λ)𝑘 ,

2

MIMO−EVA Channel

MIMO−AWGN Channel

−1

10

where (Λℎ )𝑘 =

QAM

10

BER

˜ =H ˜V ˜H ˜ 𝐻 + 𝜎𝑛2 I2𝑁 , H ˜ = bdiag{H, H∗ }, A ] [ ] [ 𝑟𝑥 − 𝑣¯𝑥 𝑚𝑥 − 𝑐¯𝑥 𝑣¯𝑥 𝑐¯𝑥 Φ𝑥 = , Γ= , 𝑐¯∗𝑥 𝑣¯𝑥 𝑚∗𝑥 − 𝑐¯∗𝑥 𝑟𝑥 − 𝑣¯𝑥 ˜ = Φ𝑥 ⊗ I𝑁 , 퓗 ˜ 𝑘 = bdiag{He𝑘 , H∗ e𝑘 }, V

} (36)

𝑃 (ˆ 𝑥𝑘 ∣𝑠𝑎 ) , 𝑥𝑘 ∣𝑠𝑎 ) 𝑠𝑎 ∈𝔖 𝑃 (ˆ

(38)

which is used to calculate LLR of coded bits same as conventional soft demodulators [10]. V. S IMULATION R ESULTS The performances of the proposed turbo equalizers are evaluated by Monte Carlo simulations. The main parameters used for simulation are following UTRA LTE Uplink [15] context given in Table I. The Extended Vehicler A (EVA) channel model is used in the simulation. The channel information and IQ imbalance parameters are perfect known at the receiver. In Fig.3 and Fig.4 the average BER performance of MUMIMO SC-FDMA systems versus average receiver SNR is shown for the conventional [10] and the proposed turbo equalizer. There are 4 receive antennas in the BS. And each clusters contains 4 UEs. In Fig.3, there is 1 bad UE with

0

0

10

10 MIMO−AWGN Channel

MIMO−EVA Channel −1

10

−1

10

−2

BER

BER

10 −2

10

−3

10

EVA Conv. It1 EVA Conv. It6 EVA Prop. It1 EVA Prop. It6 EVA Conv. It6 No IQI AWGN Prop. It6 AWGN Conv. It6 No IQI

−3

10

EVA Channel −4

10

Conventional It1 Conventional It6 Proposed It1 Proposed It6 No IQI Convent. It6

−4

10

Fig. 4.

0

1

2

3 SNR (dB)

4

5

6

BER performance. 4 Tx 4 Rx, all UE with Tx IQI 𝜖 = 0.3, 𝜙 = 5

Tx IQI 𝜖 = 0.3, 𝜙 = 5 in each cluster and other UEs are ideal IQ matched. We can see that in the MIMO-EVA channel the performance of conventional equalizer is about 0.5dB deteriorated from no Tx IQI case, while the proposed equalizer is about 0.1 dB ameliorated. It can be explained as that because the combined received signals are linear in the combined packet symbols, the IQI effect can be equivalent to 2 frequency domain paths. The proposed receiver obtains multipath diversity gains [2] by joint LMMSE turbo equalization. When the frequency selective channel is replaced by the MIMO-AWGN channel, that diversity gains disappear. In Fig.4 all UEs are IQ imbalanced with 𝜖 = 0.3 and 𝜙 = 5. We can see in MIMO-EVA channel the conventional turbo equalizer is about 2dB deteriorated compared to the no Tx IQI case while the proposed equalizer is about 0.5dB ameliorated, and that extra performance gains also disappear in MIMOAWGN channel. Those results are agree with Fig.3. Fig.5 shows the performance of conventional LMMSE and proposed WLMME turbo equalizers in the special scenario where there is only one UE with Tx IQI 𝜖 = 0.3, 𝜙 = 5 and it occupies all sub-carriers. The receiver has 1 antenna and the system frame length is 0.14ms. We can see that the proposed equalizer obtain about 1 dB performance gains compared to the conventional turbo equalizer, but it is still about 0.6 dB inferior to the no Tx IQI case. That is different from the results in Fig.3 and Fig.4. It is because in this case the receive signals are not linear in packet symbols, the interference caused by Tx IQI can not be equivalent to multi paths. When the parameters of Tx IQI are increased from zero, the MSE is still increased even though the WLMMSE equalization is adopt [6], but it is smaller compared to the conventional LMMSE equalization. VI. C ONCLUSION In this paper, we investigate the turbo receiver for LTE uplink systems in presence of Tx IQI. A low complexity LMMSE turbo equalizer based on the joint linear system model is proposed for MU-MIMO SC-FDMA systems. A low complexity WLMMSE turbor equalizer for the special scenario of single user systems is also proposed. Simulation results show that both equalizers outperform the conventional LMMSE turbo equalizers.

1.5

Fig. 5.

2

2.5

3 SNR (dB)

3.5

4

4.5

BER performance. 1 Tx 1 Rx, 1 UE with Tx IQI 𝜖 = 0.3, 𝜙 = 5

ACKNOWLEDGMENT The work was supported by National Natural Science Foundation of China under Grants 60925004, and 60902009, the China High-Tech 863 Plan under Grant 2006AA01Z264, National Basic Research Program of China under Grant 2007CB310603, and National Science and Technology Major Project of China under Grants 2009ZX03003-005 R EFERENCES [1] B. Razavi, “RF microelectronics,” Prentice Hall, pp. 118-138. 1998. [2] Y. Jin, J. Kwon, Y. Lee, J. Ahn, “Obtaining diversity gain coming from IQ imbalance in OFDM receivers,” IEEE VTC07, pp. 2175-2179, Apr. 2007. [3] Q. Zou, A. Taright, A. H.Sayed, “Joint compensation of IQ imbalance and phase noise in OFDM wireless systems,” IEEE Trans. Commun., vol. 57, no. 2, pp. 404-414, Feb. 2009. [4] L. I.Tai, C. Jiang, “Joint transmitter and receiver IQ imbalance estimation and compensation for OFDM systems,” IEEE RWS10 , pp. 476-479, Jan. 2010 [5] R. Cherukuri, P. T.Balsara,“Iterative (TURBO) IQ Imbalance Estimation and Correction in BICM-ID for Flat Fading Channels,” IEEE VTC07, pp. 2070-2074, Oct. 2007. [6] M. Lipardi, D. Mattera, F. Sterle, “MMSE equalization in presence of transmitter and receiver IQ imbalance,” IEEE IWDDC07, pp. 165-168, Oct. 2007. [7] D. Mattera, L. Paura, F. Sterle, “MMSE WL equalizer in presence of receiver IQ imbalance,” IEEE Trans. Signal Process., vol. 56, no. 4, pp. 1737-1738, Apr. 2008. [8] Y. Youshida, K. Hayashi, H. Sakai, W. Bocquet, “Analysis and compensation of transmitter IQ imbalances in OFDMA and SC-FDMA systems,” IEEE Trans. Signal Process., vol. 57, no. 8, pp. 3119-3129, Aug. 2009. [9] H. A.Mahmoud, H. Arslan, M. Kemal, “IQ imbalance correction for OFDMA uplink systems,” IEEE ICC’09, pp.1-5, Aug. 2009. [10] D. M. Wang, W. J. Wang, X. Y. Wang, B. J. Jeong, S. C. Kim, et al.,“Link level performance of SC-FDMA with SDMA, turbo equalization and real channel estimation,” CHINACOM09, pp. 1-5. May. 2009. [11] T. Li, W. J.Wang, X. Q.Gao,“Turbo equalization for LTE uplink under imperfect channel estimation,” IEEE PIMRC’09, PP.330-334, Sept. 2009. [12] B. Picinbono, P. Chevalier, “Widely linear estimation with complex data,” IEEE Trans. Signal Process, vol. 43, no. 88, pp. 2030-2033, Aug. 1995. [13] M. Tchler, A. C. Singer, ”Minimum mean squared error equalization using a priori information,” IEEE Trans. Signal Process., vol. 50, no. 3, pp. 673-683, Mar. 2002. [14] P. J. Schreier, L. L. Scharf, “Second-order analysis of improper complex random vectors and processes,” IEEE Trans. Signal Process. , vol. 51, no. 3, pp.714-725, Mar. 2003. [15] “Evolved universal terrestrial radio access (E-UTRA): Physical channels and modulation (release 8),” 3GPP TS36.211, 2008.