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Single-Carrier (SC) transmission with the same bandwidth as Multi-Carrier (MC) transmission (such as OFDM) may have far shorter symbol duration and is considered to be more robust against time selective fading. In this paper, we proposed the novel equalization and signal separation schemes in time domain for short block length transmission, i.e., Block Linear Equalization (BLE) and Block Nonlinear Equalization (BNLE) on MIMO frequency selective fading channels. The proposed BLE uses the MMSE based inverse matrix in time domain and the BNLE utilizes the QRD-M (QR Decomposition with M algorithm) with appropriate receiver complexity. We compared the computational complexity among the conventional SC-FDE (Frequency Domain Equalization) scheme and the proposed equalizers. We also used the Low-Density Parity Check (LDPC) decoder concatenated to the proposed BLE and BNLE.

For high speed mobile communications with wide transmission bandwidth, Inter-Symbol Interference (ISI) due to multipath propagation must be compensated. For the ISI compensation, the channel equalization such as FDE with Guard Interval (GI) in OFDM can effectively equalize the frequency selective fading [1,2]. However, with increasing number of subcarriers in OFDM, the symbol duration becomes large and the long symbol duration causes the problem of durability against time selective fading. Although it is considered that OFDM can cope with time selective fading with some compensation technique such as V-BLAST detection for each subcarrier [

In [8-10], the authors developed the time domain signal separation and equalization schemes for both cases when the Channel State Information (CSI) is available at the transmitter and only available at the receiver on MIMO frequency selective channels. Those papers are preliminaries for this paper. In [

In this paper, we propose the novel linear and nonlinear block equalization schemes in time domain for the SC transmission with short block length, where the CSI is only available at the receiver. We firstly defined the extended MIMO channel matrix H over the space and time domain for MIMO frequency selective channels. By carefully designing the transmit signals, we proposed several novel signal separation and equalization receiver structures using block processing in time domain with small number of transmit symbols. From the simulation results, the proposed receivers showed the excellent BER characteristics compared with the conventional SC FDE receiver, especially when the block size is small. In addition, we compare the computational complexity among SC-FDE, BLE, BNLE QRD-M and MLD, where we employ the similar steps in the case of BNLE [QRD-M] [

This paper is organized as follows. In Section 2, we introduce the system model on SISO and MIMO frequency selective channels. The proposed block equalization schemes will be described in Section 3, where we proposed and compared the several linear and nonlinear block equalization receivers. In Section 4, the simulation results are shown. In Section 5, the conclusions are given.

Throughout the paper, we illustrate some of the notations as follows; vectors and matrixes are expressed by the bold italic letters. and denote the expectation, pseudo-inverse, transpose and conjugate transpose of matrices, respectively.

We first introduce the SISO channel model on the frequency selective fading channel. The receive signal can be written as

where and are the transmit signal point at time k, overall channel impulse response including pulse shaping and transmit filters, the symbol duration and the AWGN noise, respectively. The matched filter is employed for the receive signal and the symbol time sampling is made. The sampled signal is then passed through the noise whitening filter to get the receive signal point at time k

where is the tap coefficient and is the independent complex Gaussian noise variable. Equation (2) is referred to as the tapped delay line model [

We consider the spatial multiplexing system with transmit and receive antennas. We define the spacetime extended matrix to apply the block processing for the multipath channel model.

The extended matrix H with the size can be written as follows.

where L is the number of multipath and is the element matrix, which follows the quasi-static flat fading with the size of, i.e.,

where is the l-th delay path gain from the j-th transmit antenna to the i-th receive antenna, and follows the i.i.d. Rayleigh amplitude distribution. It is also assumed, i.e., the average channel gain is normalized to one. The channel is assumed quasi-static, i.e., the channel gains are static within a block length, but change independently from block to block.

In this paper, we assumed that the extended channel matrix H is known at the receiver, i.e., the receiver has the perfect Channel State Information (CSI) by sending training sequences just before the data block.

Using H, we obtain the extended input and output re-

lationship given by

where Y is the extended receive vector expressed as

and is the element receive signal vector at time k. X is the extended transmit signal vector expressed as

and is the element transmit signal vector at time k. N is the extended receive noise vector expressed as

where is the element additive white Gaussian noise vector at time k. It also holds the following equations and.

This paper focuses on the signal separation and equalization of MIMO SC transmission schemes. In order to illustrate the BER performances of proposed schemes clearly, it is necessary to give a brief introduction of MIMO SC-FDE as the reference scheme.

In

The MIMO SC-FDE-CP scheme is illustrated in

where is the receive signal at time k from the transmit antenna j to the receive antenna i, is the complex channel gain of the l-th delay path, is the transmit symbol at time k from the transmit antenna j and is the complex Gaussian noise at time k. We have set the block length to N symbols. The channel matrix in (9) becomes the circulate convolution matrix and the FDE can be done using FFT.

The MIMO SC-FDE-ZP scheme is illustrated in

where each element in (10) is almost the same as in (9), but as shown in (10) the channel matrix now becomes the linear convolutional matrix. Also the size of receive vector is extended to due to the zero padding.

For the transmit signal vector in (7), we can expand its size arbitrary, i.e.

where N is again the number of symbols in a block. Accordingly, the sizes of extended transmit vector, channel matrix, receive vector and noise vector are expressed as

, and, respectively. The receive vector can be written as

In order to avoid the Inter-Block Interference (IBI) at the receiver, the transmit vector is processed by using Zero-Padding as shown in the

We insert zero symbols before and after the signal vector separately. The receive vector in (12) can be written as

where is the extended channel matrix with .

The transmit and receive models of the proposed time domain block equalizers are shown in

When the channel matrix is given as in (13), we can compensate the ISI and the IAI simultaneously by using Moore Penrose inverse matrix in the analogy of nulling operation as in MIMO flat fading channels.

The inverse matrix based on Zero Forcing (ZF) criterion is given by (14)

where denotes the pseudo-inverse matrix of H. The pseudo-inverse matrix exists when the number of columns is less than or equal to the number of rows. The estimates of X, which is denoted as, is given by

Equation (15) means that the ISI and IAI due to the channel are completely equalized. However, the disadvantage of ZF is that it suffers from the noise enhancement.

In order to circumvent the noise enhancement, the inverse matrix solution with the MMSE (Minimum Mean Square Error) criterion is obtained as

By multiplying by Y from the left hand side of (13), the estimates of based on MMSE criterion can be obtained. The MMSE solution in (16) trades off the signal separation quality for the noise enhancement reduction through the small term of.

Next we simply discuss the BLE in SISO and SIMO cases. In SISO case, the channel input and output equation is given by

where each element in (17) becomes scalar value and this corresponds to for the element in (4).

In SIMO case, the proposed BLE can also be applied, where the element in (13) becomes the vector with and the channel input and output expression is given by

It is necessary to discuss a special case in SIMO with the block size of. When the block size equals 1, Equation (13) can be written as

where the elements in (19) are represented as

From (19) and (20), the Maximum Radio Combining (MRC) is available and the weight vector is written as

The received signal after the MRC is expressed as

It is observed that the term achieves the sum of all the squared channel gains over all the receive antennas, and the full diversity with the order is obtained in this case.

In [6,7], the author proposed the idea of imaginary transmit antenna and employed the layered MLD detection to avoid the ISI. However, the computational complexity greatly increases when the number of receive antennas is large. Here we propose the QR decomposition with an M-algorithm (QRD-M). It is expected that the great reduction in computational complexity can be achieved without degrading the performance when compared with full MLD.

The QR-decomposition of channel matrix H is obtained as

where Q is the unitary matrix, R is the upper triangular matrix and. After multiplying by the received signal Y from the left hand side, we get

and the estimate is given by

From Equation (25), it is observed that by applying the tree search with the decoding order from to, we can calculate the Euclidian distance based branch metric and path metric of (26) and (27) respectively, for all the possible values of.

The accumulated path metrics are then ordered. Only M nodes with the smallest accumulated path metrics are retained and the rest is deleted. The same procedure is applied to the nodes in the next layer and this process continues up to the first layer. In QRD-M algorithm, the parameter M is used as the number of limit to the maximum survived branches in the breadth-first tree algorithm. By setting M equal to different values, it can provide different tradeoff solutions between the system performance and the complexity. The larger M value, the better performance is obtained, however the larger complexity is required.

In this section, we show the BER performance of the proposed schemes in MIMO, SIMO, and SISO quasistatic frequency selective channels. First we have compared the BER characteristics of BLE with the existing SC-FDE. The simulation conditions are shown in

As shown in _{s} and each tap gain follows the independent complex Gaussian random variable with equal power, i.e., the quasi-static multipath Rayleigh fading channel with equal delay path power has been assumed. Under the same simulation condition, we have made the comparison among the MIMO, SIMO, and SISO channels. _{T} = n_{R} = 4, L = 8 and we set the transmit block length to. In order to fairly compare the SC-FDE with the proposed schemes, we assume that the size of zero padding is 7 and the length of CP is, and by the Section 3.2, we insert zero symbols before and after the transmit signal vector. Since ZF-FDE with CP scheme can completely eliminate the inter block interferences, it performs better than ZF-FDE with ZP, the latter suffers from noisy prefix copy and sometime leads to a loss in frequency diversity [

Average E_{b}/N_{0} [dB]/receive antenna

full multipath diversity (the SISO case in equation (17) well supports this characteristic) and obtain the better BER performance by 5 - 8 dB at BER = 10^{−5} than the SC-FDE based on MMSE criterion. We can say, for the short block length like, BLE and QRD-M show the excellent performances among the compared schemes. In the case of BNLE with QRD-M with M = 1, we can obtain the better BER performance than BLE with MMSE by about 1 dB. From the complexity aspects of QRD-M algorithm, we only consider the case of M = 1 in the followings.

In ^{−5}.

In ^{−5}. How-

Average E_{b}/N_{0} [dB]/receive antenna

Average E_{b}/N_{0} [dB]/receive antenna

ever, these improvements are mainly brought by the effect of MRC, which is only usable for the size of N = 1. On the other hand, the SC-FDE with CP or ZP only uses the nulling operation, thus the receive powers from multiple receive antennas are not fully combined for the signal separation and equalization.

Figures 8 and 9 shows the BER performance for different blocks length of N on SISO and MIMO quasistatic frequency selective channels. The simulation result shows that the shorter block length can get more diversity gain in BLE and BNLE receivers. BLE suffers from the problem of exponential growth of computational complexity for calculating the inverse matrix when the

Average E_{b}/N_{0} [dB]/receive antenna

Average E_{b}/N_{0} [dB]/receive antenna

block size N becomes large. However, this problem does not exist for SC-FDE because of the FFT processing in frequency domain. Accordingly the merit that the shorter block length exhibits the better BER in BLE seems beneficial to the short time block transmission under the fading channel with time selectivity, because the short time block transmission is not very much influenced by the rapid time variation of the channel.

We compare the computational complexity among the proposed schemes. The total numbers of real products and additions required for detecting one transmit symbol are derived as follows:

MIMO SC-FDE:

MIMO BLE with ZF:

MIMO BLE with MMSE:

MIMO BNLE with QRD-M:

MIMO BNLE with full MLD:

where N is the block length in BLE or BNLE and also is the number of sample point in one block of MIMO SC-FDE. Q is the number of modulation levels and in this paper we set Q = 4 (QPSK) and M = 1. The computational complexities under different parameters are shown as in

From

In this section, we employed the LDPC encoding process in the transmitter to verify the feasibility of applying LDPC codes for BLE and BNLE. The sum-product algorithm is used to decode the LDPC code at the receiver. The simulation conditions are shown in

Average E_{b}/N_{0} [dB]/receive antenna

schemes with n_{T} = n_{R} = 4 for MIMO quasi-static frequency selective fading. The simulation results demonstrate that the proposed LDPC coded equalization schemes can obtain the great improvements in BER characteristics. As shown in ^{−5} compared with the corresponding uncoded schemes.

In this paper, we have proposed the time domain block linear and nonlinear equalizers for the signal separation and equalization of single carrier transmission on quasistatic frequency selective MIMO, SIMO, and SISO channels. We have compared the BER characteristics of the proposed time domain equalizers with the conventional frequency domain SC-FDE. As a result, the proposed time domain block equalizers which utilize the zero symbol insertion and short block length transmission exhibit the better BER characteristics than the conventional SC-FDE. The proposed block nonlinear equalizer using QRD-M exhibits the best BER performance. From the computational complexity analysis, we have found that the proposed BNLE with QRD-M can greatly reduce the computational burden at the receiver compared with the one using full MLD. We also applied the LDPC code to the proposed block equalization schemes and the large coding gains were observed. The single carrier transmission used in the proposed scheme is superior to the multicarrier transmission like OFDM in PAPR and the signal separation and equalization in time domain for the proposed schemes is especially suited to the short block length transmission. As the single carrier transmission is robust against the non-linear amplifiers due to the low PAPR and the short block length transmission is not very much affected by the rapid channel variation, the proposed time domain equalizers is supposed to be useful for the high speed uplink transmissions under rapidly time varying frequency selective channels.

This study has been supported by the Scientific Research Grant-in-aid of Japan No. 24560454.