Multiple-input multiple-output (MIMO) signaling is a powerful technique to enhance the system spectral efficiency and/or to achieve spatial diversity gain . For years there has been a lasting interest on its combination with orthogonal frequency-division multiplexing (OFDM). One of the main reasons for this is that OFDM is capable of converting a time-invariant (TI) frequency-selective channel into a parallel set of frequency-flat channels, thus enabling low-complexity equalization to mitigate inter-symbol interference (ISI). However, the high peak-to-average power ratio (PAPR) of OFDM may also lead to a substantial penalty in the power efficiency of MIMO systems .
, where the influence of the PAPR is more pronounced. Compared to conventional OFDM, where the data block is treated as a whole and modulated by a single full-length inverse discrete Fourier transform (IDFT), OSDM has a similar signal structure as vector OFDM in, which splits the data block into vectors and performs several component-wise IDFTs among them. By adjusting the vector length, flexible tradeoffs can be achieved between the PAPR and the bandwidth management ability, with two extremes being conventional OFDM and single-carrier block transmission (SCBT) [7, 5].
Due to this appealing feature, there have been some newly emerging works on MIMO-OSDM. For example, in  simple Alamouti-like space-time and space-frequency block coding systems were investigated over TI channels, while in  a spatial multiplexing scheme was designed for time-varying (TV) UWA channels. The latter is particularly attractive for high-rate UWA communications, where the channels are both bandwidth-limited and Doppler-distorted. However, since the equalizer in  performs direct channel matrix inversion, it incurs a cubic complexity. The aim of this paper is to design low-complexity equalizers for MIMO-OSDM by exploiting the channel matrix structure. Our contributions are as follows.
Over TI channels, only intra-vector ISI exists. As such, a per-vector equalizer is proposed based on the block-diagonal channel matrix structure.
Over TV channels, inter-vector interference (IVI) arises. A block equalizer is proposed based on the block-banded channel matrix approximation by using the complex exponential basis expansion model (CE-BEM).
Both equalization algorithms are performed in the transformed domain as their single-input single-output (SISO) counterparts in ; however, they require a judicious (post-)pre-processing of (de)interleaving to enable a linear complexity.
Notation: stands for conjugate, for transpose, and for Hermitian transpose. We define as the subvector of from entry to , and as the submatrix of from row to and from column to , where all indices are starting from 0. Moreover, represents a diagonal matrix with on its diagonal, and represents a block-diagonal matrix created with the submatrices . Also, stands for the unitary discrete Fourier transform (DFT) matrix; and refer to the identity matrix and its th column, respectively; denotes the all-zero vector; is the permutation matrix defined as
where denotes the Kronecker product.
Ii Equalization over TI Channels
We consider a MIMO-OSDM system with transmitters and receivers. The symbol block at the th transmitter is denoted by and its length is assumed to be . Unlike OFDM modulation which uses a single length- IDFT, OSDM modulation can be expressed as
for . A PAPR reduction is achieved, since it contains (shorter) IDFTs among symbol vectors
for . Then, after a cyclic prefix (CP) insertion, the block is transmitted through channels.
Let us first assume that all the channels are time invariant and denote the channel impulse response (CIR) between the th transmitter and the th receiver by , where is the channel order. In this case, the received signal block at the th receiver after CP removal has the form
for , where is the circulant channel matrix with its first column being ; is the received noise.
Subsequently, OSDM demodulation is performed as
where is termed the composite channel matrix; is the demodulated noise. It can be easily verified from (1) and (4) that, when () and (), the signal model of MIMO-OSDM is equivalent to that of MIMO-OFDM and MIMO-SCBT, respectively. In this sense, MIMO-OSDM can be deemed as a more generalized MIMO scheme.
Moreover, for TI channels, it is known that the composite channel matrix has the block-diagonal structure 
and . Given this matrix structure, we partition and into vectors of length as in (2). Since only intra-vector ISI exists in this case, the channel equalization can be decoupled on each vector. Specifically, by defining and for as the th demodulated vector and noise vector, respectively, it can be readily obtained that
where we have stacked all the th vectors, i.e, , , , and
Throughout this paper, we assume that the input symbols on all transmitters are independent and identically distributed (i.i.d.) with unit power, while the noise samples on different receivers are zero mean with the same variance. Therefore, based on (8), the minimum mean-square error (MMSE) equalization algorithm can be written as
where ; has a similar structure as in (9) with its blocks replaced by the diagonal matrices . Furthermore, as illustrated in Fig. 1, can actually be interleaved into a block-diagonal matrix, i.e.,
with being blocks of size .
), the symbol estimation in (10) can be rewritten as
where . As shown in Fig. 2(a), (13) actually corresponds to a low-complexity implementation of MIMO-OSDM equalization over TI channels, which consists of five steps:
Transform the demodulated vector by .
Interleave the transformed vector by .
Equalize the interleaved vector (denoted by ) to obtain the transformed symbol estimate (denoted by ), i.e.,
Deinterleave the output of the equalizer by .
Perform the inverse transform to finally obtain .
In this implementation, steps 1 and 5 involve DFTs and IDFTs of length , respectively, resulting in a complexity of . As for step 3, the transformed-domain equalization in (14) may look similar to that in (10); however, its computation is more tractable. To be specific, it can be seen that is a block-diagonal matrix with blocks of size on its diagonal, so the inversion in (14) has only a complexity of , which is linear in the vector length . Given the fact that the values of and are typically not large, the total complexity of (13) will be easy to handle.
Iii Equalization over TV Channels
We proceed to consider the equalization of MIMO-OSDM over TV channels. The CIR between the th transmitter and the th receiver is now denoted by , where the index is added to embody the time dependence of the CIR. And in this case no longer have the block-diagonal structure as in (5); instead, they are generally full matrices. As a result, IVI arises in OSDM, which is a counterpart of inter-carrier interference (ICI) in OFDM.
for , where is the discrete Doppler spread and are the BEM coefficients. With this model, the number of channel parameters on each delay tap is reduced from to . Moreover, based on (15), we have the TV channel matrix
where ; is a circulant matrix with its first column equal to appended by zeros. Note that, as shown in the left of Fig. 3, the composite channel matrix corresponding to (16) is (cyclically) block-banded with block semi-bandwidth (BSB) (see [6, Proposition 3] for a proof), which lays the foundation for our low-complexity equalization algorithm in this section.
Specifically, the presence of IVI excludes the use of the per-vector equalization algorithm previously designed for TI channels. We thus consider a block equalization for TV channels, which jointly estimates all symbol vectors in an OSDM block. Moreover, to achieve a low-complexity implementation, we again resort to matrix factorization of the blocks in . Let be the th block of . It has been shown in [6, Proposition 4] that only blocks in the main band of can be diagonalized. More specifically, only when , we have
and for .
To eliminate the blocks in the bottom-left and top-right corners of (which cannot be diagonalized), at each transmitter we place zero vectors at both edges of the symbol block, i.e., , where contains the middle payload vectors with . Accordingly, at each receiver the demodulated block is truncated as . Then, it can be obtained that
where and is the noise term. As shown in Fig. 3, is a standard (not cyclically) block-banded matrix. Based on (17), it can be further factorized into
where ; has the same matrix structure as , but with all its nonzero blocks being diagonal (see Fig. 3).
where , with and
As illustrated in the right of Fig. 3, the matrix structure of can be further simplified by interleaving, i.e.,
where , and the resulting matrix is block-banded with block size and BSB .
Therefore, the MMSE equalization of MIMO-OSDM over TV channels can be written as
where and . The above equations (24) and (25) represent the direct and low-complexity implementations, respectively. While (24) suffers from a cubic complexity of , as shown in Fig. 2(b), (25) actually takes the same strategy as (13) to reduce the computational burden. Specifically, here is first transformed and interleaved into , on which the TV channel equalization is then performed as
and finally the estimate of the symbol blocks is produced by . It is easy to see that is a block-banded matrix with block size and BSB . As a result, we can use the block LDL algorithm in  to compute the matrix inversion in (26), and the complexity is only .
Iv Numerical Results
In this section, the bit-error rate (BER) performances of the proposed equalization algorithms are evaluated by numerical simulations. We here consider a MIMO-OSDM system in a UWA communication scenario. At each transmitter, OSDM blocks are composed of quaternary phase-shift keying (QPSK) symbols with symbol period ms, and thus the block duration is ms. The MIMO channel is assumed to have an order of , corresponding to a multipath delay spread of ms, with all the taps Rayleigh distributed and generated from a uniform power delay profile. We first focus on TI channel equalization in Fig. 4, where , and transmissions are investigated and the vector length is set to (i.e., MIMO-OFDM), and . As expected, a better system performance is achieved with a larger due to the increased spatial diversity. Meanwhile, it is seen that the BER also improves as the vector length gets longer. This can be attributed to the intra-vector frequency diversity specific in OSDM systems [7, 5].
In Fig. 5, we further introduce the channel time variation and simulate it by a U-shaped Doppler spectrum with normalized Doppler spread and . Here, the MIMO configuration is adopted and the OSDM vector length is fixed to . It can be seen that, compared to the direct equalization algorithm in (24) using the full channel matrix, the proposed low-complexity equalizer in (25) leads to a error floor due to its block-banded channel matrix approximation based on the CE-BEM. However, their performance gap can be narrowed by increasing . Moreover, a significant reduction in complexity can be achieved. As an example, when and , the complexity of the proposed equalizer is only of that of the direct equalization method.
Low-complexity equalization algorithms of MIMO-OSDM are proposed in this paper for TI and TV channels (Fig. 2). Compared to the direct equalization method of cubic complexity, they have only a linear complexity in the transformed domain and thus are promising for practical use.
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