- Research Article
- Open Access
Time-Varying FIR Equalization for MIMO Transmission over Doubly Selective Channels
© I. Barhumi and M. Moonen. 2010
- Received: 2 November 2009
- Accepted: 8 April 2010
- Published: 17 May 2010
We propose time-varying FIR equalization techniques for spatial multiplexing-based multiple-input multiple-output (MIMO) transmission over doubly selective channels. The doubly selective channel is approximated using the basis expansion model (BEM), and equalized by means of time-varying FIR filters designed according to the BEM. By doing so, the time-varying deconvolution problem is converted into a two-dimensional time-invariant deconvolution problem in the time-invariant coefficients of the channel BEM and the time-invariant coefficients of the equalizer BEM. The timevarying FIR equalizers are derived based on either the matched filtering criterion, or the linear minimum mean-square error (MMSE) or the zero-forcing (ZF) criteria. In addition to the linear equalizers, the decision feedback equalizer (DFE) is proposed. The DFE can be designed according to two different scenarios. In the first scenario, the DFE is based on feeding back previously estimated symbols from one particular antenna at a time. Whereas, in the second scenario, the previously estimated symbols from all transmit antennas are fed back together. The performance of the proposed equalizers in the context of MIMO transmission is analyzed in terms of numerical simulations.
- Carrier Frequency Offset
- Selective Channel
- Decision Feedback Equalizer
- Error Floor
- Basis Expansion Model
The wireless communication industry has experienced a rapid growth in recent years, and digital cellular systems are currently designed to provide high data rates at high terminal speeds. High data rates give rise to intersymbol interference (ISI) due to multipath fading. Such ISI channels are described as frequency-selective. On the other hand, due to the user's mobility and/or receiver carrier frequency offset (CFO) the received signal is subject to frequency shifts. CFO in conjunction with the Doppler shift give rise to time-selectivity characteristics of the mobile wireless channel. Therefore, the mobile wireless communication channel is generally characterized as time- and frequency selective channel or the so-called doubly selective, which in turns causes degradation in the system performance. This motivates the search for efficient and robust equalization techniques to improve on the information transmission reliability.
The data rate provided by the wireless communication system can be increased substantially by using multiple antennas at the transmitter and at the receiver. It is well-known that multiple-input multiple-output (MIMO) systems can provide an increase in the system capacity by a factor linear in the minimum number of the antennas used at either the transmitter or the receiver [1, 2]. In this paper, we address the problem of equalization for spatial multiplexing-based MIMO transmission over doubly selective channels. Linear and nonlinear decision feedback equalizers are proposed based on the minimum mean-square error (MMSE) and the zero-forcing (ZF) criteria.
For slowly time-varying channel, adaptive techniques for channel estimation or equalization are developed to combat the problem of ISI. These algorithms range from the least mean-squares (LMS) algorithm , to the recursive least squares (RLS) algorithm or Kalman filtering algorithm . For fast flat-fading channels, polynomial fitting of the 1-tap time-varying channel is used to predict the channel as proposed in . Extending the polynomial fitting over the whole packet (or using a sliding window approach) to time-varying frequency-selective channels is investigated in . For single-input multiple-output (SIMO) transmission over doubly selective channels, ZF and MMSE time-varying FIR equalization techniques were proposed in . The extension to the DFE equalizer was presented in . In the MIMO context, the authors in  propose block linear filters to mitigate intercarrier interference (ICI) for OFDM transmission over time-varying multipath fading channels. A Kalman filter based MMSE interference suppression is proposed in  for MIMO transmission over doubly selective channels. In there, a two-stage suppression technique is proposed; one stage is used to mitigate ISI due to channel frequency selectivity, and another stage is used to mitigate ICI due to channel time selectivity. For estimation of MIMO doubly selective channels, an MMSE pilot-aided transmission is proposed in  for cyclic prefix (CP) based block transmission scheme. Optimal training design is proposed in . Adaptive estimation of doubly selective channels is proposed in . There in a subblock tracking scheme for the basis expansion model (BEM) coefficients of the doubly selective channel using periodically transmitted training symbols.
In this paper, we propose matched filter (MF), ZF, MMSE, and DFE time-varying FIR equalizers for MIMO transmission over doubly selective channels. Spatial multiplexing, where independent data streams are assumed on different transmit antennas, is considered for the MIMO transmission. Considering other MIMO transmission techniques, for example, space-time block coding (STBC)  is out of the scope of this paper. In the above mentioned schemes, perfect channel state information (CSI) is assumed to be known at the receiver. The basis expansion model (BEM) [15, 16] is used to approximate the underlying communication channel, and to model and design the equalizers. In this sense, a large complex 1D time-varying deconvolution problem is turned into a lower complexity 2D deconvolution problem in the BEM coefficients of the channel and the BEM coefficients of the equalizers.
This paper is organized as follows. In Section 2, the system model is introduced. The time-varying FIR linear equalizers are developed in Section 3. The time-varying FIR DFEs are investigated in Section 4. Our findings are confirmed by numerical simulations introduced in Section 5. Finally, conclusions are drawn in Section 6.
We use upper bold face letters and lower bold face letters to denote matrices and vectors, respectively. Superscripts , and represent Hermitian, transpose, and conjugate, respectively. To simplify notations and save space, the double summation over the subscripts and is denoted as , where the ranges of and should be clear from the context. We denote the identity matrix as , the all-zero matrix as . Finally, denotes Kronecker product, denotes the direct sum, and denotes the diagonal matrix with vector on its diagonal.
where , , and is similarly defined as . The diagonal matrix representing the th basis function is defined as , and the Toeplitz matrix is defined as .
3.1. Matched Filter Equalizer
where , and , with and are the signal and noise autocorrelation matrices, respectively. For short we define .
This is a special case of (22). However, for , and provided that the source and the noise are white, the matrices , and are not scalar multiple of the identity matrix.
3.2. MMSE and ZF Equalizers
where , with is a dimensional unity vector with one at position . Note that (28) is obtained from (27) by applying the matrix inversion lemma ( ), and using the fact that .
For the ZF solution to exist, the matrix has to be of full column rank. A necessary condition for to be of full column rank is that the inequality is satisfied. For sufficiently large , and , this inequality is satisfied when the number of receive antennas is larger than the number of transmit antennas, that is, . The MMSE equalizer always exists regardless of the number of receive antennas. However, the performance of the MMSE equalizer is largely improved if the above inequality is satisfied.
Obtaining the linear MMSE and ZF equalizers involves matrix inversion of size with . Therefore, the computational complexity of these equalizers requires Multiply-Add (MA) operations.
In this section we extend the results for linear equalization to decision feedback equalization. The DFE consists of two filters, a feed-forward filter and a feedback filter. The feed-forward and feedback filters are again designed to be time-varying FIR filters. The time-varying FIR filters in the forward path are identical to the linear equalizers described in Section 3.2 (see Figure 2). The feedback filters have as their input the sequence of decisions on previously detected symbols. Given the extra degrees of freedom offered by the MIMO systems, we can devise two different scenarios. In one scenario, only previously estimated symbols from one transmit antenna are fed back to cancel/reduce ISI in the data stream of that particular transmit antenna. This scenario is referred to as DFE-I. In the other scenario, previously estimated symbols from all transmit antennas are fed back together to cancel/reduce ISI in the data stream of one particular transmit antenna. This scenario is referred to as DFE-II.
The computational complexity of the feed-forward filter coefficients requires matrix inversion of size , and hence the complexity is . Computing the feedback filter coefficients requires matrix inversion of size , which requires complexity . In most of the cases, the feedback filter order and number of time-varying basis functions are smaller than the feed-forward filter order and number of basis functions, that is, . Hence, the computational complexity of the DFE of this scenario is dominated by the computation of the feed-forward filter , which is exactly the same computational complexity of the linear MMSE or ZF filters. However, the overall complexity is slightly larger for this DFE than the linear equalizers.
The computational complexity of the feed-forward filter coefficients of this scenario requires matrix inversion of size , which requires MA operations. This complexity is the same complexity associated with computing the feed-forward filter coefficients of DFE-I scenario and the linear equalizer. Computing the feedback filter coefficients requires matrix inversion of size , which requires MA operations. In this sense, the computational complexity of DFE-II for the feedback part is times the computational complexity of DFE-I. Provided that , the computational complexity is still dominated by the computation of the feed-forward filter. However, the overall computational complexity for DFE-II is the highest among all devised equalizers.
In this section we present some simulation results for the proposed equalization techniques for MIMO transmission over doubly selective channels. In the simulations, uncoded Quadrature Phase Shift Keying (QPSK) modulation is used. The channel is assumed to be perfectly known at the receiver. The BEM coefficients are then obtained by least-squares (LS) fitting the true underlying channel with the BEM. The performance of the proposed equalization techniques under channel estimation errors is outside the scope of this paper, and a topic of further investigation. The channel taps are simulated as i.i.d random variables, correlated in time according to Jakes' model with correlation function , where is the zeroth-order Bessel function of the first kind, is the maximum normalized Doppler spread. Two channel setups are used. One with order , and , and another setup with order , and normalized maximum Doppler spread . A BEM window size is considered all over the simulations. The BEM resolution is chosen such that and . We measure the performance in terms of bit error rate (BER) versus signal-to-noise ratio (SNR). The SNR is defined as , where is the transmitted symbol energy, and is the additive white Gaussian noise variance. In all the simulations the decision delay is taken as , and the approximation , and are used. It is worth mentioning here that the BEM approximated channel is used for the equalizers design, but the true channel is used for BER simulations, which will be subject to channel modeling error.
5.1. Channel Setup-I
In this setup the number of basis functions is for , and for . Two MIMO setups are considered; a first setup with , and a second setup with . In the first setup, a linear ZF equalizer may exist, and therefore it can be evaluated against the linear MMSE equalizer and DFE. In the second setup, only the linear MMSE equalizer and the DFE are evaluated.
5.2. Channel Setup-II
In this paper, time-varying FIR equalization techniques have been proposed for spatial multiplexing-based MIMO transmission over doubly selective channels. The time-varying FIR equalizers are designed considering the matched filter, MMSE and ZF criterion for linear and nonlinear decision feedback equalizers. The BEM is used to approximate the doubly selective channel, and to model and design the time-varying FIR feed-forward and feedback filters. By doing so, the one-dimensional time-varying deconvolution problem is reduced to a two-dimensional time-invariant deconvolution problem in the time-invariant coefficients of the channel BEM coefficients, and the time-invariant coefficients of the BEM equalizers. Using the BEM, and for a sufficient number of BEM parameters, the ZF solution exists for , which extends a well-known result for the time-invariant MIMO equalization of frequency-selective channels. Using the extra degrees of freedom offered by the MIMO system, a DFE that feeds back the previously estimated symbols on all data streams to cancel/reduce ISI on a particular data stream can be obtained, which is shown to outperform the linear MMSE equalizer and the DFE that feeds back only the previously estimated symbols from one particular data stream. Block equalizers can be derived, but in general require high computational complexity. Hence, time-varying FIR equalization allows for lower complexity equalization techniques.
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