4.3 Proposed Channel Estimation Method
4.3.3 Refined DF DFT-Based Channel Estimation
q(j)v = F(j)H∆¯(j)v (4.11)
M(j)v = M(j)v−1− F(j)E(j)−1q¯(j)v (4.12)
where ¯M(j)v−1 = Π−1M(j)v−1and ¯∆(j)v = Π¯δ(j)v . According to [47], the purpose of calculating the difference between ¯M(j)v−1 and ˆC−1v XˆHv R(j) in (4.10), followed by the IDFT matrix F(j)H in (4.11), is to form the gradient vector ¯q(j)v in Newton’s method, as observed in (4.12). Furthermore, it is also proved in [47]
that the role of the weighting matrix E(j)−1 in (4.12) is in fact the inverse of the Hessian matrix in Newton’s method.
4.3.3 Refined DF DFT-Based Channel Estimation
Through the equivalence relation described in Section 4.3.2, it is concluded that the classical DF DFT-based channel estimation method (called method I, hereafter) is limited by the local search capability of Newton’s method and only applicable to very slowly time-varying channels. In the previous studies [36] [37], pilot tones as well as decision data symbols are simultaneously adopted to perform channel estimation at each iteration. This is, however, not a good solution in time-varying channels because decision data symbols easily induces the error propagation effect, whereas pilot tones are much more reliable than decision data symbols. From the viewpoint of optimization, the pilot tones inserted in each OFDM data symbol can play a more important role in providing a global search direction at the first iteration of the method I. Figure 4.3 shows the refined DF DFT-based channel estimation method in the tracking stage. With the help from a few pilot tones to form a gradient
!
Figure 4.3: Block diagram of the refined DF DFT-based channel estimation method in the tracking stage. (The subscript ”p” is to indicate that the calculation is only associated with the pilot subcarrier set.)
vector according to (4.10) and (4.11), the refined channel tracking method (called method II, hereafter) is proposed here by only modifying the first iteration (v = 1) of (4.12) as follows
M(j)1 = M(j)0 − B(j)F(j)¯q(j)p (4.13)
where ¯q(j)p is the gradient vector calculated according to (4.10) and (4.11) by only utilizing the set J, instead of Θ, B(j) is a block diagonal matrix defined as diag©
β(j,1)I|Θ|, . . . , β(j,NT)I|Θ|ª
, and β(j,i) is a real-valued step size which can be determined by minimizing the following ML cost function over the pilot subcarrier indices: in the definition. Note that M(j)p is the estimated channel frequency obtained from the previous time slot. Let M(j,i)[k] and ξ(j,i)[k] for k = Jm denote the ((i − 1) |J| + m) th entries of M(j)p and F(j)p q¯(j)p , respectively. Furthermore, we define γ(j)[t, k] = R(j)[t, k] − PNT
i=1M(j,i)[k] X(i)[t, k] and φ(j,i)[t, k] = ξ(j,i)[k] X(i)[t, k]. By expanding (4.14) and taking ∂Ω(β)/∂β = 0, after straightforward manipulations, the optimum value of β(j) is given by
β(j)opt = −Φ(j)−1Γ(j) (4.15)
where the (m, u) th entry of the matrix Φ(j) is calculated by The notation <e (·) takes the real part of (·). Note that, after the first itera-tion, we execute the channel tracking process of (4.10)–(4.12) for the second and subsequent iterations until a stopping criterion holds. The stopping cri-terion is to check whether the absolute value of each entry in F(j)E(j)−1q¯(j)v
is less than a prespecified threshold ε or the iteration number v reaches the maximum value of V . The channel tracking process for the current time slot will be stopped when either of the above two conditions holds.
In order to reduce computational complexity in the method II, we fur-ther propose method III to avoid the matrix inverse of the weighting ma-trix, E(j)−1, by taking into account the strongly diagonal property of E(j) which is originally proposed for reducing the complexity of multiuser detec-tion in code division multiple access (CDMA) systems [49]. Define E(j) =
|Θ| (Iκ(j)+ Oof f), where κ(j) = PNT
i=1κ(j,i) and Oof f is a zero-diagonal ma-trix. Then, it follows that if |Θ| is large enough, an approximate weighting matrix of E(j)−1 takes the form:
4.4 Computational Complexity
Now let us look at the computational complexity of the three methods, in terms of the number of real multiplications per transceiver antenna pair. In general, the operations of K-point IDFT and K × K matrix inversion need 4K log2K and 4K3 real multiplications, respectively. Besides, the weight-ing matrix E(j)−1 only needs to be calculated once in each OFDM frame as it is only related to the multipath delays ˆτl(j,i). Therefore, the complex-ity considered in the initialization stage is mainly due to the operations of CRP˘ [τ ] and E(j)−1. The calculation and update of CRP˘ [τ ] require at most 4(|Q| + |J|)/NT + 4K log2K + 4Nh|Wb| real multiplications. Moreover, the calculation of E(j)−1 needs at most 4Nh3 real multiplications, but it needs at most 2Nh2 real multiplications if the approximate weighting matrix of E(j)−1 in (4.18) is used instead. In the tracking stage, the computation for each iter-ation of the method I (or each of the second and subsequent iteriter-ations of the method II and the method III) involves the calculation of F(j)E(j)−1q¯(j)v , in total requiring at most |Θ|(4T +2)+2|Θ|T /NT+8K log2K +4Nh2 real multi-plications. For the method II and the method III, an optimal gradient vector is formed at the first iteration, in which the computation of B(j)F(j)q¯(j)p and the optimum β(j) at most requires |J|(4T + 2) + 8K log2K + 2|Θ| + 10|J|T + NT2 + NT(2|J|T + 1) real multipliations. The computational complexity for the system parameters given in Section 4.5 is listed in Table 4.1. The values of the parameters |Wb|, Nh, T , NT, K, |Q|, and |J| are set as 109, 8, 2, 2,
256, 192, and 8, respectively, while the value of the parameter |Θ| can be 48, 96, or 192. For a fair performance comparison, both the data subcarrier set Θ and the pilot subcarrier set J are used for tracking channel variations, except that only the pilot subcarrier set is adopted at the first iteration of the method II and the method III. Hence, we use |Θ| + |J| to replace |Θ| in the calculation of the complexity in Table 4.1. As observed in Table 4.1, the complexity of the method II and the method III is a little bit lower than that of the method I. It can also be noticed that the complexity in the tracking stage is mostly due to the operations of the DFT and the IDFT which in total require 8K log2K = 16384 real multiplications. Some complexity gain can be achieved by using partial DFT processing such as in [37]. If |Θ| is larger than Nh, the computation of the partial DFT processing mainly depends on the size of Θ. As a result, the complexity of the tracking stage is basically dominated by |Θ| since |J|, NT and T are usually much smaller than |Θ|.
4.5 Computer Simulation
We demonstrate the performance of the proposed channel estimation meth-ods through computer simulation of an STBC-OFDM system with two trans-mit antennas and a single receive antenna. The parameters are set the same as those in the 802.16e OFDM standard [43] and summarized in Table 4.2.
The system occupies a bandwidth of 5MHz and operates in the 2.3GHz frequency band. The entire bandwidth is divided into K = 256 subcarriers among which |Q| + |J| = 200 subcarriers are used to transmit data symbols and pilot tones, and the remaining M = 56 subcarriers are used as virtual
Table 4.1: Computational complexity for the system parameters given in Section 4.5.
Initialization Stage
Method I and Method II 14128
Method III 12208
Tracking Stage (|J| = 8)
|Θ| 48 96 192
The first iteration of the Method II and the Method III
16806 16902 17094
Each iteration of the Method I or each of the second and subsequent iterations of the Method II and the Method III
17312 17888 19040
Table 4.2: Simulation parameters.
Parameter Value
Carrier frequency 2.3GHz
Bandwidth 5MHz
FFT size (K) 256
Length of CP 64
Number of data and pilot subcarriers 200 Number of virtual subcarriers (M) 56 Modulation scheme for data subcarriers QPSK, 16QAM Modulation scheme for pilot subcarriers BPSK Number of OFDM data symbols per frame (ND) 40
Two-path channel Channel power profiles ITU Veh-A channel [34]
Jakes model [33]
0 ∼ 50 Channel delay profiles
(Uniform distribution) Multipath observation window (Wb) [0, 108]
Preassumed number of paths (Nh) 4, 8
subcarriers at the two edges and a DC subcarrier. In the simulation, the modulation schemes for the data symbols are QPSK and 16QAM, while the BPSK modulation scheme is adopted for the pilot tones. Each pilot subcar-rier transmits the same power as each data subcarsubcar-rier. Each OFDM frame is composed of one OFDM preamble and ND = 40 OFDM data symbols.
The length of the CP is 64 sample periods, i.e., one quarter of the useful symbol time. The preambles transmitted from the first and second antennas use even and odd subcarriers respectively with a 3dB power boost, and the values of those subcarriers are set according to [43]. Both a conventional two-path channel and an International Telecommunication Union (ITU) Veh-A channel are simulated with path delays uniformly distributed from 0 to 50 sample periods, where the relative path power profiles are set as 0, 0 (dB) for the two-path channel and 0, −1, −9, −10, −15, −20 (dB) for the ITU Veh-A channel [34]. The vehicle speed ve of 240km/hr is used to simulate mobile radio environments, for which Rayleigh fading is generated by Jakes model [33]. Moreover, the multipath observation window Wbis set as [0, 108].
The preassumed number of paths Nh is set as 4 and 8 in the two-path channel and the ITU Veh-A channel, respectively. Both the data subcarrier set Θ and the pilot subcarrier set J are used in the tracking stage. The subcarrier indices of J are uniformly assigned within the available subcarriers. The set Θ is uniformly selected from Q, and the parameter |Θ| could be 192, 96, 48, 24, 12, 6 or 3. The values of the maximum iteration number V are set as 5 and 7 for QPSK and 16QAM modulation, respectively. For the stopping criterion, the prespecified threshold ε is set as 10−4. The entire simulations
are conducted in the equivalent baseband, and we assume both symbol syn-chronization and carrier synsyn-chronization are perfect. Finally, throughout the simulation, the parameter Eb/No is defined as a ratio of received bit energy to the power spectral density of noise.
For comparison purpose, the performance curve with ideal channel esti-mation, denoted as perfect CSI, is provided for reference and served as a per-formance lower bound. We also compare the proposed methods with both the STBC-based MMSE method [48] and the Kalman filtering method [50] [51]
where the decision-feedback methodology is employed under the assumption of ideal channel estimation in the initialization stage. Some statistical in-formation such as Doppler frequency, auto-covariance of channels, and noise power is assumed to be known for these two existing methods. It is noted that the Kalman filtering method is mainly based on [50] and the received signals within a time slot are used to perform channel estimation according to the decision-feedback steps in [51].
4.5.1 NSE Performance of MPIC-Based Decorrelation