7.2 Approximate LMMSE CE with Estimations of Mean Delay and RMS Delay Spread112
7.2.5 Appilcation in WiMAX Channel Estimation
We apply the approximate LMMSE channel estimation in the WiMAX UL and DL. Unlike the classic comb-type based pilot structure in OFDM system, the WiMAX adopts the cluster structure in downlink and the tile structure in uplink. These discrete pieced structure makes the common transform based estimation not feasible in WiMAX. Thus, the per piece estimation is considered. The approximate LMMSE is one suitable approach.
WiMAX DL Channel Estimations
We consider two approaches using four taps LMMSE CE in DL channel estimation. At first approach, the delay parameters are updated in every time slots of the receptions using the pilot responses of the associated cluster for the user. The second approach estimates the parameters once per subframe adopting the preamble symbol. In complexity issue, the first approach is more complicate than the second due to the updating of the delay parameters and Wiener coefficients;
whereas, in performance issue, the second approach has poor performance than the first one due to time-varying parameters.
In the first approach, referring to Fig.7.7, we have the following process:
1. estimate noise variance using the null subscarrier in the preamble symbol;
2. do least-square channel estimation at existing pilots of the clusters;
3. do time domain average to acquire the responses of H(2, 0), H(2, 12), H(3, 4) and H(3, 8);
4. estimate RMS delay and mean delay, and generate the associated coefficients for the in-terpolator; and
5. perform the interpolation to obtain the residual responses.
For the second approach, it is similar to the first, except for skipping the step 4 and using preamble to generate the delay parameters and coefficients in step 1.
We study the proposed estimators and compare to the optimal MMSE estimator with true correlation. Four-tap estimators are used in the simulation. Besides, we also compare to phase-shifted polynomially interpolator with the adaptive selection scheme proposed in section 7.1.4.
The simulated system parameters are given as follows: FFT size = 1024, bandwidth = 10 MHz,
5 10 15 20 25 30 35 40
−45
−40
−35
−30
−25
−20
−15
−10
SNR (dB)
NMSE (dB)
MMSE with true corr. model Approx. MMSE with exp. corr. model Approx. MMSE with unif. corr. model
(a) Vehicular-A Channel Model
5 10 15 20 25 30 35 40
−45
−40
−35
−30
−25
−20
−15
−10
−5
SNR (dB)
NMSE (dB)
MMSE with true corr. model Approx. MMSE with exp. corr. model Approx. MMSE with unif. corr. model
(b) SUI-4 Channel Model
Figure 7.13: Simulation studies of the periodic update scheme for multi-carrier with comb-type pilots at 100 KM/Hr mobility speed.
center frequency = 2.5 GHz and 6 subchannels for a specified burst. The ETSI Vehicular-A [21], SUI-4 and SUI-5 channel are used in the comparisons, and the mobility is set to 100 Km/Hr.
Figs. 7.14(a), 7.14(b) and 7.14(c) show the comparisons of the approaches over ETSI Vehicular-A, SUI-4 and SUI-5 channel profile correspondingly. As shown in Figs. 7.14(a), the proposed estimators are closed to the optimal benchmark, except for the adaptive selection in low SNR region. In SUI-4 channel, the approximate MMSE approaches have good performance near the optimal at low SNR condition; however, the performances are far away from the opti-mal one at high SNR due to the correlation model error. At high SNR, the adaptive selection scheme is better than the approximate MMSE approaches, but poorer at low SNR. In SUI-5 channel, the proposed algorithms perform significantly poorer than the optimal estimator. The SUI-5 has much larger delay spread than SUI-4 and ETSI Vehicular-A. As shown, the proposed estimators have bad estimation quality in the large delay spread, especially when the SNR is high. However, the performances of the approach 1 using exponential correlation model and the adaptive selection scheme are similar and better than others.
WiMAX UL Channel Estimations
According to the pilot assignment in the tile structure, the two-tap Wiener filter is used to estimate the channel response. We assume that carrier and symbol timing synchronization are done in UL subframe during the ranging process. The proposed channel estimation works in the unit of the tile. First, the channel responses at pilot subcarrier is estimated via the least square estimator. Then, the delay parameters are estimated by the estimated channel responses associated to the user and the corresponding Wiener coefficients are generated. At third step, the LMMSE channel estimation is performed in frequency dimension to generate the responses at the first and third symbols. Finally, the time domain interpolation is performed to estimate the responses at the second symbol.
We compare the proposed estimators to others methods. The conventional linear interpo-lator and the phase-compensated linear interpointerpo-lator [32] are studied in the comparisons. Both the exponential and the uniform distributed PDPs are used for generating the approximate correlation models. Besides, the optimal MMSE estimator with true correlation model is also performed as the optimal benchmark. The system profile and channel condition is similar to previous. Besides, we consider that an additional 0.8 µseconds propagation delay occurs in the propagation, which roughly yields 9 samples in a OFDM symbol.
The simulation results according to the channel models: ETSI, Vehicular-A, 4, and SUI-5 are illustrated in Figs. 7.1SUI-5(a), 7.1SUI-5(b) and 7.1SUI-5(c) correspondingly. For all channel models, the approximate MMSE with exponential correlation model has best performance, which is nearest one to the optimal benchmark. The secondary one is the approximate approach with uniform distributed correlation model, but it does not have significant difference to the phase-compensated linear interpolator [32]. As we expect, the poorest one is the conventional linear interpolation. When compare to different channel models, the SUI-5 have poorest simulation results for all estimators. This is because the delay spread of SUI-5 is much larger than others.
7.3 Conclusions
This chapter considered the factors of the symbol timing and the channel delay parameters in the channel estimation issue. We studied two commonly used channel estimation schemes:
the polynomial interpolation and the LMMSE estimation. For polynomial interpolation, we presented the time-shifted interpolation scheme and the estimate of the associated optimal shift.
Several cost-reduced implementations were proposed to realize the WiMAX DL CE. Besides, we also proposed the scheme of adaptive selection of the interpolation order in the DL subframe to
5 10 15 20 25 30 35 40
Optimal MMSE with true corr. model Approach 1:Approx. MMSE with exp. corr. model Approach 1:Approx. MMSE with unif. corr. model Approach 2:Approx. MMSE with exp. corr. model Approach 2:Approx. MMSE with unif. corr. model Adaptive selection of phase shift interp. CE
(a) Vehicular-A Model
Optimal MMSE with true corr. model Approach 1:Approx. MMSE with exp. corr. model Approach 1:Approx. MMSE with unif. corr. model Approach 2:Approx. MMSE with exp. corr. model Approach 2:Approx. MMSE with unif. corr. model Adaptive selection of phase shift interp. CE
(b) SUI-4 Model
Optimal MMSE with true corr. model Approach 1:Approx. MMSE with exp. corr. model Approach 1:Approx. MMSE with unif. corr. model Approach 2:Approx. MMSE with exp. corr. model Approach 2:Approx. MMSE with unif. corr. model Adaptive selection of phase shift interp. CE
(c) SUI-5 Model
Figure 7.14: Simulation studies of the CE schemes for WiMAX DL path at 100 Km/Hr mobility speed.
0 5 10 15 20 25 30 35 40
Optimal MMSE with true corr. model Approx. MMSE with exp. corr. model Approx. MMSE with unif. corr. model Linear interpolative CE
Optimal MMSE with true corr. model Approx. MMSE with exp. corr. model Approx. MMSE with unif. corr. model Linear interpolative CE
Optimal MMSE with true corr. model Approx. MMSE with exp. corr. model Approx. MMSE with unif. corr. model Linear interpolative CE Phase−comp. linear interp. CE
(c) SUI-5 Model
Figure 7.15: Simulation studies of the CE schemes for WiMAX UL in 100 Km/Hr mobility speed.
minimize the estimate MSE. In LMMSE channel estimation, we proposed a simple estimation scheme of the mean delay and RMS delay spread. And, we apply this estimator to the suboptimal LMMSE channel estimator by generating the correlation function according to the predefined channel model and the estimated delay parameters. The realizations of the channel estimators for the comb-type multi-carrier system and the WiMAX are also innovated.
Appendix: Derivation of G(k)
Let G = [G(x0), G(x1), · · · , G(xN)]T. Then G = x −£
V (x0)T, V (x1)T, · · · , V (xN)T¤T X−1x
= x − X X−1x = 0. (7.111)
Thus xn is a root of G(k) ∀n ∈ {0, · · · , N }. Now that G(k) is an N + 1st-order polynomial in k by definition, we have
G(k) = K YN n=0
(k − xn) (7.112)
for some K. Further, since the kN +1 term in G(k) has unity coefficient by definition, K = 1.
Part III
Transmitter Design over Wide-band MIMO and Its Block Iterative
Receiver
Orthogonal frequency-division multiplexing (OFDM) with multiple transmit and receive an-tennas has drawn much recent attention in research on high-speed transmission over multipath fading channels. To exploit more fully the inherent diversity under multi-input multi-output (MIMO) OFDM in fading channels, one usually needs to employ time and/or space-frequency coding [92, 1]. There is now abundant literature on space-time/space-frequency coding.
Taking space-time coding as an example, the approaches can be divided into two broad cate-gories: the coding approach (as represented by space-time trellis coding and space-time block coding) and the linear preprocessing approach (as represented by linear constellation precoding for signal space diversity [102, 8]).
It has been shown that a key factor influencing the performance of space-time/frequency coded transmission is the determinant of the correlation matrix of pairwise codeword differences [105, 31] In this part, we show that the determinant is maximized when the correlation matrix is a multiple of the identity matrix; or in other words, when the codeword differences are “white.”
In attempting to use this result in system design, however, we find that, when the block size of the system is large (the meaning of “block size” will become clear later), there is difficulty in achieving such whiteness. As a result, we resort to an approximately white design.
For signal reception, we consider block-based turbo decision-feedback equalization (TDFE) [56, 13, 47] to exploit the available diversity while keeping the complexity under guard. More-over, we propose a multi-stage technique for turbo DFE to further reduce the complexity. In association with the reduced-complexity turbo DFE, we propose a particular space-frequency transform (SFT) technique for quasi-whitening of the transmitted signal. The SFT combines an orthogonal transform (for which fast computational algorithms exist) and a certain way of space-frequency interleaving (SFI) whose details will be given later.
For channel estimation, we give a pilot-data separated frame structure for the proposed scheme and present a joint channel estimation(CE) and data detection(DD) scheme to iteratively estimate channel and detect data. With the cooperation of the channel prediction scheme, we greatly reduce the pilot utilization in the frame structure. For jointly iterative CE and DD, we formulate the data reception as the constrained least square problem and adopt the jointly block gradient descent algorithm to solve the problem, which is a TDFE-like iterative solution.