9.4 Joint Channel Estimation and Data Detection
9.4.1 Frame Structure and Initial Channel Estimation
Note that in performing TDFE, the channel responses need be estimated first. For this, consider the frame structure illustrated in Fig. 9.7 which we call a pilot-data separated frame structure.
A frame consists of a number of bursts where each burst starts with an all-pilot symbol as the preamble. It is then followed by a number of OFDM symbols. The pilot symbol is used to obtain an initial estimate of the channel response. During subsequent data symbols, adaptive channel prediction is used to track channel variation. For each symbol, iterative joint CE-DD is carried out, as described in more detail below.
For the pilot symbol, an LS channel estimation (LSCE) can be obtained by minimizing the cost function
J(hi) = kri− Xhik2. (9.25)
The solution is
h∗i =¡
ZHZ¢−1
ZHri (9.26)
where Z = XG. To achieve the minimum mean-square error in the estimate, the pilot symbol should be such that [4]
XHX = σ2xI. (9.27)
For channel prediction in subsequent symbols (which yields the initial channel estimate for each data symbol), we can employ block-based linear filtering whose basic form is
ˆ α(t) =
NXtap
k=1
wkα(t − k) , wα(t) (9.28)
Figure 9.7: The pilot-data separated frame structure considered.
where α(t) represents one tap of the estimated channel response at time t and ˆα(t) is one tap of the predicted channel response. If the mobile speed is known and the fading follows Jakes’
model, then the optimal prediction coefficient can be shown to be
w = (Rα+ σ2nI)−1Rα,1 (9.29)
where Rα is the autocorrelation matrix of the channel response and Rα,1 is the crosscorrelation of the channel response and its delayed version. Under Jakes’ model, the mnth element in Rα is given by Rα(m, n) = J0(2πfm(m−n)) and the nth element in Rα,1by Rα,1(n) = J0(2πfm(n+1)), where J0(x) is zeroth-order modified Bessel function and fm is the maximum Doppler shift normalized to sample spacing.
In practice, the mobile speed may not be known and the fading may not follow Jake’s model.
In this case, we may use an adaptive prediction filter in place of the filter above. By minimizing the cost function E{|ˆα(k) − α(k)|2}, the simplest LMS adaptive algorithm yields the following adaptive prediction filter for w:
w(t + 1) = w(t) + µα(t)H(α(t) − ˆα(t)) . (9.30) 9.4.2 Jointly Iterative Channel Estimation and Data Detection
For each data symbol, following the initial channel estimation, we perform jointly iterative CE-DD. In each iteration, the TDFE output can be used to re-estimate the channel responses in the LS manner as in (9.26). However, the pseudo-inverse operation costs a very large computational effort. Hence, herein we propose performing the channel re-estimation using the projected gradient descent algorithm and combining it with the TDFE to yield the desired jointly iterative CE-DD.
For this, note first that the projected gradient descent algorithm is an iterative approach to solving the convex CLS problem such as that given in (9.25). Let hk−1i be the estimate of hi in iteration k − 1. By applying a similar recursion as in TDFE, we can obtain an iteration for channel estimation as
ˆhki = hk−1i − µXH
³
ri− Xhk−1i
´
(9.31) and
hki = G†ˆhki = Gˆhki. (9.32)
Combining the above with the TDFE, we obtain the jointly iterative CE-DD procedure as
with Π(·) denoting the combined actions of the inverse SFT, the soft-output FEC decoding, and the SFT. In the initial pass through the iterative loop, the channel estimate used is the output of the channel predictor and the TDFE output is that from linear MMSE equalization.
If shaping filtering is considered in the earlier iterations as discussed previously, (9.35) is replaced by
To reduce the complexity, one may use a constant shaping filter C calculated using the predicted channel responses.
Note also that, since the super-matrices X and H are composed of circulant matrices whose FFTs are diagonal matrices, many computations can be simplified by using this property as mentioned previous chapter.
9.4.3 Simulation Study
We illustrate the performance of the proposed SFT MIMO OFDM and compare it with that of the conventional MIMO OFDM [17] and the MIMO CP BSC [17, 90], all with FEC coding, by way of simulation. In fact, the MIMO CPBSC can be considered a particular kind of SFT MIMO OFDM. For the SFT, we use FFT for the orthogonal transform.
The profile of the simulated system is the same as described one in above simulations. In addition, a frame contains 5 bursts, with each burst made of 5 MIMO OFDM symbols following 1 short pilot symbol. The FFT size of the pilot symbol is 256 only. Therefore, the pilot-data ratio is 1/20 in terms of bandwidth consumption. The pilot symbol contains BPSK-modulated sequences that are mapped to the subcarriers of each antenna in an interleaved manner [4].
In the receiver, the adaptive channel predictor has five predictor taps. The maximum number of iterations of the joint CE-DD is 10 and the iteration may stop early if the decoder output contains no error. Shaping filtering is used in the earlier iterations of the TDFE and the switching from shaped to unshaped operation depends upon the convergence of the output log-likelihood ratio (LLR) of the FEC decoder.
Consider a 2×2 MIMO block time-varying channel with the same profile in section 9.3.2. Let the carrier frequency be 2.5 GHz and the channel bandwidth be 10 MHz. Consider three mobile speeds: 50, 100, and 200 km/h, which correspond roughly to maximum normalized Doppler shifts of 0.0125, 0.025, and 0.05, respectively.
Figs. 9.8 and 9.9 illustrate the block error rate (BLER) and the bit error rate (BER) perfor-mance of the three transmission schemes, where a block means the period of one OFDM symbol.
At the two slower mobile speeds, the channel estimation mechanism proposed in this work can track the channel variation well. Its tracking ability is reduced at the highest mobile speed, as indicated by the error floors in the plots. In all cases, the SFT MIMO OFDM performs better than the conventional MIMO OFDM and the MIMO CPBSC. The performance gain is roughly 4 dB over conventional MIMO OFDM and 2 dB over MIMO CPBSC at the lower mobile speeds.
The error floor happens when mobile velocity is 400 Km/Hr, since the prediction error floor introduces large error of the initial channel estimation and the outage occurs frequently in high
4 6 8 10 12 14 16 18 20 22
Figure 9.8: Block (symbol) error rates of different transmission schemes over block time-varying channel.
velocity, which yields poor tracking ability for the receiver. To eliminate this floor, one way is to insert pilot symbol frequently; however, this reduce the bandwidth efficiency. Another way may consider temporal domain interpolative channel estimation instead of the channel forecasting;
which can be treated as a extrapolation and yields more error due to AWGN. Besides, the more accurate estimation of the delay subspace will provide larger noise reduction factor and give more clear channel estimation results to avoid the floor due to estimation error.
9.5 Conclusions
In this part, we showed the optimal design criteria of the coded MIMO OFDM transmission and proposed the design of the MIMO OFDM system, which involves the space-frequency transform, the frame structure and the receiver design. We suggested the SFT design to achieve the optimal design criteria, which is a two-step transform. The first step performs the orthogonal transform to spread the coded symbols and the second step interleaves the spread samples over frequent and spatial dimensions. For the frame design, we presented a simple pilot-data separated design to improve the spectrum efficiency by eliminating the pilot utilities. The receiver employs the IS process to jointly estimate the channel and detect the data. The prior channel references are required in the IS process. Thus, the LSCE from the pilot symbol and the channel prediction scheme are used to obtain these references. Simulations shown that the proposed SFT is superior to the conventional MIMO OFDM and the MIMO SC with CP in terms of the BER and the BLER.
For future researches, it is interest in the design to support multiuser based on the proposed SFT MIMO OFDM system. It could be a CDMA-like or a OFDMA-like schemes in either the uplink or the downlink, which depends on the position of the SFT in the transmission structure. Besides, another design issue is the receiver structure of the multiple access system.
The performance, the complexity and the spectrum efficiency all are the design constrains in
4 6 8 10 12 14 16 18 20 22
Figure 9.9: Bit error rates of different transmission schemes over block time-varying channel.
the realization.
Appendix
Derivation of the MMSE Shaping Filter
First, let the singular value decomposition (SVD) of the channel matrix be given by
H = U ΛV. (9.37)
Since the desired C commutes with H, we may let C = VHΛcV . As in typical adaptive filter analysis, assume that ek−1 is white and uncorrelated with n. Then
J , E{kwkk2}
where λ(k) and λc(k) are the kth singular values of H and C, respectively.
Now, constraining the average of the diagonal elements of B to zero amounts to requiring tr©
Employing the Language multiplier method, we consider the modified cost function
Jm , J + µ Ã
1 − 1 KN
KNX
k=1
λc(k) |λ(k)|2
!
(9.41)
and set ∂Jm/∂λc(k) = 0, which leads to the solution λ∗c(k) = µ
|λ(k)|2+ α (9.42)
and
C∗= µ¡
HHH + αI¢−1
. (9.43)
Substituting C∗ into the constraint on B, we obtain
µ = KN
tr {(HHH + αI)−1HHH}. (9.44)
Chapter 10
Thesis Conclusions and Future Topics
Several algorithms are proposed for the design of the communication receivers in SC QAM, OFDM, and MIMO-OFDM systems. A communication receiver contains two major processing modules. The synchronization calibrates the mismatches between the transmitter and the re-ceiver, and estimates the unknown parameters. The channel distortion compensator is used to remove the channel effects. It can be divided into two parts: the equalization and the channel estimation. In SC QAM systems, we consider the issues in the carrier frequency recovery, the channel equalization, and the joint operation. Both the open-loop frequency estimation and closed-loop carrier recovery loop are studied in the carrier recovery problem. The blind adap-tive DFE is used and studied in the channel equalization. In OFDM system, we consider the synchronization issues in the estimation of the carrier frequency offset and the symbol time, and the cell search of the mobile-WiMAX system. The issue in OFDM channel estimation is studied in the work. We propose two estimation schemes and apply these algorithms in mobile-WiMAX system. In wide-band MIMO system, we study the transform design at the transmitter, propose a space-frequency transform, and use the turbo-DFE as the data detector at the receiver. In summary, the designed algorithms for the corresponding receiver issues are given in Table 10.1.
10.1 Future Research Topics
Some potential research topics are given as follows. First, the adaptive system parameters of the OFDM system can be realized with the knowledge of the channel parameters. Specifically, the CP is used to prevent the IBI effect. If the system has the delay spread of the channel profile, the CP length can be optimized. Besides, the utilization of the pilots also can be optimized if the performance of the channel estimator can be predicted. For example, consider channel estimation with polynomial interpolation in the comb-type OFDM system. If σ2ξ(N, τco) and the noise variance can be estimated in advance, the estimate MSE is a function of the pilot subcarrier spacing F . Thus, the optimization of the pilot subcarrier spacing can optimize the pilot utilization.
Secondly, what is a good interpolator of the channel estimation according to the pilot struc-ture? Specifically, consider the approximate LMMSE CE approach. The pre-defined PDP shape is constructed according the estimated delay spread and the mean delay of the channel, but it may not be the optimal approximation of the PDP event if the same profile model is used. It is interesting to estimate the optimal parameters of the profile model to match the true channel PDP or to optimize the channel estimate MSE. Besides, according to the window shift concept, what is a suitable interpolator for the given pilot assignment and what is the optimal window shift corresponding to the suitable interpolator?
Finally, the relay extension system in MIMO communication currently is a popular research topic. We may apply the transform MIMO-OFDM in such scenarios. The design of the trans-form, the receiving process, and the practical realization are potential topics.
Table 10.1: Proposed algorithms for the corresponding issues
high accuracy and low complexity fre-quency estimator
Reduced-constellation phase detector
SC QAM carrier recovery loop large frequency acquisition range
Hybrid phase detec-tor
SC QAM carrier recovery loop large frequency acquisition range and small steady-state phase noise
Dynamic loop band-width control
SC QAM carrier recovery loop good tracking stability
Boundary mean-square error
SC QAM channel equalization high accuracy MSE estimator at low decision-point SNR
Blind variable step-size algorithm
SC QAM channel equalization fast convergence
Hybrid VSS algo-rithm
SC QAM channel equalization fast convergence and soft switching of op-eration mode
Phase-irrelevant decision feedback equalizer
SC QAM channel equalization convergence of blind DFF in CFO envi-ronment
Frequency domain filtering
OFDM,
WiMAX cell search low complexity estimators in frequency domain
Phase-shift polyno-mial interpolation
OFDM,
WiMAX channel estimation MMSE optimization of the polynomial in-terpolation
Adaptive selection of interpolator
OFDM,
WiMAX channel estimation optimal interpolation-order estimation Mean delay and
RMS delay spread estimator
OFDM,
WiMAX channel estimation low complexity frequency-domain estima-tion
Approximate
LMMSE channel estimator
OFDM,
WiMAX channel estimation low complexity and high performance channel estimator
Space-frequency transform
MIMO-OFDM transmitter design maximal coding and diversity gain Separable
space-frequency interleaver
MIMO-OFDM receiver design separation of de-interleaver outside turbo-DFE loop
MMSE Shaped
turbo-DFE
MIMO-OFDM channel equalizer better convergence property of iterations in turbo-DFE
low pilot utilization and joint estimation of channel and data
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