6.3 Performance Studies of Joint Detection Schemes
6.3.2 Numerical Studies and Comparisons
In this section, we present the simulative comparisons of the proposed algorithms and the ver-ification with the analytical results. We consider the situation when the received signal only contains one significant preamble from the nearest BS. The detections of the multiple preamble sequences require a threshold mechanism; however, it is out of the scope in this discussion.
Besides, one main purpose of the initial synchronization is to find the BS that has the best communication quality, which is somehow equivalent to have the best signal strength in pream-ble symbol. Thus, we just consider detecting the preampream-ble symbol that has the largest signal strength.
In the simulation, consider an OFDMA WiMAX system with channel bandwidth = 10 MHz, FFT size = 1024, and CP length = 128. Recall that there are 114 selectable preamble sequences, which are divided into three segments, and the nonzero subcarriers in each sequence are spaced three subcarriers apart. Since we have no idea about integral carrier offset before estimation and so do the preamble code sequences with largest energy belong to which set, we need to test all the 114 candidates for each alternative integral CFO.
Study in Single Path Channels
We first adopt the single path channels, which includes the AWGN and the single path fading channel, to verify the simulated results with the analytical ones and to exam the correctness of the analysis. The adopted algorithms in this comparison is listed by: case 1, the optimal detection with awareness of the PDP; case 2, the detection with uniform PDP assumption, in which T = 128; case 3, the detection adopting the frequency domain filter, where the metric adopts the averages of the successive four samples as in (6.28); case 4, the detection adopting the differential detection.
−20 −18 −16 −14 −12 −10 −8 −6 10−8
10−7 10−6 10−5 10−4 10−3 10−2 10−1 100
SNR (dB)
False Detection Probability
Analytic and Simulated Results of Algorithms over AWGN Channel
Case 1: Known PDP, Sim.
Case 1: Known PDP, Analy.
Case 2: Unif. PDP, Sim.
Case 2: Unif. PDP, Analy.
Case 3: 4 taps FD Avg., Sim.
Case 3: 4 taps FD Avg., Analy.
Case 4: Diff. Det., Sim.
Case 4: Diff. Det., Analy.
Figure 6.4: Performance comparison of proposed algorithms over AWGN channel.
Fig. 6.4 illustrates the performances over AWGN channel. As shown in the performance curves, the simulated and analytic results are quite matching for all algorithms when SNR is
“high” (herein, high SNR is still really low enough). The SNR definition is the ration of the received preamble signal power relative to the noise power. The results show that, except case 1, case 3 using four taps average in frequency domain has the best performance; then, case 4 adopting differential detection performs better the detection with assumption of the uniform PDP.
Fig. 6.5 illustrates the results over single path Rayleigh channel. The simulated results also agree with the analytical ones in this channel conditions. However, in this channel condition, the detectors of case 3 and case 4 have no significant difference and they both perform roughly 2 dB better than detector of case 2.
Comparisons in Multipath Rayleigh Fading Channel
Now, we compare the algorithms over the multipath fading channel via the simulations. We consider jointly detecting the preamble sequence and the integer CFO in the second phase of the initially post-FFT synchronization. Assume that the true carrier set is detected, the true integer part of the CFO becomes zero after the coarse integer CFO estimation. The additional search range of the integer CFO is ±3 subcarrier spacings. However, due to the preamble structure, we only need to perform three different candidates of the integer CFO in [−3, 0, 3]. Therefore, there are total 114 × 3 = 342 combinations of the integer CFOs and the preamble sequences to be tested in one reception.
The compared algorithms are listed as follows: case 1, the detector with uniform PDP assumption, in which T = 128; case 2, the detector adopting the four samples frequency domain average as in (6.28); case 3, 1-norm version of case 2 where the detector is the same as case 2 except for replacing | · |2by | · |1in (6.28); case 4, the detector adopting the differential detection;
case 5, the detector adopting the hard-limited differential detection; case 6, the detector adopting
−20 −15 −10 −5 0 5 10 10−3
10−2 10−1 100
SNR (dB)
False Detection Probability
Analytic and Simulated Results of Algorithms over Rayleight Channel
Case 1: Known PDP, Sim.
Case 1: Known PDP, Analy.
Case 2: Unif. PDP, Sim.
Case 2: Unif. PDP, Analy.
Case 3: 4 taps FD Avg., Sim.
Case 3: 4 taps FD Avg., Analy.
Case 4: Diff. Det., Sim.
Case 4: Diff. Det., Analy.
Figure 6.5: Performance comparison of proposed algorithms over Rayleigh fading channel.
the differential detection with early dropping scheme in 5 maximal iterations; case 7, the detector same as 6 except for 10 maximal iterations. We test these algorithms over the block static Rayleigh fading channel that has the Vehicular-A PDP [21]. Furthermore, in the early dropping scheme, we set the length of the comparative samples to bNp/Kc where K is the number of the maximal iterations. For example, in case 6, the length is b284/5c = 56. Besides, the dropping threshold is arbitrarily set to 1/2 of the maximal metric in current iteration.
Fig. 6.6 depicts the simulation results. If the false detection probability in 10−3 is the comparative margin, the case 2 has the minimal requirement of the SNR. The sorting of the performances are given by: case 2 > case 3 = case 4 > case 1 > case 5 = case 6 > case 7. In the sort of the differential detection algorithms, the poor performances in case 5–7 are the penalties of the lower cost implementations. In the sort of the frequency domain filter, the | · |1version has just little performance loss than | · |2 version, whereas it has fewer cost from the multiplier-free realization.
6.4 Conclusions
This chapter presented the initial synchronization of the 802.16e OFDMA DL. We proposed the two-stage process of the initial synchronization. First stage performs before the FFT to acquire the symbol time and fractional CFO using the CP correlation. The second stage works after the FFT, which jointly estimates the integer CFO and the preamble index.
We mainly discussed the post-FFT synchronization step and specially focused on the problem of the symbol identification over multipath channel. We formulated the joint detection problem as the symbol identification. For this, we proposed the ML criteria of the symbol detection, which is the weighted sum of the cross-correlations of the reception and the candidate symbols, as well as several implementations of the joint detections in low-cost manners. One major con-tribution in this place is the concept of the frequency domain filter, which gives direct detection in frequency domain instead of the detection after translating into time domain. Additionally,
−15 −10 −5 0 10−4
10−3 10−2 10−1 100
SNR (dB)
False Detection Probability
Unif. PDP assumpt.
4 taps FD avg.
4 taps FD avg. with 1−norm Diff. Det.
Hard−limited DD.
DD. with early drop, MaxStage = 5 DD. with early drop, MaxStage = 10
Figure 6.6: Simulation results over vehicular A channel.
the well-known differential detection scheme can be treated as a special case in this concept.
Chapter 7
Channel Estimations in Pilot-Aided Multi-Carrier System
Channel estimation in pilot-aided multi-carrier system is an important and well-discussed issue.
The estimation algorithms mainly can be classified as the model-based, the statistical-dependent and the statistical-independent approaches. The subspace least-square method is one of the first class, which uses the projection onto the delay-subspace to eliminate the noise out-side the channel delay subspace. The major issue in this approach is the estimation of the channel delay subspace [103, 78, 91, 101], which takes great costs in realization, especially when the channel delays are not integer-spaced.
The Wiener MMSE estimator [18, 61, 72] is the most common statistical-dependent approach.
The MMSE channel estimation may perform in 1-D [18] if the channel is estimated individual per symbol or 2-D [61] if joint frequency and temporal channel correlation are used. The Wiener filter requires additional computation cost in the estimation of the channel correlation and the noise variance. A sub-optimal approach is to use the pre-defined correlation matrix instead of the real-time estimation. The used assumption of the pre-defined power-delay profile is usually the uniform or the exponential decayed distribution [18, 3]. However, the root-mean-square (RMS) delay of the channel is additionally required when the exponential decayed power-delay profile is used [3, 99].
The third approach uses the interpolation concept to reconstruct the missing channel re-sponses. One of this approach is the transform-based interpolation, such as the maximal likeli-hood interpolator [72]. To reduce the implementation cost, the discrete Fourier transform (DFT) based interpolator are typically used [19]. However, it is only applicable in equal-pilot-spacing multi-carrier system. Besides the transform-based interpolator, the frequency filtering interpo-lator is mostly popular scheme in low-cost consideration. The polynomial interpolation is the well-established approach in the frequency-domain interpolator [32].
This chapter presents two CE approaches. We first consider the interpolative channel esti-mation. Following the phase-shift concept in [32], we present the MMSE optimal window shift of the interpolation and its estimation. Then, we consider the approximate Wiener interpolation using predefined channel channel correlations. To generate the correlations, we investigate a simple estimation scheme of the mean delay and the root-mean-square delay spread. The pro-posed estimators are applied in the comb-type pilot-aided multi-carrier system and the 802.16e mobile WiMAX system.
7.1 MMSE Optimization of Polynomially Interpolative Channel Estimate
Pilot-aided channel estimation is widely adopted in orthogonal frequency division multiplexing (OFDM) and multiple access (OFDMA) systems. And frequency-domain polynomial interpo-lation is an often considered approach for such channel estimation. However, for channels that exhibit large delay spreads, the performance of polynomial interpolators suffers due to model-ing error. Some have proposed to remedy the problem by increasmodel-ing the interpolation order or by adding a linear phase shift to linear interpolation. The linear phase shift is equivalent to window shift in time domain. We thus derive a method to estimate the optimal window shift, in the minimum mean-square error (MMSE) sense, for polynomial-interpolative channel estimation of arbitrary order. As a practical application, we show how to apply phase-shifted in-terpolative channel estimation to Mobile WiMAX downlink transmission and propose a method to automatically select the interpolation order based on some MSE estimation.