5.2 Approximated ML Detection With DD Channel Estimates
5.2.1 AML Spatial-Modulated Signal Detectors
Following the justification given in Section 4.2 and 5.1, we can also devise the approx-imated ML detector with partial CSI obtained by decision-directed channel estimation.
Recall that the data block k is detected with the channel estimates of the previous block H(k − 1). Therefore, the likelihood function aimed to be maximized becomesˆ
P
Y(k)|X(k), ˆH(k − 1)
≈
NR
Y
r=1
P
Yr(k)|X(k), ˆHr(k − 1)
. (5.10)
Invoking the Lemma with zH1 = Yr(k) and
zH2 def= ˆHr(k − 1) = Hr(k − 1)G1(k − 1) + Zr(k − 1)G2(k − 1),
where G1(k)def= X(k) ˆX†(k) and G2(k)def= X†(k)G1(k), we have
The AML detector is then given by
X(k) =ˆ arg min
Similarly, the approximation (5.10) becomes exact when ΦT = INT and thus
X(k) =ˆ arg min
5.2.2 Complexity-Aware AML M -PSK SM Detector
First, for SM system with PSK AM, the detector can be simplified as
X(k) = arg minˆ
˜sj∈AM, ˜`j∈L
NRlog det(EsC(k))˜
+tr (
Y(k)˜SH
Es − ρT(1) ˆH(k − 1)(ΦT + σz2INT)−1ΦTL˜ ˜C−1(k)
·Y(k)˜SH
Es − ρT(1) ˆH(k − 1)(ΦT + σ2zINT)−1ΦTL˜H)
where ˜C(k)def= σE2z
sIB+ ˜LH
INT − ρT(1)2ΦT(ΦT + σ2zINT)−1
ΦTL. Again, with X(k) =˜ L(k)S(k) and DD channel estimates, the two-step complexity-reduced AML detector for SM with PSK AM mandates
L(k) = arg minˆ
L˜
NRlog det(EsC(k)) + tr{ ˜˜ M(k) ˜C−1(k) ˜MH(k)}
+ 1
Es2¯sH( ˜L)˜J(k)¯s( ˜L) − 2
Es<{˜bT(k)¯s( ˜L)} (5.13)
and
ˆ
s(k) = ¯s( ˆL(k))
where ˜M(k) = ρT(1) ˆH(k −1)(ΦT+σz2INT)−1ΦTL, ˜˜ J(k) = ˜C−1(k) (YH(k)Y(k))∗, ˜b(k) equals to the diagonal of YH(k) ˜M(k) ˜C−1(k), and ¯s( ˜L) = QAM
Es(˜bT(k)˜J−1(k))H .
5.3 Simulation Results
In this section, we compare the BER performance of the derived Approximated ML (AML) SM signal detectors and that of the mismatched detectors. We are interested in operating these detectors in two scenarios: i) time-correlated channel with
decision-directed channel estimation; ii) channel with time-spatial correlation being estimated by the decision-directed method and model-based method. Throughout the simulation in this section, we adopt the system model described in Section 2.3 with B = NT = NR= 4 and equispaced transmit and receive antennas and choose the time-spatial correlation to follow [29] with carrier frequency fc= 2 GHz and symbol time Ts = 0.1 ms. Specifically,
ρT(k − `) = J0(2πfD|k − `|BTs) (5.14) ρS(i − m, j − n) = J0(2π|(i − m)|ξ/λ)J0(2π|(j − n)|ξ/λ), (5.15)
where fD is the maximum Doppler frequency, ξ the antenna spacing, λ the wavelength, and J0(·) the zeroth-order Bessel function of the first kind. The frame structure is as depicted in Fig. 3.1, thus the effective transmission rate is (N − 1)/N the claimed rate.
Note that bit power Eb = Es/m and Ep/N0 = 1/N0.
We first compare the detector performance for time-correlated channels in Figs. 5.1 and 5.2, where BPSK modulation is used to achieve a rate of 3 bits/transmission with frame size 5, 40, respectively and ΦT = ΦR= INT. As can be seen, the proposed detector XˆDDMLT (k) outperforms ˆXMM(k) significantly in all cases, especially when the channel variation is serious. This is because in such case, the channel estimated by treating the data detected by ˆXMM(k) as pilot is not reliable and may continue to affect the following data detection. In addition, what also can be seen is that a shorter frame helps receiver to get rid of this error propagation phenomenon the decision-directed method inherits at the cost of higher rate loss. The result using a higher modulation order is given in Fig. 5.3.
Next, we consider another scenario, spatial-time correlated channel. The BER per-formance of AML detector and Mismatched detector with DD estimator are compared in Figs. 5.4-5.5. The frame size is 5 with 3 bits/transmission for using BPSK, and the antenna spacing are 1λ, 5λ, respectively. we can see that the proposed approximated ML
0 5 10 15 20
Figure 5.1: BER performance comparison of the Approx. ML and mismatched detectors using decision-directed channel estimator in time-correlated channel; 3 bits/transmission, N = 5.
0 5 10 15 20
Figure 5.2: Performance of various detectors with decision-directed channel estimate in time-correlated channel; 3 bits/transmission, N = 40.
0 5 10 15 20 10−3
10−2 10−1 100
Eb/N
0
BER
XˆM M Approx. ˆXM LT
fDT
s=0.0519
fDT
s=0.0370
fDT
s=0.0222 fDT
s=0.0667
Figure 5.3: BER of Approx. ML and mismatched detectors using decision-directed channel estimate in channel with time correlation only; 6 bits/transmission, N = 5.
is out performance the conventional detector, although the high channel correlation will cause performance degradation to either the proposed ML detector or the conventional ML.
Next, the BER performance of AML, ML and mismatched detectors with MB and DD channel estimators are compared in Figs. 5.6-5.9 with different antenna spacing, where both spatial and time channel correlation is considered. In Figs. 5.6-5.7, the model-based estimation is applied to capture channel variation. Performance improvement is observed by using the proposed AML detector. We can see that the velocity influence more on MB channel estimator than on DD one. And the performance improvement of proposed AML detector is larger when the antenna spacing is small, i.e. high channel correlation.
Furthermore, we also show the BER performance of AML detector with MB channel estimator and estimated channel spatial correlation coefficient which is obtained in [27].
0 5 10 15 20
Figure 5.4: BER comparison of the Approx. ML and mismatched detectors using decision-directed channel estimator in channel with time-spatial correlation; ξ = 1λ, 3 bits/transmission, N = 5.
0 5 10 15 20
Figure 5.5: BER comparison of the Approx. ML and mismatched detectors using decision-directed channel estimator in channel with time-spatial correlation; ξ = 5λ, 3 bits/transmission, N = 5.
0 5 10 15 20
Figure 5.6: Performance of the various detectors with model-based channel estimate in time- and spatial- correlated channel; ξ = 0.1λ, 4 bits/transmission, N = 10 in 4 × 4 SM MIMO system.
Figure 5.7: Performance of the various detectors with model-based channel estimate in time- and spatial- correlated channel; ξ = 1λ, 4 bits/transmission, N = 10 in 4 × 4 SM MIMO system.
0 5 10 15 20
Figure 5.8: Performance of the various detectors with decision-directed channel estimate in time- and spatial- correlated channel; ξ = 1λ, 4 bits/transmission, N = 10 in 4 × 4 SM MIMO system.
Figure 5.9: Performance of the various detectors with decision-directed channel estimate in time- and spatial- correlated channel; ξ = 5λ, 4 bits/transmission, N = 10 in 4 × 4 SM MIMO system.
0 5 10 15 20 25 10−2
10−1 100
Eb/No
BER
Approx. ML Mismatched Perfect CSI R Est.
fDT
s=0.0519
fDT
s=0.0370
fDT
s=0.0222
Figure 5.10: BER comparison of the approximated ML and mismatched detectors using model-based channel estimator in channel with time-spatial correlation; ξ = 0.1λ, 4 bits/transmission, N = 10.
The performance is a little degradation due to the estimation error of correlation coef-ficients, but still better than the conventional mismatched detector.
Finally, let us see the performance of the complexity-reduced AML detector. In Figs.5.13-Fig.5.15, we show the BER of complexity-reduced AML detector with MB channel estimator. We can see that the BER of complexity-reduced AML have no different from AML detector at different antenna spacing and velocity. This result is similar to 4.8 and 4.9 that the complexity-reduced method cause nearly no performance degradation to ML detector.
0 5 10 15 20
Figure 5.11: Performance of the various detectors with model-based channel estimate in time- and spatial- correlated channel; ξ = 1λ, 4 bits/transmission, N = 10.
0 5 10 15 20
Figure 5.12: Performance of the various detectors with model-based channel estimate in time- and spatial- correlated channel; ξ = 5λ, 4 bits/transmission, N = 10.
0 5 10 15 20
Figure 5.13: BER comparison of the approximated ML and the complexity-reduced detector using model-based channel estimator in channel with time-spatial correlation;
ξ = 0.1λ, 4 bits/transmission, N = 10.
Figure 5.14: BER comparison of the approximated ML and the complexity-reduced detector with model-based channel estimate in time- and spatial- correlated channel;
ξ = 1λ, 4 bits/transmission, N = 10.
0 5 10 15 20 10−7
10−6 10−5 10−4 10−3 10−2 10−1 100
Eb/N 0
BER
Mismatched Approx. ML
Approx. ML(reduced) Perfect CSI
fDT
s=0.0370 fDT
s=0.0519
fDT
s=0.0222 fDT
s=0.0667
Figure 5.15: BER comparison of the approximated ML and the complexity-reduced detector with model-based channel estimate in time- and spatial- correlated channel;
ξ = 5λ, 4 bits/transmission, N = 10.
Chapter 6
Conclusion
In this thesis we investigated some issues associated with SM MIMO systems. We first introduce the SM scheme and its optimal detector. Then, two kinds of channel estimation have been introduced. The decision-directed estimator saves the pilot signal overhead and thus retains the data rate but suffer from the error propagation problem.
We also propose two other kinds of decision-directed channel estimators that take SNR into accounts to adjust the performance of estimator, the RLS and the LMS esti-mator. These two detector update the channel coefficient, with a weighting factor called forgetting factor, which is the linearly combination of the ’old’ channel coefficients and the ’new’ channel coefficients estimated by LS method. The performance of RLS estima-tor and LMS estimaestima-tor are similar to decision-directed estimaestima-tor due to the weighting factor changing with the SNR.
To error propagation, we proposed a model-based channel estimator which uses a polynomial to catch the channel variation. Model-based channel estimator keep the pilot signal overhead ratio and prevent from the error propagation problem and update channel coefficients every time index but there is a time delay from gather the enough pilot to solve the coefficients of the polynomial. The higher the order of the polynomial is, the larger the number of the polynomial coefficients is to be estimated causing longer
delay time. But the interval of the pilot time can be adjusted to make the delay shorter.
We also analyzed the effect of imperfect CSIR over MIMO system and SM MIMO system under different receiver strategies with time- and spatial-correlated fading chan-nel.
By taking channel estimation into consideration, we derive the ML detector for spatial-time correlated channel with model-based estimator and decision-directed es-timator, and show the BER of ML detector using model-based channel estimator in conventional MIMO system and in SM MIMO system. In both systems, the proposed detectors outperform the convention mismatched detector. Furthermore, we reduced the complexity of ML detector by maximize the likelihood function separately which causing the dimension reduction in the exhausted search space, and the complexity-reduced ML detector has the similar performance to the ML detector.
We also introduce another way to lower the complexity of ML detector. We derive the approximated ML detector for time-correlated channel and spatial-time correlated channel using decision-directed estimator and derive the approximated ML detector for spatial-time correlated channel using model-based estimator. The detector for time-correlation channel case using model-based estimator which we did not derive can easily got by setting the correlation matrix R = I in the detector. All these detectors con-sider the imperfect CSIR and spatial-time correlation thus outperform the conventional mismatched detector and can be seen via the simulation results. In addition, we also show the simulation results with estimated correlation matrix which is more practical and only a little degeneration to the ideal value of correlation matrix. Note that for the detectors using model-based estimator in this thesis are general form and can be used for any spatial-time correlated channel MIMO system.
ML detector is similar to the approximated ML detector via simulation results and would degenerate to the approximated ML detector when the channel is spatial-uncorrelated.
Finally, the fact that only one antenna is active at each time makes SM a promis-ing scheme for data transmission in highly correlated channel. Nevertheless, channel de-correlation, i.e., antenna spacing increment, improves the performance of both detec-tors.
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作者簡歷
一、About Me
1988 台灣台中人，出生於沙鹿。 雖然是沙鹿人不過搬過很多次加所以常 在台中海線到處遊玩。其中沙鹿跟清水的美食小吃很棒。
1995 畢業於名人幼稚園。幼稚園的記憶就是一直玩遊戲吃點心，很爽快。
2001 畢業於鎮立沙鹿國小。國小的時候不喜歡上課都一直跟同學聊天結 果被老師罵。每天休息時間就去打籃球。開始看 NBA。
2004 畢業於市立中港高級中學。開始感受到課業壓力，也開始喜歡上讀 小說，福爾摩斯全套等偵探小說。
2007 畢業於國立台中一中。聞名全台灣的一中街有很多好吃的，也很 好玩，有一段時間下課都跑去看漫畫。
2011 畢業於國立交通大學 電信工程學系。做了很多大學生所謂的瘋狂事 蹟。大四的專題課程跟蘇育德老師做通訊研究專題。
2013 畢業於國立交通大學 電信工程研究所。因為專題研究的接續，進入 蘇老師的實驗室。經過 2 年的努力地研究，終於取得碩士學位。
二、已修畢之相關課程
檢測與估計理論 數位通訊
編碼理論 隨機程序
適應性訊號處理 計算機網路
無線通訊之矩陣理論