In this section, the performance of aforementioned estimators are simulated. While frame structure shown in Fig. 3.1 is adopted, the pilot block at the beginning of each frame is assumed to be an identity matrix; thus, B = NT. We consider the estimation of block-fading, spatially-uncorrelated 4 × 4 MIMO channels with SM signals, where the time variation follows the Jakes’ model [17]. Specifically, time correlation is assumed to be
ρT(k − `) = J0(2πfD|t1− t2|BTs), (3.24) where fD is the Doppler frequency. Throughout this section, 4- and 6-bit/transmission rate are respectively achieved with AM being BPSK and 16-QAM constellation, i.e., M = 4 and 16. Besides, frame size of 5 and 40 are used to investigate the effect of error propagation lies in the DD estimators. The rest of the simulation parameters are list in Table 3.1. While the performance of channel estimators are investigated through bit error rate (BER), canonical (or later called mismatched) detectors
X =ˆ arg min
[ ˜X]ij∈AM∪{0}
kY(k) − ˆH ˜XkF; (3.25)
Table 3.1: Simulation parameters
Parameters Values
Operating frequency 2 GHz
Symbol period 0.1 ms
Number of transmit antennas NT 4 Number of receive antennas NR 4
Block size B 4
Table 3.2: Values of forgetting parameters λ in RLS.
SNR (dB)
0 4 8 12 16 20
fDTs
0.0222 0.6400 0.5120 0.4096 0.3277 0.0467 0.0280 0.0370 0.5120 0.4096 0.3277 0.2621 0.0280 0.0168 0.0519 0.4096 0.3277 0.2621 0.2097 0.0168 0.0101 0.0667 0.3277 0.2621 0.2097 0.1678 0.0101 0.0060
which treat channel estimates as real channels are employed, where ˆH = ˆH(k) for MB estimators and ˆH(k − 1) for DD ones.
µ in LMS algorithm is 1.45.
In Fig. 3.2 and Fig. 3.4 , we compared the BER of recursive least square (RLS) and LS both using decision-directed (DD) with 3 bits/transmittion at different velocity and frame size are 5 and 40, respectively. As mentioned in Subsection 3.2.2 the RLS updated equation is similar to the LS updated equation. Both channel coefficients at next time index are the linear combination of channel coefficient at last time index and the received data in this time index.
In Fig. 3.3-3.5, we compared the BER of RLS, LMS and MB estimators with 3 bits/transmittion and frame size 5 or 40 at different velocity. It can been seen from LMS estimator that channel coefficient at next time index are also the linear combination of channel coefficient at last time index and the received data this time index, but the weighting factor does not change with SNR. Therefore, even in high SNR, the channel is updated by the ’old’ channel coefficients with certain ratio causing the poor performance than RLS. However, the benefit of LMS is low complexity compared to the RLS method.
Table 3.3: Values of regularization parameters δ in RLS.
SNR (dB) 0 4 8 12 16 20
δ 1000 0.0665 0.0186 0.0067 0.0026 0.0001
0 5 10 15 20
10−4 10−3 10−2 10−1 100
Eb/N
0
BER
DD RLS DD LS DD LMS
fDT
s=0.0667
fDT
s=0.0519
fDT
s=0.0370 fDT
s=0.0222
Figure 3.2: BER of DD estimators with 3 bits/transmission; N = 5.
and LMS estimator at frame size 5, but worse than them when the frame size become larger. This is because the polynomial order is too small to chase the channel variation if the frame size is too large. The MB estimator is sensitive to the change of the velocity.
Fig. 3.6-3.7 is the simulation result of RLS, LS, LMS and MB estimator with 16-QAM and frame size is 5.
In Fig.3.8 and Fig.3.9, we compared the Mean square Error (MSE) of RLS and LMS at different SNR and different velocity. When the iterations number is large, both algorithm converge at low velocity. Both figure shows that the RLS algorithm have lower MSE than LMS algorithm, which explains the performance of LMS is poor than the RLS algorithm in Fig.3.8 and Fig.3.9. Furthermore, when the SNR is large, the MSE of these two algorithm get closer.
0 5 10 15 20
Figure 3.3: Comparison of RLS and MB estimators with 3 bits/transmission; N = 5;
fDTs= 0.0222, 0.0370, 0.0519, 0.0667.
Figure 3.4: BER of DD estimator with 3 bits/transmission; N = 40.
0 5 10 15 20 10−3
10−2 10−1 100
Eb/N
0
BER
MB
fDT
s
Figure 3.5: BER of MB estimators with 3 bits/transmission; N = 40; fDTs = 0.0222, 0.0370, 0.0519.
0 5 10 15 20
10−3 10−2 10−1 100
Eb/N
0
BER
DD RLS DD LS DD LMS
fDT
s=0.0222 fDT
s=0.0370 fDT
s=0.0519 fDT
s=0.0667
Figure 3.6: BER of DD estimators with 6 bits/transmission, N = 5.
0 5 10 15 20 10−5
10−4 10−3 10−2 10−1 100
Eb/N
0
BER
MB DD RLS
fDT
s
fDT
s
Figure 3.7: Comparison of RLS and MB with 6 bits/transmission, N = 5; fDTs = 0.0222, 0.0370, 0.0519, 0.0667.
1 2 3 4
Figure 3.8: MSE of RLS and LMS with 3 bits/transmission, N = 5.
0 20 40
Figure 3.9: MSE of RLS and LMS 3 bits/transmission, N = 40.
Chapter 4
Spatio-Temporal Correlation and Channel Estimation Error-Aware ML Detection
As the so-claimed ML MIMO detector (2.10) maximizes likelihood and thus mini-mizes symbol error rate (SER) when full CSIR is available, it becomes strictly suboptimal when only partial CSI or channel estimate is available at the receiver and is alternatively referred to as the mismatched detector
XˆMM(k) def= arg min
[ ˜X]ij∈AM∪{0}
kY(k) − ˆH ˜XkF, (4.1)
where ˆH = ˆH(k) for MB estimators and ˆH(k − 1) for DD ones. In this chapter, the real ML detectors corresponding to LS decision-directed and model-based channel estimators are derived. While most of the existing researches assume channel to be time-invariant within a frame, i.e., pilot and data transmission over the same channel, throughout this work, a more general environment is adopted. The channel varies from block to block and is spatio-temporally correlated. In the next section, the ML detector based on MB
channel estimation is given first. The following lemma is useful for deriving our proposed detectors throughout the work.
Lemma: Let z1 and z2 be circularly symmetric complex Gaussian random vectors with zero means and full-rank covariance matrices Σij def= E{zizHj }. Then, conditioned on z2, the random vector z1 is circularly symmetric Gaussian with mean Σ12Σ−122z2 and covariance matrix Σ11− Σ12Σ−122Σ21.
4.1 ML Detection With MB Channel Estimates
4.1.1 Universal MIMO Signal Detection
While the well-known Frobenius norm-based metric fails to optimize SER, the true ML detector shall be derived. Recall that with MB channel estimate ˆH(k), the universal ML MIMO detection shall be done with the maximization of likelihood function
P
vec(Y(k))
vec(X(k)), vec( ˆH(k))
. (4.2)
Since the entries of Y(k) and ˆH(k) are all zero-mean, invoking Lemma with
z1 = vec(Y(k)) = vec(H(k)) vec(X(k)) + vec(Z(k)), z2 = vec( ˆH(k))
=h
vec(Y(kp)) vec(Y(kp+ N )) vec(Y(kp+ 2N ))i
tH(k)T−1(kp)T
(4.3)
helps us obtain (4.2). Specifically, with
Σ11= E{z1zH1 } = XT(k) ⊗ INR Φ(X∗(k) ⊗ INR) + σ2zINRNT, Σ12= E{z1zH2 } = tH(k)T−1(kp)q(k) XT(k) ⊗ INR Φ,
Σ22= E{z2zH2 } = ν(k)Φ + σz2
tH(k)T−1(kp)
2
F INRNT, (4.4)
we are able to find the conditional mean and covariance of Y(k) given X(k) and ˆH(k):
Finally, the ML detector with MB channel estimates for universal MIMO system is obtained:
Note that instead of D(k), ˜D(k) is used because D(k) is a function of X(k) and thus varies with candidate blocks.
While in the above derivation no assumption has been made on the structure of data matrix X(k), i.e., it can carry spatial-multiplexed, spatial-modulated symbols, etc., we specifically find the ML detectors for SM systems with QAM or PSK-modulated symbols
which are of lower complexity than (4.8) due to the simplicity of SM in the following.
4.1.2 An Alternative Perspective on ML Detector Derivation
From another point of view, we can express in terms of ˆH which is estimated by Model-Based estimator as
H(k) = H(k) + E(k)ˆ (4.9)
where E(k) is the channel estimation error matrix. And we can express the error matrix as follow
vec(E(k)) = vec( ˆH(k)) − vec(H(k))
= h
vec(Y(kp)) vec(Y(kp + N )) vec(Y(kp+ 2N )) i
tH(k)T−1(kp)
T
− vec(H(k)) (4.10)
Then, we have the mean and covariance matrix of E(k)
E n
vec(E(k))o
= 0 E
n
vec(E(k))vec(E(k))Ho
= (ν(k) − 2tH(k)T−1(kp)q(k) + 1)Φ + σ2z
tH(k)T−1(kp)
2
F INRNT (4.11)
At the receiver side we have
Y(k) = ˆH(k)X(k) +
Z(k) − E(k)X(k)
= ˆH(k)X(k) + W(k) (4.12)
where W(k) is the colored noise.
To derive the ML detector, we invoke Lemma with z1 = vec(Y(k)) and
and with (4.11), (4.12), we can have the following parameters
Σ11= E{z1zH1 } = XT(k) ⊗ INR Φ(X∗(k) ⊗ INR) + σ2zINRNT,
Finally, we have the universal ML detector.
XˆML(k) = arg min
which has exactly the same as (4.8).
4.1.3 ML Spatial-Modulated Signal Detectors
Because we may let candidate ˜X = ˜L˜S with ˜S = Diag(˜s1, ˜s2, · · · , ˜sB) and
the dedicated ML detector for SM signals using different AM are derived with A(k) (4.7).
I M -PSK
On the other hand, if AM denotes M -QAM, the ML decision rule becomes
X˜ML(k) = arg min
4.1.4 Complexity-Aware Near-ML M -PSK SM Detector
As SM detector (4.17) calls for the exhaustive search over all possible combination of active antenna-transmitted symbol pairs which belong to LB× ABM, the computational complexity is nontrivial. It is desirable to find a low-complexity counterpart that reduces the search dimension while keeping the performance loss to a minimum. To this end, we develop a two-step approach that detects the active antenna indices and then transmitted
symbols sequentially. Specifically, the maximization of
Since for a specific ˜L, the following sum is dominated by one term:
X
is obtained by differentiation with QAM(·) demodulating the enclosed item to its nearest constellation points in AM.
Therefore, we have
L(k) ≈ arg minˆ
L˜
1 2
log det(EsD(k)) + ˜˜ mH(k) ˜D−1(k) ˜m(k) + 1
Es2¯sT( ˜L)˜J(k)¯s∗( ˜L)
− 1
Es<n ˜bH(k)¯s∗( ˜L) o
. (4.23)
Consequently, the transmitted symbol is decided as
ˆs(k) = ¯s( ˆL(k)).
Compared to (4.17), the search dimension of detector (4.23) is effectively reduced.