= N0
B Ks, (3.49) where we have invoked the facts that (i) the training signal Xp and the noise N are independent, (ii) unitary pilot matrix is used, X∗pXTp = BI and (iii) elements of N is i.i.d.
complex white Gaussian noise with variance σ2n = N0. (3.49) implies that thermal noise induced MSE can be reduced by using a small modelling order. In Section 3.5 (Figs.
3.3 -3.5), we find that this noise-reduction effect is significant in low SNR environments where thermal noise dominates the MSE performance while the modelling error of (3.48) dominates in high SNR region.
If cW is not perfect and W = cW+ ∆W, then
Zb def= Z + ∆Z = ((WXp)TQT) ⊗ QR+ ((∆WXp)TQT) ⊗ QR. (3.50)
The coefficient vector estimation vec( bC) can be approximated up to the first order of ∆Z as [34]
vec( bC) ≃ vec(C) − Z†∆Zvec(C) + Z†vec(N) + (ZHZ)−1∆ZHPZ⊥vec(N) − Z†∆ZZ†vec(N),(3.51) where PZ⊥= I − Z(ZHZ)−1Z. The above equation indicates that, besides the terms that have to do with the noise N, the coefficient vector estimation error is determined by the projection error ∆Z. Hence, when the projection error ∆W is small (and thus ∆Z is small), vec( bC) is a good approximation of vec(C) at high SNR region.
3.5 Numerical Results and Discussion
Simulation results reported here use the reference MIMO channel model [2], the IEEE 802.11 TGn channel model [3], and the SCM model [35]. The former two are stochastic models whose spatial correlation matrices are generated by the power azimuth spectrum (PAS) at the BS and MS, respectively. The SCM model generates the channel coefficients according to a set of selected parameters (e.g., AS, AoD, AoA, etc.). It is a popular parametric stochastic model whose spatial cross correlations are functions of the joint
1e-05 0.0001 0.001 0.01 0.1
0 5 10 15 20 25 30 35 40
Normalized Mean Squared Error
Eb/N0 (dB) KT=2 (AS=2˚)
KT=3 (AS=2˚) KT=4 (AS=2˚) KT=5 (AS=15˚) KT=6 (AS=15˚) KT=7 (AS=15˚) Least Squared
Figure 3.1: MSE performance of Algorithm B as a function of SNR with different modelling orders; solid curves: AS=2◦, dotted curves: AS=15◦.
distribution of the AoD at the transmit side and the AoA at the receive side. We assume that the environment surrounding MS is rich scattering with negligible spatial correlations.
Hence, a full rank basis matrix is used to characterize the spatial correlation at the receive side. Other assumptions and conditions used in our simulation are: (i) the antenna spacings at transmit and receive arrays are both 0.5λ, (ii) an orthogonal pilot matrix is used, (iii) 10 iterations are used for all simulations (although in most cases convergence occurs in less than 3 iterations), and (iv) SNR (Eb/N0) is defined as the average signal to noise power ratio at the input of each receive antenna, (v) orthonormal polynomial basis matrices are used. Both algorithms compute bH by substituting the final result of Phase
I–estimated coefficient matrix b
C–and that of Phase II–cW–into (3.3).Solid curves in Fig. 3.1 are the MSE performance of Algorithm B in Section 3.1 for an 8×8 MIMO system with ∆ = 2◦and are based on the channel model of [2]. The channel is a block fading channel with an approximated rank of two. Since the BS spatial correlations are high, the corresponding correlation function lies in a low-dimension subspace so that a small KT is sufficient to describe the channel. Dotted curves in Fig. 3.1 show the system
1e-05 0.0001 0.001 0.01 0.1
0 5 10 15 20 25 30 35 40
Normalized Mean Squared Error
Eb/N0 (dB) KT=3, KL=4 (fdTs=0.015886)
KT=3, KL=5 (fdTs=0.015886) KT=4, KL=5 (fdTs=0.015886) KT=3, KL=4 (fdTs=0.031772) KT=3, KL=5 (fdTs=0.031772) KT=4, KL=5 (fdTs=0.031772) Least Squared
Figure 3.2: The effect of the modelling order on Algorithm B’s MSE performance in a channel generated by the model described in [1] with AS=2◦.
performance when ∆ = 15◦. It is obvious that as ∆ increases, the spatial correlations among the transmit antennas elements decrease and a higher modelling order is necessary to describe rapid-changing spatial waveforms at the transmitter side. However, as can be seen from Figs. 2-5, an optimal KT exists for any given SNR and ∆; increasing the modelling order does not necessary improves the channel estimator’s performance. As expected, we find that modelling errors dominate the MSE performance when SNR is high. Such a behavior is consistent with what the performance analysis given in Section 3.4 has predicted and is similar to those observed in other model-based approaches [9]-[13].
The MSE performance of Algorithm B of Section 3.2 for a time-correlated fading channel [2] are depicted in Fig. 3.2 and Fig. 3.3 using an observation window of 12 EIs and fdTs = 0.031772 or 0.015886. Similar to the single-blocked based case (Fig. 3.1), the processing dimension (KT) can be drastically reduced provided that either the spatial or time domain correlation is high enough. Performance degradation occurs when the modelling order is not large enough to capture the channel characteristics. In Fig. 3.4, we compare the theoretical MSE derived in Section 3.4 with the simulated performance
1e-05
Figure 3.3: The effect of modelling order on Algorithm B’s MSE performance in a channel generated by the model described in [1] with AS=15◦ and fdTs=0.031772.
1e-05
Figure 3.4: Comparison of theoretical and simulated MSE performance of Algorithm B in a channel generated by the model described in [1]; AS=15◦ and fdTs=0.031772.
and find that the latter is very close to the theoretical bound which assumes a perfect c
W. When used for estimating other reference channels, the proposed estimators exhibit similar performance behaviors. Fig. 3.5 depicts the MSE performance in an IEEE 802.11 TGn channel [3] with L = 12, ∆ = 15◦, and fdTs= 0.0022, while Fig. 3.6 shows the MSE performance in a 3GPP-SCM channel [35] with L = 12, ∆ = 15◦ and fdTs = 0.02844.
When KT is large enough, the time-domain modelling order needed to characterize a slow fading channel like the IEEE 802.11 TGn channel is smaller than that for a fast fading SCM channel. Note that in all cases, the performance becomes independent of the AS when the full modelling order is used (i.e., KT = 8) and is equivalent to that of the conventional LS approach.
Figure 3.5: The effect of the modelling order on the MSE performance of Algorithm B in a channel generated by IEEE 802.11 TGn channel model A; AS=15◦, and fdTs =0.0022.
The next two numerical results assume that the algorithms developed in Section 3.1 are used and, except for Fig. 3.8, the same channel model as that used for Fig. 3.1.
Fig. 3.7 compares the MSE performance of Algorithms A and B developed in Section 3.1 when ∆ = 15◦. If the maximum matching output is obtained by selecting the best one from the outputs using 100 candidate phases uniformly distributed within [−π, π),
1e-05
Figure 3.6: The effect of the modelling order (KT) on the MSE performance of Algorithm
B in a 3GPP-SCM channel; AS=15
◦ and fdTs=0.02844. Algorihtm A, Segment=20, KT=4Algorihtm A, Segment=20, KT=5 Algorihtm A, Segment=20, KT=6 Algorihtm A, Segment=20, KT=7 Algorihtm A, Segment=20, KT=8 Algorihtm A, Segment=100, KT=4 Algorihtm A, Segment=100, KT=5 Algorithm A, Segment=100, KT=6 Algorithm A, Segment=100, KT=7 Algorithm A, Segment=100, KT=8 Algorihtm B, KT=4
Algorithm B, KT=5 Algorithm B, KT=6 Algorithm B, KT=7 Algorithm B, KT=8
Figure 3.7: MSE performance comparison of Algorithm A (−−) and Algorithm B (−);
AS=15◦.
Algorithm A and Algorithm B give almost identical performance. However, if only 20
candidate phases are used, Algorithm A results in a little performance degradation with respect to that obtained by Algorithm B when SNR is high. Fig. 3.8 examines the MSE performance when cW is updated with different EI lengths for various channel settings.Smaller performance loss results if the channel is more static or less dynamic (smaller fdTs). When KT ≥ 3 for channel 1 and KT ≥ 2 for channel 2, the performance loss is negligible for all the update frequencies. Recall that more computation saving is obtained by a larger Tow. It is clear that our reduced-order modelling approach outperform the conventional LS estimator for most NEb0 when a proper KT is used.
1e-06
Figure 3.8: The effect of the update period on the MSE performance of Algorithm B.
Channel-1 is based on [2] with fdTs=0.015886 while Channel-2 is based on [3] with fdTs = 0.0022. AS=2◦, Toc = 1; both Toc and Tow are measured in EIs.
Fig. 3.9 and 3.10 illustrate the MSE performance when the target MIMO channels are time-variant and frequency-selective. We use Algorithm B developed in Section 3.3 to estimate the instant channel waveform. Fig. 3.9 is simulated under the MIMO channels generated by [2] where the power delay profile of six independent paths is given by [0, -1, -9, -10, -15, -20] dB with relative delays of [0, 310, 610, 1090, 1730, 2510] nanoseconds.
1e-05 0.0001 0.001 0.01 0.1
5 10 15 20 25 30 35 40
Normalized Mean Squared Error
Eb/N0 (dB) KT=4, KL=3, AS=15˚
KT=5, KL=3, AS=15˚
KT=6, KL=3, AS=15˚
KT=7, KL=3, AS=15˚
KT=8, KL=3, AS=15˚
KT=1, KL=3, AS=2˚
KT=2, KL=3, AS=2˚
KT=3, KL=3, AS=2˚
KT=4, KL=3, AS=2˚
KT=8, KL=3, AS=2˚
Figure 3.9: The effect of the angle spread on the MSE performance of Algorithm B in a channel generated by the model described in [2]; L = 12 and fdTs=0.015886.
Fig. 3.10 depicts the identification error under the fadings generated by [35]. In both Fig.
3.9 and 3.10, KL = 3 is used to model the time-domain correlation. Compared with the frequency-nonselective cases reported above, we can also reach similar conclusions about the relationship between the performance trend and the underlying modelling order.