5.4 Implementation: EM-based Iterative Receivers
5.5.4 FER Performance of TURBO-EM Receiver
Figure 5.11 depicts the frame error rate (FER) performance in the ITU Veh-A channel. This figure also shows that the receiver with channel estimation update performs very well in terms of FER. Hence, in order to achieve a good performance, it is necessary to refine the CSI, especially when the number of pilot tones is small and the normalized maximum Doppler frequency is large.
From Figure 5.9 to Figure 5.11, we also observe that the receivers with G = 2 and G = 4 have nearly identical BER performance.
12 15 18 21 24 27 30 10−5
10−4 10−3 10−2
Eb/No (dB)
BER
Initialization Iter. 1 (CE update) Iter. 3 (CE update) Iter. 3 (w.o. CE update) Iter. 3 (CSI known)
Iter. 3 (CSI and data known) Quasi−static channel (CSI known)
Figure 5.6: BER performance of the ML-EM receiver in the two-path channel (NM L = 3 and [G, Q] = [4, 4]).
12 15 18 21 24 27 30 10−5
10−4 10−3 10−2
Eb/No (dB)
BER
Initialization Iter. 1 (CE update) Iter. 3 (CE update) Iter. 3 (w.o. CE update) Iter. 3 (CSI known)
Iter. 3 (CSI and data known) Quasi−static channel (CSI known)
Figure 5.7: BER performance of the ML-EM receiver in the ITU Veh-A channel (NM L = 3 and [G, Q] = [4, 4]).
12 15 18 21 24 27 30 10−4
10−3 10−2
Eb/No (dB)
BER
Two−Path channel [G,Q]=[1,19]
Two−Path channel [G,Q]=[2,9]
Two−Path channel [G,Q]=[4,4]
Veh−A channel [G,Q]=[1,19]
Veh−A channel [G,Q]=[2,9]
Veh−A channel [G,Q]=[4,4]
Figure 5.8: BER performance of the ML-EM receiver with channel estimation update for various [G, Q] (NM L = 3).
5 7 9 11 13 10−5
10−4 10−3 10−2 10−1
Eb/No (dB)
BER
Initialization
Iter. 3 (w.o. CE update) Iter. 1 (CE update) Iter. 3 (CE update)
Iter. 3 (CSI and data known) Iter. 3 (CE update) [G,Q]=[2,9]
Figure 5.9: BER performance of the TURBO-EM receiver in the two-path channel (NT B = 3 and [G, Q] = [4, 4]).
5 7 9 11 13 10−6
10−5 10−4 10−3 10−2 10−1
Eb/No (dB)
BER
Initialization
Iter. 4 (w.o. CE update) Iter. 1 (CE update) Iter. 4 (CE update)
Iter. 4 (CSI and data known) Iter. 4 (CE update) [G,Q]=[2,9]
Figure 5.10: BER performance of the TURBO-EM receiver in the ITU Veh-A channel (NT B = 4 and [G, Q] = [4, 4]).
5 7 9 11 13 10−3
10−2 10−1 100
Eb/No (dB)
FER
Initialization
Iter. 4 (w.o. CE update) Iter. 1 (CE update) Iter. 4 (CE update)
Iter. 4 (CSI and data known) Iter. 4 (CE update) [G,Q]=[2,9]
Figure 5.11: FER performance of the TURBO-EM receiver in the ITU Veh-A channel (NT B = 4 and [G, Q] = [4, 4]).
5.6 Summary
In this chapter, we have investigated two EM-based iterative receivers for OFDM and BICM-OFDM systems in doubly selective channels. Based on the proposed EM algorithm for data detection, both receivers use groupwise processing with ICI cancellation to reduce computational complexity and to explore time diversity inherent in time-variant channels. For OFDM sys-tems, the ML-EM receiver significantly outperforms the conventional one-tap equalizer, and its BER performance even approaches the BER performance without Doppler effect. Compared with the matched-filter bound, an Eb/No gap appears because of the error propagation effect. For BICM-OFDM sys-tems, a TURBO-EM receiver, which iterates between the MAP EM detector and the SOVA decoder, is then introduced. This receiver effectively solves the error propagation problem, and it attains a performance close to the low bound in terms of both BER and FER. Simulation results indicate that in order to attain a good performance, the channel estimation update is re-quired when we use low-density pilot tones at high Doppler frequencies. As a final remark, a group size of two to four is large enough to guarantee an acceptable performance under practical channel conditions.
Chapter 6 Conclusions
In this dissertation, we have studied channel estimation and data detection methods for OFDM systems in time-varying multipath channels. The scope of our research encompasses the design of pilot signals for MIMO channels, channel estimation and tracking for low-mobility channels as well as data detection for high-mobility channels. There are four main contributions in our works. First of all, we design CC pilot signals for optimal channel es-timation in MIMO systems. It is worth noting that the CC pilot signals not only exhibit the properties of both impulse-like auto-correlation and zero cross-correlation, but also have the characteristic of minimum PAPR in time domain. We also propose a CC pilot-based STBC-OFDM system to achieve bandwidth-efficient transmission at high vehicle speed. A receiver architecture for channel estimation and data detection is developed and its BER performance is analyzed and simulated. The second contribution is to present the equivalence between the DF DFT-based channel estimation method and the Newton’s method. Thus, the relationship between them
is well established. We clarify that the DF DFT-based channel estimation method can be further improved through the use of this equivalence. As to the third contribution, a refined channel estimation method is investigated for STBC-OFDM systems. The proposed channel estimation method is ac-complished in two stages. In order to reduce computational complexity and improve channel estimation performance, we propose an MPIC-based decor-relation method to catch significant channel paths in the initialization stage.
The tracking stage considers the use of a gradient vector derived from pilot tones to track temporal channel variation at the first iteration, followed by the DF DFT-based channel estimation method at the subsequent iterations.
In addition, an analytic formula for adaptively determining the optimum step size is investigated. The final contribution of this dissertation is the devel-opment of OFDM receivers that enable to deal with the ICI in the presence of Doppler spread in multipath channels. In this work, the EM algorithm is derived and performed for data detection using ML criterion. By inte-grating the groupwise ICI cancellation with the proposed EM algorithm, we study the design of a low-complexity iterative ML-EM receiver for OFDM systems. Based on the turbo processing principle, a TURBO-EM receiver, for joint detection and decoding in BICM-OFDM systems, is proposed to further improve system performance.
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Appendix A
Complementary Sequences
Complementary sequences are defined as a pair of sequences having the sum of their autocorrelation values equal to a Kronecker delta function. Binary complementary sequences, also widely named Golay sequences, were first introduced by Marcel J. E. Golay in 1949 [30, 31]. Later, complementary se-quences were generalized to polyphase or multilevel complementary sese-quences by other authors [81].
Let {α [0] , . . . α [N − 1]} and {β [0] , . . . β [N − 1]} be a pair of binary complementary sequences, i.e., the values of the two sequences are either
”+1” or ”-1”. The two sequences are complementary if we have
γ [n] ≡
N −1X
m=0
{α [m] α∗[((m − n))N] + β [m] β∗[((m − n))N]}
= 2N · δ [n]
=
2N, for n = 0
0 , for n 6= 0 (A.1)
where δ [n] is a Kronecker delta function. This ideal autocorrelation property
makes complementary sequences attractive for many applications like radar pulse compression and spread spectrum communication. From (A.1), the complementary sequences in frequency domain representation have comple-mentary power spectrum as follows:
Γ [k] =
N −1X
k=0
|A [k]|2+ |B [k]|2 = 2N (A.2)
where {A [k]} and {B [k]} are the DFT of {α [n]} and {β [n]}, respectively.
For the simplest example, we have binary complementary sequences of length two, given by {α [n]} = [+1, +1] and {β [n]} = [+1, −1]. Given a pair of complementary sequences {α [n]} and {β [n]}, a new pair of complementary sequences can be generated from the following rules if
1. Any of the two sequences is multiplied by eφ. 2. Any of the two sequences is time-reversed.
3. Any of the two sequences is circular-shifted.
4. The two sequences are interchanged.
5. Both sequences are decimated in time by K.
6. Both sequences are multiplied by eπkn/N, where k is a constant.
7. The two sequences are concatenated to form [α [0] , . . . , α [N − 1] , β [0] , . . . , β [N − 1]] and [α [0] , . . . , α [N − 1] , −β [0] , . . . , −β [N − 1]].
8. The two sequences are concatenated and interleaved to form [α [0] , β [0], . . . , α [N − 1] , β [N − 1]] and [α [0] , −β [0] , . . . , α [N − 1] , −β [N − 1]].
9. The two sequences are added and subtracted to form [α [0] + β [0] , . . . , α [N − 1] + β [N − 1]] and [α [0] − β [0] , . . . , α [N − 1] − β [N − 1]].
Appendix B
Proof of (2.37) and (2.38)
By substituting the definition of ˜De in (2.36) into E hD˜e
i
, we can obtain
E hD˜e
i
= (1 − BERC(ζ)) (1 − BERC(ζ)) E h
Λ1XˆF C +Λ2XˆSC
i
+ (1 − BERC(ζ)) BERC(ζ) E h
Λ1XˆF C+ 2H2XS +Λ2XˆSC
i
+BERC(ζ) (1 − BERC(ζ)) E h
2H1XF + Λ1XˆF C
+Λ2XˆSC i
+BERC(ζ) BERC(ζ) E h
2H1XF + Λ1XˆF C +2H2XS+ Λ2XˆSCi
= 2BERC(ζ) (H1XF + H2XS) (B.1)
Moreover, we can calculate the second moment of ˜De as follows:
Appendix C
E(j,i)−1 holds the same structure as the matrix E(j,i), given byE(j,i)−1 =
Appendix D
Explanation of Hessian Matrix
In this appendix, we provide an explanation of ˜E(j,i)−1. For simplicity, we assume that the DF data symbols are all correct, i.e., ˆX[k] = X[k], and neglect noise terms. Therefore, the LS estimate in (3.20) for channel H(j,i)[k]
given in (3.1) becomes
C [k] H(j,i)[k] = C [k]
XG l=1
˘
µ(j,i)l e−2π(k−1)(l−1)
K (D.1)
for k ∈ Θ, where C[k] = PNL
t=1|X(i)[t, k]|2 and ˘µ(j,i)l is the complex gain of the lth path. Taking the IDFT of (D.1), we get the estimate for the l0th channel path gain as follows
ˆ ηl(j,i)0 =
XG l=1
˘
µ(j,i)l X
k∈Θ
C [k] e2π(k−1)K(l0−l) (D.2)
where 1 ≤ l0 ≤ G. By rewriting (D.2) in a vector form, we have ˆ
η(j,i) = 1
2E˜(j,i)µ˘(j,i) (D.3)
where ˜E(j,i) is defined as in (3.21), ˆη(j,i) = [ˆη1(j,i), . . . , ˆηG(j,i)]T, and ˘µ(j,i) = [˘µ(j,i)1 , . . . , ˘µ(j,i)G ]T. As can be seen in (D.2) and (D.3), ˘µ(j,i)l from other paths
causes interference in ˆηl(j,i)0 due to the effect of aliasing, and ˜E(j,i)−1 acts as a path decorrelator to mitigate this effect.
Appendix E
Review of EM Algorithm
Consider a general ML estimate problem in a missing data model. Suppose that we observe a vector y generated from P (y| x, θ), where θ is the param-eter vector and x represents the missing data vector (or called unobserved latent data vector), and want to compute the ML estimate
θ = arg maxˆ
θ P (y| θ) (E.1)
The EM algorithm seeks to find the ML estimate of (E.1) by iteratively applying the following two steps:
E - step:
Compute
Ω
³
θ| y, ˆθ(m−1)
´
= Ex|y,θˆ(m−1)[L (y, x| θ)] (E.2) where the expectation is with respect to P
³
x| y, ˆθ(m−1)
´ . M - step:
Maximize
ˆθ(m) = arg max θ Ω³
θ| y, ˆθ(m−1)´
(E.3)
We refer to L (y, x| θ) as the complete log-likelihood function which cor-responds to the complete data of both y and x. The theoretical core of the EM algorithm is that the likelihood function L (y| θ) is guaranteed to increase monotonically at each iteration by maximizing Ω
³
θ| y, ˆθ(m−1)
´ at the M-step. In other word, we can ensure that
´ at the M-step. In other word, we can ensure that