Chapter 4 The ML-EM Receiver
4.5 Group-wise Method
Fig. 4.2 An illustration of group-wise detection.
The aforeme complexity, and
rs.
ntioned ML-EM algorithm has high computational
hence we use the group-wise method to provide a practical implementation for the ML-EM receiver. N subcarriers are partitioned into R groups, and each group contains G subcarrieS The j th group of subcarriers is given by Gj =
{
jG,…,(
j+1)
G−1}
, for j=0,…,R−1.e of the data groups e.g. the kth data group and illustr in Fig.4.2. The energy of x spreads over the adjacent 2 +1k Q groups (involving k
R . In order to sim on
= j Q− plify the interpretation, the concept of
ated
nsists of the interference resulted from the spreading energy made b ata groups denoted as
y a set of
ne hbo
{
xj: j∈Bk\{ }
k}
.Outside the EM detector, there is an outer iteration loop iterating between the
ML-EM detector and the ICI canceller as depicted in Fig.4.3. During an outer iteration, we diminish the ICI term by subtracting the leakage of other groups from the set of observation
groups and then obtain the ICI-reduced signals:
detector at the p vious fr the initialization in the beginning. After eliminating the ICI effect, the EM detector is performed by using EM-based channel estimation together with the group-wise method.
When Gibbs sampling is executed, samp re outer iteration or om
les are drawn by applying a ZF sampler.
d Every data group, which belongs to the sampler, takes turn by iteration loops to be calculate with the corresponding observation groups
{
yj: j∈D , and thus a large number of samples k}
are acquired. Then xk(m-1) is the decided data group at the
(
m 1−)
th EM iteration defined by an average of the collected samples drawn for the group xk. And xk(m-1) is sent back to be an initial data group for the ZF sampler at the mth EM iteration, while xk( )0 given from the initial setting is an initial data group at the first EM iteration. Going ugh the iteration o Gibbs sampling, we obtain d− sets of detected data groups (for neglecting n0 n sets of 0burn-in samples).
With the o
thro f
btainment of all the groups
{
xk(m-1):k∈{
1,…,R} }
, we can form d− n0vectors of N transmitted data symbols entireS ly and store up a sequence of probabilities of the final drawing within the
(
m 1−)
th EM iteration denoted as P(m-1). Because of theprobabilities P(m-1), the expected values required in (4.22) can b ulated, and hence derive the channel information ˆ(m 1)
e calc
a − for the
(
m 1−)
th EM iteration. aˆ(m 1−) and x(m-1) both provided for the given knowledge at next EM iteration. Besides, a ndare
( )m
ˆ a x( )m e
yielded as the way described above and become the output of the EM detector at the mth EM iteration. After cancelling the ICI effect,
ar
M can be modified by substituting the updated data
( )m
x into (4.29) via the unit of channel estimation update shown in Fig.4.3. M is replaced by M, and then
H
is calculated as M ΦFa+ ˆ( )m . Therefore, another outer iteration startsbelow. The data group x contributes most of energy to the ICI-reduced observation groups k
{
yj: j∈D ; thus, we can obtain diversity gains and draw samples for the k}
kth group data by nly through x and kperforming o
{
yj: j∈D . And it is obvious that the f l diversity gain k}
is attainable when the ICI iminated as well as the value of Q is large enough. Because spreading energy of a group mostly affects the neighboring 2Q groups, i.e the observation group (( ))
so the data groups used for the ICI cancellation should be a set of
,
experimental trials.
Y Y
( )m
x M
M
( )0
x
( )m
a
ˆFig. 4.3 The ML-EM receiver for OFDM systems.
Chapter 5 Residual ICI Power
oup-wise ICI canceller within an outer s
ta
ly, the quantity of the residual ICI power contained in the ICI-reduced signals v
When the received signals pass through a gr
iteration, the ICI effect in those signals are cancelled by using the decided data at the previou outer iteration. If CE is perfect and the decided data are detected correctly, we assume that there is no ICI power after ICI cancellation. However, it is almost impossible to make all da correct; thus, some ICI power would be left and turns to be the so-called residual ICI power discussed as follows.
And according
aries with the correctness of the detected data used in the ICI canceller. Furthermore, the residual ICI power differs from the ICI power,σICI2 , which also means the variance of ICI, because the latter represents the original power of the ICI effect without any subtraction of power from the received signals. The accuracy of the residual ICI power we estimate plays a important role in Gibbs sampling for the reason that the appropriate residual ICI power makes the a posteriori probabilities more reliable.
Following the group-wise EM-based
n
data detection described in section 4.5, the s calculation of the residual ICI power can be performed through the group-wise method a well and then illustrated with the subsequent case by employing the detected data groups
{
xj: j∈Bk \{ }
k}
and the observation groups{
yj: j∈D that are shown in Fig.4.2. Eak}
consists of G subcarriers wh m different levels of residual ICI ch
observation group ich suffer fro
power. For a subcarrier in the kth observation group, we calculate its residual ICI power by summing up the interference power caused by the incorrectness of the adjacent subcarriers in
the set
{
xj: j∈Bk \{ }
k}
. The observation groups of received signals are expressed as. Now it is assumed that an initi
where j∈ D and k W=Fa al channel estimate M and the
channel information Wˆ are perfectly estimated, and the ICI-reduced signals are given by
{ }
uses the residual ICI.
+ z
j(5.2)
The first two terms of (5.2) represent the signal energy of the kth data group, and the third term and the fourth term are caused by the mismatch between the transmitted data and the decided data and induce the residual ICI.
Supposing that
x
are the same as the transmitted signals, the ICI-reduced signals are free from the ICI effect and presented as, , ,
j = j k k + j k j k k + j
y M x Φ W x z
(5.3)where we find that the kth data group produces the remainin e the g energy spreading over th ICI-reduced signals. In case that some of the detected signals are wrong, there is not only
energy of the kth data group but also the residual ICI power resulted from the incorrect
cancelling in the ICI-reduced signals. According to (5.2), the residual ICI power is defin
2 RIP
ed by σ and calculated as below.
he first outer iteration:
At t
channel variable at the first outer iteration, and thus We have no information about the Wˆ
(5.2) is rewritten as
( )
if no prior probability is given.
outer iterations:
uter iteration, we obtain P x
(
=1)
After executing the previous o and P x
(
= −1)
from thetion Wˆ fr d
last drawing in Gibbs sampling and have the estimated channel informa om the output of the EM detector. (5.2) shows the ICI-reduced signals in this case an σRIP2 is provided by
spreading energy of one subcarrier through the computer simulation. And (5.6) is rewritten as
( ) ( )
A subcarrier spreads the energy over the neighboring subcarriers, and the spreading energy becomes smaller when the distance is long between two subcarriers. The percentages of spreading energy resulted from one subcarrier are generated according as the normalized maximized Doppler frequencies are 0.05 and 0.1, and depicted in Fig.5.1 and Fig.5.2 respectively. As shown in these two figures, it is found that the percentages are very much alike. Varying with the accuracy of the detected data, σRIP2 is used to be the adaptive ICI power for the a posteriori probabilities in Gibbs sampling.
Fig. 5.1 The ICI power percentage for the normalized MDF=0.05.
Fig. 5.2 The ICI power percentage for the normalized MDF=0.1.
Chapter 6 Computer Simulation
6.1 System Parameters
Results of computer simulation in this section demonstrate the performance of the ML-EM receiver. Based on the parameters defined in the 802.16e OFDM standard [12], we know that the system occupies a bandwidth of 5MHz and the carrier frequency is 2.3GHz. The entire bandwidth is divided into 256 sub-bands for N = 256 subcarriers among which J = 8 subcarriers carry the pilot tones, = 192 subcarriers transmit data and the remaining 56 subcarriers are used as virtual subcarriers. Those pilot subcarriers transmit the pilot tones adopting the BPSK modulation scheme and each having the same power as the data carried by a data subcarrier. The length of guard interval is = 64 (i.e. one quarter of 256 for the cyclic prefix). Each OFDM frame consists of = 40 OFDM data symbols and one OFDM symbol used for the CP-added preamble. Besides, the parameter of Q is set to 4 through the observation from experimental trials. The numbers of the EM iteration and the outer iteration are selected as = 2 and = 4.
NS
OL
NG
NF
NEM N
A two-path channel and an International Telecommunication Union (ITU) Veh-A channel are simulated with the path delays uniformly distributed from 0 to 50 sample periods.
The relative path power profiles are set as 0, 0 (dB) for the two-path channel and set as 0, -1, -9, -10, -15, -20 (dB) for the ITU Veh-A channel, where the fading channel can be generated with Jake’s Model by setting the normalized MDF equal to 0.05 and 0.1. And it is assumed
that both symbol synchronization and carrier synchronization are perfect and the receivers have some of the statistical information like noise power, power delay profiles and Doppler frequency. In addition, the parameter Eb N represents a ratio of received bit energy to the 0 power spectral density of noise.
6.2 Simulation Results
The following three cases are used in the simulation for comparison.
(A) CSI and data known: The curves with ideal CSI initialization and perfect initial data can be regarded as a performance lower bound.
(B) CSI known: This kind of curves is generated by using ideal CSI initialization and initial data given from the one tap equalizer.
(C) CSI est: The curves labeled as “CSI est” are made by setting initial CSI estimated and initial data given from the one tap equalizer.
The ways to produce initial data make difference between (A) and (B), and the modes of CSI initialization make (B) perform better than (C) if other conditions remain the same.
Most of the figures come from the simulation in the two-path channel, while Fig.6.9 and Fig.6.10 are given by the simulation in both the two-path channel and the ITU Veh-A channels.
The group size of the ML-EM receiver must be decided first of all, and the BER
performance curves are compared with each other in the case of “CSI and data known”. As depicted in Fig.6.1 and Fig.6.2 for the normalized MDF=0.05 and 0.1 respectively, joint detection of more subcarriers improves the performance, so the receiver with group sizes of 1 and 2 are worse than with group sizes of 4 and 8 which have nearly identical performance, and we choose the group size of 4 for the reason that a smaller group size takes less
computational time. Next, the BER performance makes progress by applying the residual ICI power described in chapter 5 to the Gibbs sampling. Based on the case of “CSI known”, it can be shown in both Fig.6.3 and Fig.6.4 for comparison between the performance of the ML-EM receiver with ICI power update and without ICI power update, and then the former is proved to be better. We subsequently develop the receiver combined with ICI power update.
Fig.6.5 and Fig.6.6 demonstrate the BER performance of the ML-EM receiver with ICI power update in the cases of “CSI known” and “CSI est”. It is observed that the curves with perfect CSI initialization perform better than the curves with CSI initialization estimated by zero forcing criteria and the performance improves as the outer loop iterates more times.
Moreover, Fig.6.7 and Fig.6.8 show that the ML-EM receiver with CE refinement has better performance than without CE refinement in the case of “CSI est”. And the “CSI est” curve with CE refinement is quite close to the “CSI known” curve, which means the receiver with CE refinement decrease the gap between the two modes of CSI initialization. CE refinement can obviously make the performance better for a channel with more paths, i.e. the ITU Veh-A
channel, and the BER performance is illustrated in Fig.6.9 and Fig.6.10.
Fig.6.11 and Fig.6.12 demonstrate the BER performance of the ML-EM receiver with the group size of 4 and CE refinement. Compared with the “CSI and data known” curve for the normalized MDF=0.05, the “CSI and data known” curve for the normalized MDF=0.1 has lower BER at the same Eb N , because time-variant channels introduce more diversity gains 0 for higher speed when the initial CSI and data are both perfect. And it is seen from Fig.6.11 that the three curves in different cases are rather close; however, there is a gap between the performance lower bound and the two curves else due to the error propagation effect observed from Fig.6.12. Finally, the number of samples required for Gibbs sampling affects the BER performance shown in Fig.6.13 and Fig.6.14, and thus we find that the receiver with more samples attain better performance, and the improvement is gradually saturated as the number of samples increases.
12 14 16 18 20 22 24 26 28 30 10-6
10-5 10-4 10-3 10-2 10-1
Eb/No(dB)
BER
FdT=0.05
One Tap EQ (CSI known) EM (G=1)
EM (G=2) EM (G=4) EM (G=8)
Fig. 6.1 BER performance of the ML-EM receiver with different group sizes for the normalized MDF=0.05.
12 14 16 18 20 22 24 26 28 30 10-6
10-5 10-4 10-3 10-2 10-1
Eb/No(dB)
BER
FdT=0.1
One Tap EQ (CSI known) EM (G=1)
EM (G=2) EM (G=4) EM (G=8)
Fig. 6.2 BER performance of the ML-EM receiver with different group sizes for the normalized MDF=0.1.
12 14 16 18 20 22 24 26 28 30 10-5
10-4 10-3 10-2 10-1
Eb/No(dB)
BER
FdT=0.05
One Tap EQ (CSI known) EM (w.o. ICI Power updated) EM (with ICI Power updated)
Fig.6.3 BER performance of the ML-EM receiver with/w.o. ICI power update for the normalized MDF=0.05.
12 14 16 18 20 22 24 26 28 30 10-5
10-4 10-3 10-2 10-1
Eb/No(dB)
BER
FdT=0.1
One Tap EQ (CSI known) EM (w.o. ICI Power updated) EM (with ICI Power updated)
Fig.6.4 BER performance of the ML-EM receiver with/w.o. ICI power update for the normalized MDF=0.1.
12 14 16 18 20 22 24 26 28 30 10-6
10-5 10-4 10-3 10-2 10-1
Eb/No(dB)
BER
FdT=0.05
One Tap EQ (CSI known) One Tap EQ (CSI est)
ICI cancel (iter.=1) (CSI known) ICI cancel (iter.=1) (CSI est) ICI cancel (iter.=2) (CSI known) ICI cancel (iter.=2) (CSI est) ICI cancel (iter.=3) (CSI known) ICI cancel (iter.=3) (CSI est) ICI cancel (iter.=4) (CSI known) ICI cancel (iter.=4) (CSI est)
Fig.6.5 BER performance of the ML-EM receiver for the normalized MDF=0.05.
12 14 16 18 20 22 24 26 28 30 10-6
10-5 10-4 10-3 10-2 10-1
Eb/No(dB)
BER
FdT=0.1
One Tap EQ (CSI known) One Tap EQ (CSI est)
ICI cancel (iter.=1) (CSI known) ICI cancel (iter.=1) (CSI est) ICI cancel (iter.=2) (CSI known) ICI cancel (iter.=2) (CSI est) ICI cancel (iter.=3) (CSI known) ICI cancel (iter.=3) (CSI est) ICI cancel (iter.=4) (CSI known) ICI cancel (iter.=4) (CSI est)
Fig.6.6 BER performance of the ML-EM receiver for the normalized MDF=0.1.
12 14 16 18 20 22 24 26 28 30 10-5
10-4 10-3 10-2 10-1
Eb/No(dB)
BER
FdT=0.05
CSI known (iter.=4)
CSI est (iter.=4) (w.o CE refined) CSI est (iter.=4) (with CE refined)
Fig.6.7 BER performance of the ML-EM receiver with/w.o. CE refinement for the normalized MDF=0.05.
12 14 16 18 20 22 24 26 28 30 10-4
10-3 10-2 10-1
Eb/No(dB)
BER
FdT=0.1
CSI known (iter.=4)
CSI est (iter.=4) (w.o CE refined) CSI est (iter.=4) (with CE refined)
Fig.6.8 BER performance of the ML-EM receiver with/w.o. CE refinement for the normalized MDF=0.1.
12 14 16 18 20 22 24 26 28 30 10-5
10-4 10-3 10-2 10-1
Eb/No(dB)
BER
FdT=0.05
CSI known (iter.=4)
CSI est (iter.=4) (w.o CE refined) CSI est (iter.=4) (with CE refined)
Fig.6.9 BER performance of the ML-EM receiver with different cases in the ITU Veh-A channel for the normalized MDF=0.05.
12 14 16 18 20 22 24 26 28 30 10-4
10-3 10-2 10-1
Eb/No(dB)
BER
FdT=0.1
CSI known (iter.=4)
CSI est (iter.=4) (w.o CE refined) CSI est (iter.=4) (with CE refined)
Fig.6.10 BER performance of the ML-EM receiver with different cases in the ITU Veh-A channel for the normalized MDF=0.1.
12 14 16 18 20 22 24 26 28 30 10-6
10-5 10-4 10-3 10-2 10-1
Eb/No(dB)
BER
FdT=0.05
EM(CSI and data known) EM (CSI known)
EM (CSI est with CE refined)
Fig.6.11 BER performance of the ML-EM receiver in different cases for the normalized MDF=0.05.
12 14 16 18 20 22 24 26 28 30 10-6
10-5 10-4 10-3 10-2 10-1
Eb/No(dB)
BER
FdT=0.1
EM(CSI and data known) EM (CSI known)
EM (CSI est with CE refined)
Fig.6.12 BER performance of the ML-EM receiver in different cases for the normalized MDF=0.1.
12 14 16 18 20 22 24 26 28 30 10-5
10-4 10-3 10-2 10-1
Eb/No(dB)
BER
FdT=0.05
sample no.=10 sample no.=20 sample no.=30 sample no.=40 sample no.=50
Fig.6.13 BER performance of the ML-EM receiver with various numbers of samples for the normalized MDF=0.05.
12 14 16 18 20 22 24 26 28 30 10-4
10-3 10-2 10-1
Eb/No(dB)
BER
FdT=0.1
sample no.=10 sample no.=20 sample no.=30 sample no.=40 sample no.=50
Fig.6.14 BER performance of the ML-EM receiver with various numbers of samples for the normalized MDF=0.1.
Chapter 7 Conclusions
In this paper, we investigate an EM-based iterative receiver under the ML criterion for OFDM systems in doubly selective channels. The receiver consists of an initialization unit, an ICI canceller, a CE update unit and an ML-EM detector. First, the initial setting of CSI and data can be executed through the use of the MMSE-based CE method and a decision-directed approach with the one-tap equalizer. Next, the ML-EM algorithm is proposed for channel variable estimation by using the samples given from Gibbs sampling. Incorporated with the group-wise processing, the ICI cancellation is developed to reduce the computational complexity and to exploit the time diversity inherent in time-variant channels. And the CE update unit provides better CSI to improve the performance at high Eb N especially. 0 Simulation results indicate that the ML-EM receiver significantly outperforms the one-tap equalizer.
References
[1] T. S. Rappaport : Wireless Communications – Principles and Practice, 2nd Ed., Prentice-Hall, 2002.
[2] Juha Heiskala and John Terry : OFDM Wireless LANs : A Theoretical and Practical Guide, 1st Ed, Sams, 2003.
[3] Y.-S. Choi, P. J. Voltz, and F.A Cassara, ”On channel estimation and detection for multicarrier signals in fast and selective Rayleigh channels,” IEEE Trans. On Wireless Communications., vol.49, pp. 1375-1387, Aug. 2001.
[4] J. Armstrong, P. M. Grant, and G. Povey, ”Polynomial cancellation coding of OFDM to reduce intercarrier interference due to Doppler spread,” in Proc. IEEE Golbal Telecommunications Conf., vol.5, pp.2771-2776, 1998.
[5] A. Seyedi, and G.J. Saulnier, ”General self-cancellation scheme for mitigation of ICI in OFDM systems,” IEEE Communications Conf., vol.5, pp. 2653-2657, June 2004.
[6] Y. Mostofi and D. C. Cox, ”ICI Mitigation for Pilot-Aided OFDM Mobile Systems,” IEEE Trans. On Communications, vol.4, No.2, pp. 765-774, Mar. 2005.
[7] Chorong-Ren Sheu, Ming-Chien Tseng, Ching-Yung Chen and Hsin-Piao Lin , ”A Low-Complexity Concatenated ICI Cancellation Scheme for High-Mobility OFDM Systems,” IEEE Communications Society, pp. 1525-3511, July 2007.
[8] Hongmei Wang, Xiang Chen, Shidong Zhou and Yan Yao, ”A Low-Complexity ICI cancellation Scheme in Frequency Domain for OFDM in Time-varying Multipath Channels,” IEEE, 2005, pp. 1234-1237.
[9] Xiaodong Cai and Georgios B. Giannakis, ”Bounding Performance and Suppressing
Intercarrier Interference in Wireless Mobile OFDM,” IEEE Trans. on Communications, vol.51, No.12, pp. 2047-2056, Dec. 2003.
[10] Meir Feder and Ehud Weinstein, ”Parameter Estimation of Superimposed Signals Using the EM Algorithm,” IEEE Tran., vol.36, No.4, pp. 477-489, April 1998.
[11] Xiaoqiang Ma, Hisashi Kobayashi and Stuart C. Schwartz, ”EM-Based Channel Estimation Algorithms for OFDM,” Euraship Journal on Applied Signal Processing, pp.
1460-1477, Dec. 2004.
[12] IEEE Std 802.16e-2005 and IEEE 802.16-2004/Cor1-2005,“Part 16: Air Interface for Fixed and Mobile Broadband Wireless Access Systems,＂IEEE-SA Standards Boards, Tech. Rep., 2006.
[13] P. Schniter, ”Low-Complexity Equalization of OFDM in Doubly Selective Channels,”
IEEE Trans. Signal Processing, vol. 52, No. 4, pp. 1002-1011, Apr. 2004.
[14] Xiaoqiang Ma, Hisashi Kobayashi and Stuart C. Schwartz, ”EM-Based Channel Estimation Algorithm for OFDM,” Euraship Journal on Applied Signal Processing, , pp.
1460-1477, Dec. 2004.
[15] O. Edfors, M. Sandell, J.van de Beek, S. K. Wilson, and P. O. Borjesson,“Analysis of DFT-based Channel Estimators for OFDM,＂Wireless Pers. Commun., vol. 12, pp. 55-70, Jan. 2000.
[16] Y. Li and L. J Cimini, Jr.,“Bounds on the interchannel interference of OFDM in time-varying impairments,＂IEEE Tran. Commun., vol. 49, pp. 401-404, Mar. 2001.
[17] 陳文娟,“雙選擇性通道中基於期望值最大化之正交分頻多工系統接收機的設 計與模擬,＂交通大學電信工程學系碩士論文.
[18] Xiaodong Wang and Rong Chen, “ Adaptive Bayesian Multiuser Detection for
Synchronous CDMA with Gaussian and Impulsive Noise,＂IEEE Tran. Signal Process. , vol. 47, No.3, July 2000.
[19] Behrouz Farhang-Boroujeny, Haidong Zhu and Zhenning Shi,“Markov Chain Monte Carlo Algorithm for CDMA and MIMO Communication Systems,＂IEEE Tran. Signal Process. , vol. 54, No. 5, May 2006.
[19] Behrouz Farhang-Boroujeny, Haidong Zhu and Zhenning Shi,“Markov Chain Monte Carlo Algorithm for CDMA and MIMO Communication Systems,＂IEEE Tran. Signal Process. , vol. 54, No. 5, May 2006.