Chapter 3 Decision Directed Channel Estimation
3.6 Simulation
3.6.3 Vehicular A
Decision directed in 240 km/hr
SNR
Fig. 3-19 SER performance for DD in 240 km/hr
10 15 20 25 30 35 40
Decision directed in 100 km/hr
SNR
Fig. 3-20 SER performance for DD in 100 km/hr
Accordimg to the simulations, we can see that the Decision directed-AR estiamtor can not work in high speed for SUI-3 and SUI-5. The Decision directed-Pilot Symbol Aided has the best performance of the symbol error rate (SER), but it has the highest pilot number. The Decision directed-Linear Interpolation can work in each environments, but it need to use one block data to estimate channel.
Chapter 4
Expectation Maximization Channel Estimation
4.1 Introduction of Expectation Maximization (EM) [16]
The EM algorithm [28, [29] is a technique for finding maximum likelihood (ML) estimates of system parameters in a broad range of problems where observed data are incomplete. The EM algorithm consists of two iterative steps: the expectation (E) step and the maximization (M) step. The E-step is performed with respect to the unknown underlying parameters, using current estimates of the parameters, conditioned upon the incomplete observations.
TheM-step then provides new estimates of the parameters that maximize the expectation of the log-likelihood function defined over complete data, conditioned on the most recent observation and the last estimate. These two steps are iterated until the estimated values converge.
The EM algorithm [28], [29] is an iterative method to find the ML estimates of parameters in the presence of unobserved data. The idea behind the algorithm is to augment the observed data with latent data, which can be eithermissing data or parameter values, so that the likelihood function conditioned on the data and the latent data has a form that is easy to manipulate. The algorithm can be broken down into two steps: the E-step and the M-step. We assume that the data Z (“complete” data) can be separated into two components,
(
,)
Z = X Y , where Y are the observed data (“incomplete” data) and X are the missing data. We denote θ as the unknown parameter we try to estimate from X .
The E-step finds Q
(
θ θ| ( )p)
, the expected value of the log-likelihood of θ , logf Z(
|θ ,)
where the expectation is taken with respect to X conditioned on Y and the latest estimate of θ , θ( )p
(
( )) { ( )
( )}
Q θ θ| p =E log f Z|θ | ,Y θ p .
The M-step then finds θ(p+1), the value of θ that maximizes Q
(
θ θ| ( )p)
over allpossible values of θ :
(p 1) arg max Qθ
(
| ( )p)
θ + = θ θ
This procedure is repeated until the sequence θ , ( )0 θ , ( )1 θ ,…converges. The EM ( )2 algorithm is constructed in such a way that the sequence of θ ’s converges to the ML ( )p estimate of θ .
Under our system model:
( ) ( ) ( ) ( )
Y m =H m X m +N m
Z is “unobserved data” or “complete data”.
Y is “observed data” or “incomplete data”.
H is unknown parameter we try to estimates.
X is “latent data” or “missing data”.
4.2 Iteration- Expectation Maximization (EM)
Fig. 4-1 EM Algorithm iteration process
Due to the Gaussian noise assumption, the probability density function of Y given X and H is given by
By assuming that all C symbols are equally likely and averaging the condition pdf over the variable X , we obtain the pdf of Y given H as follows
We assume that the frequency-domain signal X of a given sub-carrier represents a QPSK signal with constellation size C (=4).
(2 1)
consists of the following two steps E-step
Before doing Expectation Maximization (EM) Algorithm, the initial channel estimation H ( )0 is the input of the EM step, the output of the EM Algorithm H is shown as ( )1
( )1
( ) ( )
4 ( )0( )
The final result of the detection signal is ˆX(p+1). The iteration stop until the channel estimation ˆH(p+1) is stable. The final result ˆH(p+1) is shown as
4.3 Estimate Noise Power
Base on the Expectation Maximization (EM) Algorithm, we know that the noise power
2
σn is necessary. We can obtain the parameter easily. When we received one symbol signal, the receiver signal are shown as
( ) ( ) ( ) ( )
Y m =H m X m +N m (4.9)
We can obtain the noise power as
4.4 Expectation Maximization Channel Estimator
Base on the last discussion, we have three methods to obtain the initial channel, and iteration method.
Initial channel
4. Pilot Symbol Aided 5. Linear Prediction 6. Linear Interpolation
Iteration- Expectation Maximization
We combine the methods to be six channel estimators 4. EM Algorithm Pilot Symbol Aided (EM Pilot)
5. EM Algorithm Linear Prediction (EM AR) 6. EM Algorithm Linear Interpolation (EM LI)
4.5 Channel System Environment
The channel system environments are the same as Chapter 3.5 .
Table 4-1 SUI-3 Channel Model
Relative delay (μs or sample number) Average power
Tap ( sμ ) (normal) (dB) (normalized)
1 0 0 0 0.7061
2 0.4 4 -5 0.2233
3 0.9 10 -10 0.0706
Table 4-2 SUI-5 Channel Model
Relative delay (μs or sample number) Average power
Tap ( sμ ) (normal) (dB) (normalized)
1 0 0 0 0.7061
2 4 45 -5 0.2233
3 10 112 -10 0.0706
Table 4-3 ETSI “Vehicular A” Channel Model in Different Units
Relative delay (μs or sample number) Average power
Tap ( sμ ) (normal) (dB) (normalized)
1 0 0 0 0.4850
2 0.31 3 -1 0.3852
3 0.71 8 -9 0.061
4 1.09 12 -10 0.0485
5 1.73 19 -15 0.0153
6 2.51 28 -20 0.0049
4.6 Simulation
We show the simulation result as in Fig. 4-2~Fig. 4-7, there are four cases to be discussing,
SUI3 for v=100km/hr, v=240km/hr
SUI5 for v=100km/hr, v=240km/hr
Vehicular A for v=100km/hr, v=240km/hr
Ts is the sampling time, and F is the Doppler frequency, where d
d c
F =vF c
F Doppler d frequency
F carrier c frequency
v velocity of mobile
c velocity of light
This simulation are base on SUI-3 channel, SUI-5 channel and Vehicular A. The subcarrier number M is 1024, cyclic prefix length Ncp =M 8 . Carrier frequency
c 3.5
F = GHz. Bandwidth BW =11.2 MHz, block length B= symbols in Linear 4 Interpolation. When Linear Interpolation is used, block length B should not too large, the symbol channels of this block are still linear, so the Linear Interpolation can work. If B is too large, the channels are not linear, so the Linear Interpolation can not work effective.
4.6.1 SUI3
Fig. 4-2 SER performance for EM in 240 km/hr
10 15 20 25 30 35 40
Fig. 4-3 SER performance for EM in 100 km/hr
4.6.2 SUI5
Fig. 4-4 SER performance for EM in 240 km/hr
10 15 20 25 30 35 40
Fig. 4-5 SER performance for EM in 100 km/hr
4.6.3 Vehicular A
Fig. 4-6 SER performance for EM in 240 km/hr
10 15 20 25 30 35 40
Fig. 4-7 SER performance for EM in 100 km/hr
According to the simulations, we can see that the performance of the EM-Pilot Symbol Aided has the best SER performance. Just like Decision directed- Pilot Symbol Aided, EM-Pilot Symbol Aided has the highest pilot number. The EM-AR can work very well in each environments, no matter high speed or low speed. The EM-AR method does not use much pilot. When we use Pilot Symbol Aided to obtain the first symbol channel correctly, we can predict next symbol channel without any pilot. The EM-AR can work on time and dose not use any buffer to save data. The EM-Linear Interpolation can work as well as EM-AR, but it has to use some buffer. The buffer is the cost.
Chapter 5 Conclusion
Orthogonal Frequency Division Multiplexing (OFDM) is a popular technique in modern wireless communications. There are many systems adopting the OFDM technique, such as IEEE 802.11 a/g/n, IEEE 802.16, IEEE 802.20, Digital Video Broadcasting, etc. On the other hand mobile transmission is a trend in future wireless communications. For example, IEEE 802.16-2005 supports vehicle speed up to 120 km/hr, IEEE 802.20 supports vehicle speed up to 250 km/hr. Some channel estimator use much pilot to estimate channel, or use buffer to obtain channel. In this thesis, we show one predict-iteration method without pilot and buffer.
Finally, we evaluate the performance of the proposed system under mobility using IEEE 802.16-2005 standard and confirm that it achieves good SER performance.
In this thesis, we presented several channel estimation techniques and applied them on Orthogonal Frequency Division Multiplexing (OFDM) system. In Chapter 3, the channel environments and Decision directed Channel Estimation are introduced. We choose SUI-3, SUI-3 and Vehicular A to run the computer simulations. In Chapter 4, the Expectation Maximization Channel Estimation is introduced.
For the result, the EM-AR can work on time and dose not use any buffer to save data.
The EM-Linear Interpolation can work as well as EM-AR, but it has to use some buffer. The buffer is the cost. No mater in high speed or low speed, the Expectation Maximization
Channel Estimation can work very well.
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自 傳
陳錫祺, 西元 1979 年生於台北市。 西元 2004 畢業於台灣新竹國立交通大 學工業工程與管理學系，之後進入交通大學電子研究所攻讀碩士學位，於 2007 年取得碩士學位。研究方向為無線通訊、通道估測。