Time- frequency- selective channel estimation of ofdm systems

Time- Frequency- Selective Channel Estimation of OFDM Systems

A Thesis

Submitted to the Faculty

of

Drexel University

by

Wei Chen

in partial fulfillment of the

requirements for the degree

of

Doctor of Philosophy

October 2005

ii

Dedications

Dedicated to my parents,

Guangxi Chen, Jimei Xu,

and my brother,

Bing Chen.

iii

Acknowledgements

It is my pleasure to acknowledge the individuals who were by my side both profession-

ally and personally during the years of my Ph.D. experience at Drexel University.

I owe my gratitude to my thesis supervisor, Dr. Ruifeng Zhang. The unwavering sup-

port and encouragement that I received from him helped me keep up my spirits and become

a better researcher. He was very informal and friendly through all the years of this research.

He provided me many valuable suggestions in both professional and everyday life. His ex-

citing advice and critical comments on reports, articles, presentations etc. improved my

oral and written communication skills.

I present, with pleasure, my sincere thanks to my doctoral committee members, Dr.

Harish Sethu, Dr. Stanislav Kesler, Dr. Kapil Dandekar, andDr. Eric J. Schmutz for their

valuable suggestions on my thesis and for serving in the committee of my Ph.D. defense.

I would like to thank Dr. Leonid Hrebien and Dr. Mohana Shankar for the guidance

and help during my Ph.D. experience at Electrical and Computer Engineering (ECE) de-

partment.

I would like to thank all my friends at Drexel University for their help.

Finally, I would like to give special thanks to my parents andbrother for their constant

love and support.

iv

Table of Contents

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 1

1.1 Basic Principles of OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 History and State-of-the-Art of OFDM . . . . . . . . . . . . . . . .. . . . 6

1.3 Channel Estimation of OFDM Systems . . . . . . . . . . . . . . . . . .. 7

1.3.1 Review of Previous Work . . . . . . . . . . . . . . . . . . . . . . 9

1.3.2 Kalman Filter for Channel Estimation . . . . . . . . . . . . . .. . 16

1.3.3 Contributions of This Thesis . . . . . . . . . . . . . . . . . . . . .18

1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Chapter 2. OFDM System and Channel Model . . . . . . . . . . . . . . . . .. . . 20

2.1 OFDM System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 DFT-Implemented OFDM Signal Model . . . . . . . . . . . . . . . . . .. 25

2.3 WSSUS Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Chapter 3. Kalman Filter Based Channel Estimation Algorithms . . . . . . . . . . . 32

3.1 State Space Model for OFDM . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Kalman Filter Channel Estimator . . . . . . . . . . . . . . . . . . . .. . . 33

3.3 Per-Subcarrier Kalman Filter Estimator with MMSE Combiner . . . . . . . 35

3.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

v

Chapter 4. Sequential Monte Carlo Method and Mixture KalmanFilter . . . . . . . 47

4.1 Sequential Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . .. 47

4.2 Mixture Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.1 Mixture Kalman Filter Algorithm . . . . . . . . . . . . . . . . . .52

4.2.2 Application of Mixture Kalman Filter to OFDM Receiver. . . . . 53

4.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Chapter 5. Conclusion and Future Research . . . . . . . . . . . . . . .. . . . . . . 70

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

vi

List of Tables

1.1 Applications of OFDM Technique . . . . . . . . . . . . . . . . . . . . .. 8

3.1 Simulation parameters of Kalman filter based algorithms. . . . . . . . . . 39

4.1 Simulation parameters of mixture Kalman filter based algorithms . . . . . . 63

vii

List of Figures

1.1 Signals transmitted through frequency-selective channels . . . . . . . . . . 2

1.2 Multi-carrier modulation scheme . . . . . . . . . . . . . . . . . . .. . . . 2

1.3 Frequency Division Multiplexing system . . . . . . . . . . . . .. . . . . . 3

1.4 An OFDM system withN sub-carriers over a bandwidthW . . . . . . . . . 4

1.5 Comparison between conventional multi-carrier technique and OFDM . . . 5

1.6 DFT Implementation of transmitted waveform . . . . . . . . . .. . . . . . 5

2.1 Base-band continuous-time OFDM systems model . . . . . . . .. . . . . 21

2.2 Transmitted signal of OFDM systems . . . . . . . . . . . . . . . . . .. . 21

2.3 CP prevents ISI of OFDM symbols . . . . . . . . . . . . . . . . . . . . . .23

2.4 Equivalent model of OFDM systems . . . . . . . . . . . . . . . . . . . .. 24

2.5 Lattice points of OFDM systems . . . . . . . . . . . . . . . . . . . . . .. 25

2.6 Sampling of one OFDM symbol . . . . . . . . . . . . . . . . . . . . . . . 26

2.7 DFT-Implemented OFDM systems . . . . . . . . . . . . . . . . . . . . . .26

2.8 Jake’s Doppler power spectrum . . . . . . . . . . . . . . . . . . . . . .. . 29

2.9 Relationship of wireless channel parameters . . . . . . . . .. . . . . . . . 31

3.1 Vector Kalman filter and per-subcarrier Kalman filter with MMSE combiner 37

3.2 Transmission symbol format . . . . . . . . . . . . . . . . . . . . . . . .. 39

3.3 Normalized channel estimate error versus SNR . . . . . . . . .. . . . . . 41

3.4 Bit-error-rate versus SNR . . . . . . . . . . . . . . . . . . . . . . . . .. . 42

3.5 Learning curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

viii

3.6 Mismatch performance (NMSE) of Doppler frequency (SNR=10dB) . . . . 43

3.7 Mismatch performance (BER) of Doppler frequency (SNR=10dB) . . . . . 43



3.10 Mismatch performance (NMSE) of delay spread (SNR=10dB) . . . . . . . 45

3.11 Mismatch performance (BER) of delay spread (SNR=10db). . . . . . . . . 45


3.13 Mismatch performance (BER) of delay spread (SNR=20dB). . . . . . . . 46

4.1 Flowchart of mixture Kalman filter algorithm . . . . . . . . . .. . . . . . 54

4.2 Comparison of Kalman filter and mixture Kalman filter . . . .. . . . . . . 58

4.3 Mixture Kalman filter solution . . . . . . . . . . . . . . . . . . . . . .. . 61

4.4 Normalized channel estimate error versus SNR . . . . . . . . .. . . . . . 64

4.5 Bit-error-rate versus time . . . . . . . . . . . . . . . . . . . . . . . .. . . 65

4.6 Learning curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.7 Mismatch performance (NMSE)of Doppler frequency (SNR=10dB) . . . . 66








5.1 L-subcarrier Kalman filter with MMSE combiner . . . . . . . . . . . . .. 71

ix

AbstractTime- Frequency- Selective Channel Estimation of OFDM Systems

Wei ChenDr. Ruifeng Zhang

Communications through frequency-selective fading channels suffer inter-symbol inter-

ference (ISI), which limits the data rate. Complex equalizers are usually needed to compen-

sate for the channel distortion. Orthogonal Frequency Division Multiplexing (OFDM) di-

vides the channel spectrum into many sub-bands, each of which carries low-rate data. Since

each sub-channel is narrow band, communications through itexperience only flat-fading.

The sub-channels are separated with the minimum space required by channel orthogonal-

ity. Therefore, a large number of low-rate data streams can be transmitted in parallel and

aggregate to a high-rate one.

For coherent detection of the information symbols, the channel gain of each sub-carrier

is needed. This problem is further complicated by the time-varying nature of the channel

fading and the correlation between the sub-channels due to Doppler frequencies.

The contribution of this dissertation is a Kalman filter based framework of channel

equalizer for OFDM systems in a time-frequency-fading environment. The gains of the

sub-channels are defined as the state variables, and an AR process is used to model the

dynamics of the channel. Then the Kalman filter can be appliedto estimate the channel

from the received OFDM signals when pilot symbols are available. This Kalman filter is

of a dimension equal to the product of the number of sub-carriers and the order of the AR

model, which can be very large. To reduce its complexity, per-subcarrier Kalman filter

scheme is proposed, that is, the Kalman filter channel estimator is applied to obtain the

gain of each sub-carrier independently, and then a minimum mean-square-error (MMSE)

combiner is used to refine the estimates. The per-subcarrierKalman estimator explores the

x

time-domain correlation of the channel, while the MMSE combiner explores the frequency-

domain correlation. This two-step solution offers a performance comparable to the much

more complicated Kalman joint estimator.

The Kalman filter method is also extended to give a blind channel estimation algorithm

based on the mixture Kalman filter (MKF). The MKF uses Monte Carlo simulations to do

filtering. By simulating the transmitted symbol sequences according to theira posteriori

probabilities, so called importance sampling, and feedingthem to the Kalman filter chan-

nel estimator, a Bayesian estimate of the channel can be obtained through averaging the

Kalman filter output of each simulation. The MKF method applies to the joint estimation

of all sub-carriers and also works in the per-subcarrier fashion with a much reduced com-

plexity. In addition to the channel estimate, the MKF can directly give symbol detection.

1

Chapter 1. Introduction

Orthogonal Frequency Division Multiplexing (OFDM) originated from the need of effi-

cient communications through frequency-selective fadingchannels. A channel is frequency-

selective if the frequency response of the channel changes significantly within the band of

the transmitted signal [1]. While, a constant frequency response is called flat fading. Fig.

1.1 (a), (b) exemplifies the frequency-selective and flat fading channels. Digitally mod-

ulated signals going through a frequency-selective channel will be distorted, resulting in

inter-symbol-interference (ISI). To mitigate the ISI, a complex equalizer [20] is usually

needed to make the frequency response of the channel flat within the bandwidth of interest;

or the symbol duration must be long enough so that the ISI-affected portion of a symbol can

be negligible. From the frequency-domain viewpoint, the latter approach means to transmit

a narrow-band signal within whose bandwidth the channel canbe well considered to be flat

fading, as shown in Fig. 1.1 (d). This fact gives the idea thatone can transmit several

low-rate data streams, each at a different carrier frequency through the channel in parallel,

and each data stream is ISI-free and only a simple one-tap equalizer is need to compensate

the flat fading. This idea is illustrated in Fig. 1.2. That is actually the idea of Frequency

Division Multiplexing (FDM).

However, this multi-carrier transmission scheme may suffer inter-carrier interference

(ICI), i.e., the signals of neighboring carriers may interfere each other. To avoid the ICI,

guarding bands are employed in FDM to separate different sub-carriers. This results in a

waste of the spectrum. OFDM follows the very similar multi-carrier modulation strategy.

However, it employs the orthogonality among sub-carriers to eliminate the ICI without the

need of the guarding bands.

2

��

� � � � � � � � � � � � � � � � � ��

��

� � ��

� ��

��

bandwidth bandwidth

bandwidthbandwidth

(a) A frequency-selective fading channel (b) A flat fading channel

f

ff

f

(c) Modulated signal (d) Narrow band signal

Figure 1.1: Signals transmitted through frequency-selective channels

... ...

��

� � ��

� ��

� � ��

� ��

� � ��

� ��

channel bandwidth

f

Figure 1.2: Multi-carrier modulation scheme

3

1.1 Basic Principles of OFDM

......

d�t �

ej2π f0t

ej2π f1t

ej2π fN�1t

e� j2π f0t

e� j2π f1t

e� j2π fN�1t

s0�l

s1�l

sN�1�l

y0�l

y1�l

yN�1�l

channel

Figure 1.3: Frequency Division Multiplexing system

As illustrated in Fig. 1.3, frequency-multiplexed digitally modulated signals in one

symbol duration are of the form

d�t � � N1

∑k0

sk�lej2π fkt � lT t �l �1�T (1.1)

whereN information symbolssk�l �k � 0� � � � �N �1 are transmitted simultaneously and are

considered a block,l indicates block,fk is thekth sub-carrier, andT is the symbol duration.

In OFDM signaling, the following orthogonality condition is satisfied,

� T

0ej2π fite j2π f j t �

� T

0ej2π �fif j �t � 0 (1.2)

That means the space between the frequencies of the sub-carriers should be

∆ f � fi � f j � mT

(1.3)

4

...

N

1�Tsub-carriers

Spacing

Figure 1.4: An OFDM system withN sub-carriers over a bandwidthW

wherem can be any positive integer. The smallest space for orthogonality is equal to the

symbol rate 1�T.

With the orthogonality, each sub-carrier can be demodulated independently without

ICI. It should be noted that the passbands of the sub-carriers may overlap in OFDM, as

shown in Fig. 1.4. This allows one to pack the sub-carriers into a given spectral band in a

densest fashion, so a high spectral efficiency is achieved. Fig. 1.5 illustrates the difference

between the conventional non-overlapping multi-carrier technique and the OFDM. OFDM

signald�t � in (1.1) can also be obtained using a digital method, as shownin Fig. 1.6, if

we note that theN�T-rate samples ofd�t � is the IDFT of the information symbolssk�l �k �0� � � � �N �1,

d�t ��tnTN� N1

∑k0

sk�lej2π kT nT

N � N1

∑k0

sk�l ej2π nkN � n � 0� � � � �N �1 (1.4)

Then, the FFT algorithm makes the implementation of the OFDMscheme very efficient.

In addition to the high spectral efficiency and simple equalization, the advantages of

OFDM include:

5

�a�

�b�

f

f

savings in bandwidth

conventional multi-carrier

OFDM

1T

Figure 1.5: Comparison between conventional multi-carrier technique and OFDM

N Point

IDFT D/Ad�t�d�t �

ej2π f0t

ej2π f1t

ej2π fN�1t

s0�ls0�l

s1�ls1�l

sN�1�lsN�1�l

�

fi � f j� i� j

T

Figure 1.6: DFT Implementation of transmitted waveform

6

� OFDM can easily achieve optimal “bit-loading” [103], i.e.,assign different power

and constellation size to each sub-carrier to enhance system capacity.

� OFDM is robust against narrow-band interference because such interference affects

only some of the sub-carriers [69].

� OFDM allows efficient FFT implementation.

1.2 History and State-of-the-Art of OFDM

Early forms of multi-carrier modulation date back to the 1950s and early 1960s with

military high frequency (HF) radio links. In the mid-60s, R.W. Chang [2] presented the

basic principles of OFDM. The innovative feature of the system developed by Chang com-

pared to traditional systems is that the spectra of the sub-carriers overlap under the condi-

tion that they are all mutually orthogonal. This property ofOFDM systems makes unnec-

essary the steep bandpass filters used in older multi-carrier modulation systems to separate

the spectra of the individual sub-carriers.

Shortly after Chang’s paper, Saltzberg [3] analyzed the performance of OFDM and

made an important conclusion that “ the strategy of designing an efficient parallel system

should concentrate more on reducing crosstalk between adjacent channels than on per-

fecting the individual channels themselves, since the distortions due to crosstalk tend to

dominate”.

In 1971, Weinstein and Ebert [4] first proposed to apply discrete Fourier transform

(DFT) and inverse discrete Fourier transform (IDFT) to perform baseband modulation and

demodulation in OFDM systems. Their contribution lies in the elimination of the banks

of sub-carrier oscillators and the introduction of the efficient processing. This innovation

makes OFDM technology more practical. Currently, OFDM systems apply fast Fourier

transform (FFT) and inverse FFT (IFFT) to perform modulating and demodulating of the

7

information data.

When transmitted through a frequency-selective channel, the orthogonality of the sub-

carriers will be destroyed, causing inter-carrier interference (ICI). To combat the ICI, Peled

and Ruiz [5] introduced the concept of a cyclic prefix (CP) in 1980. Rather than using

an empty guard space in time between OFDM symbols, they filledthe guard space with

a cyclic extension of OFDM symbols. This effectively simulates a channel performing

circular convolution, which implies orthogonality of sub-carriers when the CP is longer

than the impulse response of the channel. The penalty of using a CP is loss of signal

energy proportional to the length of the CP. However, the benefits of using a CP generally

outweighs any loss of signal energy.

Over the last two decades, OFDM has been used in a wide varietyof applications.

When applied in a wired environment, OFDM is often referred to discrete multi-tone

(DMT). DMT has been adopted as the standard for the Asymmetric Digital Subscriber Line

(ADSL), which provides digital communication at several Mb/s from a telephone company

central office to a subscriber.

OFDM has been particularly successful in numerous wirelessapplications, where its

superior performance in multi-path environments is desirable. OFDM is currently applied

in several high rate wireless communications standards such as European digital audio

broadcasting (DAB) [10], digital video broadcasting (DVB-T) [13], 802.11a [15], 802.11g

[16], and 802.16 [17]. Several DAB systems proposed in NorthAmerica are also based on

OFDM [6]. A list of OFDM applications is shown in Table 1.1.

1.3 Channel Estimation of OFDM Systems

Channel estimation is an important component of a communication system. With the

information of the channel impulse response, and/or model parameters of a channel, one

8

Table 1.1: Applications of OFDM Technique

Fix-wire network Asymmetric Digital Subscriber Line (ADSL) [7]

High bit rate Digital Subscriber Line (HDSL) [8]

Very high speed Digital Subscriber Line (VDSL) [7]

Broadcasting Digital Audio broadcasting (DAB) [9,10]

Digital Video Broadcasting (DVB) [11]

High definition television terrestrial broadcasting [12,13]

WLAN HIPERLAN2 (European) [14]

IEEE 802.11a (U.S.A.) [15]

IEEE 802.11g (U.S.A.) [16]

WMAN IEEE 802.16-2004 (WiMax) [17]

Others A candidate of 4G mobile communication [18]

A candidate of high rate wireless PAN (802.15.3a) [19]

can perform the optimal symbol detection or construct a equalizer, or predict the channel.

In OFDM systems, modulated bits are distorted during transmission through the chan-

nel since the channel introduces amplitude and phase shiftsdue to frequency selective and

time-varying nature of the wireless channel. In order for the receiver to acquire the origi-

nal bits, it needs to take into account these unknown changes. The receiver applies either

coherent detection or non-coherent detection to recover the original bits. Coherent detec-

tion [20] uses reference values that are transmitted along with data bits. The receiver can

estimate the channel only at reference value locations. Theentire channel can then be esti-

mated by using several interpolation techniques [20, 51]. Non-coherent detection [20], on

the other hand, does not use any reference values but uses differential modulation where

9

the information is transmitted in the difference of two successive symbols. The receiver

uses two adjacent symbols in time or two adjacent sub-carriers in frequency to compare

one with another to acquire the transmitted symbol.

Without channel estimation, OFDM systems have to use differential phase-shift key-

ing (DPSK), which has a 3-dB signal-to-noise ratio (SNR) loss compared with coherent

phase-shift keying (PSK) [20]. DPSK is appropriate for relatively low data rates, such as

the European digital-audio broadcast (DAB) system [10]. Toimprove OFDM systems per-

formance, channel estimation is needed. Accurate channel estimation algorithms can be

applied in OFDM systems to allow coherent detection, thereby improving system perfor-

mance.

1.3.1 Review of Previous Work

Channel estimation in OFDM systems has been well studied [40–45, 53–58, 67, 68].

Based upon whether those channel estimation algorithms apply training symbols, we can

divide those algorithms into three categories: training (pilot) based algorithms [40–45],

blind algorithms [53–58] and semi-blind algorithms [67, 68]. We will discuss these three

groups of algorithms separately. Training-based algorithms [40–45] assume known sym-

bols (training or pilot symbols) are inserted in the transmitted signals. It is then possible to

identify the channel at the receiver through exploiting knowledge of these known symbols.

Blind algorithms [53–58] estimate the channel based on properties of the transmitted sig-

nals (finite alphabet properties [55], cyclo-stationarity[53]). Semi-blind algorithms [67,68]

can improve the performance of blind algorithms by exploiting the knowledge of both

known symbols and properties of the transmitted signals. The objective of semi-blind chan-

nel estimation algorithms is to get better performance thanblind algorithms while requiring

fewer known symbols than training based channel estimationalgorithms.

10

1. Training (pilot) methods

The channel estimation algorithms proposed in [40–45] are training based algo-

rithms. The idea of the algorithms in [40–43] is to exploit OFDM channel corre-

lation, thus the statistics (mean, variance) of the channelmust be known to make

these algorithms work. In [44, 45], channel statistics are not needed. The idea is to

build a model with unknown coefficients, then exploit the information of the output

symbols and training symbols to estimate the coefficients ofthe model.

Beek [40] proposed two OFDM channel estimation algorithms,the minimum mean

square error (MMSE) algorithm [94] and the least squares (LS) [94] algorithm. The

MMSE algorithm uses only the correlation of the channel in frequency domain (i.e.,

the correlation between sub-channels) and fails to addresstime-domain dynamics.

The MMSE algorithm performs better than LS algorithm at the cost of higher com-

plexity. Additionally, MMSE algorithm requires knowledgeof channel covariance

and noise variance, which are assumed to be known asa priori knowledge. The

LS algorithm does not need any knowledge about the channel. Beek [40] also took

advantage of the finite impulse response property of the channel to simplify the pro-

posed MMSE and LS algorithms.

Li [42] proposed a MMSE OFDM channel estimation algorithm inwhich coarse

channel estimates from several successive OFDM symbols arefurther combined op-

timally in the MMSE [94] sense to get an updated channel estimate. The algorithm

exploits both time domain and frequency domain channel correlation, and makes

use of the fact that the OFDM channel correlation can be written as the product

of time domain channel correlation and frequency domain channel correlation [39].

The proposed MMSE channel estimator first exploits the frequency domain channel

correlation, and then exploits the time domain channel correlation. Information of

11

channel statistics and operating SNR are needed to make thisalgorithm work. More-

over, the proposed MMSE algorithm can work in a mismatch mode. However, its

performance degrades much if the Doppler frequencies and delay spreads applied in

the MMSE algorithm are smaller than the ones of the practicalchannel. However, the

initial coarse channel estimates are obtained independently from one OFDM symbol

to another without taking advantage of the time domain dynamics of the channel.

Morelli [46] gave a comparison between two training based OFDM channel esti-

mation algorithms, Maximum Likelihood (ML) algorithm and MMSE algorithm.

The difference between these two estimators is in their assumptions of the chan-

nel state information (CSI). The ML algorithm regards the channel as a deterministic

but unknown vector, whereas MMSE algorithm regards the channel as a random vec-

tor, whose particular realization is to be estimated. The MLalgorithm achieves the

Cramer-Rao Lower Bound (CRLB) [94], therefore it is a Minimum Variance Unbi-

ased (MVU) estimator. Minimum MSE is achieved on the condition that the CSI is

considered deterministic and the estimator is unbiased. With the aid of prior chan-

nel information, the MMSE algorithm outperforms the CRLB [46], because CRLB

is a bound for deterministic CSI. With more available information about the chan-

nel, MMSE algorithm obtains a better performance. The MMSE algorithm [46]

assumes that the channel is zero mean and Gaussian, and is uncorrelated with the

noise. With the Gaussian assumption of the channel, MMSE estimator becomes

linear (LMMSE). MMSE algorithm becomes ML algorithm at highSNRs or if no

prior information on the channel is available. Overall, MMSE algorithm exploits in-

formation about the channel and performs better than the ML algorithm. However,

the MMSE algorithm is more complex and more difficult to implement compared

with ML algorithm. Specifically, 1)ML algorithm does not require knowledge of

12

the channel statistics and the SNR, and it is simple to implement; 2) MMSE algo-

rithm performs better than ML algorithm at low SNR; 3) MMSE algorithm and ML

algorithm have comparable performance at intermediate andhigh SNRs.

Edfors [41] applied the theory of optimal rank-reduction [47] to LMMSE estima-

tor, and presented a low-rank channel estimation algorithm, which exploits only the

frequency domain channel correlation. Although channel correlation and signal-to-

noise ratio (SNR) are needed in the channel estimation algorithm proposed in [41],

its performance is robust to changes in channel correlationand SNR. There is an irre-

ducible error floor for this low-rank estimator due to the fact that part of the channel

does not belong to the subspace. Sufficiently large rank is needed to control the error

floor up to a given SNR. Specifically, the smaller the rank is, the lower the algorithm

complexity, and the higher the estimation error.

Wang and Chang [44,45] proposed training assisted OFDM channel estimation algo-

rithms based on nonlinear two-dimensional (2-D) regression channel models. Due to

the fact that the polynomial model can be used to approximatethe fading multi-path

channel if it is viewed as a smoothly varying function [48], Borah [49,50] applied the

polynomial approximation in the time domain, and [52] applied the polynomial ap-

proximation in the frequency domain. Wang and Chang [44,45]applied polynomial

approximation in both time and frequency domains. Unlike the algorithms proposed

in [40–43], the model based channel estimation algorithms do not need to know the

channel statistics (correlation of the channel) and the operating SNR. There are three

steps of the proposed model based channel estimation algorithms. The first step is

to obtain an initial estimate of the channel at the pilot locations by a simple channel

estimation algorithm, such as the LS method, and the next step is to model the chan-

nel of the OFDM system. The time-frequency plane is divided into small blocks of

13

the same polynomial structure. For each block, a 2-D surfacefunction is built with

unknown coefficients, which can be estimated based on the criteria such as the min-

imized weighted Euclidean distance to the LS-estimated CSIat the pilot locations.

The third step is to estimate the CSI at other locations (CSI corresponding to data

symbol) according to the estimated regression surface function.

Dong [51] considered the problem of optimally placing pilotsymbols for tracking

time-varying frequency selective fading channels in an OFDM system. The time-

varying channel is approximated as a Gauss-Markov process,and a Kalman filter is

used for updating channel states in time domain. For a fixed percentage of pilots,

assuming pilot symbols are periodically sent throughL �1 sub-channels, the goal

is to optimize the placement of pilot symbols by minimizing the average steady-

state MSE of the channel estimator. It is shown that single pilot periodic placement

achieves the minimum MSE with the assumption that channel fading is relatively

slow compared to the symbol rate, i.e. the channel state remains unchanged over the

duration of each OFDM block, but changes from block to block.

2. Blind algorithms

In wireless systems where bandwidth is the most precious resource, periodic train-

ing symbols can significantly reduce overall system capacity. Blind channel estima-

tion algorithms are well motivated when the overhead of training symbols becomes

high, which is true when performing channel estimation of fast time varying wireless

channels. There are many blind algorithms for estimating OFDM channels [53–58].

Some of them [53,54,57,58] need to average over a number of OFDM symbols dur-

ing which the channel must be static. Others [55, 56] work in asymbol-by-symbol

manner, so can deal with fast time-varying channels. However, they assume inde-

pendence of the channel for different OFDM symbols, and do not explore the time-

14

domain correlation of the channel.

Due to the cyclo-stationarity introduced by the cyclic prefix (CP), Heath [53] pro-

posed a blind OFDM channel estimation algorithm using cyclic correlations at the

OFDM receiver. The CP is formed by copying the lastTcp-long part of the sym-

bol waveform to the beginning, which will be discussed in detail in chapter 2. This

algorithm does not require the cyclic prefix to be longer thanthe channel impulse re-

sponse, but it is helpful to reduce the error floor present in the unshortened scenario

if this algorithm is combined with impulse response shortening.

Muquet [54] proposed a blind OFDM channel estimation algorithm via subspace

decomposition [47]. Thanks to the inherent redundancy introduced by the CP of

the OFDM systems, it is possible to apply the subspace methodproposed in [60]

to blindly estimate the channel of OFDM systems. The advantages of the subspace

method [54] are: 1) it is applicable to any signal constellation; 2) it does not need to

know the signal constellation used for transmission.

Zhou [55] proposed a blind OFDM channel estimation method exploiting the finite

alphabet property of transmitted symbols. Compared with the subspace method pro-

posed in [54, 59], the finite alphabet method [55] has two advantages: 1) channel

identifiability is guaranteed even when the channel is zero on sub-carriers; 2) fewer

received symbols are required to make the finite alphabet method work. Channel

estimation is possible even from a single OFDM symbol at highSNR when the con-

stellation of transmitted symbols is PSK, therefore the finite alphabet method can be

applied to fast channel variations at high SNR. Additionally, due to the property of

finite alphabet, the scalar ambiguity which is inherent to all blind channel estimation

algorithms can be restricted to a finite set of values, which make it easy to solve

the scalar ambiguity. Various channel estimation algorithms derived from the finite

15

alphabet method become available by trading off complexityfor performance.

Yang [56] considered the design of a blind Bayesian receiverfor wireless OFDM

systems over unknown frequency-selective fading channels. The designed receiver

is based on a Bayesian formulation of the problem and the sequential Monte Carlo

method, and it works with the observations over one OFDM symbol duration, but it

does not exploit the time domain channel correlation of the channel.

Petropulu [57] proposed a linear precoding based blind OFDMchannel estimation

algorithm. It consists of a simple linear transformation applied to blocks of symbols

before they enter the OFDM system, which enables blind channel estimation at the

output of the OFDM system via cross-correlation operations. Unlike other coded

OFDM schemes, the proposed algorithm does not introduce redundancy to the block,

nor does it change the block length. The proposed algorithm is computationally

simpler than subspace algorithm [54].

3. Semi-Blind algorithms

Blind algorithms can also be used in cooperation with training symbols in order to

achieve better performance, which are referred to as semi-blind methods. Semi-blind

channel estimation algorithms are discussed in [62–66], and Al-Naffouri [67, 68]

proposed semi-blind channel estimation algorithms applicable for OFDM systems.

Semi-blind algorithms allow to outperform blind techniques by exploiting the knowl-

edge of known symbols and properties of the transmitted signals. The objective

of semi-blind channel estimation algorithms is to obtain better performance than

blind algorithms while require fewer number of training symbols needed for train-

ing based channel estimation algorithms. In training basedalgorithms, the training

symbols must be placed closely enough in time and frequency to accurately track a

channel [40, 43]. However, semi-blind algorithms works effectively even when the

16

frequency-domain training symbol spacing is greater than the Nyquist spacing [68],

thereby the resulting training symbols can be significantlyreduced, especially when

the channel varies rapidly both in time and frequency domains.

The semi-blind algorithm proposed in [67] takes advantage of the channel and data

information to get channel estimation and reduces the number of the training symbols

needed to achieve this task. Specifically, the knowledge [67] exploited includes:

1) maximum delay spread of the channel; 2) a prior channel statistics (mean and

covariance); 3) redundancy of the input introduced due to the cyclic prefix; 4) the

finite alphabet property of transmitted signals due to the constellation limit. Through

exploiting these knowledge collectively, the channel estimation problem becomes a

least-square (LS) problem [67]. The algorithm first estimates the channel based on

training symbols and then iterates between channel estimation and data detection.

Thomas [68] proposed a multi-user semi-blind channel estimator for OFDM based

on the Maximum-Likelihood (ML) criteria.

1.3.2 Kalman Filter for Channel Estimation

The Kalman filter is an efficient recursive algorithm which estimates the state of a dy-

namic system from a series of noisy measurements [95,96]. Kalman filter has been applied

in communication systems since 1970s [83–85]. Lawrence andKaufman [83] first applied

Kalman filter in the equalization of digital communication systems. The proposed algo-

rithm in [83] do not exploit any prior information about the channel state, and it can only

be applied in time invariant channel. The state equation is built based on the information

symbol, with the driving noise as the information symbol. The state equation said nothing

more than that each succeeding component at timei is equal to the previous component

at time i �1. With known channel state, the state variable can be estimated based on the

17

observation. When the channel state is not known, the state model needs to be extended

in which all the unknown parameters, the information symboland the channel state, are

included.

Godard [84] showed how a Kalman filter was applied to identifytap gains of transversal

equalizers to minimize mean-square distortion. In [84], the state variables are defined as

the tap gains of transversal equalizers, and this algorithmworks in pilot mode or decision-

feedback mode. As in [83], no prior knowledge of the channel is available, and this algo-

rithm can only be applied in time invariant channels or slow varying channels.

Mulgrew [86] developed a linear IIR equalizer to combat inter-symbol interference.

The IIR filter is implemented in a Kalman filter architecture of [83], and this algorithm can

only be applied in time invariant channel. Grohan [87] gave acomparison of the Kalman

filter structures of [83,84,86].

Kalman filter could also be applied to track the states of timevarying channels [88–91].

The state variable is defined as the channel states, which canbe modeled in an AR model

[88]. The state equation is then built based on the AR model, which reflects the statistics

of time varying channel. Iltis [88] proposed a Kalman filter to track multi-path channel

in a direct-sequence, spread-spectrum communication system. Iltis and Fuxjaeger [88, 89]

modeled the time-varying tap coefficients as autoregressive (AR) processes with respect

to time, and then employed an extended Kalman filter to adaptively estimate them. Their

methods take advantage of the channel’s time domain statistics, but no method is provided

to estimate the AR model parameters. In addition, the assumption of uncorrelated tap

coefficients further limits the applicability of the method.

Tsatsanis [90] proposed a least square method to estimate the parameters of AR model

based on channel correlation. The channel correlation is estimated from training data by

exploring a joint fourth-order statistic between the inputand the output. Zhou [91] proposed

a precoding scheme to estimate the channel correlation fromthe second-order statistics of

18

the output and training symbols are not necessary. Kalman filter algorithm is then applied

to track the time varying channel [90,91].

For Wide Sense Stationary Uncorrelated Scattering (WSSUS)channel, the channel cor-

relation can be derived from Jakes’ model [72]. The parameters of AR model can then be

solved through Yule-walker equation [93]. Bulumulla [92] applied Kalman filter in the

channel estimation of OFDM systems, where the state variable is defined as the frequency

domain channel state of OFDM systems. The proposed algorithm [92] exploits time and

frequency domain channel correlation simultaneously, andachieves optimum performance.

However, the proposed algorithm has high complexity due to the high dimension of the

state variable.

1.3.3 Contributions of This Thesis

Noticing the fact that channel correlation of WSSUS channelin time and frequency

domains can be separated, we built an AR model for each sub-carrier of OFDM systems

by only exploiting the time domain channel correlation. Obviously the AR model built for

each sub-carrier is not as precise as the AR model built for OFDM systems, because only

the time domain channel correlation is exploited. With the AR model for each sub-channel

and input-output relationship of OFDM systems, a state-space model for each sub-channel

can be built, and Kalman filter can then be applied to track thestate of sub-channels. Due

to the fact that all the sub-carriers have the same time domain channel correlation, the same

AR model can be shared by all the sub-carriers. This greatly simplifies the channel estima-

tor. The proposed algorithm that exploits only the time domain channel correlation does not

perform as good as the algorithm that exploits time and frequency domains channel corre-

lation simultaneously [92]. Through exploiting the frequency domain channel correlation,

we then proposed a algorithm to refine the “rough” channel estimation in per-subcarrier

19

Kalman filter. The main idea of this algorithm is to exploit the time and frequency domain

channel correlation separately, and it has the following advantages compared with Kalman

filter algorithm which exploits the time and frequency domain channel correlation simulta-

neously [92]: 1) its complexity is greatly reduced; 2) its performance is comparable.

We also proposed a blind channel estimation algorithm through replacing the per-

subcarrier Kalman filter with Mixture Kalman filter [73], which does not need training

symbols to track channel states. Mixture kalman filter algorithm [73] was proposed by R.

Chen in 2000. In Mixture Kalman filter, the unknown transmitted symbols are considered

as “missing”, and multiple symbols are drawn from the constellation of transmitted signals

according to the transmission probability of each element in the constellation. Through

putting the drawn symbols back to the state model, we can apply Kalman filter algorithm

to track the channel state. Moreover, Mixture Kalman filter algorithm can be applied to

achieve joint channel estimation and symbol detection blindly.

1.4 Organization

The rest of this dissertation is organized as follows. Chapter 2 presents the OFDM sig-

nal model. WSSUS channel and its properties are also discussed in chapter 2. Chapter 3

presents the Kalman filter based channel estimation algorithms and their performance, in-

cluding vector Kalman filter algorithm and per-subcarrier Kalman filter with MMSE com-

biner. Chapter 4 discusses the sequential Monte Carlo method and mixture Kalman filter,

which is a blind solution. In chapter 4 we also discuss the application of mixture Kalman

filter in OFDM systems. Finally, chapter 5 presents the conclusion and discusses the future

research work.

20

Chapter 2. OFDM System and Channel Model

2.1 OFDM System Model

An OFDM system is a multiplexing of several (N) digital modulation systems over a

bandwidth ofW Hz [69], as is shown in Fig. 2.1 in a baseband form. The modulation

filters of the sub-systems are rectangular pulses modulatedon the carriers of frequencies

kW�N�k � 0� � � � �N �1,

φk�t � ��

1�TTcp

ej2π WN k�tTcp� if t � �0�T�

0 otherwise(2.1)

whereT �NTs�Tcp is the symbol duration,Ts �1�W, andTcp is the length of the guarding

signals called cyclic prefix (CP). The CP is formed by copyingthe lastTcp-long part of the

symbol waveform to the beginning, as shown in Fig. 2.2,

φk�t � � φk�t � NW

� � φk�t �NTs�� t � �0�Tcp�

The CP plays a key role in OFDM by maintaining the orthogonality of the sub-carriers in

frequency-selective channels, which will be elaborated inlater discussions.

In the period�lT � �l �1�T �, the modulated signal is

dl �t � �N1

∑k0

sk�l φk�t �lT � (2.2)

It is called thel th OFDM symbol, wheresk�l �k � 0� � � � �N �1 are the information symbols.

The output of the OFDM modulator can be written as

d�t � �∑l

dl �t � �∑l

N1

∑k0

sk�l φk�t �lT � (2.3)

The channel is modeled as a linear filter with impulse responseg�t �τ �. It is assumed that

21

Ts0�l

s1�l

sN�1�l

y0�l

y1�l

yN�1�l

φ0�t �

φ1�t �

φN�1�t �

ψ0�t �

ψ1�t �

ψN�1�t �

g�t �τ �d�t � r �t �n�t �

Figure 2.1: Base-band continuous-time OFDM systems model

� � ��

� � ��

� � � ��

� � � ��

Tcp

T �NTs �Tcp

NTs

Figure 2.2: Transmitted signal of OFDM systems

22

1. The impulse response of the channel is shorter than the CP duration, i.e.g�t �τ � � 0

for τ �Tcp.

2. The channel is time invariant during one OFDM symbol durationT, i.e.,�Tcp0 g�t �τ �d�t �

τ �dτ � �Tcp0 g�lT �τ �d�t �τ �dτ for lT t �l �1�T.

The received signal is obtained as the convolution of OFDM modulated signal and the

channel plus the additive white Gaussian noise ˜n�t �.

r �t � �� Tcp

0g�t �τ �d�t �τ �dτ � n�t �� (2.4)

The OFDM receiver is a bank of filtersψk�t � matched to the non-CP part of the symbol

waveformsφk�t �,

ψk�t � ��

φ �k ��t � if t � ��T��Tcp�0 otherwise

(2.5)

whereφ �� indicates the conjugate ofφ ��.According to (2.3), (2.4), (2.5), the sampled output of the OFDM system receiver is

yk�l � r �t ��ψk�t ��tlT �� ∞

∞r �t �ψk�lT �t �dt (2.6)

If the channel length is shorter than the CP duration, as is assumed previously, the ISI only

affects the corresponding CP part of the received signal, which is illustrated in Fig. 2.3.

The corresponding CP part of the received signal is discarded to avoid ISI. Then the output

of each receiving filter (sayψk�t �) is

yk�l � r �t ��ψk�t ��tlT �� ∞

∞r �t �ψk�lT �t �dt

� N1

∑k�0

sk� �l � �l�1�TlT�Tcp

�� Tcp

0g�lT �τ �φk� �t �lT �τ �dτ�

�φ �k �t �lT �dt �� l�1�T

lT�Tcp

n�t �φ �k �t �lT �dt (2.7)

23

...

......

...� � ��

� � ��

� � ��

� ��

� ��

symbol i symbol i �1Transmitter

Receiver

t(+)

cyclic prefix

channel length

Figure 2.3: CP prevents ISI of OFDM symbols

the inner integral of (2.7) can be written as

� Tcp

0g�lT �τ �φk� �t �lT �τ �dτ

�� Tcp

0g�lT �τ �ej2πk� �tτlTTcp�W�N

T �Tcpdτ

� ej2πk� �tlTTcp�W�N T �Tcp

� Tcp

0g�lT �τ �e j2πk�τW�Ndτ (2.8)

It is noted that�Tcp0 g�lT �τ �e j2πk�τW�Ndτ is the frequency response of the channelg�lT �τ �

sampled at thek�th sub-carrier frequencyf � k�W�N. Define the notationHk� �l as

Hk� �l �G

�lT �k�W

N��

� Tcp

0g�lT �τ �e j2πk�τW�Ndτ (2.9)

whereG�lT � f � is the Fourier transform ofg�lT �τ �. Then (2.7) can be rewritten as

yk�l � N1

∑k�0

sk� �l � �l�1�TlT�Tcp

ej2πk� �tlTTcp�W�N T �Tcp

Hk� �lφ �k �t �lT �dt

�� l�1�T

lT�Tcp

n�t �φ �k �t �lT �dt

� N1

∑k�0

sk� �lHk� �l � T

Tcp

φk� �t �φ �k �t �dt�nk�l � (2.10)

where

nk�l �� l�1�T

lT�Tcp

n�t �φ �k �t �lT �dt (2.11)

24

which is white Gaussian noise. Due to the orthogonality of the modulation filters

� T

Tcp

φk� �t �φ �k �t �dt �� T

Tcp

ej2πk� �tTcp�W�N

T �Tcp

e j2πk�tTcp�W�N

T �Tcpdt

� δ �k�k�� (2.12)

whereδ �k� is the Kronecker delta function, we can simplify (2.10) and obtain a compact

OFDM signal model

yk�l �Hk�lsk�l �nk�l � k � 0�1�2� � � � �N �1 (2.13)

Therefore, the OFDM system can be considered as a system witha set of parallel transmis-

sions through flat fading sub-channels, as illustrated in Fig. 2.4. Another useful view of the

OFDM is that the information symbols are transmitted through a time-frequency lattice.

Fig. 2.5 shows the time-frequency lattice. The values of∆t and∆ f are

∆t �T� ∆ f �W�N � 1��NTs�

... ...

s0�l

s1�l

sN�1�l

y0�l

y1�l

yN�1�l

n0�l

n1�l

nN�1�l

H0�l

H1�l

HN�1�l

Figure 2.4: Equivalent model of OFDM systems

25

. . . . . .

.

.

.

.

. . . . ... . . .

. . . . .

. . . . .

∆t

∆ f

Frequency

Time

Figure 2.5: Lattice points of OFDM systems

2.2 DFT-Implemented OFDM Signal Model

An OFDM system as described above is usually implemented digitally using DFT. Con-

sider the samples of one OFDM symbol waveform at a rate of 1�Ts, as shown in Fig. 2.6,

dn�l � dl �t ��tnTs �N1

∑k0

sk�lej2π kNTs

�nTsGTs� � N1

∑k0

sk�l ej2π knN � n � �G� � � � �N �1

where we assumeTcp � GTs, and the subscripts�G� � � � ��1 represent the CP. It is inter-

esting to note thatdn�l �n � 1� � � � �N �1 are the IDFT coefficients of the symbols sequence

sk�l �k � 0� � � � �N �1, anddn�l � dn�N�l � n � �G� � � � ��1. Fig. 2.7 shows an IDFT based

implementation of the OFDM modulator.

At the receiver side, the rate-1�Ts samples of the received signalr �t � is

rn�l � r �t ��tlT�nTs �� Tcp

0g�lT �τ �d�lT �nTs �τ �dτ � nn�l

26

. . ....

� � ��

� � ��

� � � ��

� � � ��

Tcp

T �NTs �Tcp

NTs

0�G N �1

Figure 2.6: Sampling of one OFDM symbol

... ... ...

... ... ...

IDFT

DFT

sk�l

yk�l

dn�lAdd

P/S D/Ad�t�

Channel g�t �τ �

Ts

S/PRemove

CP

CP

Figure 2.7: DFT-Implemented OFDM systems

27

�� Tcp

0g�lT �τ �N1

∑i0

si �lφk�lT �nTs �τ �dτ � nn�l

� N1

∑i0

si �lej2π niN �Hi �l � nn�l � n � 0� � � � �N �1 (2.14)

where

Hi �l �G�lT �iWN

� �� Tcp

0g�lT �τ �e j2π i W

N τdτ (2.15)

Therefore, a DFT can be employed to demodulate the received signal,

yk�l � 1N

N1

∑n0

rn�l e j2π nkN � 1

N

N1

∑n0

N1

∑i0

si �lej2π niN �Hi �l �e j2π nk

N �nk�l

� 1N

N1

∑i0

si �lHi �lN1

∑n0

ej2π niN e j2π nk

N �nk�l� sk�l Hk�l �nk�l � k � 0� � � � �N �1 (2.16)

2.3 WSSUS Channel

A time-varying frequency-selective wireless channel is usually modeled as a wide-sense

stationary uncorrelated scattering (WSSUS) process [70–72]. The impulse response of a

WSSUS channel is expressed as [70]

g�t �τ � �∑k

γk�t �δ �τ �τk� (2.17)

which describes the propagation of waves through multiple paths of different delaysτk and

attenuationγk�t �. γk�t � are wide-sense stationary (WSS) complex Gaussian processes, and

are uncorrelated for different paths. Therefore, the autocorrelation function of the impulse

responseg�t �τ �E �g�t �τ �g��t �∆t �τ �∆τ ��Pg�∆t �τ �δ �∆τ � (2.18)

which implies that there is no correlation on theτ axis, but some correlation may exist on

the time axis. The functionPg�∆t �τ � is the autocorrelation of the impulse response at the

28

delayτ with the time difference∆t. The Fourier transform of (2.17) with respect toτ is

G�t � f � �� ∞

∞g�t �τ �e j2π f τdτ �∑

k

γk�t �e j2π f τk (2.19)

It can be shown that in a WSSUS channel the transfer function is also WSS with respect

to the frequency variable [70]. The Fourier transform ofPg�∆t �τ � with respect to the time

difference∆t is the scattering functionS�fd �τ � of the channel,

S�fd �τ � �� ∞

∞Pg�∆t �τ �e j2π fd∆td∆t (2.20)

where fd can be explained as the Doppler frequency. The scattering function is a measure

of the average power output as a function of the time delayτ and the Doppler frequency

fd. The delay power spectrum is defined as

ρg�τ � �� ∞

∞S�fd �τ �d fd (2.21)

and the Doppler power spectrum

SG�fd� �� ∞

∞S�fd �τ �dτ (2.22)

The width of the delay power spectrum is referred to as the maximum delay spread, and the

width of the Doppler power spectrum the maximum Doppler frequency. The relationship

of wireless channel parameters are shown in Fig. 2.9.

A typical approximation for the delay power spectrum is exponential [72]

ρg�τ � � 1τmax

exp��τ�τmax� (2.23)

whereτmax is the maximum delay spread of the channel. A typical approximation for the

Doppler power spectrum is [72]

SG�fd� ��

1π fD

1�1�fd � fD�2 � �fd �� fD

0� else(2.24)

29

fD�fD fd

S G

�

f d

�

0

Figure 2.8: Jake’s Doppler power spectrum

where fD is the maximum Doppler frequency, which is defined as

fD � vc

fc�

v is the velocity of the mobile user,c is the velocity of the radio waves, andfc is the carrier

frequency. (2.24) is often referred to as Jake’s Doppler power spectrum [72], as shown in

Fig. 2.8. The correlation of thekth path in (2.17) is

rγk �E�γk�t �∆t �γ �k �t �� σ2k rt �∆t � (2.25)

whereσ2k is the power of thekth path. According to Jakes’ model [72],

rt �∆t � � J0�2π fD∆t � (2.26)

whereJ0�x� is the zeroth-order Bessel function of the first kind. With (2.23), (2.26), we

can rewrite (2.25) as

rγk � ρg�τk�J0�2π fD∆t � (2.27)

30

According to (2.19), the time and frequency domain channel correlation can then be

obtained as

rG�∆t �∆ f � � E�G�t � f �G�t �∆t � f �∆ f �H �� E�∑

k∑k� γk�t �γk� �t �∆t �e j2π∆ f τk�

� J0�2π fD∆t �∑k

σ2k e j2π∆ f τk

� J0�2π fD∆t �� ∞

0ρg�τ �e j2π∆ f τdτ

� J0�2π fD∆t � 1� j2π∆ f τmax

1� �2π∆ f τmax�2 (2.28)

From (2.28), we observe that the time and frequency domain channel correlation can be

separated as the product of the time domain channel correlation and the frequency domain

channel correlation. Additionally, the channel correlation of the OFDM system can be

obtained if we replace the parameters∆t �∆ f in (2.28) with the parameters of the OFDM

system.

31

g�t �τ � G

�t � f �

Pg�∆t �τ �δ �

∆τ � rG�∆t �∆ f �

SG�fd �τ �

SG�fd� ρg

�τ �

� �τ �

��1 �f �

� �∆t��1 �fd�

��1 �∆t �∆ f ��1� �

fd �τ �

Figure 2.9: Relationship of wireless channel parameters

��

G�t � f � � �∞∞ g�t �τ �e j2π f τdτ

g�t �τ � � �∞∞ G�t � f �ej2π f τd f

Pg�∆t �τ �δ �∆τ � � E �g�t �τ �g��t �∆t �τ �∆τ ��rG�∆t �∆ f � � E�G�t � f �G�t �∆t � f �∆ f �H �S�fd �τ � � �∞∞ Pg�∆t �τ �e j2π fd∆td∆t

Pg�∆t �τ � � �∞∞ S�fd �τ �ej2π fd∆td fd

S�fd �τ � � �∞∞�∞∞ rG�∆t �∆ f �e j2π fd∆tej2π∆ f τd�∆t �d�∆ f �

rG�∆t �∆ f � � �∞∞�∞∞ S�fd �τ �ej2π fd∆te j2π∆ f τd fddτ

SG�fd� � �∞∞ S�fd �τ �dτ

ρg�τ � � �∞∞ S�fd �τ �d fd

32

Chapter 3. Kalman Filter Based Channel Estimation Algorithms

The OFDM system in a time-varying frequency-selective environment can be modeled

as a dynamic system. The time- frequency- domain channel correlations can be explored

to build the state equation, and the input-output relationship of the OFDM system is used

to build the observation equation. Thus Kalman filter can be applied to estimate the state

variable, i.e., the time varying channel.

3.1 State Space Model for OFDM

Replacing the parameters∆t, ∆ f in (2.28) with the parameters∆t � T, ∆ f � 1��NTs�in OFDM systems, we obtain the correlation of the OFDM systemchannel gainHk �n�s as

rk�l �m� � E�Hk �n�H �l �n�m�� J0�2π fDmT� 1� j2π �l �k�τmax��NTs�

1� �2π �l �k�τmax��NTs��2� (3.1)

whereT is the OFDM symbol duration, 1�Ts is the sampling rate. With the channel cor-

relation, we can apply auto-regressive (AR) model to model the channelHk �n�s [92, 93].

Defineh�n�� H1 �n�� HN �n��T . A pth-order AR model forh�n� is presented as

h�n� � � p

∑i1

A�i�h�n�i��Qv�n� � (3.2)

whereA�1�� A�p� andQ areN �N matrices andv�n� is the driving noise of AR pro-

cess, which is anN �1 vector white Gaussian process.A�1�� A�p� andQ are the model

parameters which are obtained by solving a Yule-Walker equation [93].

Based on the AR model of the channel, a state space model for OFDM system can be

built. Define

x�t �� h�t �H � � � � �h�t �p�1�H �H (3.3)

33

we obtain the state equation

x�t � � Cx�t �1��Gv�t � (3.4)

where the matrixC andG are defined as

C��

��

�A�1� �A�2� � � � �A�p�1� �A�p�IN 0N

� � � 0N 0N

0N IN� � � 0N 0N

......

. . ....

...

0N 0N� � � IN 0N

��

G�� IN

�0N� � � � �0N�T

IN and 0N areN by N identity matrix and all zero matrix, respectively. According to the

input-output relationship of OFDM system, we have

y�t � � D �t �x�t ��w�t � (3.5)

with

D �t ��S�t ��IN

�0N� � � � �0N�N�pN; (3.6)

whereS�t � is anN �N diagonal matrix withsk �t � being itskth diagonal entry.

It can be seen that the matrixesC andG are time invariant, while the matrixD depends

on the transmitted signalssk �t ��s and this dependence is made explicit by usingt as an

argument in matrixD.

3.2 Kalman Filter Channel Estimator

The standard Kalman filter can be applied to solve the state-space model (3.4), (3.5) to

obtain the estimate of the channel [92].

34

1. Initialize the Kalman Filter with µ �0� � 0pN and Σ0 � Σ, where Σ is the station-

ary covariance of x�t � and can be computed analytically from (3.1).

2. For each t, do the Kalman filter update according to

Prediction:

x�t ��Cµ �t �1� (3.7)

Minimum Prediction MSE Matrix:

Mt �CΣt1CH �GQGH � (3.8)

Kalman Gain Matrix Kt:

Γt � D �t �MtDH �t ��σ2

wIN (3.9)

Kt � MtDH �t �Γ1

t� (3.10)

correction:

µ �t � � x�t ��Kt �y�t ��D �t �x�t �� (3.11)

Minimum MSE Matrix:

Σt � �I pL �KtDt �Mt� (3.12)

3. Channel estimate at instance t is

h�t �� IN�0N

� � � � �0N�µ �t �� (3.13)

Note that the algorithm needs the information symbolssk �t ��s, so it is working in the

training or decision-feedback mode. The vector Kalman-filter algorithm gives the optimal

linear estimate of the channel. Its drawback is its high complexity. Considering that the

dimension of the state vector ispN, which can be significantly high when the number of

sub-carriers is large.

35

3.3 Per-Subcarrier Kalman Filter Estimator with MMSE Combiner

One way to reduce the complexity of the Kalman-filter channelestimator is to imple-

ment it at a per-subcarrier fashion. Consider thekth sub-carrier gainHk�n�. It can be

modeled as an one-dimensional (stillpth order) AR process:

Hk �t � � � p

∑i1

aiHk �t �i��σvk �t � (3.14)

the coefficientai andσ can be computed from the correlationrk�k�0�according to the Yule-

Walker equation [93]. Notice that there is no indexk in the coefficient because all the sub-

carriers have the same statistics and fit in the same AR model.This can greatly simplify the

channel estimator for many components are shared by the estimator for each sub-channel.

Follow the similar procedure as the previous section, we obtain the state-space model for

thekth sub-channel

xk �t � � Cxk �t �1��gvk �t � (3.15)

yk �t � � dk �t �xk �t ��wk �t � (3.16)

where the state vector is

x�t �� Hk �t �H � � � � �Hk �t �p�1�H �H (3.17)

C��

��

�a1 �a2� � � �ap1 �ap

1 0 � � � 0 0

0 1 � � � 0 0...

..... .

......

0 0 � � � 1 0

��

g�� σ �0� � � ��0�T

dk �t �� sk �t ��1�0� � � ��0�1�p

36

We still use the alphabetC from (3.4) for the sake of uniformity. Now we can apply

the Kalman filter to state-space model (3.15) and (3.16) to adaptively track the sub-channel

gainHk �t �. Notice that we have replaced thepN-dimensional Kalman filter used in previous

section withN p-dimensional Kalman filters. The algorithm is as follows:

1. Initialize the Kalman Filter with µ �0��0pL and Σ0 �Σ, where Σ is the stationary

covariance of xk �t � and can be computed analytically from (3.1).

2. For each t, do the Kalman Filter update according to

xk �t � � Cµk �t �1��Mt � CΣt1C

H �ggH �Γk�t � dk �t �Mtd

Hk �t ��σ2

w�

Kk�t � MtdHk �t �Γ1

k�t �µk �t � � xk �t ��Kk�t �yk �t ��dk �t �xk �t ��

Σt � �I p �Kk�tdk �t ��Mt�

3. Channel estimate at instance t is

Hk �t �� 1�0� � � � �0�µk �t �� (3.18)

However, the per-subcarrier Kalman filter only explores thetime-domain correlation of

the channel fading and fails to take advantage of the frequency-domain correlation. There-

fore, further improvement of the estimates of (3.18) is possible. Our proposal is to intro-

duce a linear combiner to combineH1 �t �� HN �t � to refine them. Specifically, the refined

channel estimate is

h�t ��Th�t � � (3.19)

37

� � � � � ��

� � � � ��

Kalman

p−Dim KF

p−Dim KF

p−Dim KF

y1�t �

y1�t �

y2�t �

y2�t �

yN�t �

yN�t �

H1�t �

H2�t �

HN�t �

H1�t �

H2�t �

HN�t �

H1�t �

H2�t �

HN�t �

pN-Dim

Filter

Post-processing

T

(a)

(b)

Figure 3.1: Vector Kalman filter and per-subcarrier Kalman filter with MMSE combiner

whereh�t �� H1 �t �� HN �t ��T andT is the combining matrix. We optimizeT in an MMSE

sense, that is

T � argminE��h�t ��h�t ��2�� argminE��h�t ��Th�t ��2�� E�h�t �hH �t ��E�h�t �hH �t ��1 � (3.20)

Sinceh is the estimate ofh�t � from the Kalman filter in (3.18), we can write

h�t � � h�t ��e�t � � (3.21)

38

wheree�t � is the estimation error which is zero-mean Gaussian and independent ofh�t �. The

covariance matrix ofe�t � (also the covariance matrixh�t �) P � E�eeH � is a diagonal matrix

with thekth diagonal entry being the covariance ofHk �t � in (3.18), which can be obtained

from the Kalman filter updating procedure:

P�k�k� � �1�0� � � ��0�Σk�t �1�0� � � � �0�T � (3.22)

i.e, P�k�k� is the�1�1�st entry ofΣk�t . Consequently, we have

E�h�t �hH �t �� E�h�t �hH �t ��R�0� �E�h�t �hH �t �� R�0��P � (3.23)

whereR�0� is anN �N matrix with its �l �k�th entry beingr l �k�0� of (3.1). The expression

T �R�0��R�0��P�1 (3.24)

then follows. Fig. 3.1 illustrates the difference between the vector Kalman filter algorithm

and the per-subcarrier filter with MMSE combiner algorithm.

3.4 Simulations

In this section, we provide computer simulation results to demonstrate the performance

of the algorithms discussed above. We consider an OFDM system with the parameters as

shown in Table 3.1 For the Kalman filter based algorithms, 30 training OFDM symbols are

used at the beginning of the transmission, then for every 45 OFDM symbols, 5 training ones

are inserted. During the transmission of the information symbols, the channel estimators

work in decision-feedback mode. For comparison purpose, wealso implemented the SVD

method [41] and the finite alphabet method [55]. To be fair, weuse the same percent of

training symbols in the SVD receiver, as shown in Fig. 3.2. Channel estimation is obtained

at the places with training symbols, and then be considered invariant over the adjacent 9

39

OFDM symbols. For the finite alphabet method, we average every 10 OFDM symbols to

obtain channel estimation.

Table 3.1: Simulation parameters of Kalman filter based algorithms

Modulation Scheme BPSK

Carrier Frequency 800 MHz

Number of sub-carriers 16

Normalized Max. Doppler frequencyfDT 0.045

Normalized Max. delay spreadτmax��NTs� 0.08

AR model order for the channel 2

� � � ��

� � � ��

� ��

� ��

��

��

��

��5 45

Training

91

Data

Figure 3.2: Transmission symbol format

The performance measures that we use are the normalized meansquare error (NMSE)

and the bit-error-rate (BER). The NMSE is defined as

NMSE� 1M

M

∑m1

� Hm �Hm �2

� Hm �2 (3.25)

40

whereM is the averaged number of the channel estimation. In our simulation, 3000 OFDM

symbols are transmitted to obtain NMSE.

Fig. 3.3 shows the normalized mean square error (NMSE) of thechannel estimation

versus the receiver signal-to-noise ratio (SNR). We can seethat the vector Kalman filter of-

fers the best performance in terms of the channel estimationerror. While, the per-subcarrier

Kalman filter with MMSE combiner algorithm performs comparable with vector Kalman

filter. Compared with the per-subcarrier Kalman filter without MMSE combiner algorithm,

the per-subcarrier Kalman filter with MMSE combiner algorithm provides a 6 dB improve-

ment in SNR, from which we can see the MMSE combiner plays an important role for

the channel estimation. The SVD and finite alphabet methods are not comparable with the

Kalman filter based algorithms because they do not exploit the time domain statistics of the

channel. The same conclusion can be drawn from the BER performance comparison, as

shown in Fig. 3.4.

Fig. 3.5 shows the learning curve of the Kalman filter based channel estimation al-

gorithms, from which we observe the Kalman filter based algorithms converge fast, i.e.,

converge after about 30 OFDM symbols.

Our proposed Kalman filter based channel estimation algorithms needs the channel pa-

rameter, maximum Doppler frequencyfD and maximum delay spreadτmax, to build the

state-space model. However, the precise parameters are notalways available. Therefore,

it is important to investigate the performance degradationwhen only the estimated values

of those parameters are available, i.e., there is a model mismatch. Fig. 3.6, 3.7 show the

mismatch performance of maximum Doppler frequency with 10 dB SNR. We fix the re-

ceiver parameter of maximum Doppler frequency with 80Hz, and simulate the performance

of different channels with different maximum Doppler frequencies. We observe both the

per-subcarrier Kalman filter with MMSE combiner algorithm and per-subcarrier Kalman

filter without MMSE combiner algorithm perform robustly when the channel parameter is

41

smaller than the receiver parameter. However, when the channel parameter is bigger than

the receiver parameter, both algorithms degrades seriously. The same conclusion can be

drawn from Fig. 3.8, 3.9, which show the mismatch performance of maximum Doppler

frequency with 20 dB SNR.

Fig. 3.10, 3.11 show the mismatch performance of maximum delay spread with 10 dB

SNR. Similarly, we fix the receiver parameter maximum delay spread with 45µs, and sim-

ulate the performance of different channels with differentmaximum delay spreads. we can

see the proposed Kalman filter based algorithms perform robust with mismatched param-

eter maximum delay spread. We can see the same conclusion from Fig. 3.12, 3.13, which

show the mismatch performance of maximum delay spread with 20 dB SNR.

10 12 14 16 18 20 22 24 26 28 3010

−4

10−3

10−2

10−1

100

101

SNR

NM

SE

Finite Alph −− MMDScalar KFSVDPer−sub KFVector KF

Figure 3.3: Normalized channel estimate error versus SNR

42

10 12 14 16 18 20 22 24 26 28 3010

−5

10−4

10−3

10−2

10−1

100

SNR

BE

R

Finite Alph −− MMDScalar KFSVDPer−sub KFVector KFPerfect Chan

Figure 3.4: Bit-error-rate versus SNR

0 10 20 30 40 50 60 70 80 90 10010

−3

10−2

10−1

100

Time

NM

SE

SNR = 20 dB

Vector KF Scalar KF Per−sub KF

0 10 20 30 40 50 60 70 80 90 10010

−3

10−2

10−1

100

Time

NM

SE

SNR = 20 dB

Vector KFScalar KFPer−sub KF

Figure 3.5: Learning curve

43

40 50 60 70 80 90 100 110 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

fD

NM

SE

SNR = 10 dB

Scalar KFPer−sub KF

Figure 3.6: Mismatch performance (NMSE) of Doppler frequency (SNR=10dB)

40 50 60 70 80 90 100 110 1200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

fD

BE

R

SNR = 10 dB


Figure 3.7: Mismatch performance (BER) of Doppler frequency (SNR=10dB)

44

40 50 60 70 80 90 100 110 1200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

fD

NM

SE

SNR = 20 dB



40 50 60 70 80 90 100 110 1200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

fD

BE

R

SNR = 20 dB



45

25 30 35 40 45 50 55 60 650

0.05

0.1

0.15

0.2

0.25

0.3

σmax

NM

SE

SNR = 10 dB


Figure 3.10: Mismatch performance (NMSE) of delay spread (SNR=10dB)

25 30 35 40 45 50 55 60 65

0.04

0.06

0.08

0.1

0.12

0.14

0.16

σmax

BE

R

SNR = 10 dB


25 30 35 40 45 50 55 60 65

0.04

0.06

0.08

0.1

0.12

0.14

0.16

σmax

BE

R

SNR = 10 dB


Figure 3.11: Mismatch performance (BER) of delay spread (SNR=10db)

46

25 30 35 40 45 50 55 60 65

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

σmax

(µs)

NM

SE

SNR = 20 dB



25 30 35 40 45 50 55 60 65

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

σmax

(µs)

BE

R

SNR = 20 dB


Figure 3.13: Mismatch performance (BER) of delay spread (SNR=20dB)

47

Chapter 4. Sequential Monte Carlo Method and Mixture Kalman Filter

In wireless systems where bandwidth is the most precious resource, periodic training

symbols can significantly reduce overall system capacity. Blind channel estimation algo-

rithms are well motivated when the overhead of training symbols becomes high, which is

true when performing channel estimation of fast time varying wireless channels. We pro-

posed a blind OFDM channel estimation algorithm based on mixture Kalman filter (MKF),

which is a special type of sequential Monte Carlo (SMC) methods.

4.1 Sequential Monte Carlo Method

The principle of SMC methods may date back to the “growth Monte Carlo” method

known in molecular physics in the 1950s [78, 79]. A complete theoretical framework for

the SMC appeared only recently [76]. It has been applied and shown a great promise in

solving a wide class of non-linear filtering problems [73–77]. SMC methods are a set of

techniques that use Monte Carlo simulations to solve “on-line” estimation and prediction

problems in dynamic systems. From a Bayesian perspective, SMC methods allow one to

compute the posterior probability distributions of interest “on-line” through Monte Carlo

simulations.

Consider a generic model for dynamic systems

��

zt � ft �zt1�ut �

yt � gt �zt�vt �

(4.1)

z : state variable y : observation

u : driving noise v : observation noise

48

DefineZt � �z0�z1

� � � � �zt � �Yt � �y0�y1

� � � � �yt � � to represent the accumulated state variables

from the beginning to timet and the accumulated observations from the beginning to timet,

respectively. Our objective is to estimate the state variablesZt or more generally a function

of themh�Zt � based on the observationsYt . The Bayesian estimate ofh�Zt � is

E�h�Zt ��Yt ��

h�Zt �p�Zt �Yt �dZt (4.2)

The computation of (4.2) is usually prohibitive due to the high-dimensional integral. One

can use the Monte Carlo simulation to compute the expection.The method is: drawmsam-

plesZ�j �t , j � 1� � � � �m according to distributionp�Zt �Yt �, and compute the sample average

to approximate the expection,

E�h�Zt ��Yt ��

h�Zt �p�Zt �Yt �dZt �� 1m

m

∑j1

h�Z�j �t � (4.3)

However, the distributionp�Zt �Yt � usually does not offer a simple close-form expres-

sion and the computation of it is hardly possible [73]. Moreover, the dimension ofZt

increase infinitely with time, which requires an increasingly huge population of samples.

To overcome these difficulties, the importance sampling technique can be used [76].

The importance sampling method does not draw samples directly from the target dis-

tribution p�Zt �Yt �. Instead, it draws samples from a trial distributionq�Zt �Yt �, which is

usually easy to compute. Then a weighted average of those samples is computed to ap-

proximate the Bayesian estimation,

E�h�Zt ��Yt � �� 1Wt

m

∑j1

h�Z�j �t �w�j �

t (4.4)

wherew�j �t is called the importance weight associated with sampleZ�j �

t drawn from the

trial distributionq�Zt �Yt �, and is defined as the ratio of the target distribution and thetrial

distribution

w�j �t � p�Z�j �

t �Yt �q�Z�j �

t �Yt � (4.5)

49

andWt is defined as

Wt �m

∑j1

w�j �t (4.6)

The pair�Z�j �t

�w�j �t �� j � 1� � � � �m, is called aproperly weighted samplewith respect to the

target distributionp�Zt �Yt �.A widely used choice of the trial distributionq�Zt �Yt � for the dynamic system model

(4.1) is of the form

q�zt �Z�j �t1

�Yt � � p�zt �Z�j �t1

�Yt � (4.7)

An important feature of this trial distribution is that the importance weight can be computed

recursively.

According to Bayes equation, (4.7) can be transformed to theform


�Yt � � p�yt �zt�Z�j �

t1�Yt1�p�zt �Z�j �

t �Y�t1��p�yt �Z�j �

t1�Yt1� (4.8)

With the relationship of the state variables and observation in dynamic system model (4.1),

we have��

p�yt �zt�Z�j �

t1�Yt1� � p�yt �zt �

p�zt �Z�j �t �Y�t1�� p�zt �z�j �t1�

p�yt �Z�j �t1

�Yt1� � p�yt �z�j �t1�Thus (4.8) can be simplified as


�Yt � � p�yt �zt �p�zt �z�j �t1�p�yt �z�j �t1�

(4.9)

The importance weight can then be computed recursively [76]

w�j �t � w�j �

t1� p�Z�j �

t �Yt �p�Z�j �

t1 �Yt1�p�z�j �t �Z�j �t1

�Yt � (4.10)

50

Notice the fact

p�Z�j �t �Yt � � p�z�j �t

�Z�j �t1 �Yt �

� p�z�j �t�Z�j �

t1�Yt �Yt �

� p�z�j �t�Z�j �

t1�Yt �

p�Yt �� p�z�j �t

�Z�j �t1

�Yt �p�Z�j �

t1�Yt �

p�Z�j �t1

�Yt �p�Yt �

� p�z�j �t �Z�j �t1

�Yt � �p�Z�j �t1 �Yt �

(4.10) can be written in the form

w�j �t � w�j �

t1� p�Z�j �

t1 �Yt �p�Z�j �

t1 �Yt1�� w�j �

t1� p�yt

�Z�j �t1

�Yt1�p�Yt1�p�Z�j �

t1�Yt1�p�Yt �

� w�j �t1

�p�yt �Z�j �t1

�Yt1� � p�Yt1�p�Yt �

� w�j �t1

�p�yt �z�j �t1� � p�Yt1�p�Yt � (4.11)

For the last part of (4.11) has no relationship with the drawnsamplesz�j �t , we get

w�j �t ∝ w�j �

t1�p�yt �z�j �t1� (4.12)

From (4.4),(4.12), we observe that we do not need to compute the specific value of the

importance weight, i.e. a proportional value with the importance weight will be enough.

For the operation of SMC methods, we first decide the trial distribution, then draw

samples of the state variable from the trial distribution, and compute the corresponding

importance weight, thus we can obtain the desired value through the weighted average.

� Sequential Monte Carlo Methods [R.Chen,2000]

FOR j � 1� � � � �m

51

1. Draw a sample z�j �t from the trial distribution q�zt �Z�j �t1� and let Z�j �

t ��Z�j �

t1�z�j �t �

2. Compute the importance weight w�j �t ∝ w�j �

t1�p�yt �z�j �t1�

END

This algorithm works recursively to get on-line estimationof the state variable.

4.2 Mixture Kalman Filter

The MKF [73] is a special type of SMC methods applied in conditional dynamic linear

models (CDLM). The CDLM is a direct generalization of the dynamic linear model (DLM)

[97], and has been widely used in practice [98]. Given the trajectory of indicator variables,

the CDLM is Gaussian and linear, thus the Kalman filter can be applied to track the state

variables of the CDLM. MKF generates samples of the indicator variables recursively based

on sequential importance sampling (SIS) and integrates outthe linear and Gaussian state

variables conditioned on these indicators.

The earlier work similar to the MKF were proposed by Ackersonand Fu [99], Akashi

and Kumanmoto [100] and Tugnait [101], and the recent work ofLiu and Chen [82] and

Doucet [102] are closely related with the MKF. The complete mathematical description of

the MKF was proposed by R. Chen in 2000 [73], and MKF has been applied in digital

communications [80]. Compared with SMC methods, which dealwith both state variables

and indicator variables, the MKF draws Monte Carlo samples only in the indicator space

and uses a mixture of Gaussian distributions to approximatethe target distribution. The

MKF is substantially more efficient than the SMC methods, i.e., it produces more accurate

results with the same computational resources.

52

4.2.1 Mixture Kalman Filter Algorithm

Consider a conditional dynamic linear model (CDLM)��

xt � Fλtxt1 �Gλt

ut

yt � Hλtxt �Kλt

vt

(4.13)

whereλt is a random indicator variable,ut andvt are random variables indicating driv-

ing noise and observation noise respectively, andut �� 0�I �, vt �

� �0�I �. The dif-

ference between CDLM and DLM lies in the unknown random indicator variableλt . If

the value ofλt is known, the coefficientsFλt�Gλt

�Hλt�Kλt

become constant and CDLM be-

comes DLM. For example, considering a wireless communication system, we can model

the wireless channel as state variable, and the transmittedsignal as indicator variable. For

training based channel estimation, the state-space model is called a DLM. For blind chan-

nel estimation, the state-space model is called a CDLM, i.e.the model coefficients “condi-

tioned” on the unknown transmitted signal.

Define

Yt � �y0�y1

� � � � �yt �Λt � �λ0

�λ1� � � � �λt �

κ �j �t �

�µ �j �

t�Σ�j �

t �The MKF algorithm can be summarized as

� Mixture Kalman Filter Algorithm [73]

FOR j = 1,. . . ,m

1. Draw a sample λ �j �t from the trial distribution q�λt �Λ�j �

t1�κ �j �

t1�Y �t ��;

2. Run a one-step Kalman filter based on λ �j �t

�κ �j �t1 and yt to obtain κ �j �

t ;

53

3. Compute the weight w�j �t �w�t1�

t�p�Λ�j �

t1�λ �j �

t �Yt ��1� �p�Λ�j �

t1 �Yt1�q�λ �j �t �Λ�j �

t1�κ �j �

t1�Yt ��

END

As stated before that MKF is a special type of SMC methods, andthere are two steps if

we apply SMC methods to the state model (4.13):

1. Rewrite the state equation. Group the state variablext and indicator variableλt as the

new state variable;

2. Rewrite the observation equation based on the new state variable;

Then (4.13) becomes the form of (4.1), we can apply SMC methods as stated above.

4.2.2 Application of Mixture Kalman Filter to OFDM Receiver

In this section, we will tailor the MKF for the state-space model of OFDM receivers.

Consider the state-space model of thekth sub-carrier of wireless OFDM systems

��

xk �t � � Cxk �t �1��gvk �t �yk �t � � sk �t �dxk �t ��wk �t �

(4.14)

where

� xk �t � is the state variable, representing wireless channel;

� sk �t � is the transmitted signal, which is in the place of indicatorvariable;

� yk �t ��vk �t ��wk �t �are the observation, driving noise, observation noise respectively, and

C�g� andd are model coefficients.

54

compute symbol detection

channel estimationKalman filter

draw

compute

yk

�

t

�

P

�

sk

�

t

� � ai

�

P

�

sk

�

t

� � ai

�

S

�

j

�

k �t �1 �Yk �t

�

w

�

j

�

k �t �1

m

j �1

w

�

j

�

k �t

m

j �1

µ

�

j

�

k �t �1 � Σ

�

j

�

k �t �1

m

j �1

µ

�

j

�

k �t � Σ

�

j

�

k �t

m

j �1

w

�

j

�

k �t

s

�

j

�

k

�

t

�

ρ

�

j

�

k �t � i : i � 1 � � � � �� m

j �1

w

�

j

�

k �t

m

j �1

s

�

j

�

k

�

t

� mj �1

µ

�

j

�

k �t � Σ

�

j

�k �t

m

j �1

P

�

sk

�

t

� � ai

�

Yk �t

�

E

�

xk �t

�

Yk �t

�

Figure 4.1: Flowchart of mixture Kalman filter algorithm

55

As in the scenario of sequential Monte Carlo method, our purpose here is to estimate the

state variable (channel coefficient)xk �t � and transmitted signalsk �t � based on observation

Yt . According to Bayesian theorem, we have

E�h�xk �t ��sk �t ��Yk �t ��

h�xk �t ��sk �t ��p�xk �t ��sk �t � �Yk �t ��dxk �t �dsk �t �(4.15)

whereh�� represents a general function.

For the posterior distributionp�xk �t ��sk �t � �Yk �t ��, we have

p�xk �t ��sk �t � �Yk �t ��

p�xk �t ��Sk �t � �Yk �t ��dSk �t �1�

��

p�xk �t ��Sk �t ��Yk �t ��p�Sk �t ��Yk �t ��

� p�Sk �t ��Yk �t ��p�Yk �t �� dSk �t �1�

��

p�xk �t � �Sk �t ��Yk �t ��p�Sk �t � �Yk �t ��dSk �t �1� (4.16)

The reason why we transform the posterior distributionp�xk �t ��sk �t � �Yk �t �� to (4.16) is: with

a givenSk �t �, the state model (4.14), (4.14) become a linear Gaussian model and then we

have

p�xk �t � �Sk �t ��Yk �t �� µk�t �Sk �t ��Σk�t �Sk �t �� (4.17)

Notice that we can get the value ofµk�t andΣk�t through running Kalman filter, because

now we have a givenSk �t �. (4.15) can then be written as

E�h�xk �t ��sk �t ��Yk �t ��

h�x�sk �t ��p�x�Sk �t ��Yk �t ��dx�� ξ �Sk

�t��p�Sk �t � �Yk �t ��dSk �t �� 1

Wk �t �m

∑j1

ξ �S�j �k �t ��w�j �k �t � (4.18)

Whereξ �� has different expression with differenth��.

56

� For channel estimation,

h�xk �t ��sk �t �� d �xk �t �� ξ �Sk �t �� d �µk�t �Sk �t �� (4.19)

� For symbol detection,

h�xk �t ��sk �t �� 1�sk �t �� ai �� ξ �Sk �t �� 1�sk �t � � ai � (4.20)

where 1�� is the indicator function��

1�sk �t �� ai � � 1 i f sk �t �� ai

1�sk �t �� ai � � 0 else(4.21)

The reason why we select the form of (4.20) comes from

sk �t � � arg maxai�� P�sk �t � � ai �Yk �t ��

� arg maxai�� E�1�sk �t �� ai ��Yk �t ��

� 1Wk �t �

m

∑j1

1�s�j �k �t �� ai �w�j �k �t � (4.22)

The diagram of mixture Kalman filter algorithm is shown in Fig. 4.1. Specifically, the

mixture Kalman filter algorithm applied in wireless OFDM systems is described below. For

thekth sub-carrier,k � 1�2� � � � �N, we apply the mixture Kalman filter algorithm proposed

by R.Chen [73], [80] as follows:

1. Initialize the Kalman filter with µ �0��j � � 0pL and Σ�j �0 � Σ, j � 1� � � � �m, where

m is the number of random samples, and Σ is the stationary covariance of

xk �t � and can be computed analytically from (3.1). All importance weights are

initialized as w�j �0 � 1, j � 1� � � � �m;

Then for j=1,2,. . . ,m, do the following steps:

57

2. Compute the one-step predictive update of each Kalman filter

M �j �k�t � CΣ�j �

k�t1CH �ggH (4.23)

γ �j �k�t � dM�j �k�t �d�H �σ2

w (4.24)

ϕ �j �k�t � dHCµ �j �

k�t1 (4.25)

The first two steps are exactly the same as Kalman filter algorithm because

information about transmitted signal at time t is not needed till this step.

According to the Kalman filter algorithm, the transmitted signal at time t is

needed when implementing Kalman filter at the next step. But for MKF, the

transmitted signal at time t is not available. Then MKF computes the trans-

mission probability of each element within the constellation of the transmitted

signal, draw samples according to the transmission probability, and consider

the drawn samples as the “missing” transmitted signal. Through putting the

drawn samples back to the system model, the Kalman filter algorithm can be

implemented. The difference between Kalman filter and MKF is shown in Fig.

4.2.

3. Compute the trial sampling density: For each ai �� , which is the transmitted

signal constellation, compute

ρ �j �k�t �i

�� P�sk �t � � ai �S�j �k�t1

�Yk�t �� p�yk �t ��sk �t � � ai

�S�j �k�t1�Yk�t1�

p�S�j �k�t1�Yk�t �

� p�yk �t � �sk �t � � ai�S�j �k�t1

�Yk�t1�P�sk �t �� ai �S�j �k�t1�Yk�t1�

p�S�j �k�t1�Yk�t ��p�S�j �k�t1

�Yk�t1�∝ p�yk �t � �sk �t � � ai

�S�j �k�t1�Yk�t1�P�sk �t �� ai �S�j �k�t1

�Yk�t1�

58

.

.

.... .

.

....

...

Kalman filter algorithm for each subcarrier

mixture Kalman filter algorithm for each subcarrier

,

,

,

,

yk

�

t

�

yk

�

t

�

yk

�

t

�

yk

�

t

�

yk

�

t

�

sk

�

t

�

Hk

�

t

�

s

�

1

�

k

�

t

�

s

�

2

�

k

�

t

�

s

�

m

�

k

�

t

�

s

�

1

�

k

�

t

�

s

�

2

�

k

�

t

�

s

�

m

�

k

�

t

�

w

�

1

�

k

�

t

�

w

�

2

�

k

�

t

�

w

�

m

�

k

�

t

�

¯Hk�

1

��

t�

¯Hk

�2

��

t

�

¯Hk

�

m

��

t

�

¯Hk

�

t

�

Kalman filter

Kalman filter

Kalman filter

Kalman filter

sample

sample

sample

weight

weight

weight

weighted average

Figure 4.2: Comparison of Kalman filter and mixture Kalman filter

59

(4.26)

� p�yk �t � �sk �t � � ai�S�j �k�t1

�Yk�t1�P�sk �t �� ai � (4.27)

we get (4.26) by removing the part having no relationship with ai, and (4.27) is

due to the fact that the transmitted symbol at time t is independent of received

symbol and transmitted symbol before time t, i.e. Sk�t1 and Yk�t .According to state model (4.14),(4.14), we have

p�yk �t � �sk �t �� ai�S�j �k�t1

�Yk�t1�� aiϕ

�j �k�t �γ �j �k�t �

where ϕ �j �k�t and γ �j �k�t can be got from the previous step (step 2) of mixture

Kalman filter algorithm.

4. Impute the symbol sk �t �: Draw samples �s�j �k �t ��mj1 from the set

�with prob-

ability

P�s�j �k �t �� ai �∝ ρ �j �k�t �i � ai �� (4.28)

Append s�j �k �t � to S�j �k�t1, then we get S�j �k�t � �s�j �k �t ��S�j �k�t1�;

5. Compute the importance weight:

w�j �k�t � w�j �

k�t1�p�yk �t � �S�j �k�t1

�Yk�t1�� w�j �

k�t1� ∑ai��

P�sk �t �� ai � �p�yk �t � �sk �t � � ai�S�j �k�t1

�Yk�t1�(4.29)

Based on (4.27), we can rewrite (4.29) in the form of

w�j �k�t ∝ w�j �

k�t1� ∑ai��

ρ �j �k�t �i (4.30)

A relatively small weight implies that the sample is drawn far from the main

body of the posterior distribution and has a small contribution in the final

estimation.

60

6. Compute the one step filtering update of kalman filter: Based on the imputed

signal s�j �k �t � and the observation yk �t �, complete the Kalman Filter update ac-

cording to

K �j �k�t � M �j �

k�t dH �γ �j �k�t �1 (4.31)

µ �j �k�t � Cµ �j �

k�t1 �K �j �k�t �yk �t ��s�j �k �t �ϕ �j �

k�t � (4.32)

Σ�j �k�t � �I �K �j �

k�t d�j �t �M �j �k�t (4.33)

7. Compute the Bayes channel estimate

xk�t �E�xk�t �Yk�t � �� 1Wk�t

�m

∑j1

µ �j �k�t w�j �

k�t � (4.34)

¯Hk �t � � �1�0� � � ��0�xk�t (4.35)

where Wk�t �∑mj1w�j �

k�tThus the channel estimate of OFDM system at timet is obtained as

¯h�t � � � ¯H1 �t �� ¯HN �t ��T � (4.36)

Similar as discussed in the above section, we use MMSE combiner here to refine the chan-

nel estimate. Suppose the refined channel estimate is

¯h�t �� ¯H1 �t �� ¯HN �t ��T � T ¯h�t � � (4.37)

Then

T � argminE��h�t �� ¯h�t ��2�� argminE��h�t ��T ¯h�t ��2�� E�h�t �¯hH �t ��E�¯h�t �¯hH �t ��1

� R�0��R�0��P�1 (4.38)

61

.

.

....

...

......

.

.

.

.

.

.

......

Kalman filter

Kalman

Kalman

Kalman

Kalman

Kalman

filter

filter

filter

filter

filter

post

proc

essi

ng

,

,

,

,

,

,

y0�t �

yN�1�t �

s�1�0

�t �

s�2�0

�t �

s�m�0

�t �

s�1�N�1

�t �

s�2�N�1

�t �

s�m�N�1

�t �

w�1�0�t

w�2�0�t

w�m�0�t

w�1�N�1�t

w�2�N�1�t

w�m�N�1�t

1W

0

�t

� ∑m j

�1µ

� j

�

0

�tw

� j

�

0

�t

�

1W

N

�1�t

� ∑m j

�1µ

� j

�

N

�1

�tw

� j

�

N

�1

�t

�

¯H0�t �

¯HN�1�t �

¯H0�t �

¯HN�1�t �

Figure 4.3: Mixture Kalman filter solution

62

whereP is a diagonal matrix with its�k�k�th entry

P�k�k� � �1�0� � � ��0�min�Σ�j �k�t ��1�0� � � ��0�T � j � 1� � � � �m (4.39)

i.e. P�k�k� is the minimum value of�1�1�st entry ofΣ�j �k�t � j � 1� � � � �m.

The system structure of mixture Kalman filter based channel estimation algorithm is

shown in Fig. 4.3. A relatively small weight implies that thesample is drawn from the

main body of the posterior distribution and has a small contribution in the final estima-

tion. We consider the samples associated with a small weightis ineffective samples. A

useful method to reduce ineffective samples and enhance effective samples is re-sampling,

which is very important for the performance of sequential Monte Carlo method and mix-

ture Kalman filter and is suggested in [81], [82]. The main idea of re-sampling is to discard

those samples associated with small importance weights andreplicate those samples as-

sociated with large importance weights. Re-sampling can bedone at any time. However,

re-sampling may also sometimes result in loss of efficiency,because it decreases “diver-

sity” of the Monte Carlo filter, i.e. it decreases the number of distinctive filters and loses

information. In addition, re-sampling also adds computational burden.

4.3 Simulation

To compare the performance with training based Kalman filteralgorithms, we select the

same simulation parameters, which is shown in Table 4.1, anddraw 50 samples for each

sub-carrier.

In counting the NMSE of the MKF based algorithm, the first 50 samples were discarded

to allow the algorithm to reach the steady state. Fig. 4.4 shows the NMSE of the chan-

nel estimation versus the receiver SNR. We observe the MKF based algorithm performs

better than vector Kalman filter algorithm when SNR is higherthan 11 dB, because the

MKF based algorithm works in “filter” mode while vector Kalman filter algorithm works

63

Table 4.1: Simulation parameters of mixture Kalman filter based algorithms

Modulation Scheme BPSK

Carrier Frequency 800 MHz

Number of sub-carriers 16

Normalized Max. Doppler frequencyfDT 0.045

Normalized Max. delay spreadτmax��NTs� 0.08

AR model order for the channel 2

Number of parallel sampling 50

in “decision-feedback” mode. Fig. 4.5 shows the BER versus the receiver SNR when the

estimated channel is supplied to the coherent detector. From Fig. 4.5, we can draw the

same conclusion as from Fig. 4.4.

Figure 4.6 shows the learning curve of presented channel estimation algorithms. It takes

about 50 OFDM symbols for the MKF based algorithm to converge.

Fig. 4.7, 4.8 show the mismatch performance of maximum Doppler frequency with 10

dB SNR. We fix the receiver parameter of maximum Doppler frequency with 80Hz, and

simulate the performance of different channels with different maximum Doppler frequency.

Similar as the training based Kalman filter algorithms, the MKF based algorithm performs

robust when the channel parameter is smaller than the receiver parameter, and degrades

seriously when the channel parameter is bigger than the receiver parameter. The same

conclusion can be drawn from Fig. 4.9, 4.10, which show the mismatch performance of

maximum Doppler frequency with 20 dB SNR.

Fig. 4.11, 4.12 show the mismatch performance of maximum delay spread with 10

dB SNR. Similarly, we fix the receiver parameter maximum delay spread with 45µs, and

64

simulate the performance of different channels with different maximum delay spread. We

can see the MKF based algorithm performs robust with the mismatch of maximum delay

spread. We can see the same conclusion from Fig. 4.13, 4.14, which show the mismatch

performance of maximum delay spread with 20 dB SNR.

10 12 14 16 18 20 22 24 26 28 3010

−4

10−3

10−2

10−1

100

101

SNR

NM

SE

Finite Alph −− MMDScalar KFSVDPer−sub KFVector KFScalar MKFCombine MKF

Figure 4.4: Normalized channel estimate error versus SNR

65

10 12 14 16 18 20 22 24 26 28 3010

−5

10−4

10−3

10−2

10−1

100

SNR

BE

R

Finite Alph −− MMDScalar KFSVDPer−sub KFVector KFScalar MKFCombine MKFPerfect Chan

Figure 4.5: Bit-error-rate versus time

0 10 20 30 40 50 60 70 80 90 10010

−4

10−3

10−2

10−1

100

Time

NM

SE

SNR = 20 dB

Vector KFScalar KFPer−sub KFMixture KF

Figure 4.6: Learning curve

66

40 50 60 70 80 90 100 110 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

fD

NM

SE

SNR = 10 dB

Scalar KFPer−sub KFScalar MKFCombine MKF

Figure 4.7: Mismatch performance (NMSE)of Doppler frequency (SNR=10dB)

40 50 60 70 80 90 100 110 1200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

fD

BE

R

SNR = 10 dB



67

40 50 60 70 80 90 100 110 1200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

fD

(Hz)

NM

SE

SNR = 20 dB



40 50 60 70 80 90 100 110 1200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

fD

(Hz)

BE

R

SNR = 20 dB



68

25 30 35 40 45 50 55 60 650

0.05

0.1

0.15

0.2

0.25

0.3

0.35

σmax

NM

SE

SNR = 10 dB



25 30 35 40 45 50 55 60 65

0.04

0.06

0.08

0.1

0.12

0.14

0.16

σmax

BE

R

SNR = 10 dB

Scalar KFPer−sub KFScalar MKFComine MKF


69

25 30 35 40 45 50 55 60 65

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

σmax

(µs)

NM

SE

SNR = 20 dB

Scalar KF Per−sub KF Scalar MKF Combine MKF

25 30 35 40 45 50 55 60 65

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

σmax

(µs)

NM

SE

SNR = 20 dB



25 30 35 40 45 50 55 60 65

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

σmax

(µs)

BE

R

SNR = 20 dB



70

Chapter 5. Conclusion and Future Research

Kalman-filter based channel estimation algorithms are proposed for OFDM systems in

a time-varying frequency-selective environment. Kalman filter solution works with training

symbols or works in decision-feedback mode, and mixture Kalman filter solution is a blind

solution.

Though the per-subcarrier Kalman filter with MMSE combiner algorithm is a two-step

solution: filtering in time and frequency domain successively, the performance of it is

comparable to the much more complicated vector Kalman estimator. The reason behind

this may be that the time and frequency components of the Jakes’ model is separable in

nature.

Mixture Kalman filter solution outperforms training based Kalman filter solution in that

mixture Kalman filter works in “filtering” mode while Kalman filter works in “decision-

feedback” mode. Through the comparison with finite alphabet, SVD channel estimation

algorithms, we can find the proposed channel estimation algorithms perform better in time-

frequency- selective environment, which makes them more suitable for wireless applica-

tion.

In addition, due to the fact that it is difficult for the receiver to obtain the exact parame-

ters, fD, σmax, of the wireless channel, the mismatch performance of the designed receiver

is very important. Through the simulation of the mismatch performance of the proposed

algorithms, we can see the proposed algorithms are robust tothe channels with mismatched

parameters.

The length of channel taps,L, is usually much smaller than the number of sub-carriers,

N, in OFDM systems. It seems promising if we can use only part ofthe sub-carriers, such

71

...

......

...

y0�t �

yN�L�t �

yN�L�1��L�t �

H0�t �

HN�L�t �

HN�L�1��L

H0�t �

H1�t �

HN�L�t �

HN�L�1�t �

HN�t �

T

p-Dim KF

p-Dim KF

p-Dim KF

post-processing

Figure 5.1:L-subcarrier Kalman filter with MMSE combiner

asL, to achieve comparable channel estimation performance with the algorithm which ex-

ploits all the sub-carriers, as shown in Fig. 5.1 In this way the complexity of the algorithms

can be greatly reduced. In addition, we can apply channel coding to improve the perfor-

mance of proposed channel estimation algorithms.

72

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81

Vita

Wei Chen was born in Zibo, P. R. China in 1976. He received his B.S. degree in 1998

from Shandong University of Technology, Jinan, China, the M.S. degree in 2001 from In-

stitute of Automation, Chinese Academy of Science, Beijing, China, both in Automation

Engineering. Since 2001, he has been with the Electrical andComputer Engineering de-

partment at Drexel University, pursuing his Ph.D. degree. He has worked as a research

assistant under the supervision of Dr. Ruifeng Zhang duringhis years at Drexel University.

He focuses his research interests on communication theory and signal processing, includ-

ing multicarrier communication system, OFDM, wireless system, blind channel estimation,

adaptive channel estimation (tracking), and equalizationalgorithms. He is a student mem-

ber of IEEE. Publications include:

� W. Chen and R. Zhang, “An SVD Method for Blind Channel Estimation of OFDM

and Related Multi-carrier Systems”,PROC. CISS, Baltimore, MD, March 2003

� W. Chen and R. Zhang, “Kalman Filter Channel Estimator for OFDM Systems in

Time and Frequency Selective Fading Environment,”IEEE PROC. ICASSP 04, Vol.

4, pp. 17-21, May 2004.

� W. Chen and R. Zhang, “Estimation of time and frequency selective channels in

OFDM systems: a Kalman filter structure,”IEEE GLOBECOM ’04, Volume 2, pp.

800-803, Nov. 2004.

� R. Zhang and W. Chen, “A Mixture Kalman Filter Approach for Blind Channel

Estimation,” 38th Asilomar Conference on Signals, Systems and Computers, CA,

November 2004.

Time- frequency- selective channel estimation of ofdm systems

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