final year thesis - university of hong konglixiao/thesis.pdf · final year thesis submitted to the...

Superimposed Pilot Sequence Assisted Estimation

for Rapidly Time-Varying OFDM Channels Based

on Basis Expansion Model

Final Year Thesis

Submitted to the Department of Electronics and Communication Engineering

Sun Yat-sen University

in partial fulfillment of the requirements for the degree of

BACHELOR OF ENGINEERING

by

Student’s Name: Li, Xiao

Student Number: 033523029

Supervisor: Dai, Xianhua

2007, April 30

Communication Engineering

Contents

Abstract 1

Chinese Abstract 2

Acknowledgements 3

1 Introduction 1

2 System Model 3

2.1 Orthogonal Frequency Division Multiplexing . . . . . . . . . . . . . . . 3

2.2 Channels in Mobile Environment . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Statistical Channel Modeling . . . . . . . . . . . . . . . . . . . . 6

2.2.2 Jake’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.3 Inter-Carrier Interference Analysis . . . . . . . . . . . . . . . . . 8

2.3 Channel Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Deterministic Channel Modeling . . . . . . . . . . . . . . . . . . 9

2.3.2 Channels Modeled by CE-BEM . . . . . . . . . . . . . . . . . . . 10

2.4 Mathematical Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 Input-Output Relationship . . . . . . . . . . . . . . . . . . . . . 11

2.4.2 Matrix Interpretation . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Channel Estimation 14

3.1 Introduction of OFDM Channel Estimation . . . . . . . . . . . . . . . . 14

3.2 Superimposed Pilot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2

3.2.1 Pilot and Proposed Pilot Cluster . . . . . . . . . . . . . . . . . . 16

3.2.2 Data and Pilot Structure . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Estimation Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Optimal Pilot Design 20

4.1 Optimal Condition by MSE Analysis . . . . . . . . . . . . . . . . . . . . 20

4.2 Pilot Design Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2.1 Constraint 1 : Pilot Energy Allocation and Placement . . . . . . 21

4.2.2 Constraint 2 : Pilot Phase Orthogonality . . . . . . . . . . . . . 22

4.3 Proposed Pilot Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3.1 Energy Allocation and Placement . . . . . . . . . . . . . . . . . . 23

4.3.2 the Number of Pilot Clusters . . . . . . . . . . . . . . . . . . . . 23

4.3.3 Phase Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Simulation 25

5.1 Parameter Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2 Investigation on System Parameters . . . . . . . . . . . . . . . . . . . . 27

5.2.1 Different Pilot Energy Proportions . . . . . . . . . . . . . . . . . 28

5.2.2 Different Pilot Patterns . . . . . . . . . . . . . . . . . . . . . . . 29

5.2.3 Different Doppler Frequency Shifts . . . . . . . . . . . . . . . . . 30

5.2.4 Different values of Symbol Lengths K . . . . . . . . . . . . . . . 31

6 Concluding Remarks 32

Bibliography 33

List of Figures

2.1 A typical OFDM system block diagram . . . . . . . . . . . . . . . . . . 4

3.1 Block-type and Comb-type Pilot Illustration . . . . . . . . . . . . . . . . 15

3.2 Proposed Pilot-Data Structure Illustration . . . . . . . . . . . . . . . . . 17

4.1 Proposed Pilot Pattern Illustration . . . . . . . . . . . . . . . . . . . . . 24

5.1 MSE performance when 60% power is allocated to pilots . . . . . . . . . 26

5.2 MSE performance when no data is superimposed onto the pilots . . . . 27

5.3 MSE performance under different pilot energy proportions . . . . . . . . 28

5.4 MSE performance under different pilot patterns . . . . . . . . . . . . . . 29

5.5 MSE performance under different Doppler shifts . . . . . . . . . . . . . 30

5.6 MSE performance with different symbol lengths . . . . . . . . . . . . . . 31

Abstract

Orthogonal Frequency Division Multiplexing (OFDM) technique is a powerful so-

lution for future wireless communications. It has received considerable interests for

its enhanced performance in terms of spectral efficiency and robustness to frequency-

selectivity. However, in modern wireless communications, the mobility of transmitter

or receiver introduces both frequency and time selectivity into the system due to the

so-called multi-path fading as well as the Doppler effect. Although estimation al-

gorithms of time-invariant (TI) systems have been widely studied through different

perspectives by various research communities, the estimation of time-varying (TV)

channels for OFDM systems remains challenging. This thesis hereby approximates

the time variation of the channel by applying the Complex Exponential Basis Ex-

pansion Model (CE-BEM) and puts forward a channel estimator in MMSE sense.

Moreover, in consideration of bandwidth efficiency, this thesis approaches the problem

by using superimposed pilot sequence as an investigation upon the channel estimator

performance. The proposed channel estimator is shown to have satisfactorily good

performance in terms of Mean Square Error (MSE) by utilizing the proposed pilot

pattern through computer simulations.

Keywords : OFDM, linear time-variant (LTV) channel, channel estimation, basis

exponential model

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Acknowledgements

First and foremost, I wish to express my heart-felt thanks to my advisor Prof.

Dai Xianhua and the senior student Mr.Zhang Han. They have provided me with

invaluable guidance and support, throughout the process of my thesis preparation.

But for their keen interest and support, this thesis would not have taken the present

form.

I am also indebted to my parents who have supported me and given me constant

love all the time and I would not have accomplished what I achieve today if it were

not for their endless support.

Chapter 1

Introduction

Nowadays, OFDM technology, as a multi-carrier modulation scheme, has been

demonstrated to offer a desirably high spectral efficiency and efficient implementa-

tion, especially the so-called single tap equalization in the frequency domain. In con-

trast to the traditional time-domain equalizer, the reduction in complexity greatly

simplifies the implementation of OFDM wireless applications. Moreover, OFDM is

capable of combating Inter-symbol Interference (ISI) brought by multi-path fading (or

namely frequency selectivity). In this context, OFDM technique emerges as a promis-

ing candidate and has become an attractive solution for next-generation wireless local

area networks (WLANs), wireless metropolitan area networks (WMANs), and fourth-

generation (4G) mobile cellular wireless systems.

However, although OFDM technique displays desirable robustness against ISI, the

single tap equalization still becomes inapplicable in a time-varying environment since

the time variation violates the subcarrier orthogonality and leads to Inter-carrier In-

terference (ICI). If the OFDM symbol duration is larger than the channel’s coherence

time, time-selectivity starts to become dominant and degrade the system performance.

On the other hand, channel estimation, as one of the most significant processing

module at the receiver, provides critical information about the channel to recover the

original transmitted sequence, especially in coherent detection. In a time-varying chan-

nel, the task becomes even more challenging and vital because the channel coefficients

1

CHAPTER 1. INTRODUCTION

are changing all the time and the channel estimates need to be updated frequently in

the tracking process for improving accuracy. Therefore, a robust estimator for time-

frequency selective channels is essential for apply the powerful OFDM technology into

mobile communications.

Generally speaking, channel estimation can be accomplished by using data statis-

tics (blind) or inserting pilots that are a priori known (pilot-assisted). Nevertheless,

blind estimation algorithm usually employs high-order statistics of the received sam-

ples and requires long data records to attain reliable accuracy, which however lowers

its applicability in a time-varying channel. As a result, only pilot-assisted estima-

tions are considered in this thesis. More specifically, the suggested channel estimation

scheme here is designed by superimposing pilots onto the transmitted signal instead of

multiplexing them in time or frequency domain for the sake of spectral efficiency. For

simplicity, the original transmitted signals before pilot superimposition are treated as

noise-Gaussian process with zero mean.

Based on the foregoing discussion, this thesis mainly derives a robust channel es-

timator in SISO-OFDM system for time-varying channels using superimposed pilot

sequence. Conventionally, the available algorithms on time-variant channel estimation

assume the channels to vary in a linear fashion when the doppler frequency shift is

relatively small. From a different angle, this thesis uses a deterministic modeling ap-

proach (Basis Expansion Model) instead of statistical channel modeling. In contrast,

the proposed algorithm is applicable to even large doppler frequency shift.

The thesis is organized as follows: Chapter 2 provides background information

about OFDM systems and subsequently characterizes the communication system model

in a mobile wireless context. Chapter 3 goes on to briefly introduce major categories

of channel estimation approach and proposes a channel estimator followed by perfor-

mance analysis. Chapter 4 further discusses the design of pilot pattern to achieve

optimal performance of the proposed estimator while Chapter 5 explores how different

system parameters affect the estimator’s performance through computer simulations.

Finally, the thesis is concluded in Chapter 6.

2

Chapter 2

System Model

This section concentrates on the characterization of a time-varying channel while

providing fundamentals of OFDM modulation. Channel modeling is introduced in

detail and a time-variant system model is deduced afterwards for further discussion

on OFDM channel estimation in a changing environment.

2.1 Orthogonal Frequency Division Multiplexing

An overview of MCM (especially for OFDM) has been provided by T. Keller and L.

Hanzo in [3], which demonstrates the advantages and disadvantages of MCM through

simulations, while highlighting the adaptive loading concept in OFDM. By using

tightly packed orthogonal subcarriers, OFDM greatly mitigates the effects caused by

intercarrier interference (ICI) as well as ISI, and achieves desirable spectral efficiency.

Besides, another reason for OFDM’s popularity lies in its efficient implementation

by using DFT (Discrete Fourier Transform), suggested by Weinstein and Ebert in

1971. The underlying principle is to use the set of harmonically related orthogonal

basis functions in DFT as subcarriers, as is discussed later. A typical OFDM system

is as the block diagram below:

In OFDM, signals are sampled at frequency fs = 1/Ts and modulated by K-point

IFFT modulator and the output is discrete with sampling interval of T . The signal

3

CHAPTER 2. SYSTEM MODEL

ModulationPilot

Insertion

Cyclic Prefix

Insertion

Inverse

FFT

Time-varying

Channel

Cyclic Prefix

RemovalFFT

Channel

EstimationDemodulation

Binary

Data X(k) x(n)

h(n,l)

Noise

y(n)Y(k)Recovered

Data

Figure 2.1: A typical OFDM system block diagram

is then transmitted in the form of a “time domain” K-length data block with K

subcarriers, also called a OFDM symbol.

x(n) = IFFTX(k), k = 0, · · · , K − 1 (2.1)

where n is the sample index within a symbol.

A cyclic prefix (CP), which is padded with the values from the last part of the

OFDM symbol, is pre-appended to the data signals n = −G, · · · ,−1. Thus, the signal

X(k) is intuitively transmitted in the time domain form of x(k) as:

x(n) =1

K

K−1∑

k=0

X(k)ej 2πkK n (2.2)

where G is the length of CP, and n = −G, · · · , 0, · · · , K − 1 is the CP-inserted index

within the symbol.

The CP eliminates ISI by accommodating the maximum sampled channel delay

spread L = ⌈ τmax

T ⌉(G ≥ L). Moreover, it turns the linear convolution between the

transmitted sequence and channel impulse response to a circular convolution. Then

4


the received time domain signal is:

y(n) =L−1∑

l=0

h(l)x(n − l) + w(n)

=1

K

L−1∑

l=0

h(l)K−1∑

k=0

X(k)ej 2πkK (n−l) + w(n) (2.3)

where l is the multi-path index and w(n) is time-domain noise. After rearranging the

terms,

y(n) =1

K

K−1∑

k=0

X(k)ej 2πkK n

L−1∑

l=0

h(l)e−j 2πkK l

︸︷︷︸H(k)=FFTh(l)

+w(n)

=1

K

K−1∑

k=0

[X(k)H(k)]ej 2πkK n

︸︷︷︸IFFTX(k)H(k)

+w(n) (2.4)

Finally, the received CP is discarded (n = 0, · · · , K − 1) and after FFT demodula-

tion, the original information appears to be transmitted on K decoupled independent

subchannels experiencing less fading:

Y (k) = FFTy(n) = H(k)X(k) + W (k) (2.5)

where H(k) and W (k) are FFTs of h(l) and w(n) respectively and k = 0, · · · , K − 1.

Rewrite the above relation in matrix form, we have:

Y = HX + W (2.6)

where Y = [Y (0), · · · , Y (K − 1)]T and X = [X(0), · · · , X(K − 1)]T . In particular, H

is the frequency response matrix of the channel, which is diagonal in this case:

H =

H(0) 0

. . .

0 H(K − 1)

K×K

(2.7)

The deduction above is based on the assumption that all OFDM subchannels are

independent with each other, or namely the channel remains unchanged over the trans-

5


mission period. For time-varying environment in mobile communications, the mathe-

matical interpretation of the input-output relationship is not as straightforward as the

one above and H is no longer diagonal.

2.2 Channels in Mobile Environment

The most intrinsic characteristic of the multi-path channel in mobile communi-

cations is its time-varying nature. This time variation comes into being due to the

mobility of the transmitter or the receiver. In consequence, the location of reflectors

in the transmission path, which give rise to multi-path, will vary as time goes by,

particularly when the transmitter or receiver is moving at a hight speed. Thus, the

Channel Impulse Response (CIR) is not only a function of multi-path delay tap l, but

also a function of time index n. We denote the the new CIR as h(n, l) mathematically.

2.2.1 Statistical Channel Modeling

In order to simulate the channel variation in practical situations, a lot of mathemat-

ical work has been done to approximate the fading nature of wireless communication

channels. The most traditional and usual way to represent a wireless communication

channel is the statistical channel modeling, which treats different channel coefficients

as uncorrelated, Gaussian stochastic process when there exist abundant reflectors and

scatters:

h(n, l) =

I(n)∑

i=1

αi(n)e−jφi(n)δ(n − l(n)) (2.8)

where i is the transmitted path index, I(n) is the total path number at time n. Also,

αi(n) represents the random path gain and φi(n) = 2πfcl(n) − φDn − φ0 interprets

the phase shift due to random scattering, reflection and so on as well as the Doppler

frequency shift.

Based on the previous statistical model, it has been well established that the prob-

ability density function (PDF) of the envelope A(t) of a transmitted carrier is Rayleigh

distributed when there is no line-of-sight (LOS) component:

fA(a) =a

σe−

a2

2σ2 , a ≥ 0 (2.9)

6


and the PDF of the phase of the carrier is uniformly distributed as:

fΘ(θ) =1

2π, 0 ≤ θ < 2π (2.10)

Moreover, the correlation of one CIR delay tap in the time domain can be charac-

terized as:

Eh(ni, l)h∗(nj , l) = σ2

l J0(2π|ni − nj |fDTs) (2.11)

and σ2l = E|h(n, l)|2 (2.12)

where Ts = 1/fs and J0(·) is the 0th order Bessel function of the first kind,

J0(x) =

∞∑

k=0

(−1)k

22k(k!)2x2k (2.13)

These formulations have been verified by both measurement and theory over the

decades. Usually, if the normalized Doppler frequency fDTs is less than 0.01, the

channel can be considered as constant within one OFDM frame.

2.2.2 Jake’s Model

In order to approximate the mobile environment due to Doppler phenomenon, the

famous mathematical model suggested by Jake has been employed to simulate Rayleigh

fading for a long time.

In Jake’s model, the channel consists of many scatterers densely packed with re-

spect to angle. The received continuous low-pass signal is a sum of randomly-phased

sinusoids:

r(t) = Er

M−1∑

m=0

αmej(2πfDt cos θm) (2.14)

where m is the transmitted path index, M is the total path number, fD is the maximum

Doppler frequency shift, and also:

αm =1√M

, m = 0, · · · , M − 1 (2.15)

which represents the random path gain, and

θm =2πm

M, m = 0, · · · , M − 1 (2.16)

7


which interprets the angle of incoming path. As M → ∞, the received envelope |r(t)|is Rayleigh distributed and the phase is uniformly distributed. Through sending a

training pulse, the CIR h(n) could be obtained after sampling the received signal r(t)

at frequency fs = 1/Ts:

h(n) = r(t) |t=nTs (2.17)

Moreover, the power spectral density (PSD) Sl(f) of each channel tap in Jake’s

model is given in [1] as:

Sl(f) =

Er

4πfD

1r1−

(|f−fc|

fD

)2, |f − fc| ≤ fD

0, |f − fc| > fD

(2.18)

In the final simulation, the channel is generated according to the Jake’s model.

2.2.3 Inter-Carrier Interference Analysis

In the previous deduction, the frequency domain transfer matrix of a time-invariant

OFDM system is diagonal, implying the orthogonality between different subcarriers.

However, in a TV environment, ICI between subcarriers is brought to the scenario and

H is no longer diagonal. Denote the transfer matrix in a TV system as G, then we

have the general form [4]:

Y = GX + W (2.19)

where

G =

G(0, 0) . . . G(0, K − 1)...

. . ....

G(K − 1, 0) . . . G(K − 1, K − 1)

K×K

(2.20)

The elements of the matrix G are independent and identically distributed (i.i.d.)

Gaussian random variables. Additionally, G(i, j) is evaluated in [4] as:

G(i, j) =1

K

K−1∑

n=0

L−1∑

l=0

h(n, l)ej2πn(j−i)/Ke−j2πlj/K (2.21)

Then equivalently, the expression of received signal Y (k) incorporates ICI terms as

8


follows:

Y (k) = G(k, k)X(k) + W (k)︸︷︷︸time-invariant case

+

K−1∑

s=0

s6=k

G(k, s)X(s)

︸︷︷︸ICI term Z(k)

(2.22)

The ICI term Z(k) can be further analyzed to have the following statistics [4]:

EZ(k)Z∗(k) = E K−1∑

i=0

i6=k

K−1∑

j=0

j 6=k

G(k, i)X(i)X∗(j)G(j, k)

=

K−1∑

i=0

i6=k

K−1∑

j=0

j 6=k

G(k, i)RXX(i, j)G(j, k) (2.23)

As can be seen, the covariance of ICI terms is involved with the autocorrelation of

transmitted signals X(k). Due to the existence of ICI, the system model is no longer

straightforward as the previous TI system one. In order to derive a robust channel

estimator, the following discussion is based on the assumption that the channel is

changing rapidly.

2.3 Channel Modeling

Channel modeling is necessary to sufficiently reduce the number of unknowns.

2.3.1 Deterministic Channel Modeling

As mentioned in the preceding paragraphs, statistical modeling is well motivated

when TV path delays arise due to a large number of scatterers. But recently deter-

ministic basis expansion models have gained popularity for cellular radio applications,

especially when the multi-path is caused by a few strong reflectors and path delays

exhibit variations due to the kinematics of the mobiles [5, 6, 7, 25].

The reason is that the channel parameters can then be expressed by a small number

of coefficients by projecting them onto a subspace spanned by a finite set of basis

functions (namely the Basis Expansion Model, BEM). Instead of estimating the channel

parameters directly, the projected coefficients associated with the basis functions are

9


estimated. Then, the channel is reconstructed with these estimated coefficients and

the corresponding basis functions.

Through deterministic modeling, a relatively small number of coefficients is often

sufficient to accurately characterize the channel, depending on the channel variation

speed. The problem now is to find a good set of basis functions. It should be efficient,

which means that it requires as few coefficients as possible. On the other hand, it

should be accurate enough, which means that the difference between the modeled

channel and the real channel response should be sufficiently small.

2.3.2 Channels Modeled by CE-BEM

Generally, the time-varying channel could be regarded as a linear changing process

[23, 6, 7] when the normalized Doppler frequency shift is relatively small (KTsfD <

0.1), where fD is the maximum doppler frequency shift. However in the case of rapidly

changing environment, linear approximation no longer holds.

In [5], a complex exponential BEM is employed to estimate the channel and the

channel model approximates the real random channel well even when KTsfD < 1. It

has been shown in [5] that if fD and L are already known, the TI coefficients of the

basis functions can be inferred mathematically. The model is given as:

h(n, l) =

Q∑

q=−Q

hq(l)ejωqn (2.24)

where q represents the number of transmitted paths for one tap and 2Q+1 is the total

number of transmitted paths.

Compared to the statistical model, the TI coefficients hq(l) is actually equivalent

to:

hq(l) = αq(l)ejφl

where ejφl is the random phase shift and αq(l) the path gain of each multi-path

component. On the other hand, the time-varying basis function

ejωqn = ej2πfD cos (2πq/Q)n (2.25)

models the time-variation caused by Doppler phenomenon.

10


For mathematical simplicity, [6, 7, 25] further developed CE-BEM to a more sim-

plified expression:

h(n, l) =

Q∑

q=−Q

hq(l)ej 2πqn

K (2.26)

where Q = ⌈fDKTs⌉ = ⌈K fD

fs⌉ (2.27)

with fD indicating the maximum Doppler frequency shift, n is the block index which

represents the nth realization of hq(l), and ⌈·⌉ stands for the ceiling integer.

In practice, it is assumed that fD and Q are known because these two parameters

can be measured experimentally anyway(K and Ts depend on personal choice). Time

variation in the above CE-BEM is captured by the complex exponential basis functions,

while the basis coefficients remain invariant over each block containing K symbols. A

fresh set of BEM coefficients is considered every KTs seconds.

Furthermore, the generalized model shows that if fD

fs≤ Q

K , a basis expansion

at order Q is capable of approximating the LTV channel’s coefficients. Similar ap-

proximations can be also found in [28, 5]. Thus in subsequent sections, CE-BEM is

employed as the channel model for its improved accuracy.

2.4 Mathematical Manipulation

Before designing the channel estimator, a mathematical view towards the whole

system is vital. Hence, the input-output relationship in a CE-BEM based OFDM

channel is hereby derived for further analysis. Moreover, reasonable simplification is

made to the relationship by considering the time-varying nature in practice.

2.4.1 Input-Output Relationship

As discussed earlier in Section 2.2.3, the received signal Y (k) is calculated as fol-

lows:

Y (k) =K−1∑

s=0

G(k, s)X(s) + W (k) (2.28)

11


where

G(k, s) =1

K

K−1∑

n=0

L−1∑

l=0

h(n, l)ej2πn(s−k)/Ke−j2πls/K (2.29)

Since CE-BEM is used in this thesis, the channel h(n, l) is modeled by CE-BEM

as:

h(n, l) =

Q∑

q=−Q

hq(l)ejωqn =

Q∑

q=−Q

hq(l)ej 2πqn

K (2.30)

Substitute all the expressions into the received signal expression, we have:

Y (r) =1

K

K−1∑

s=0

K−1∑

n=0

L−1∑

l=0

Q∑

q=−Q

hq(l)ejωqnej2πn(s−r)/Ke−j2πls/KX(s) + W (r) (2.31)

By rearranging the summing procedure, the above equation becomes interesting

after combining certain terms:

Y (r) =1

K

Q∑

q=−Q

K−1∑

s=0

X(s)[ L−1∑

l=0

hq(l)e−j2πls/K

]

︸︷︷︸Hq(s)=FFT[ hq(l)]

[ K−1∑

n=0

(ejωqnej2πns/K

)e−j2πnr/K

]

︸︷︷︸Eq(r−s)=FFT[ ejωqnej2πns/K ]

+W (r)

=1

K

Q∑

q=−Q

K−1∑

s=0

X(s)Hq(s)Eq(r − s) + W (r) (2.32)

where Eq(r) is the FFT of ejωqn. By evaluating this K-point FFT transform, we have:

Eq(r) = FFT[ejωqn] = δ(r − rq) and rq =ωq

2π· K = q, q = −Q, · · · , Q (2.33)

where rq = q is the discrete normalized frequency shift ranging from −Q to Q in

contrast to the whole OFDM bandwidth K. After all these manipulations, Y (r)

becomes:

Y (r) =1

K

Q∑

q=−Q

K−1∑

s=0

X(s)Hq(s)δ(r − q − s) + W (r) (2.34)

According to the shifting property of δ function, the final expression of Y (r) can

be simplified as:

Y (r) =1

K

Q∑

q=−Q

X(r − q)Hq(r − q) + W (r) (2.35)

=1

KX(r)H0(r) +

1

K

Q∑

q=−Q

q 6=0

X(r − q)Hq(r − q) + W (r) (2.36)

12


Intuitively, a specific Y (r) is involved with the preceding Q as well as the following

Q values of X(r), resulting that if a pilot at frequency r is inserted, the adjacent 2Q

subcarriers should also be assigned as pilots.

2.4.2 Matrix Interpretation

In signal processing, it is always more convenient to write the input-output rela-

tionship in matrix form. Hence, a matrix form of the expression deduced in section

2.4.1 is developed here for channel estimation.

As shown in section 2.4.1, the estimation of h(n, l) is reduced to estimating (2Q +

1)L coefficients hq(l). Group it in a (2Q + 1)L × 1 column vector and denote it as:

h = [hT−Q, · · · ,hT

0 , · · · ,hTQ]T (2.37)

where hq = [hq(0), · · · , hq(L − 1)]T . Define a 1 × L row vector (or equivalently FFT

transform submatrix) F(r) as

F(r) =[1 · · · e−j 2πr

K l · · · e−j 2πrK (L−1)

], r = 0, · · · , K − 1 (2.38)

Then Y (r) can be formulated in matrix form as:

Y (r) =[X(r + Q)F(r + Q) · · ·X(r)F(r) · · ·X(r −Q)F(r −Q)

]h + W (r) (2.39)

Now include P consecutive samples of Y (r) into the above input-output relation-

ship, each denoted as Y (kp) and p is an integer ranging from 1 to P , a more general

matrix interpretation for Y = [Y (k1), · · · , Y (kP )]T is obtained:

Y =

X(k1 + Q)F(k1 + Q) · · · X(k1 − Q)F(k1 − Q)...

. . ....

X(kP + Q)F(kP + Q) · · · X(kP − Q)F(kP − Q)

h + W (2.40)

13

Chapter 3

Channel Estimation

During the equalization at the receiver, it requires perfect channel knowledge.

Many equalization algorithms for wireless communications are developed under the

premise that channel estimation has been successfully performed. However, this is not

realistic in practice, especially when the channel is rapidly changing.

The channel estimation problem for OFDM in a fast fading environment has been

brought to the attention of many researchers recently. Unlike those WLANs using

OFDM, (IEEE 802.11a/g, established for fixed wireless access), the channels keep

changing from time to time in mobile communications. Estimating a time-varying

channel is one of the most challenging tasks associated with OFDM research.

3.1 Introduction of OFDM Channel Estimation

There are two main problems in designing channel estimators for wireless OFDM

systems. The first problem is the arrangement of pilot information, where pilot means

the reference signal used by both transmitters and receivers. The second problem is

the design of an estimator with both low complexity and good channel tracking ability.

The two problems are interconnected.

In general, the fading channel of OFDM systems can be viewed as a two-dimensional

(2D) signal (time and frequency). The optimal channel estimator in terms of mean-

14

CHAPTER 3. CHANNEL ESTIMATION

square error is based on 2D Wiener filter interpolation. Unfortunately, such a 2D

estimator structure is too complex for practical implementation. The combination

of high data rates and low bit error rates in OFDM systems necessitates the use of

estimators that have both low complexity and high accuracy, where the two constraints

work against each other and a good trade-off is needed.

The one-dimensional (1D) channel estimations are usually adopted in OFDM sys-

tems to accomplish the trade-off between complexity and accuracy [9, 15]. The two

basic 1D pilot-assisted channel estimations are block-type pilot channel estimation and

comb-type pilot channel estimation, in which the pilots are inserted in the frequency

direction and in the time direction, respectively.

Figure 3.1: Block-type and Comb-type Pilot Illustration

The estimations for the block-type pilot arrangement can be based on least square

(LS), minimum mean-square error (MMSE), and modified MMSE. The estimations

for the comb-type pilot arrangement includes the LS estimator with 1D interpolation,

the maximum likelihood (ML) estimator, and the parametric channel modeling-based

(PCMB) estimator. Other channel estimation strategies were also studied [8, 10, 11,

12, 13, 14], such as the estimators based on parametric channel modeling, or iterative

filtering and decoding, as well as estimators for the OFDM systems with multiple

transmit-and-receive antennas, and so on.

15


When it comes to time-varying channels, the estimation problem becomes more

complicated. Due to the time-variations of the channels, block-type pilots are no

longer practical because the pilots are multiplexed in the time domain and it is highly

possible that the channel estimates obtained from this set of pilots differ greatly in

the next time interval, especially when the channel experiences relatively severe time-

selectivity.

In order to estimate the varying channels more effectively and efficiently, paramet-

ric channel modeling is always employed to simplify the random time-variations in a

“deterministically random” way. Such as in [8], a polynomial basis expansion model

is used to estimate the rapidly changing channels. Similarly, [17, 18] developed the

LMMSE and LS solutions respectively based on a windowed OFDM channel modeled

by CE-BEM. Different algorithms for rapidly changing channels can also be found in

[20, 19].

3.2 Superimposed Pilot

Other than the semi-blind methods or namely the pilot-assisted methods, a super-

imposed training based approach has been recently. The main idea is to arithmetically

add the pilot information to the information data. If the training sequence is periodic,

the received signal will exhibit cyclostationary statistics. Since the superimposed pi-

lots are directly added upon the data, there is no loss in data transmission rate, which

in other words improves bandwidth efficiency. However on the other hand, some power

is wasted in allocating the pilots rather than imposing the power on data.

Here, the thesis tries to implement the proposed algorithm by utilizing superim-

posed pilots, making a comparison between how different pilot energies affect the

estimator’s performance in latter simulation.

3.2.1 Pilot and Proposed Pilot Cluster

Because of the foregoing discussion of input-output relationship in section 2.4.1, it

has been demonstrated that a specific Y (r) is involved with the preceding Q as well

16


Figure 3.2: Proposed Pilot-Data Structure Illustration

as the following Q values of X(r).

Therefore, the pilots are grouped in a clustered form of 2Q + 1 consecutive pilots

for each cluster in the proposed channel estimation scheme. There are altogether P

pilot clusters, and the center of each pilot cluster is determined as:

kp = K · p − 1

P, p = 1, · · · , P uniformly distributed pilot spacing (3.1)

This placement is proved to be optimal in chapter 4.

3.2.2 Data and Pilot Structure

The input of the expression in section 2.4.1 is X(kp − q), p = −Q, · · · , Q. Now

it should be changed into the superimposed version given by the addition of data

D(kp − q) and pilots S(kp − q), which is X(kp − q) = D(kp − q)+ S(kp − q). In matrix

form it becomes:

Xq = Dq + Sq, q = −Q, · · · , Q (3.2)

Using the above pilot placement, the superimposed structure of data and pilot is

illustrated in Figure 3.2.

17


3.3 Estimation Scheme

Based on the previous discussion, the estimation of g is turned to a estimation

problem in linear systems. Recall the input-output matrix interpretation in section

2.4.1, we have:

Y =

X(k1 + Q)F(k1 + Q) · · · X(k1 − Q)F(k1 − Q)...

. . ....

X(kP + Q)F(kP + Q) · · · X(kP − Q)F(kP − Q)

h + W (3.3)

For clarity of channel estimation procedure, we separately define:

Xq = diag[X(k1 − q), · · · , X(kP − q)] (3.4)

Fq = [FT (k1 − q), · · · ,FT (kP − q)]T (3.5)

where Xq is a P × P signal matrix and Fq is a P × L FFT-alike matrix.

Then the above matrix expression is further elaborated as:

Y = [X−QF−Q · · ·X0F0 · · ·XQFQ]h + W (3.6)

Since Xq is made up of the data sequence Dq and the superimposed pilot sequence

Sq, we rewrite the above expression:

Y = AD · h + AS · h + W (3.7)

and

AS = [S−QF−Q · · ·S0F0 · · ·SQFQ] (3.8)

AD = [D−QF−Q · · ·D0F0 · · ·DQFQ] (3.9)

where AD and AS are both of dimensions P × (2Q + 1)L.

For the sake of simplicity, AD · h is treated as noise in the following analysis

because of the data’s random nature. Then the equation can be further simplified by

incorporating AD · h into a new noise term Ω = AD · h + W, as follows:

Y = AS · h + Ω (3.10)

18


Finally, the least square (LS) estimate of the channel coefficient can be obtained

by:

h = A†S ·Y = h + A

†S ·Ω (3.11)

where A†S is the pseudoinverse of the matrix AS . By observing the above estimation

problem, it can be easily found that there are (2Q + 1)L variables to be estimated.

Thus, the observation or namely Y, should contain no less than (2Q + 1)L samples

in order for the P × (2Q + 1)L matrix A†S to be full column rank (2Q + 1)L. This

requirement is mathematically equivalent to P ≥ (2Q + 1)L.

19

Chapter 4

Optimal Pilot Design

4.1 Optimal Condition by MSE Analysis

Here we analyze the performance of the proposed channel estimator mentioned

above in terms of Mean Square Error (MSE).

The MSE of the channel estimator M(h) can be calculated as:

M(h) =1

(2Q + 1)L· E

∣∣∣ h− h

∣∣∣2

=1

(2Q + 1)L· tr

(A

†S · EΩΩH · A†H

S

)

=σ2

Ω

(2Q + 1)L· tr

(A

†S · A†H

S

)

=σ2

Ω

(2Q + 1)L· tr

[(AH

S · AS)−1]

(4.1)

It has been well established in [23] that the minimum MSE can be achieved when

AHS · AS = PI(2Q+1)L, where P is a constant indicating the power. Thus, according

to this condition, we can derive the optimal pilot sequence in the following.

20

CHAPTER 4. OPTIMAL PILOT DESIGN

4.2 Pilot Design Constraint

From the discussion in chapter 3, we can recall the expression for AS :

AS = [S−QF−Q · · ·S0F0 · · ·SQFQ] (4.2)

Thus we can write AHS · AS as:

AHS ·AS =

C−Q,−Q · · · C−Q,Q

.... . .

...

CQ,−Q · · · CQ,Q

(2Q+1)L×(2Q+1)L

(4.3)

where Ci,j is an L × L submatrix calculated as:

Ci,j = FHi SH

i SjFj , i, j = −Q, · · · , Q (4.4)

As mentioned in section 4.1, the minimum MSE is achieved under the constraint

AHS · AS = PI(2Q+1)L. Therefore, by applying this constraint to the submatrix Ci,j

we have:

Ci,j =

PIL×L, i = j

0, i 6= j(4.5)

4.2.1 Constraint 1 : Pilot Energy Allocation and Placement

In the case i = j = q:

Cq,q = FHq SH

q SqFq = PIL×L, q = −Q, · · · , Q (4.6)

Furthermore, since Sq = diag[S(k1 − q), · · · , S(kP − q)

], thus

SHq Sq = diag

[P(k1−q), · · · ,P(kP−q)

](4.7)

where P(kp−q) is the energy of each pilot.

Then Cq,q is expressed as:

Cq,q = FHq diag

[P(k1−q), · · · ,P(kP −q)

]Fq (4.8)

21


The (x, y)th element (Cq,q)x,y of the L × L submatrix Cq,q can be evaluated by

doing the above matrix multiplication as follows:

(Cq,q)x,y =

P∑

p=1

Pkp−q · exp[j2π(kp − q)

K(x − y)

], x, y = 0, · · · , L − 1 (4.9)

According to the requirement in equation 4.5, only the diagonal elements (x, x) are

non-zero, hence the only option for pilot energy and placement is (for q = −Q, · · · , Q):

P(k1−q) = · · · = P(kP −q) = P/P uniformly distributed pilot energy (4.10)

kp = K · p − 1

P, p = 1, · · · , P uniformly distributed pilot spacing (4.11)

4.2.2 Constraint 2 : Pilot Phase Orthogonality

Since the optimal energy allocation scheme has been identified, all the pilots are

equi-powered and equi-spaced as mentioned in constraint 1. Now we denote each pilot

as S(kp − q) =√P/P · exp (jϕkp−q), where exp (jϕkp−q) is the phase information of

each pilot for p = 1, · · · , P and q = −Q, · · · , Q.

In the case i 6= j:

Ci,j = FHi SH

i SjFj = 0, i, j = −Q, · · · , Q (4.12)

Now according to the above constraint, the relationship between different pilots’

phase information can be inferred. Since

Si = diag[S(k1 − i), · · · , S(kP − i)

](4.13)

= diag√P/P ·

[exp (jϕk1−i), · · · , exp (jϕkP −i)

](4.14)

thus

SHi Sj = diag

PP

·[exp (j∆1

i,j), · · · , exp (j∆Pi,j)

](4.15)

where ∆pi,j = ϕkp−j − ϕkp−i is the phase difference between the ith and jth pilot in

the same pilot cluster kp.

Then Ci,j is expressed as:

Ci,j =PP

· FHi diag

[exp (j∆1

i,j), · · · , exp (j∆Pi,j)

]Fj (4.16)

22


The (x, y)th element (Ci,j)x,y of the L × L submatrix Ci,j can be evaluated by

doing the above matrix multiplication as follows:

(Ci,j)x,y =PP

·P∑

p=1

exp (∆pi,j) · exp j 2π

K[x(kp − i) − y(kp − j)] (4.17)

=PP

· exp [y(i − j)]

P∑

p=1

exp (∆pi,j) · exp j 2π

K[(kp − i)(x − y)] (4.18)

In order to satisfy constraint 2, the above expression must equal to 0 for ∀ i, j ∈−Q, · · · , Q and ∀x, y ∈ 0, · · · , L − 1. Hence, according to mathematics, the only

feasible choice for the phase difference ∆pi,j is:

∆pi,j =

2π

K[(kp − i)µ], p = 1, · · · , P and i, j = −Q, · · · , Q (4.19)

and µ is an arbitrary integer outside the range of −(L − 1), · · · , L − 1.

4.3 Proposed Pilot Pattern

Based on the above generalization of pilot design constraints, an optimal pilot

pattern is proposed here.

4.3.1 Energy Allocation and Placement

Recall the pilot placement introduced in section 3.2 and its optimality verified in

section 4.2.1, there is no doubt that the pilots should be evenly distributed over all the

samples from 0 to K in a clustered form consisting of 2Q + 1 pilots with each energy

of P/P , which is separately centered at frequencies

kp = K · p − 1

P, p = 1, · · · , P (4.20)

4.3.2 the Number of Pilot Clusters

The value of P is the number of pilot clusters and it should satisfy the full column

rank requirement P ≥ (2Q+1)L. On the other hand, for convenience of mathematics,

P should divide K which is a power of 2. Hence, it is better that P be a power of 2.

Altogether, P is here chosen as P = 2⌊log2K

2Q+1⌋.

23


Figure 4.1: Proposed Pilot Pattern Illustration

4.3.3 Phase Distribution

As the phase constraint developed in section 4.2.2 illustrates, the phase difference

∆pi,j between the ith and jth pilot in the same pilot cluster kp is:

∆pi,j =

2π

K[(kp − i)µ], p = 1, · · · , P and i, j = −Q, · · · , Q (4.21)

and µ is an arbitrary integer outside the range of −(L− 1), · · · , L− 1. For simplicity,

µ is taken to be L. Then we start to approach the phase design of the center pilot kp

in the cluster. Denote the phase of the pilot at kp as ϕkp , and similarly:

ϕkp−q = ϕkp +2πkpL

K· q, q = −Q, · · · , Q (4.22)

The above is the phase expression for each pilot at frequencies from kp + Q to

kp − Q.

24

Chapter 5

Simulation

The proposed channel estimator is carried out by computer simulation in this chap-

ter. As an investigation, the computer simulation provided here is a comprehensive

one regarding the influences of different system parameters, such as the OFDM symbol

size K, Doppler frequency shift fD, different pilot-data power ratios as well as different

pilot patterns.

5.1 Parameter Setting

In this simulation, a QPSK-OFDM system with following parameters is analyzed

against the ideal case (where data are not superimposed onto the pilots):

1. OFDM Symbol Size : K = 512

2. Pilot-Data Power Ratio : λ = 0.6

3. Transmission Carrier Frequency : fc = 2 GHz

4. Sampling Frequency : fs = 1 MHz

5. Speed of the Vehicle : v = 200 km/h

6. Maximum Doppler Frequency Shift : fD = vfc

c = 370 Hz

25

CHAPTER 5. SIMULATION

7. Normalized Doppler Frequency Shift: fN = KfD

fs= 0.19

According to Q = ⌈fDKTs⌉ = ⌈K fD

fs⌉, the approximation order is obtained as

1. Hence the P is calculated to be 170. The channel is generated and estimated by

CE-BEM with each channel tap as independent, standardized and complex Gaussian

random variable. The multi-path intensity profile is chosen as Φc(τ) = e−0.1τ/Ts and

for any q, the doppler spectrum is specified as Sc = 1

π√

f2D−f2

when f < fD and

otherwise 0.

A general comparison is illustrated is as follows. Although the superimposed

scheme degrades the performance in contrast to the traditional pilot-assisted way,

it is practically satisfactory because it has much better bandwidth efficiency while

reaching a stably low MSE at a level of 10−2 after the SNR rises above 14dB.

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 3010

−2

10−1

100

X: 30Y: 0.06157

Signal−Noise Ratio (dB)

Mea

n S

quar

e E

rror

MSE Performance of the Superimposed Pilot Scheme

60% Pilot Energy

Figure 5.1: MSE performance when 60% power is allocated to pilots

26


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 3010

−5

10−4

10−3

10−2

10−1

100


Mea

n S

quar

e E

rror

MSE Performance of the Ideal Case

Ideal Case

Figure 5.2: MSE performance when no data is superimposed onto the pilots

5.2 Investigation on System Parameters

Here, we change several system parameters and investigate how they affect the

estimation performance. Unless specifically stated, other system parameters follow

the same setting as in section 5.1. The parameters include different pilot-data power

ratios, different OFDM symbol size K, different Doppler frequency shifts fD, as well

as different pilot patterns.

27


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 3010

−3

10−2

10−1

100

X: 30Y: 0.09021

X: 30Y: 0.05821

X: 30Y: 0.03824

X: 30Y: 0.02228

X: 30Y: 0.01048

X: 30Y: 0.004703


Mea

n S

quar

e E

rror

MSE Performance vs SNR under Different Pilot Energy Proportions (K=512, fD=370Hz )

95% Pilot Energy90% Pilot Energy80% Pilot Energy70% Pilot Energy60% Pilot Energy50% Pilot Energy

Figure 5.3: MSE performance under different pilot energy proportions

5.2.1 Different Pilot Energy Proportions

We can see from the above graph that the proposed channel estimator performs

better when the pilot energy is increased. The MSEs all reach a plateau after a certain

SNR at around 14dB due to data superimposition, because noise does not affect the

estimator’s performance in high SNR regions whereas the data serve as noise. Also,

the larger energy the pilots occupy, the slower the performance curve reaches the

plateau. Hence, the MSEs will not continue to decrease unless the data are recovered

and removed from the observation.

The worst performance belongs to the curve with 50% pilot power. However, it is

still around 10−1 which provides satisfactorily reliable estimate.

28


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 3010

−2

10−1

100

X: 30Y: 0.5364

X: 30Y: 0.05826


Mea

n S

quar

e E

rror

MSE Performance under different Pilot Patterns (K=512, fD=370Hz)

Proposed Optimal Pilot PatternRandom Pilot Pattern

Figure 5.4: MSE performance under different pilot patterns

5.2.2 Different Pilot Patterns

As is demonstrated above, the optimal pilot designed in this thesis has much better

performance than that of a random pilot pattern, nearly 10 times better.

29


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 3010

−2

10−1

100

X: 30Y: 0.1452

X: 30Y: 0.06181


Mea

n S

quar

e E

rror

MSE Performance vs SNR under Different Doppler Shifts (K=512)

fD=370Hz and f

N=0.19

fD=1100Hz and f

N=0.57

fD=1850Hz and f

N=0.95

fD=2600Hz and f

N=1.33

Figure 5.5: MSE performance under different Doppler shifts

5.2.3 Different Doppler Frequency Shifts

Doppler shift is a key factor in the algorithm because it is specifically proposed to

address the problem of time-varying channel estimation caused by doppler shift. On

the other hand, the proposed channel estimator is shown to have stable performance

when the normalized doppler shift is smaller than fN ≤ 1, however, its performance

degrades when fN reaches beyond that bound since the value Q is no longer enough

for the approximation and the estimator gradually fails to estimate the channel with

the same accuracy.

30


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 3010

−2

10−1

100

X: 30Y: 0.1197

X: 30Y: 0.05891

X: 30Y: 0.02877


Mea

n S

quar

e E

rror

MSE Performance under Different Symbol Lengths K (fD=370 Hz)

K=256 and fN=0.09

K=512 and fN=0.19

K=1024 and fN=0.38

Figure 5.6: MSE performance with different symbol lengths

5.2.4 Different values of Symbol Lengths K

The influence of K on the estimator is two-folded. One is that it increases the

normalized Doppler shift which degrades the performance, while the other is that it

provides more observation samples for estimation which improves the accuracy. Hence,

even under larger normalized Doppler shift, an estimator with system parameters

K = 1024 and fN = 0.39 still outperforms the previous one with K = 512 and

fN = 0.19.

31

Chapter 6

Concluding Remarks

In this thesis, relying on the basis expansion channel model, a superimposed pilot-

based estimation scheme for rapidly changing OFDM channels is proposed. The de-

sign of optimal pilot pattern is also studied and verified through simulations. It is

shown that the proposed estimation scheme performs satisfactorily under relatively

high doppler shifts while keeping the data rate unchanged, which improves bandwidth

efficiency. In contrast to those algorithms designed for LTI systems, this algorithm is

efficient and accurate in most practical scenarios.

As a development of this thesis, how to remove the corruption caused by data

superimposition can serve as a direction of research. Because in this thesis, data

is treated as noise without any further processing, which leads the ”MSE plateau”

displayed in the listed graphs. If this problem can be solved skillfully through some

powerful decoding techniques and iterative algorithms, the observation samples can be

further processed to obtain the pure pilot information after the decoding. By doing

this, the channel estimator should be robust to pilot-data power ratio as well as the

noise corruption, and also exhibits desirably high bandwidth efficiency.

32

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