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
CHAPTER 2. SYSTEM MODEL
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
CHAPTER 2. SYSTEM MODEL
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
CHAPTER 2. SYSTEM MODEL
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
CHAPTER 2. SYSTEM MODEL
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
CHAPTER 2. SYSTEM MODEL
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
CHAPTER 2. SYSTEM MODEL
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
CHAPTER 2. SYSTEM MODEL
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
CHAPTER 2. SYSTEM MODEL
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
CHAPTER 2. SYSTEM MODEL
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
CHAPTER 3. CHANNEL ESTIMATION
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
CHAPTER 3. CHANNEL ESTIMATION
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
CHAPTER 3. CHANNEL ESTIMATION
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
CHAPTER 3. CHANNEL ESTIMATION
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
CHAPTER 4. OPTIMAL PILOT DESIGN
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
CHAPTER 4. OPTIMAL PILOT DESIGN
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
CHAPTER 4. OPTIMAL PILOT DESIGN
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
CHAPTER 5. SIMULATION
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
Signal−Noise Ratio (dB)
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
CHAPTER 5. SIMULATION
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
Signal−Noise Ratio (dB)
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
CHAPTER 5. SIMULATION
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
Signal−Noise Ratio (dB)
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
CHAPTER 5. SIMULATION
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
Signal−Noise Ratio (dB)
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
CHAPTER 5. SIMULATION
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
Signal−Noise Ratio (dB)
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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