On Optimum Pilot Design for CombType OFDM Transmission over DoublySelective Channels.
ABSTRACT We consider combtype OFDM transmission over doublyselective channels. Given a fixed number and total power of the pilot subcarriers, we show that the MMSEoptimum pilot design consists of identical equallyspaced clusters where each cluster is zerocorrelationzone sequence.

Conference Paper: Multielement antenna with close spacing for highly mobile OFDM systems
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ABSTRACT: In this paper, we consider employing a multielement antenna (MEA) with close spacing to tackle the challenging channel estimation (CE) in highly mobile OFDM systems. Instead of large spacing for diversity, we propose to place the adjacent elements with one symbol distance in the moving direction to observe the quasiduplicated channels in temporal difference of one symbol period. In exploiting the quasiduplicated channels, we developed a novel CEsymbol detection (SD) iteration that cooperates with the standardized combtype pilots to track fast varying channels. From simulation results, we show the proposed system outperforms the conventional receiver of two antennas with spatial diversity in highly mobile channels as long as the mutual coupling effects with the closespaced elements are restricted.Wireless Communications and Networking Conference (WCNC), 2013 IEEE; 01/2013  SourceAvailable from: Yu T. Su[Show abstract] [Hide abstract]
ABSTRACT: In this paper, we propose three classes of systematic approaches for constructing zero correlation zone (ZCZ) sequence families. In most cases, these approaches are capable of generating sequence families that achieve the upper bounds on the family size (K) and the ZCZ width (T) for a given sequence period (N). Our approaches can produce various binary and polyphase ZCZ families with desired parameters (N,K,T) and alphabet size. They also provide additional tradeoffs amongst the above four system parameters and are less constrained by the alphabet size. Furthermore, the constructed families have nestedlike property that can be either decomposed or combined to constitute smaller or larger ZCZ sequence sets. We make detailed comparisons with related works and present some extended properties. For each approach, we provide examples to numerically illustrate the proposed construction procedure.IEEE Transactions on Information Theory 08/2013; 59(8):49945007. · 2.62 Impact Factor
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IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION1
On Optimum Pilot Design for CombType OFDM Transmission over
DoublySelective Channels
K. M. Zahidul Islam, Student Member, IEEE, Tareq Y. AlNaffouri, and Naofal AlDhahir, Fellow, IEEE
Abstract—We consider combtype OFDM transmission over
doublyselective channels. Given a fixed number and total power
of the pilot subcarriers, we show that the MMSEoptimum pilot
design consists of identical equallyspaced clusters where each
cluster is zerocorrelationzone sequence.
Index Terms—Pilot optimization, Doppler, ICI, OFDM, ZCZ
sequence.
I. INTRODUCTION
U
their orthogonality resulting in performancelimiting Inter
Carrier Interference (ICI). ICI makes channel estimation more
challenging since both the subcarrier frequency responses
and the interference caused by each subcarrier into other
subcarriers in each OFDM symbol have to be estimated.
Recently, we proposed in [1] a frequencydomain high
performance computationallyefficient OFDM channel estima
tion algorithm in the presence of severe ICI. We exploited
the channel correlations in the time and frequency domains
to enhance the channel estimation accuracy and reduce its
complexity (by performing most of the computations offline).
In most OFDMbased wireless systems, pilot subcarriers are
inserted in each OFDM symbol for channel estimation and
tracking. When the channel is fixed over each OFDM sym
bol, the optimum pilot structure consists of equallyspaced
individual pilot subcarriers [2], [3]. On the other hand, when
the channel varies within the OFDM symbol, [4] argued that
the pilot subcarriers should be grouped into equallyspaced
clusters. However, [4] did not optimize the pilot subcarrier
clusters which is the subject of this paper.
The main contributions of this Letter are
NDER high mobility, the subcarriers of an orthogonal
frequencydivision multiplexing (OFDM) symbol lose
∙ Proving that the MMSEoptimum pilot design for OFDM
over doublyselective channels consists of identical
equallyspaced frequncydomain pilot clusters.
∙ Proving that ZCZ sequences (see [5] and references
therein) are MMSEoptimal designs for the frequency
domain pilot clusters (see Fig. 1 and Appendix B).
Paper approved by H. Arslan, the Editor for Cognitive Radio and OFDM
of the IEEE Communications Society. Manuscript received June 24, 2010; no
revision.
K. M. Z. Islam and N. AlDhahir are with the Department of Electrical
Engineering, The University of Texas at Dallas, Richardson, TX, USA (e
mail: zislam@student.utdallas.edu, aldhahir@utdallas.edu).
T. Y. AlNaffouri is with the Electrical Engineering Department, King
Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia (email:
naffouri@kfupm.edu.sa).
This work was supported by King Abdulaziz City for Science and Tech
nology (KACST), Saudi Arabia, Project no. AR 2798.
Digital Object Identifier 10.1109/TCOMM.2011.020411.100151
∙ A new proof (more rigorous than the one in [1]) that the
MMSEoptimal OFDM channel estimation error covari
ance matrix over doublyselective channels is diagonal
(see Appendix A).
Reference [6] proposed a frequencydomain clustered pilot
pattern where each cluster has an impulsive structure made
of a single pilot subcarrier padded with zero subcarriers as
guard band on both sides to eliminate the ICI. This impulsive
pilot design ignores signal energy dispersed into the adjacent
subcarriers. The novelty of the pilot designs we propose in this
paper lies in designing MMSEoptimal nonimpulsive periodic
pilot clusters which exploit the banded structure of the CFR
matrix to increase the accuracy of channel estimation.
This paper is organized as follows. In Section II, we
present the doublyselective channel model and assumptions
and briefly review the channel estimation algorithm in [1].
The formulation and solution of the pilot cluster optimization
problem are given in Section III. Performance comparisons of
our proposed pilot design with the impulsive design are given
in Section IV followed by conclusions. For the convenience
of the reader, we summarized the key variables used in the
paper in Table I.
II. PRELIMINARIES AND BACKGROUND
A. System Model
We start with the following frequencydomain representa
tion of an OFDM system with 푁 subcarriers over a doubly
selective channel
퓨 ≜ QHQ퐻퓧 + 퓩 = G퓧 + 퓩
(1)
where Q is the 푁point FFT matrix and (.)퐻is the Hermitian
operator. 퓧 is a pilotdatamultiplexed OFDM symbol where
certain subcarriers are allocated as pilots surrounded by data
subcarriers. We refer to such a multiplexed OFDM symbol
structure as combtype OFDM symbol hereafter. H is the
푁 × 푁 timedomain channel matrix which corresponds to
convolution with the timevarying CIR coefficients ℎ푛(푙) at
lag 푙 (for 0 ≤ 푙 ≤ 퐿 − 1) and time instant 푛 and 퓩
is the frequencydomain noise vector. Over doublyselective
channels, the CFR matrix G ≜ QHQ퐻is not diagonal as
in timeinvariant channels. Rather, the energy of the main
diagonal is dispersed into adjacent diagonals depending on
the severity of the Doppler spread. We approximate G as a
banded matrix and set all elements of G outside of 푀 main
diagonals to zero where 푀 is odd integer.
00906778/11$25.00 c ⃝ 2011 IEEE
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2IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION
TABLE I
LIST OF KEY VARIABLES
VariableDescription
푁
푓푑
L
푁푇
푁푝
H
G
푁푑
푀
퓧
C휖
퐿푐
푁푐
FFT size
Doppler frequency
Number of channel impulse response taps
Total number of pilot subcarriers
Number of subcarriers in each pilot cluster
(푁 × 푁) Timedomain channel matrix
(푁 × 푁) Frequencydomain channel matrix
Number of dominant R퐻eigenvalues for each tap
Number of diagonals in banded G
(푁 × 1) Frequencydomain combtype input vector
(푁푑퐿 × 푁푑퐿) Channel estimation errorcovariance matrix
Period of the pilot clusters in 퓧
Total number of pilot clusters in 퓧
B.
Channel Estimation
In [1], we derived a
decompositions
R퐻 ≜ 퐸[vec(H)vec(H)퐻]. Assuming Jakes’s model with
퐸[ℎ푚(푙)ℎ∗
is the Doppler frequency and 퐽0(⋅) is the zeroorder Bessel
function of the first kind, we derived the eigendecomposition
of R퐺 in closed form in terms of the 푁 × 푁 symmetric
Toeplitz Bessel function matrix J whose (푚,푛)th element
is given by 퐽(푚,푛) = 퐽(∣푚 − 푛∣) = 퐽0(2휋푓푑∣푚 − 푛∣푇푠).
Let G푝 denote the matrices formed by unvectorizing the
푁퐿 eigenvectors of R퐺. We showed in [1] that G푝 can be
expressed in terms of the eigenvectors of J as follows
ReducedComplexity FrequencyDomain MMSE OFDM
relation between the eigen
≜
퐸[vec(G)vec(G)퐻]
of
R퐺
and
푛(푙)] = 퐽0(2휋푓푑(푚 − 푛)푇푠) ≜ 퐽(푚 − 푛) where 푓푑
G푝= Qdiag(v푛)B푙Q퐻;0 ≤ 푙 ≤ (퐿−1) and 1 ≤ 푝 ≤ 푁퐿
(2)
where v푛,푛 = 1,2,⋅⋅⋅ ,푁 are the dominant eigenvectors
of J and B is a circulant shift matrix whose first column
is
010
⋅⋅⋅
eigenvectors of R퐺, (1) can be approximated as follows
[
0
]푇. Considering the 푁푑퐿 dominant
퓨 = G퓧 + 퓩 ≈
푁푑퐿
∑
푝=1
훼푝G푝퓧
? ?? ?
퓔푝
+퓩 =
푁푑퐿
∑
푝=1
훼푝퓔푝+ 퓩
(3)
where the 훼푝’s are unknown independent random variables.
Considering only the 푇 output subcarriers that result in input
output equations free of unknown data subcarriers in (3), we
arrive at the following linear system of 푇 equations in 푁푑퐿
unknowns
퓨 =
푁푑퐿
∑
푝=1
훼푝퓔푝+ 퓩 = E푝휶 + 퓩
(4)
where E푝=
This is a Bayesian estimation model since the unknown
random vector 휶 is assumed zero mean with covariance
matrix R훼 = diag([훾1휆1,...,훾푁푑퐿휆푁푑퐿]) where 훾푝 and
휆푝,푝 = 1,2⋅⋅⋅ ,푁푑퐿 are the channel powerdelay profile
(PDP) path variances and the dominant eigenvalues of R퐺,
respectively. Hence, we can estimate 휶 using the following
[
퓔1⋅⋅⋅퓔푁푑퐿
]
and 휶 =
[
훼1⋅⋅⋅훼푁푑퐿
]푇.
linear minimum mean square error (LMMSE) estimator [7]
[
?
where 휎2
are i.i.d. samples). Given 푁, 푓푑 and 휎2
in (5) can be precomputed and stored in lookup tables to
reduce the realtime implementation complexity significantly.
The performance of this channel estimator is measured by
the error vector 흐 = 휶 − ˆ 휶 which has zero mean with the
following covariance matrix
⎡
⎣R−1
Hence, the MSE in estimating 훼푖is MSE(ˆ 훼푖) = C휖(푖,푖).
ˆ 휶 =
1
휎2
푧
R−1
훼 +
1
휎2
푧
??
E퐻
푝E푝
]−1
E퐻
푝
?
≜W
퓨 = W퓨
(5)
푧is the noise variance (assuming the 풵(푘)’s in (4)
푧and the 푃퐷푃, W
C휖=
⎢⎢
훼 +
1
휎2
푧
E퐻
? ?? ?
푝E푝
≜R퐸
⎤
⎦
⎥⎥
−1
=
[
R−1
훼 +
1
휎2
푧
R퐸
]−1
(6)
III. MAIN RESULTS
A. Problem Formulation
Consider 퓧 to be a combtype OFDM symbol with data
subcarriers masked out by zeros. Our objective is to design
a frequencydomain pilot structure for the LMMSE channel
estimator in (5) to minimize the trace of C휖 in (6). In
Appendix A, we show that this is achieved by making R퐸a
diagonal matrix. Using (3), (4) and (6), it is clear that making
R퐸diagonal is equivalent to designing 퓧 such that
퓧퐻G퐻
and 퓧퐻퓧 = 푐 where 푐 is a constant which depends on the
total pilot power constraint.
푖G푗퓧 = 0,
for 푖 ∕= 푗;푖,푗 = 1,2,⋅⋅⋅ ,푁푑퐿 (7)
B. Asymptotic Analysis
Using (2), the (푚,푛)th element of R퐸 can be written as
follows
푅퐸(푚,푛) = 퓧퐻G퐻
= 퓧퐻QB퐻(푗1)diag(v푖1)퐻diag(v푖2)
?
= x퐻B퐻(푗1)Λ푖1푖2B(푗2)
?
where 푚 = (푖1− 1)푁푑+ 푗1, 푛 = (푖2− 1)푁푑+ 푗2 for
푖1,푖2 = 1,2,⋅⋅⋅ ,푁푑 and 푗1,푗2 = 1,2,⋅⋅⋅ ,퐿. We can
gain further insight into the pilot optimization problem by
approximating the Toeplitz matrix J defined in Section IIB
by a circulant matrix for large 푁 using Szego’s theorem
[8]. Hence, the eigenvectors and eigenvalues of J converge
to the FFT columns and FFT transform of the first column
of J, respectively. Based on this circulant approximation of
J, there are 4 possible values of 푅퐸(푚,푛) in (8) as listed
in Table II. Now, as long as 푗1 = 푗2, I푐(푖1,푖2,푗1,푗2) is a
diagonal matrix whose entries are real if 푖1= 푖2or complex
otherwise. If 푗1 ∕= 푗2, I푐(푖1,푖2,푗1,푗2) has zero diagonal
elements and a nonzero 푑푗th superdiagonal or subdiagonal
푚G푛퓧
???
Λ푖1푖2
x = x퐻I푐(푖1,푖2,푗1,푗2)x
B(푗2)Q퐻퓧
???
I푐(푖1,푖2,푗1,푗2)
(8)
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ISLAM et al.: ON OPTIMUM PILOT DESIGN FOR COMBTYPE OFDM TRANSMISSION OVER DOUBLYSELECTIVE CHANNELS3
TABLE II
OFFDIAGONAL ELEMENTS OF R퐸
Case
푖1= 푖2
푗1= 푗2
푅퐸(푚,푛)
and 푚 ∕= 푛
0
0
Comments
1
2
Yes
Yes
Yes
No
R퐸= ∥퓧∥2I푁푑푙
I푐(⋅) is upper/lower
shifted diagonal matrix
Z푢 is linear uppershift matrix,
Z푙is linear lowershift matrix
I푐(⋅) is upper/lower
shifted diagonal matrix
(assuming pilot structure in Fig. 1)
푐푖′퓧퐻
or 푐푖′퓧퐻
0
(assuming pilot structure in Fig. 1)
3 NoYes
푝Z푑푖
푝Z푑푖
푢퓧푝
푙퓧푝
4No No
Fig. 1.Optimized pilot structure for our channel estimation algorithm in [1].
where 푗1−푗2= +푑푗 or −푑푗, respectively. Since, our objective
is to make R퐸 diagonal, we have to force the offdiagonal
elements of R퐸in Table II to zero.
Proposition: If the frequencydomain pilot vector 퓧 has
the periodic clustered structure shown at the top of Fig. 1 with
푁푝adjacent subcarriers in each pilot cluster and the number
푁푐and period 퐿푐of the pilot clusters satisfy the relation 푁 =
푁푐퐿푐, then the timedomain pilot vector x will be sparse as
shown at the bottom of Fig. 1. The proof is given in Appendix
B.
Consider a sparse x of the form shown at the bottom of
the Fig. 1, then x퐻I푐(.)x will be a weighted sum of the
elements of I푐(.) that correspond to the positions of ‘1’s in
the puncturing matrix PI푐whose (푚,푛)th element is given
by
{
where, 푘1,푘2= 0,1,⋅⋅⋅ ,(퐿푐−1) and (퐿−1) is the highest
index of the super or sub diagonal of I푐(.) that is nonzero.
Hence, the I푐(.)’s in Case 2 and Case 4 of Table II can be at
most (퐿−1) shifted upper or lower diagonal matrices. If the
sparse x has at least 퐿 zeros between any two of its nonzero
elements, all Case 2 and Case 4 offdiagonal elements of R퐸
will be zero. In other words, we have to design the number of
pilot clusters in the frequency domain to be greater than the
length of the CIR vector, i.e.
푃I푐(푚,푛) =
1,
0,
푚 = 푘1푁푐+ 1, 푛 = 푘2푁푐+ 1
otherwise
(9)
푁푐> 퐿
(10)
In [1], we showed that the pilot cluster size must satisfy
푀 ≤ 푁푝≤ 2푀 − 1;푀 = 3,5,⋅⋅⋅
(11)
In addition, the periodic clustered structure of 퓧, as shown in
Fig. 1, implies that the pilot clusters must be equally spaced.
Hence, the period of pilot clusters 퐿푐is given by
퐿푐=푁푁푝
푁푇
(12)
where 푁푇 = 푁푐푁푝 is the total number of pilot subcarriers.
Since G푝is assumed to be a banded matrix with 푀 diagonals,
to include all diagonals in the inputoutput equations at pilot
locations, the first and last푀−1
2
OFDM symbol cannot be assigned as pilots. We can avoid
making these edge subcarriers pilots by placing푀−1
at the start of each period of 퓧 and inserting 푁푝adjacent pilot
subcarriers followed by 퐿푐−(푁푝−푀−1
퐿푐≥ (푁푝+ 푀 − 1)
Using (10)(13), we arrive the following design guideline on
푁푐
max
2푀 − 1,퐿
subcarriers of the combtype
2
zeroes
2
)zeroes implying the
following lower bound
(13)
(
푁푇
)
≤ 푁푐≤푁푇
푀
(14)
C. Pilot Cluster Optimization
All we are left with now is the 3rd case in Table II ; i.e.
we have to make x퐻I푐(푖1,푖2,푗1,푗2)x = 0 when 푖1 ∕= 푖2
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4IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION
and 푗1 = 푗2. Note that due to the periodic structure of
퓧, all pilot clusters are identical. Hence, we only need
to optimize one pilot cluster. Next, we will show how to
make the Case 3 R퐸(푚,푛) elements in Table II equal to
zero with periodic clustered pilot designs. From (8), we see
that each Case 3 R퐸(푚,푛) element in Table II corresponds
to the case when 푖1 ∕= 푖2 and 푗1 = 푗2, i.e. when the
eigenvectors are different for the same CIR tap. Under this
scenario, I푐(푖1,푖2,푗1,푗2) becomes a diagonal matrix whose
diagonal is a scaled, circularlyshifted FFT vector. Let a푖
contain these modified FFT vectors when 푖1 ∕= 푖2 and
푗1= 푗2 where 푖 = (−(푁푑− 1),⋅⋅⋅ ,−1,1,⋅⋅⋅ ,(푁푑− 1))푁
denotes the FFT column index and (.)푁 is the 푚표푑푢푙표 − 푁
operation. In [1], we chose 푁푑 dominant eigenvectors of J
to reduce computational complexity. The column indices of
the FFT vectors chosen as dominant eigenvectors are given
by(−푁푑−1
of (푁푑− 1)푁푑nondiagonal elements in R퐸to be forced to
zero.
Towards this objective, the timedomain sparse vector x is
given by
x = Q퐻˜I퓧푝
where 퓧푝is an individual frequencydomain pilot cluster of
length 푁푝and˜I = 1푁푐⊗˜I푝
˜I푝=0푁푝×푀−1
1푁푐is the length푁푐allones column vector and ⊗ denotes
the Kronecker product. Now, from (8), by using (15) and the
sparse structure of x as shown at the bottom of Fig. 1, we can
restate our pilot optimization objective as finding 퓧푝 such
that
x퐻I푐(푖1,푖2,푗1,푗2)x
=퓧푝퐻˜I퐻QB퐻(푗1)Λ푖1푖2B(푗1)
?
⇒퓧푝퐻˜I퐻Q diag(a푖)Q퐻˜I
?
푖 = (−(푁푑− 1),⋅⋅⋅ ,−1,1,⋅⋅⋅ ,(푁푑− 1))푁
Since the a푖’s are scaled, circularlyshifted FFT vectors,
Q diag(a푖)Q퐻can be written as 푐푖Z푖
scalar and Z푐is the 푁 ×푁 circular uppershift matrix whose
first column is
0
⋅⋅⋅
of R푖 will be a weighted sum of the elements of 푐푖Z푖
correspond to the positions of ‘1’s in the puncturing matrix
PR푖(푚,푛)given by
{
where, 푘3,푘4= 0,1,⋅⋅⋅ ,(퐿푐−1). Let 푑푖denote 푖, as defined
in (16), without the 푚표푑푢푙표N operation. If 푑푖 is negative,
it can be shown that R푖 = 푐푖′Z푑푖
scalar and Z푢is the 푁푝×푁푝linear uppershift matrix whose
first row is[
whose first column is[
2
,⋅⋅⋅ ,푁푑−1
2
)
푁. For each dominant eigenvector,we
have (푁푑−1) Case 3 offdiagonal elements resulting in a total
(15)
[
2
I푁푝
0푁푝×(퐿푐−푀−1
2
−푁푝)
]푇
,
???
diag(a푖)
Q퐻˜I퓧푝= 0
???
R푖
퓧푝≜ 퓧푝퐻R푖퓧푝= 0
(16)
푐where 푐푖is a complex
]푇. The (푚,푛)th element
[
01
푐that
푃R푖(푚,푛)(푞,푟) =
1,
0,
푞 = 푘3푁푐+ 푚, 푟 = 푘4푁푐+ 푛
otherwise
(17)
푢 where 푐푖′is a complex
]. On the other hand, if 푑푖is
⋅⋅⋅
010
⋅⋅⋅
0
positive, R푖= 푐푖′Z푑푖
푙
where Z푙as a linear lowershift matrix
0100
]푇. Since the range
of 푑푖is [−(푁푑− 1),−1,1,⋅⋅⋅ ,(푁푑− 1)], there are (푁푑−1)
distinct R푖’s associated with Z푢and another (푁푑−1) distinct
R푖’s associated with Z푙. Therefore, (16) is equivalent to
퓧푝퐻R푗퓧푝= 0
Separating the R푖’s in (18) corresponding to Z푢and Z푙yields
퓧푝퐻Z푑푖
퓧푝퐻Z푑푖
Using the frequencydomain pilot cluster notation shown in
Fig. 1, for each 푑푖, (19) can be written as
:푗 = 1,2,⋅⋅⋅2(푁푑− 1)
(18)
푢퓧푝
푙퓧푝
=0:푑푖= 1,2,⋅⋅⋅(푁푑− 1) (19)
=0
(20)
푁푝−1
∑
푛=휏풫푛풫푛−휏∗)∗
satisfies (20), we focus on solving (19) only. From (19), we
see that the aperiodic autocorrelation of the optimum pilot
cluster sequence must be zero at lag 푑푖 which is the design
criterion for a zerocorrelation zone (ZCZ) sequence [5] with
푍푐 ≜ 푁푑− 1 zero lags. To be specific, for 푁푝 = 5 and
푁푑= 3, the MMSEoptimal pilot cluster is a ZCZ sequence
of length 5 with 푍푐 = 2. In Table III, we present 3 such
sequences obtained through numerical search under a total
power constraint of 푁푝= 5 with 푀 = 3. The aperiodic auto
correlation sequence of these optimized sequences are also
given in Table III. The inputs to the numerical optimization
algorithm are 푁푝 and 푁푑. Hence, the optimization can be
performed offline and the optimum pilot sequences of different
sizes are stored in lookup tables.
푛=휏
풫푛풫푛−휏∗= 0:휏 = 1,2,⋅⋅⋅(푁푑− 1)
(21)
Similarly, for the same 푑푖, (20) can be written as
(∑푁푝−1
= 0. Since the solution to (19) also
IV. SIMULATION RESULTS
In our simulations, we assume the SUI3 channel model
with a rate1
of 10% (normalized to the subcarrier spacing) with 푁 = 1024
and 푀 = 3.
Assuming a pilot cluster size 푁푝 = 2푀 − 1 = 5, Fig.
2 depicts the 퐵퐸푅 of our channel estimation algorithm in
[1] with the optimized pilot clusters (shown in Table III)
along with the 퐵퐸푅 of perfect CSI under full and banded
G assumptions. While the 3 optimized pilot cluster sequences
achieve MMSE and make R퐸 diagonal, their 퐵퐸푅 perfor
mance is different at high 푆푁푅 where ICI dominates noise.
It can be seen that sequence ‘a’ which has a higher aperiodic
autocorrelation at lags larger than 푍푐, performs worse in ICI
limited (high 푆푁푅) scenarios than sequence ‘b’ and ‘c’. As
a benchmark, the 퐵퐸푅 of the impulsive pilot cluster design
[
V. CONCLUSION
In combtype OFDM transmission over doublyselective
channels, we showed that the channel estimation mean square
error is minimized by dividing the available pilot subcarriers
into periodic (i.e. identical and equallyspaced) clusters. Under
a fixed total pilot power budget, we exploited the banded
structure of the CFR matrix to show that the optimum pilot
cluster is a ZCZ sequence. Simulation results demonstrated
significant 퐵퐸푅 improvement over impulsive pilot designs
which ignore the banded CFR structure.
2convolutional code, a high Doppler frequency
00500
]푇suffers from an irreducible error floor.
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Page 5
ISLAM et al.: ON OPTIMUM PILOT DESIGN FOR COMBTYPE OFDM TRANSMISSION OVER DOUBLYSELECTIVE CHANNELS5
TABLE III
SAMPLE SIZE5 MMSEOPTIMUM PILOT CLUSTERS FOR LARGE 푁
Sequence
a
Aperiodic
autocorrelation
Sequence
b
Aperiodic
autocorrelation
Sequence
c
Aperiodic
autocorrelation
0.7790 + 0.3011i
0.2792 + 0.7106i
0.0002  0.0013i
0.0031 + 0.0002i
5.0 (zero lag)
0.0031  0.0002i
0.0002 + 0.0013i
0.2792  0.7106i
0.7790  0.3011i
0.0112  0.0084i
0.0180 + 0.0053i
0.0001 + 0.0001i
0.0000  0.0000i
5.0 (zero lag)
0.0000 + 0.0000i
0.0001  0.0001i
0.0180  0.0053i
0.0112 + 0.0084i
0.0006  0.0037i
0.0009 + 0.0040i
0.0001 + 0.0001i
0.0001  0.0001i
5.0 (zero lag)
0.0001 + 0.0001i
0.0001  0.0001i
0.0009  0.0040i
0.0006 + 0.0037i
0.2008  0.8820i
0.4227  0.2853i
0.7785  1.5159i
0.3268 + 0.2416i
0.5157 + 0.7658i
0.0083  0.1174i
0.0256 + 0.0768i
2.1629  0.5304i
0.0198 + 0.0752i
0.0645  0.0997i
0.0107 + 0.0576i
0.4252  0.2802i
1.6952  1.2656i
0.1461  0.4865i
0.0638  0.0007i
5101520 25
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
SNR (dB)
BER
Perfect CSI
Perfect CSI, banded G (M=3)
Optimized Sequence a
Optimized Sequence b
Optimized Sequence c
Impulsive pilot cluster
Fig. 2.
with perfect CSI for 푁 = 1024, 푀 = 3 and 푁푝= 5.
퐵퐸푅 comparison of proposed and impulsive pilot cluster designs
APPENDIX A
PROOF THAT OPTIMUM R퐸IS DIAGONAL
We start by presenting the following three useful matrix
derivative identities
∂Tr(Y−1)
∂Y
= −Y−2푇(see (57) in [9])
I.2
∂Y
= Y∗(see (225) in [9])
I.3 If A is a function of F = C+B퐻Y퐻YB where C ≥ 0,
then
∂Y
Proof:Let
D=B퐻Y퐻.
퐶(푖,푗) +∑
퐷(푖,푘)퐵(푚,푗). Hence,
I.1
∂Tr(Y퐻Y)
∂Tr(A)
= Y∗B∗∂Tr(퐴)
∂FB푇.
Hence,
퐹(푖,푗)=
푘
∑
푚퐷(푖,푘)푌 (푘,푚)퐵(푚,푗). Differentiat
ing 퐹(푖,푗) with respect to 푌 (푘,푚), we get
∂퐹(푖,푗)
∂푌 (푘,푚)=
∂Tr(A)
∂푌 (푘,푚)=
∑
∑
푗
∑
∑
푖
∂Tr(A)
∂퐹(푖,푗)
∂퐹(푖,푗)
∂푌 (푘,푚)
=
푗푖
∂Tr(A)
∂퐹(푖,푗)퐷(푖,푘)퐵(푚,푗)
∴∂Tr(A)
∂Y
= D푇∂Tr(A)
∂F
B푇=(B퐻Y퐻)푇 ∂Tr(A)
∂F
∂F
B푇
= Y∗B∗∂Tr(A)
B푇
(22)
Our pilot optimization objective is to minimize the trace
of C휖=R−1
휎2
where 퐸푡표푡is the total pilot energy in an OFDM symbol. We
form the cost function using Lagrangian multipliers as follows
(
Next, we compute∂푓푐표푠푡
E푝
(KuhnTucker) condition on E푝. From I.3 with F = R−1
1
휎2
(
∂E푝
(
훼+
1
푧R퐸
)−1
subject to∑
푖푅퐸(푖,푖) = 퐸푡표푡
푓푐표푠푡= Tr
R−1
훼 +
1
휎2
푧
E퐻
푝E푝
)−1
+ 휆
(
e퐻
푖E퐻
푝E푝e푖
)
− 퐸푡표푡
(23)
and set it to 0 to get the optimality
훼 +
푧E퐻
푝E푝, C = R−1
훼, B =
1
휎푧I and A = F−1, we have
∂Tr
R−1
훼 +
1
휎2
푧E퐻
푝E푝
)
=
1
휎2
−1
푧
E∗
푝
∂
∂FTr(F)−1
(24)
=
휎2
푧
E∗
푝F−2푇
(Using I.1)
Moreover,
E∗
and
∂
∂E푝
(
e퐻
푖E퐻
푝E푝e푖
)
=
∂
∂E푝
(
E퐻
푝E푝e푖e퐻
푖
)
=
푝e∗
푖e푇
∂tr(A)
∂A
푖where we used I.3 with B = e푖, Y = E푝, C = 0
= I. From (24), we get
−1
휎2
푧
E∗
푝
(
R−1
훼 +
1
휎2
푧
E퐻
푝E푝
)−2푇
+ 휆
(∑
푖
E∗
푝e∗
푖e푇
푖
)
= 0
Transposing both sides yields
[
=⇒
I − 휆휎2
푧
(
R−1
훼 +
1
휎2
푧
E퐻
푝E푝
)2(∑
푖
e∗
푖e푇
푖
)]
E퐻
푝= 0
(25)
Since E퐻
푝is a tall fullcolumn rank matrix, we have
)2
=
(
R−1
훼 +
1
휎2
푧
E퐻
푝E푝
=
1
휆휎2
1
휎2
푧
?
Λ
푧
(e∗
푖e푇
(1
푖
)−1
휆,1
??
(26)
)
?
(27)
diag
휆,⋅⋅⋅ ,1
휆
≜Λ
=⇒
(
R−1
훼 +
1
휎2
푧
E퐻
푝E푝
)
=
1
2
Hence, the trace of C휖 will be minimized when R퐸 ≜
E퐻
푧
Λ
훼
> 0. Since Λ and R훼are diagonal
matrices, the optimum R퐸is also a diagonal matrix.
푝E푝= 휎2
(
1
2 − R−1
)
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