BLIND PERIODICALLY TIMEVARYINGMMOECHANNEL SHORTENINGFOROFDMSYSTEMS
D. Darsena
Parthenope University, Napoli, Italy
darsena@uniparthenope.it
G. Gelli, L. Paura, F. Verde
University Federico II, Napoli, Italy
[gelli,paura,f.verde]@unina.it
ABSTRACT
In this paper, the problem of synthesizing a blind channel shorteningalgorithm for orthogonal frequencydivision multiplexing (OFDM)systems is addressed. In particular, a commonly adopted assumption in the channel shortening framework is discussed, showing that,when it is violated, the data statistics usually needed for the synthesis of the shortening algorithms turn out to be periodically timevarying (PTV) rather than timeinvariant. Elaborating on this point,and considering in particular the recently proposed minimummeanoutputenergy (MMOE) blind channel shortening algorithm, it isshown how its synthesis must be modi
Þ
ed in order to account forthe PTV nature of the data statistics. Numerical results assessing theperformance of the proposed blind PTVMMOE channel shorteningalgorithm are reported.
Index Terms
—
Blind channel shortening, orthogonal frequencydivision multiplexing (OFDM), periodically timevarying (PTV)
Þ
ltering.
1. INTRODUCTION
Orthogonal frequencydivision multiplexing (OFDM) is a convenient and
ß
exible choice to achieve high datarate transmission overdispersive channels, without the need to resort to complicated equalization strategies. OFDM systems counteract channel dispersion byinserting a cyclic pre
Þ
x (CP) at the beginning of each OFDM block.Under the assumption that the CP length
L
cp
obeys
L
cp
≥
L
h
,where
L
h
is the maximum channel order, performing CP removal atthe receiver assures perfect interblock (IBI) suppression. In highlydispersive channels, however, ful
Þ
lling condition
L
cp
≥
L
h
mightbe impractical. A viable alternative is to preprocess the receivedsignal, before CP removal, by means of a time equalizer (TEQ)aimed at
shortening
the channel up to a length
L
eff
≤
L
cp
.Several channel shortening algorithms have been proposed in theliterature, both
nonblind
ones (like e.g. the pioneering contribution[1]), which assume knowledge or estimation of the channel impulseresponse (CIR) to be shortened, as well as
blind
ones [2, 3, 4, 5, 6].Channel shortening algorithms are generally implemented as
Þ
niteimpulse response (FIR)
Þ
lters of order
L
e
, whose weights are calculated by solving quadratic optimization problems, which involveonly second order statistics (SOS) of the received data. In particular,the commonly considered assumption
L
g
L
e
+
L
h
< M
assuresthat the relevant SOS are timeinvariant, which entails some simpli
Þ
cation in the synthesis and analysis of the shortening algorithms.However, the latter assumption is likely to be violated in systemswith high channel dispersion (large
L
h
) and/or smalltomoderatenumber
M
of subcarriers.In this paper, we consider in particular the blind channel shortening algorithm [6], which is based on the minimummeanoutputenergy (MMOE) criterion. We show that, when
L
g
≥
M
, the SOSneeded for its synthesis exhibit a periodically timevarying (PTV)behaviour, which enforces a PTV structure on the resulting channelshortening
Þ
lter. The obtained PTVMMOE channel shortening algorithm can be conveniently implemented by resorting to the Fourierseries representation [7].
Notations:
The
Þ
elds of complex and real numbers are denoted with
C
and
R
, respectively; matrices [vectors] are denoted with upper[lower] case boldface letters (e.g.,
A
or
a
); the
Þ
eld of
m
×
n
complex [real] matrices is denoted as
C
m
×
n
[
R
m
×
n
], with
C
m
[
R
m
]used as a shorthand for
C
m
×
1
[
R
m
×
1
];
{
A
}
i
1
i
2
indicates the
(
i
1
+1
,i
2
+1)
th element of matrix
A
∈
C
m
×
n
, with
0
≤
i
1
≤
m
−
1
and
0
≤
i
2
≤
n
−
1
; the superscripts
∗
,
T
,
H
, and
−
1
denote the con jugate, the transpose, the Hermitian (conjugate transpose), and theinverse of a matrix, respectively;
0
m
∈
R
m
,
O
m
×
n
∈
R
m
×
n
, and
I
m
∈
R
m
×
m
denote the null vector, the null matrix, and the identitymatrix, respectively; rep
P
{
f
[
n
]
}
=
+
∞
i
=
−∞
f
[
n
−
iP
]
denotes thereplication of period
P
of
f
[
n
]
;
R
N
[
n
]
is the rectangular window of length
N
, i.e.,
R
N
[
n
] = 1
for
0
≤
n
≤
N
−
1
and zero otherwise;E
[
·
]
and
<
·
>
denote ensemble and time averaging, respectively;
Þ
nally,
j
√ −
1
denotes the imaginary unit.
2. OFDMSYSTEMMODELANDSOSANALYSIS
Following mostly the framework and notation of [6], we consider anOFDM system with
M
subcarriers, CP length
L
cp
, and symbol period
T
. The data symbols
s
[
n
]
, modeled as an iid zeromean circularrandom sequence with variance
σ
2
s
, are subject to (s.t.) conventionalOFDM precoding, encompassing inverse discrete Fourier transform(IDFT) and CP insertion, obtaining thus the transmitted OFDM sequence:
u
[
n
] =
M
−
1
m
=0+
∞
i
=
−∞
s
[
iM
+
m
]
b
m
[
n
−
iP
]
where the
precoding
Þ
lters
[8] are given by
b
m
[
n
]
1
√
M
e
j
2
πM mn
,
0
≤
m
≤
M
−
1
,
0
≤
n
≤
P
−
1;0
otherwise.(1)The sequence
u
[
n
]
undergoes digitaltoanalog conversion withsampling period
T
c
T/P
, where
P
M
+
L
cp
, followed by upconversion for transmission over the physical channel, modeled as alinear timeinvariant
Þ
lter.At the receiver, after downconversion, and assuming perfect timing and frequency synchronization, the signal is sampled with period
35769781457705397/11/$26.00 ©2011 IEEE ICASSP 2011
T
c
/N
, where
N >
1
is an oversampling factor
1
and the resulting
N
samples are collected in
r
[
n
]
∈
C
N
, which can be expressed as
r
[
n
] =
L
h
ℓ
=0
h
[
ℓ
]
u
[
n
−
ℓ
] +
w
[
n
]
(2)where
h
[
n
]
∈
C
N
contains samples of the baseband equivalent CIR,with
L
h
denoting the channel order, and
w
[
n
]
∈
C
N
is the noisevector, modeled as a zeromean circular complex Gaussian randomvector, with autocorrelation matrix
R
ww
E
[
w
[
n
]
w
H
[
n
]] =
σ
2
w
I
N
, independent from the data symbols.In the sequel, we assume that
L
cp
< L
h
(insuf
Þ
cient CP length);in this case, the effect of channel dispersion cannot be completelyeliminated by CP removal. To overcome this drawback, a TEQ canbe employed to shorten the CIR up to a certain length
L
eff
≤
L
cp
,thus allowing the use of CP removal to eliminate IBI. The inputoutput relationship of a FIR TEQ can be expressed as
y
[
n
] =
f
H
[
n
]
z
[
n
]
(3)where
f
[
n
]
∈
C
N
(
L
e
+1)
represents the (possibly timevarying)weight vector, with
L
e
denoting the equalizer order, and
z
[
n
]
[
r
T
[
n
]
,
r
T
[
n
−
1]
,...,
r
T
[
n
−
L
e
]]
T
∈
C
N
(
L
e
+1)
collects the dataat the TEQ input. It results that
z
[
n
] =
H
u
[
n
] +
v
[
n
]
(4)where
H
h
[0]
h
[1]
...
h
[
L
h
]
0
N
...
0
N
0
N
h
[0]
h
(1)
...
h
[
L
h
]
...
0
N
...... ... ... ... ...
0
N
0
N
...
...
h
[0]
h
[1]
...
h
[
L
h
]
(5)is the
N
(
L
e
+ 1)
×
(
L
g
+ 1)
block Toeplitz channel matrix, with
L
g
L
e
+
L
h
,
u
[
n
]
[
u
[
n
]
,u
[
n
−
1]
,...,u
[
n
−
L
g
]]
T
∈
C
L
g
+1
,and
v
[
n
]
[
w
T
[
n
]
,
w
T
[
n
−
1]
,...,
w
T
[
n
−
L
e
]]
T
∈
C
N
(
L
e
+1)
.In the following section, we will choose
f
[
n
]
according to theMMOE criterion [6], which involves only SOS of the received data.To this respect, observe that, as widely recognized in the literature (see, e.g., [9, 10]), the multirate
Þ
lterbank structure (1) of theOFDM transmitter induces
cyclostationarity
[7] at the transmittedsequence
u
[
n
]
, and possibly at the TEQ input
z
[
n
]
. In order tounderstand if
z
[
n
]
is actually
widesense cyclostationary
(WSCS),we calculate the autocorrelation matrix
2
R
zz
[
n
]
E
[
z
[
n
]
z
H
[
n
]]
∈
C
N
(
L
e
+1)
×
N
(
L
e
+1)
. Relying on (4), one has:
R
zz
[
n
] =
HR
u
u
[
n
]
H
H
+
σ
2
w
I
N
(
L
e
+1)
(6)with
R
u
u
[
n
]
E
[
u
[
n
]
u
H
[
n
]]
∈
C
(
L
g
+1)
×
(
L
g
+1)
. Observe that
R
zz
[
n
]
is periodic if and only if
R
u
u
[
n
]
is periodic, with the sameperiod. Hence, the problem boils down to studying the periodicityof
R
u
u
[
n
]
, whose entries are given by
{
R
u
u
[
n
]
}
i
1
i
2
=
R
u
[
n
−
i
1
,i
2
−
i
1
]
,
0
≤
i
1
,i
2
≤
L
g
(7)
1
Oversampling [6] is necessary for performing blind channel shortening,and can be substituted by the use of multiple antennas at the receiver.
2
The conjugate correlation matrix
R
zz
∗
E
[
z
[
n
]
z
T
[
n
]]
is zero due tothe assumption of circularity of the data symbols.
where
R
u
[
n,m
]
E
[
u
[
n
]
u
∗
[
n
−
m
]]
denotes the autocorrelationof
u
[
n
]
. By accounting for (1), it can be shown that
R
u
[
n,m
] =
σ
2
s
rep
P
M
−
1
k
=0
b
k
[
n
]
b
∗
k
[
n
−
m
]
=
σ
2
s
, m
= 0;
σ
2
s
rep
P
{
R
P
[
n
]R
P
[
n
−
m
]
}
, m
=
±
M
;0
,
otherwise
.
(8)It turns out that
R
u
[
n,m
]
, as a function of
n
, is periodic (of period
P
) only for
m
=
±
M
(provided that
P > M
, that is,
L
cp
>
0
).Hence, taking into account that

m

=

i
2
−
i
1
≤
L
g
in (7), it resultsthat, when
L
g
≥
M
,
R
u
u
[
n
]
is periodic with period
P
, hence
u
[
n
]
is WSCS; on the contrary, when
L
g
< M
,
R
u
u
[
n
] =
σ
2
s
I
L
g
+1
does not depend on
n
, hence
u
[
n
]
is widesense stationary. The hypothesis
L
g
< M
was considered in [6] and is commonly adopted inthe channel shortening literature(see, for example, [4]): the resultingTEQs turn out to be timeinvariant. However, in highly dispersivechannels (characterized by large values of
L
h
) and/or for OFDMsystems with smalltomoderate number
M
of subcarriers, this inequality is likely to be violated; in this case, the resulting TEQs turnout to be periodically timevarying (PTV).
3. THEPTVMMOETEQ
To shorten the channel up to a certain length
L
eff
≤
L
cp
, theMMOE TEQ minimizes, with respect to
f
[
n
]
, the meanoutputenergy (MOE) at its output:MOE
{
f
[
n
]
}
E
[

y
[
n
]

2
] =
f
H
[
n
]
R
zz
[
n
]
f
[
n
]
.
(9)In order to avoid the trivial solution
f
[
n
] =
0
N
(
L
e
+1)
, after expressing the TEQ input (4) as
z
[
n
] =
h
δ
u
[
n
−
δ
] +
L
g
d
=0
,d
=
δ
h
d
u
[
n
−
d
] +
v
[
n
]
(10)where
h
d
∈
C
N
(
L
e
+1)
is the
(
d
+ 1)
th column of
H
, and
0
≤
δ
≤
L
g
is a design delay, one can impose the constraint
f
H
[
n
]
h
δ
=
γ
δ
[
n
]
∈
C
, which, however, would require knowledgeof
h
δ
and, hence, of the CIR to be shortened. In order to imposea fully
blind
constraint, instead, we rely on parameterization
P
1
of
H
(see [6]), which limits
0
≤
δ
≤
min[
L
e
,L
h
]
, and set theconstraint
Θ
T δ
f
[
n
] =
γ
δ
[
n
]
, where
Θ
δ
∈
R
N
(
L
e
+1)
×
N
(
δ
+1)
is a
known
fullcolumn rank matrix [6] satisfying
Θ
T δ
Θ
δ
=
I
N
(
δ
+1)
,and
γ
δ
[
n
]
∈
C
N
(
δ
+1)
is a constraint vector, whose choice willbe discussed later. Therefore, the blind PTVMMOE optimizationproblem is given by
min
f
[
n
]
∈
C
N
(
Le
+1)
f
H
[
n
]
R
zz
[
n
]
f
[
n
]
s.t.
Θ
T δ
f
[
n
] =
γ
δ
[
n
]
.
(11)By resorting to the method of Lagrange multipliers, optimization(11) reduces to solving the following couple of equations:
R
zz
[
n
]
f
[
n
] =
Θ
δ
λ
[
n
]
,
Θ
T δ
f
[
n
] =
γ
δ
[
n
]
(12)where
λ
[
n
]
∈
C
N
(
δ
+1)
collects the Lagrange multipliers. Since, asdiscussed in the previous section, for
L
g
≥
M
,
R
zz
[
n
]
is a periodicfunction of period
P
, it can be inferred that also
f
[
n
]
,
λ
[
n
]
, and
γ
δ
[
n
]
must all be periodic functions in
n
of the same period
P
. By
3577
expanding all the functions in terms of their discrete Fourier series(DFS), and substituting in (12), by straightforward calculations andexploiting the linear independence of complex exponentials, system(12) can be restated as
1
P
P
−
1
ℓ
=0
R
zz
[
m
−
ℓ
]
P
f
[
ℓ
] =
Θ
δ
λ
[
m
]
(13)
Θ
T δ
f
[
m
] =
γ
δ
[
m
]
(14)where
R
zz
[
m
]
,
f
[
m
]
,
λ
[
m
]
, and
γ
δ
[
m
]
, for
0
≤
m
≤
P
−
1
,represent the
m
th DFS coef
Þ
cient of
R
zz
[
n
]
,
f
[
n
]
,
λ
[
n
]
, and
γ
δ
[
n
]
,respectively. By vertically stacking all the DFS coef
Þ
cients
f
[
m
]
,
λ
[
m
]
, and
γ
δ
[
m
]
, for
0
≤
m
≤
P
−
1
, in
f
∈
C
NP
(
L
e
+1)
,
λ
∈
C
NP
(
δ
+1)
, and
γ
δ
∈
C
NP
(
δ
+1)
, respectively, (13) and (14) can becompactly expressed as
Φ
zz
f
= (
I
P
⊗
Θ
δ
)
λ
(15)
(
I
P
⊗
Θ
T δ
)
f
=
γ
δ
(16)where
Φ
zz
[
Ψ
T
zz
[0]
,
Ψ
T
zz
[1]
,...,
Ψ
T
zz
[
P
−
1]]
T
∈
C
NP
(
L
e
+1)
×
NP
(
L
e
+1)
(17)with
Ψ
zz
[
m
]
P
−
1
[
R
zz
[
m
]
P
,
R
zz
[
m
−
1]
P
,...,
R
zz
[
m
−
P
+1]
P
]
∈
C
N
(
L
e
+1)
×
NP
(
L
e
+1)
. Solving (15) with respect to
f
andsubstituting the solution back into (16), one obtains:
f
=
Φ
−
1
zz
(
I
P
⊗
Θ
δ
)[(
I
P
⊗
Θ
T δ
)
Φ
−
1
zz
(
I
P
⊗
Θ
δ
)]
−
1
F
δ
∈
C
NP
(
Le
+1)
×
NP
(
δ
+1)
γ
δ
.
(18)The analysis of the channel shortening capabilities of the PTVMMOE TEQ is omitted for brevity, since it closely follows thatreported in [6, Theorem 1]. It can be shown that a necessary condition to asymptotically (as
σ
2
w
→
0
) achieve
ideal
channel shorteningwith
L
eff
≤
L
cp
is that
L
h
≤
(
N
−
1)(
L
e
−
δ
)
, from which it isapparent that increasing values of
N >
1
allows one to shortenlonger channels.The PTVMMOE TEQ admits an alternative form, known asFourier series representation [7], which can be obtained by substituting into (3) the expression of
f
[
n
]
in terms of its DFS. One has:
y
[
n
] =
f
H
ζ
[
n
]
(19)where
ζ
[
n
]
e
[
n
]
⊗
z
[
n
]
∈
C
NP
(
L
e
+1)
collects frequencyshifted versions of
z
[
n
]
, with
e
[
n
]
P
−
1
[1
,e
−
j
2
π
(
n/P
)
,...,e
−
j
2
πn
(
P
−
1)
/P
]
T
∈
C
P
. By substituting (4) and (18) into (19), andassuming ideal channel shortening, one obtains
y
[
n
] =
y
u
[
n
] +
y
v
[
n
]
(20)where
y
u
[
n
]
γ
H δ
F
H δ
(
I
P
⊗
H
win
)(
e
[
n
]
⊗
u
win
[
n
])
and
y
v
[
n
]
γ
H δ
F
H δ
(
e
[
n
]
⊗
v
[
n
])
represent the useful and noise components,
H
win
[
h
0
,
h
1
,...,
h
L
eff
]
∈
C
N
(
L
e
+1)
×
(
L
eff
+1)
, and
u
win
[
n
]
[
u
[
n
]
,u
[
n
−
1]
,...,u
[
n
−
L
eff
]]
T
∈
C
L
eff
+1
. This expression allows one to optimize the constraint vector
γ
δ
containing the DFScoef
Þ
cients of
γ
δ
[
n
]
, by maximizing the timeaveraged shorteningsignaltonoise ratio (TASSNR):TASSNR
<
E
[

y
u
[
n
]

2
><
E
[

y
v
[
n
]

2
>
=
σ
2
s
γ
H δ
F
H δ
(
I
P
⊗
H
win
H
H
win
)
F
δ
γ
δ
σ
2
w
γ
H δ
F
H δ
F
δ
γ
δ
=
γ
H δ
F
H δ
Φ
zz
F
δ
γ
δ
σ
2
w
γ
H δ
F
H δ
F
δ
γ
δ
−
1
(21)whose solution is the eigenvector corresponding to the maximumeigenvalue of the matrix pencil
(
F
H δ
Φ
zz
F
δ
,σ
2
w
F
H δ
F
δ
)
.As regards OFDM symbol recovery, after channel shortening
y
[
n
]
undergoes a polyphase decomposition of order
P
, i.e.,
y
p
[
n
]
y
[
nP
+
p
]
, for
0
≤
p
≤
P
−
1
. Since
L
eff
≤
L
cp
, after removingthe CP from the vector
y
[
n
]
[
y
0
[
n
]
,y
1
[
n
]
,...,y
P
−
1
[
n
]]
T
, oneobtains the IBIfree model
y
[
n
]
R
cp
y
[
n
] =
R
cp
GT
cp
W
IDFT
s
[
n
] +
d
[
n
]
,
(22)where
R
cp
∈
C
M
×
P
and
T
cp
∈
C
P
×
M
are the CP removal andinsertion matrices [6], respectively,
W
IDFT
∈
C
M
×
M
represents theunitary symmetric IDFT matrix,
s
[
n
]
∈
C
M
and
d
[
n
]
∈
C
M
denotethe vertical stacking of the data symbols
s
m
[
n
]
s
[
nM
+
m
]
andof the disturbance
d
m
[
n
]
f
H
[
nM
+
m
]
v
[
nM
+
m
]
, for
0
≤
m
≤
M
−
1
, respectively, and,
Þ
nally, the channel matrix
G
∈
C
P
×
P
is
G
=
g
∗
0
[0] 0 0
...
0
...
g
∗
0
[1] 0
...
0
g
∗
L
eff
[
L
eff
]
...
...
...
.........
...
...
00
... g
∗
L
eff
[
P
−
1]
... g
∗
0
[
P
−
1]
,
(23)with
g
ℓ
[
n
]
being the
(
ℓ
+ 1)
th entry of the
effective
channel vector
g
[
n
]
H
H
f
[
n
]
. Observe that, unlike conventional OFDM equalization, as a consequence of the timevarying TEQ
Þ
ltering,
G
losesits Toeplitz structure and, thus, matrix
R
cp
GT
cp
is no more diagonalized by the FFT. Nevertheless, provided that
R
cp
GT
cp
isnonsingular, the data symbol vector
s
[
n
]
can still be recovered byquantizing to the nearest information symbol block (in the Euclideanmetric) the vector
W
DFT
(
R
cp
GT
cp
)
−
1
y
[
n
]
. It should be noted,however, that this operation entails a higher complexity comparedto conventional OFDM decoding. Such a complexity, which is typical of OFDM systems operating in timevarying channels, can bereduced by resorting to lowcomplexity equalization schemes [11].
4. NUMERICALRESULTS
In this section, numerical results, obtained via Monte Carlo computer simulations, are presented. The considered OFDM system employs
M
= 16
subcarriers with QPSK signaling, and a CP length
L
cp
= 4
. The order of the FIR channel is
L
h
= 7
, and the CIRsamples are modeled as iid zeromean complex circular Gaussianrandom variables. As a performance measure, we adopt the averagebiterrorrate (ABER), which is calculated as the arithmetic averageof the
M
subcarrier BERs.We considered two different versions of the proposed PTVMMOE TEQ implemented with
N
= 2
and
L
e
= 11
: the
Þ
rstone is based on (18) and employs all the
P
= 20
DFS coef
Þ
cients,which are needed to exactly represent
f
[
n
]
; the second one (labeled
3578
Fig.1
. Average BER versus SNR.with the suf
Þ
x “low” in the
Þ
gures) is a lowcomplexity version thatconsiders only the
5
dominant DFS coef
Þ
cients of
R
zz
[
n
]
for thesynthesis of
f
[
n
]
. We compared the performances of the proposedreceiver with those of its timeinvariant counterpart (TIMMOE).Allreceivers (PTVMMOEand TIMMOE)are implemented both intheir exact version (labeled with “exact”), i.e., by assuming perfectknowledge of the weight vectors, as well as in their datadependentversion (labeled with “dataestimated”), i.e., by estimating the required SOS on the basis of a samplesize of
K
OFDM symbols. Asa reference, we reported also the performance of the exact OFDMreceiver without channel shortening (labeled with “w/o TEQ”).In Fig. 1, the ABER is reported as a function of the signaltonoise ratio (SNR), de
Þ
ned as SNR
σ
2
s
/σ
2
w
, with a samplesize
K
= 500
for the dataestimated versions of the receivers. Theresults show that, when implemented exactly, the proposed PTVMMOE TEQ outperforms the TIMMOE by about 45 dB for almost all values of SNR, whereas its low complexity version performs signi
Þ
cantly worse. Turning to the dataestimated versions,however, the low complexity PTVMMOE TEQ performs comparably to its complete version, and both receivers outperform the dataestimated version of the TIMMOE. Note, however, that due to
Þ
nitesamplesize effects, all the performance curves of the dataestimatedreceivers, unlike theirexact counterparts, exhibit adistinct BER
ß
oorwhen the SNR increases. Finally, observe that the performances of the receiver without TEQ are completely unsatisfactory in this scenario.In Fig. 2, the ABER for the datadependent receivers is reportedas a function of the sample size
K
ranging from
500
to
3000
andfor SNR = 30 dB. The results con
Þ
rm the advantage of the datadependent PTVMMOE (both versions) over the TIMMOE one,whose performances are quite insensitive to the increase of the sample size. On the contrary, the performances of the complete PTVMMOE TEQ exhibit a marked improvement with the samplesize,while those of its lowcomplexity version exhibit an intermediatebehavior. In particular, the lowcomplexity receiver performs comparably or better than its complete counterpart for small values of
K
, due to the reduced number of parameters to be estimated in itssynthesis, whereas, for larger values of
K
, it shows a marked performance degradation compared to its complete version.
Fig.2
. Average BER versus samplesize
K
.
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Þ
, “Optimum
Þ
nitelength equalization for multicarrier transceivers,”
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Þ
lterbank precoders and equalizers. Part I: uni
Þ
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Þ
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Þ
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