IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 9, SEPTEMBER 2002
1525
Blind OFDM Symbol Synchronization
in ISI Channels
Rohit Negi, Member, IEEE, and John M. Cioffi, Fellow, IEEE
Abstract—We present a new algorithm for blind symbol
synchronization in orthogonal frequency division multiplexing
(OFDM) systems. The new algorithm declares symbol synchronization when a certain autocorrelation matrix, constructed from
the received signal, achieves minimum rank. Unlike previously
proposed blind algorithms, the new rank method guarantees
correct symbol synchronization, even in the presence of intersymbol interference. Also, it does not assume that the OFDM
time samples are i.i.d. In particular, the rank method works even
with OFDM systems that employ pulse shaping. The increased
complexity of the algorithm would be acceptable for systems,
such as fixed-receiver broadcast systems, that require guaranteed
synchronization under all conditions.
Index Terms—Blind algorithm, intersymbol interference,
orthogonal frequency division multiplexing, symbol synchronization.
I. INTRODUCTION
R
ECENTLY, there has been considerable interest in using
orthogonal frequency division multiplexing (OFDM)
systems for wireless transmission [1], such as in digital audio
broadcasting (DAB) and digital television. The resilience of
OFDM systems to frequency-selective fading can be attributed
to the cyclic prefix inserted between symbols that allows
decomposition of the channel into independent subchannels
by use of the fast Fourier transform (FFT). However, a consequence of this “frame” structure of an OFDM symbol is that
it becomes important for the receiver to identify the beginning
of each new symbol. This is the problem of symbol synchronization. Usually, once the correct symbol synchronization has
been achieved, tracking this position is a simpler problem.
Symbol synchronization can be achieved by transmitting pilot
symbols. However, this is an unnecessary waste of bandwidth,
especially in broadcast systems, where the transmitter would
have to keep transmitting pilot symbols periodically, to allow
new users to synchronize. Therefore, various schemes have been
proposed [3]–[6] that use only the transmitted symbol statistics
for symbol synchronization. These blind methods essentially
exploit the redundancy in the cyclic prefix, and therefore, do
not require additional pilot symbols.
Paper approved by H. Liu, the Editor for Synchronization and Equalization of
the IEEE Communications Society. Manuscript received April 20, 2000; revised
June 1, 2001. This paper was presented in part at IEEE Globecom, November
8–12, 1998, Sydney, Australia.
R. Negi is with the Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213 USA (e-mail:
negi@ece.cmu.edu).
J. M. Cioffi is with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305 USA (e-mail: cioffi@dsl.stanford.edu).
Publisher Item Identifier 10.1109/TCOMM.2002.802568.
However, the blind synchronization methods proposed to date
assume a channel free of intersymbol interference (ISI) in their
analysis. Whereas, they are extremely efficient in their use of
symbol statistics, they do not guarantee correct synchronization
when the channel has ISI. Also, they assume that the OFDM
time samples are independent and identically distributed (i.i.d.).
In particular, if the symbol is shaped by a nonconstant power
profile, special techniques may be required by these algorithms
to handle the pulse shape [8]. See [7] for a comparison of some
of these methods.
This paper presents a new algorithm for blind symbol synchronization that, given sufficient signal statistics, is guaranteed
to achieve correct symbol synchronization even in the presence
of ISI. The new algorithm, which we call the “rank method,”
declares symbol synchronization when a certain autocorrelation
matrix, constructed from the received signal, achieves minimum
rank. The rank method is necessarily more complex than any
of the algorithms proposed earlier, and requires more statistics.
However, the guarantee of correctness is an attractive feature,
especially in fixed-receiver broadcast systems, where a particular user may not have much choice in the (nearly time invariant)
channel it sees. Partial results have appeared in [2].
The organization of the paper is as follows. The problem is
formulated in Section II. In Section III, we present the theoretical basis on which the rank method is based. It will be shown
that the ranks of certain autocorrelation matrices convey information about the correct symbol synchronization position. In
Section IV, we show how these ideas can be used in a realistic
scenario, where we only have a noisy estimate of the autocorrelation matrices. Section V illustrates these ideas with simulations. We finally conclude in Section VI.
II. PROBLEM FORMULATION
A. Notations
Standard notations are used in this paper. Bold letters denote
denotes
vectors and matrices. Other notations are as follows:
denotes matrix Hermitian,
denotes (any)
transpose,
idensquare root of a positive-definite matrix, denotes
denotes estimate of ,
denotes
,
tity matrix,
denotes the rank of matrix .
while
B. OFDM System
The basic baseband-equivalent OFDM system is shown
in Fig. 1. The system equations in the time domain can be
written as
0090-6778/02$17.00 © 2002 IEEE
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 9, SEPTEMBER 2002
Fig. 1. Basic OFDM system.
Fig. 2.
Notation for the transmitted and received time sample sequences.
length
similarly,
vector
are vectors of length
..
.
..
.
(1)
is the number of OFDM tones, is the cyclic prefix
where
is the maximum channel dispersion allowed, is
length,
, and
the dispersion of specific channel
denotes time.
is the channel impulse reare the time domain transmitted
sponse,
is the time domain noise,
samples,
is the time-domain noiseless output (immeasurable),
is the time-domain noisy output (meais the th column of matrix . Channel “dispersured),
sion” is one less than the number of taps in the channel impulse
are the successive
response. Note that
. The cyclic
transmitted symbols, each a vector of length
,
prefix in the symbols implies
. Fig. 2 illustrates the notation defined
above.
C. Assumptions
It is assumed that the transmitted symbol vectors
are identically distributed. Thus, the se-
quences
,
,
are cyclostationary with period
. The channel dispersion is assumed to be at least one
tap shorter than the cyclic prefix (i.e., maximum channel
). This assumption is slightly stronger than
dispersion
. This assumption is
the assumption normally used, i.e.,
satisfied in OFDM systems by appropriate choice of parameters
, .
is explicitly chosen as part of the blind estimator
of the vectors depends on the
design. The length
must be explicitly specified.
choice of , which means that
of a specific channel
On the other hand, the dispersion
could be less than . The channel will be assumed to be time
invariant (or at least slowly time varying), so that second-order
statistics of the output signal can be collected. This assumption
is valid for fixed-receiver broadcast OFDM systems.
The noise will be assumed to be additive, white, and Gaussian
will also be as(AWGN). The transmitted time samples
sumed to be Gaussian. This is a valid assumption for OFDM,
is large. The assumption is required so that the minwhen
imum description length (MDL) criterion can be used to estimate the ranks of certain matrices [10]. It is further assumed
that the transmitted OFDM time samples (except for the cyclic
prefix) is statistically nondegenerate, i.e., if we strip the cyclic
prefixes from the transmitted time sample sequence, then with
probability one, no time sample in the resulting sequence can be
expressed as a linear combination of other time samples in the
sequence. Note that no assumption is made on the resulting sequence being i.i.d., unlike [3] and [4]. In particular, we allow for
the transmitted symbols to bear any pulse shape. The assumption of nondegeneracy will be weakened in Section IV–C, where
a modified algorithm will be described that can handle certain
types of degeneracy (for example, inactive OFDM tones).
D. Problem Formulation
The problem of blind symbol synchronization in OFDM is
to use only the statistics (in our case, second-order statistics)
of the received signal to identify the correct positions where
time samples,
the OFDM symbols “begin.” Thus, the first
beginning at the identified position , can be used by the FFT
demodulator without causing ISI. For a channel of dispersion
, this amounts to identifying any one of the positions
(see Fig. 2) of the vectors . For these positions,
due to the cyclic prefix, an -point FFT can be used to decomparallel independent subchannels [9].
pose the channel into
will give the maximum range of synNote that choosing
NEGI AND CIOFFI: BLIND OFDM SYMBOL SYNCHRONIZATION IN ISI CHANNELS
chronization. However, in this paper, we will choose an
for our method.
Earlier approaches to this problem (see [7] for a survey)
have concentrated on using the autocorrelation function of the
. They use the information in the correlation
sequence
function
Though these algorithms are derived for flat-fading channels,
they work remarkably well in mild-ISI channels. However, they
break down in the presence of stronger ISI.
Our goal is to derive an algorithm that explicitly accounts
for ISI in channels. Specifically, we would like to use the
of the noisy output vectors
autocorrelation matrices
. Note that these matrices are well defined, because the
is cyclostationary with a period
. Also,
sequence
note that only
need be considered, because of the cyclostationarity.
of these
However, in practice, only an estimate
vectors;
matrices is known, by time averaging over
. The problem,
therefore, is given the estimates
(these matrices are known in the correct order, but without
), to identify any one
knowing which one corresponds to
.
of the positions
III. THEORETICAL BASIS FOR BLIND
SYMBOL SYNCHRONIZATION
In this section, we describe the basic theorems that
will be used for blind symbol synchronization. We first
assume knowledge of the exact autocorrelation matrices
, and make some fundamental
observations about the behavior of their ranks. It will be shown
that the ranks can be used to identify the correct synchronization positions. We first begin by bounding the ranks.
Lemma III.1: The rank of the autocorrelation of the noiseless
output vector, for a channel of maximal dispersion , can be
bounded as shown in the equation at the bottom of the page.
Proof: See Appendix I for the proof.
Lemma III.1 indicates that the rank of the
matrices may
be of use in determining the correct symbol synchronization.
The bounds depend on the position selected. They are min. More imimum for the correct synchronization position
where
1527
portantly, the lemma shows that the
matrices have a rank
, a fact that will be used in Section IV to simof at least
plify the computations.
Next, we state the main theorem, that describes the behavior
in detail.
of
,
has
Theorem III.2: For a channel where
the following behavior:
where
are unknown integers.
Proof: See Appendix II for the proof.
Theorem III.2 provides a clear idea of the synchronization
method that can be adopted. It shows that as the chosen position
gets closer to the correct synchronization position
from either side, the rank of
is nonincreasing, and in fact
. This shows that the correct position
decreases near
is the position of minimum rank for the matrices
. It is also clear that the rank cannot decrease
by more than .
Theorem III.2 is restrictive because it assumes a channel of
dispersion exactly . In most realistic scenarios, the channel
will be shorter than the maximum allowed. The next corollary
extends Theorem III.2 to this general case.
,
beCorollary III.3: For a channel where
haves as shown in the following equation
where
are unknown integers.
Proof: The parameter was defined earlier as
. See Appendix III for the proof.
The Corollary shows that the effect of a short channel
is to increase the number of positions of minimum
from
only, to the set
.
Noting that the positions of correct symbol synchronization are
for a channel of dispersion , we conclude
indeed the set
that the minimum-rank criterion can be used for symbol syn-
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 9, SEPTEMBER 2002
fore, the problem reduces to estimating the rank of matrices
separately, and searching for
the minimum.
of
In practice, however, we only have estimates
. Since this is invariably noisy, it becomes important to
use a mathematically plausible method for the rank estimation.
We use a modification of the well-known rank estimation
method proposed in [10], that is based on Rissanen’s MDL
criterion.
A. Estimation With
Known
It is shown in Appendix IV that the MDL criterion to compute
of the rank of
, applied to the case
an estimate
of known , reduces to
Fig. 3. Theoretical behavior of [
32, L = 26.
R (i)] for some channels: N = 128, =
chronization. The behavior of
is illustrated in Fig. 3
,
,
for channels of disperfor the case
24, 21, and 16, respectively. It is clear that the positions
sion
of the minima can be used to identify the correct synchronization position.
,
All the previous observations assumed knowledge of
which can only be obtained in the absence of noise. The next
theorem (trivially) extends these results to the case of AWGN.
Theorem III.4: In the case of a frequency-selective
and the noise variance
AWGN channel for which
is known, the correct symbol synchronization positions
can be identified by seeking those
for which the number of singular values of
that are
is maximum.
equal to
is minimum
Proof: Corollary III.3 shows that
, and only for these positions. These
for all
are the positions of correct symbol synchronization. In the
AWGN case with noise variance , we can write
By the spectral shift theorem [11], we see that
number of singular values of
are the singular values of
, arranged
in decreasing order of magnitude. Note that only the smallest
singular values need to be computed. Note again that is the
.
number of vectors averaged to calculate
Therefore, the algorithm for symbol synchronization, which
we call the rank method, is as shown below.
Algorithm Rank Method
by time
1) Compute
received signal vecaveraging over
.
tors;
to compute
2) Use the
, using the MDL criterion
described above.
3) Estimate the position(s) of correct
synchronization as the set
that equal
Therefore, the positions of correct synchronization can be identified as specified in Theorem III.4.
IV. PRACTICAL CONSIDERATIONS
The ideal method to estimate the correct position(s) of
symbol synchronization would be to jointly estimate the ranks
(from matrices
of the matrices
, making use of the spectral shift theorem), and then
use Corollary III.3 to estimate the position. The optimum
solution to this is nonobvious. But as a first attempt, we could
, and thus the correct
simply find the minimum rank
synchronization position, as specified in Theorem III.4. There-
4) Since the algorithm works with noisy
samples, we check for the validity of
the result using Corollary III.3, i.e.,
(as
we check that the function
a function of ) behaves as specified
in Corollary III.3. If not, we have
the option of applying heuristics (such
as checking for a “reasonable match”).
Alternately, we could declare a misestimation due to lack of statistics,
and repeat from Step 1 again, using a
larger value of . This is not necessarily a lot more computation, because
NEGI AND CIOFFI: BLIND OFDM SYMBOL SYNCHRONIZATION IN ISI CHANNELS
the recomputed singular values will
be close to the previous set, and so
the previous set can be used as a good
initialization point for the recomputation.
B. Estimation With
Unknown
When is unknown, one could think of using the method described in [10] (Wax–Kailath method), which jointly estimates
rank and . The problem, however, in applying the method directly to our case, is that the method essentially estimates rank
based solely on the clustering of the smallest singular values. It
does not take into account their absolute magnitude. Therefore,
even
it is possible that it would declare a low rank for
though it is technically full rank.
Therefore, our approach is to use the Wax–Kailath method
. This then gives
to compute
for each . Since the
us an estimate of the “noise variance”
smallest singular value is, ideally, equal to , we choose the
lowest of the computed noise variances, and declare it as .
Then we can use the algorithm in Section IV-A to estimate the
correct symbol synchronization position.
C. Computational Complexity
The rank method seems to be prohibitively computation inautocorrelatensive, since it requires the computation of
, each of size
tion matrices
(though consecutive
differ from each other in a single row
and column only), by averaging using received signal vectors.
computations. Further, the algorithm reThis requires
singular value decompositions, one on each of the
quires
. However, a closer look at the proof of Theorem III.2 and
Corollary III.3 shows that it is sufficient to consider matrices
in place of the matrices
, respectively, where the
are
vectors
1529
mitted signal (Section II-C). In fact, transmission is often turned
off in some OFDM tones, in which case the time samples (excluding the cyclic prefix) linearly depend on less than OFDM
tones. This would result in a degenerate time signal, since there
will be more time samples than active OFDM tones. However,
depends upon not
in the modified rank method, each vector
unique time samples of an OFDM symbol.
more than
Thus, in the modified rank method, if the number of active tones
, which implies that any
time samis at least
matrices (Apples are statistically independent, then the
pendix II) are always full rank, and thus, the various proofs hold.
Therefore, even in the presence of inactive tones (but at least
active tones), the rank method achieves the correct
synchronization.
Further savings in computation can be achieved by using the
following two observations.
1) The MDL rank estimation procedure described in
Section IV-A shows that only the smallest singular
. This is also
values need to be computed for each
. Various specialized algorithms can do
true for
this quickly [12].
can be obtained from
by re2) The matrix
placing two rows and two columns. To recompute the sinoperations
gular values, therefore, takes only
[12]. Either of these observations can be used to gain significant computational savings.
D. Consistency of Rank Method
It is shown in [10] that the rank estimation method presented
there is consistent, i.e., the estimator yields the correct rank with
increases to infinity. A
probability one, as the sample size
similar argument can be made to show that the rank method proposed in this paper is also consistent. Thus, we will always identify the symbol synchronization positions correctly, provided we
are willing to wait and collect sufficient statistics. This is in contrast to the methods presented in [3]–[5], where no guarantees
can be made for ISI channels.
V. SIMULATION RESULTS
i.e., the vectors
are formed from the top and bottom elements of the vectors , respectively. This can be seen from
is caused by the cyclic
the fact that the change is rank of
prefix, which only affects the top and bottom elements of .
satisfies Theorem III.2 and Corollary III.3, except that
instead of
. Therethe maximum rank is now
fore, the correct symbol synchronization can be identified by
in place of the maapplying Theorem III.4 on matrices
, respectively, where the vectors are defined as
trices
Thus, the modified rank method requires
singular value
matrix only. Since
decompositions (SVDs), each on a
is usually much smaller than , this offers substantial savings
in computation.
Another direct benefit of the modified rank method is that it
is no longer necessary to insist on nondegeneracy of the trans-
To demonstrate the performance of the rank method, we
,
, and
. We comchoose the case
pare the performance to that of the maximum-likelihood-based
synchronization algorithm proposed in [3] (Beek–Sandell),
which is typical of the autocorrelation-function-based algorithms (see also [4] and [5]). The Beek–Sandell algorithm
declares symbol synchronization at the position , for which
the following function is maximized:
In all cases, the signal-to-noise ratio (SNR) is defined as the
, and
is assumed
matched filter bound
known. We first illustrate how the proposed rank method
handles ISI better. We choose a strong ISI case, as in Fig. 4, at
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Fig. 4.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 9, SEPTEMBER 2002
Impulse response of SFN channels.
Fig. 6. Decision function used by the proposed rank method. Synchronization
position estimate is the minimum rank position.
C
Fig. 5. Decision functions used by Beek–Sandell algorithm. Synchronization
position estimate is the maximum of the decision function.
Fig. 7. Impulse response of channels used for simulation. Channel
more ISI than channel C .
an SNR of 30 dB. Such channels occur in single frequency networks (SFN) [6]. Frequency offset is assumed to be negligible,
due to an accurate blind-carrier recovery loop [13].
Both algorithms were run on these channels, and the estimated position of correct synchronization was found by
collecting time samples from 600 symbols. In the case of
the Beek–Sandell algorithm, the estimated position was considered to represent the center of the channel (i.e., pointing
), as suggested in [3]. In the case of
to the position
the rank method (we use the modified method described in
Section IV-C), the estimated position was found using the
algorithm described in Section IV-A (without Step 4, so as to
illustrate some limitations). Fig. 5 shows the decision functions
used by the Beek–Sandell algorithm. For both channels A and
B, the algorithm determines the position of synchronization as
the maximum of the decision function, erroneously shifted from
. In particular, notice that whereas
the correct position
both channels should have resulted in the same synchronization
position (to avoid ISI), the Beek–Sandell algorithm offers posi-
tion estimates that differ in 16 time samples between channel A
time
and B. This is far larger than the ambiguity of
samples that the short channel allows. In contrast, Fig. 6 shows
that the rank method correctly points out the synchronization
.
position (minimum-rank position) as
We next choose three cases that will further illustrate the differences between the two methods. The first two use channels
and
(Fig. 7), respectively, and i.i.d. time samples (resulting from i.i.d. data on the OFDM tones). The third case is
but statistically dependent time samples (i.e.,
that of channel
, due to correlation between tones). Tables I
nondiagonal
and II show the results for these cases. Table I lists the normalized “wall energy” (i.e., the energy of the channel taps that fall
outside the cyclic prefix, once the method has been used to synchronize the symbols) for an SNR of 30 dB, but using different
number of training symbols . Table II lists the normalized wall
symbols, but at different values of SNR.
energy for
200 Monte Carlo runs were used in this simulation. The tables show that whereas the Beek–Sandell algorithm works well
has
NEGI AND CIOFFI: BLIND OFDM SYMBOL SYNCHRONIZATION IN ISI CHANNELS
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TABLE I
NORMALIZED WALL ENERGY AFTER SYNCHRONIZATION,
USING RANK METHOD AND BEEK–SANDELL ALGORITHM,
FOR VARIOUS VALUES OF P , AND SNR = 30 dB
TABLE II
NORMALIZED WALL ENERGY AFTER SYNCHRONIZATION,
USING RANK METHOD AND BEEK–SANDELL ALGORITHM,
FOR VARIOUS VALUES OF SNR, AND P = 2400 SYMBOLS
Fig. 8. Outage probability for various P , and SNR = 20 dB, when using the
rank method or the Beek–Sandell algorithm for OFDM synchronization.
on channels with mild ISI (case 1), it can be misled when the
channel has a reasonable amount of ISI (case 2) or when the
transmitted time samples are not i.i.d. (case 3). Higher values
do not help in alleviating this problem. On the
of SNR or
other hand, the rank method is seen to work reasonably well, and
more importantly, always gives the correct result, provided either SNR or is large enough. This is because the rank method
explicitly allows for ISI and colored transmitted time samples.
The final set of simulation experiments consists of simulating
a multipath channel with four multipaths, each uniformly spaced
samples, and
within the allowed channel dispersion of
with a square-root raised cosine pulse shape. The OFDM pa,
. 200 Monte Carlo runs were
rameters were
were
used to obtain the performance plots. The SNR and
varied, and their effect on the synchronization algorithms were
observed. Figs. 8 and 9 show the probability of the signal-to-distortion-plus-noise ratio (SINR) falling below a threshold value
(i.e., the outage probability), for different , when the
SNR is fixed at 20 and 30 dB, respectively. Note that the distor.
tion is caused by the wall energy, resulting in an
Fig. 8 shows that the rank method performs poorly in relation
to the Beek–Sandell algorithm for small , resulting in a larger
. On the other hand, as
outage probability for a given
increases, the rank method outperforms the Beek–Sandell
values
method, resulting in negligible outage, even at
that are close to SNR. The Beek–Sandell algorithm, on the other
hand, does not improve significantly with increasing . Fig. 9
shows that the performance of the rank method improves with
SNR, requiring a smaller to achieve the desired outage probability. Again, the Beek–Sandell algorithm does not improve significantly with increasing .
Figs. 10 and 11 show the effect of SNR on the performance
. The figures show that at
of the algorithms, when
low SNRs, the rank method performs poorly (though the performance can be improved by increasing ). However, as the
SNR is increased, the performance of the Beek–Sandell algorithm degrades substantially, in relation to the rank method.
Fig. 9. Outage probability for various P , and SNR = 30 dB, when using the
rank method or the Beek–Sandell algorithm for OFDM synchronization.
We also simulated the case where
was used. The
, and are not
results were very close to the results with
presented here.
These experiments show that the rank method can guarantee
correct synchronization (signified by a low outage probability),
is made large enough. A larger
is required at smaller
if
SNRs, to average out the effect of the noise.
VI. CONCLUSION
This paper presented a new algorithm for blind symbol synchronization of OFDM signals, which we call the rank method,
using the rank behavior of certain autocorrelation matrices. It
was shown that the rank method is guaranteed to produce the
correct result, provided sufficient statistics are collected (or the
SNR is large), even when the channel has severe ISI, and even
when the transmitted time samples are colored. In particular,
pulse shaping of the OFDM symbols is handled by the rank
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 9, SEPTEMBER 2002
where
is the number of columns in
we can identify
get
,
. Recognizing that
, and
to
Noticing that each repeated element in
(due to the cyclic
, and also because
prefix) causes loss of rank by one in
the nonprefix time samples are nondegenerate, we obtain the
and
as specified in Lemma III.1.
bounding functions
Fig. 10. Outage probability for low SNR, and P = 500, when using the rank
method or the Beek–Sandell algorithm for OFDM synchronization.
APPENDIX II
PROOF OF THEOREM III.2
A. Proof
. Because
Case 1) Consider
of the cyclic prefix (that repeats elements),
, where
we can write (1) as
.
in terms of its columns as
We can write
. Then, since
is assumed, we can relate the
to
as
Similarly, we can write
the appropriate definitions. Again, writing
, we can relate
Fig. 11. Outage probability for high SNR, and P = 500, when using the rank
method or the Beek–Sandell algorithm for OFDM synchronization.
(2)
, with
(3)
From (2) and (3), it is clear that
method, without any changes. This is in contrast to the autocorrelation-function-based methods, that are derived under the
assumption of an ISI-free channel. The latter methods are remarkably efficient, low complexity, and work well for mild ISI
channels. But for a situation that requires guaranteed performance for a diverse range of scenarios, such as fixed-receiver
broadcast, the rank method is an attractive alternative. Another
possibility is to use the rank method for fine acquisition of synchronization, following a coarse synchronization using an autocorrelation function-based method.
APPENDIX I
PROOF OF LEMMA III.1
Since the rank of a matrix is the maximum number
of linearly independent columns, and we are losing
as compared to
at most one degree of freedom in
, hence
is a constant
i.e., rank decreases by, at most, one in going from
to . By the assumption of nondegeneracy of
and
are full rank, and so
the input,
we can write
A. Proof
We have the inequalities [11]
thus giving the result stated in Theorem III.2 for this
case.
NEGI AND CIOFFI: BLIND OFDM SYMBOL SYNCHRONIZATION IN ISI CHANNELS
Case 2) Consider
. If we flip the
vectors and upside down, then (1) will change
to a new convolution equation, but with the channel
in reverse order. The same analysis
taps
as carried out above will apply, showing that
Case 3) Consider
and
. In this case,
,
is full rank. So,
.
. As in Case 1, we
Case 4) Consider
can write
Call the first row of
as
, so that
. Due to the special structure of , its
first row is identical to its th row. By observing the
structure of , we can write
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APPENDIX IV
RANK ESTIMATION USING MDL
The MDL criterion can be used to estimate the ranks of autocorrelation matrices [10]. Here, we derive the criterion for the
case of known . According to the MDL criterion, the rank escan be obtained as
timate
number of free parameters in
where the matrix
has rank .
is the likelihood
, given underlying autocorfunction for the received signal
relation model . Now the number of free parameters in is
, obtained by noting that the rank
matrix
is , and that it is
of the size
complex orthonormal. Further, due to the Gaussian assumption
,
are also Gaussian. For ease of notation, define
of
. Define the SVD
. Then
Therefore
Now, the two observations made earlier allow us to
infer that
and so,
Case 5) Consider
. This case
can be analyzed by flipping vectors and upside
down and noting that this case reduces to Case 4.
Combining the results for the five cases proves
Theorem III.2.
APPENDIX III
PROOF OF COROLLARY III.3
A. Proof
Since
, (1) can be modified as
..
.
..
.
Notice that if we replace that pair
by the pair
,
can be analyzed in exactly the
then the case
same manner as in Theorem III.2. The result found there for this
case, will apply here, with the above replacements. Note that
prefix
due to the short ( taps) channel here, at most
.
elements can occur in any vector
, we conclude that
For the case,
because each of these
has exactly
prefix
.
elements in the corresponding vector
is similar to the
The analysis of the case
in Theorem III.2. Comanalysis of the case
bining these results, we get Corollary III.3.
number of free parameters in
Using the Hadamard inequality [11], it is seen that the mini.
mization over is partly achieved by choosing
Minimizing further over , enforcing the rank condition of
, and simplifying the expression by removing constant
terms, we obtain the MDL criterion specified in Section IV-A. It
because
is sufficient to check the minimization for
of the lower bound on the ranks, as shown in Lemma III.1.
ACKNOWLEDGMENT
The authors would like to thank the reviewers for their
comments.
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Rohit Negi (S’98–M’00) received the B.Tech.
degree from the Indian Institute of Technology,
Bombay, India, in 1995, and the M.S. and Ph.D.
degrees from Stanford University, Stanford, CA,
in 1996 and 2000, respectively, all in electrical
engineering.
Since 2000, he has been with the Electrical
and Computer Engineering Department, Carnegie
Mellon University, Pittsburgh, PA, where he is an
Assistant Professor. His research interests include
signal processing, coding for communications
systems, information theory, networking, and cross-layer optimization.
Dr. Negi received the President of India Gold Medal in 1995.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 9, SEPTEMBER 2002
John M. Cioffi (S’77–M’78–SM’90–F’96) received
the B.S.E.E. degree in 1978 from the University of
Illinois at Urbana-Champaign and the Ph.D.E.E. degree in 1984 from Stanford University, Stanford, CA.
He was with Bell Laboratories from 1978 to 1984,
and IBM Research from 1984 to 1986. Since 1986, he
has been with the Electrical Engineering Department,
Stanford University, where he is currently a Professor
of Electrical Engineering. He founded Amati Com.
Corporation in 1991 (purchased by TI in 1997) and
was Officer/Director from 1991 to 1997. He currently
is on the boards or advisory boards of BigBand Networks, Coppercom, GoDigital, Ikanos, Ionospan, IteX, Marvell, Kestrel, Teknovus, Charter Ventures, and
Portview Ventures, and is a member of the U.S. National Research Council’s
CSTB. His specific interests are in the area of high-performance digital transmission. He has published over 200 papers and holds over 40 patents, most of
which are widely licensed, including basic patents on DMT, VDSL, and vectored transmission.
Dr. Cioffi has received various awards, including Member, National
Academy of Engineering 2001; IEEE Kobayashi Medal (2001); IEEE Millennium Medal (2000); IEE JJ Tomson Medal (2000); 1999 University of Illinois
Outstanding Alumnus; 1991 IEEE Communications Magazine Best Paper
Award; 1995 ANSI T1 Outstanding Achievement Award; and the National
Science Foundation Presidential Investigator (1987–1992).