In this letter, we analyze the performance of a recently reported generalized blind channel estimation algorithm. The algorithm has a parameter called repetition index, and it reduces to two previously reported special cases when the repetition index is chosen as unity and as the size of received blocks, respectively. The theoretical performance of the generalized algorithm is derived in high-SNR region for any given repetition index. A recently derived Cramer–Rao bound (CRB) is reviewed and used as a benchmark for the performance of the generalized algorithm. Both theory and simulation results suggest that the performance of the generalized algorithm is usually closer to the CRB when the repetition index is larger, but the performance does not achieve the CRB for any repetition index

Su, Borching

Vaidyanathan, P. P.

English

Caltech Authors - Main

IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 11, NOVEMBER 2007 789
Performance Analysis of Generalized Zero-Padded
Blind Channel Estimation Algorithms
Borching Su, Student Member, IEEE, and P. P. Vaidyanathan, Fellow, IEEE
Abstract—In this letter, we analyze the performance of a recently
reported generalized blind channel estimation algorithm. The al-
gorithm has a parameter called repetition index, and it reduces to
two previously reported special cases when the repetition index is
chosen as unity and as the size of received blocks, respectively. The
theoretical performance of the generalized algorithm is derived in
high-SNR region for any given repetition index. A recently derived
Cramer–Rao bound (CRB) is reviewed and used as a benchmark
for the performance of the generalized algorithm. Both theory and
simulation results suggest that the performance of the generalized
algorithm is usually closer to the CRB when the repetition index
is larger, but the performance does not achieve the CRB for any
repetition index.
Index Terms—Blind identification, Cramer–Rao bound, repeti-
tion index, zero padding.
I. INTRODUCTION
I N this letter, we analyze the performance of a recently re-ported blind channel estimation algorithm [5], which is a
generalization of the pioneering algorithm advanced in [2] by
Scaglione, Giannakis, and Barbarosa (the SGB algorithm). The
generalized algorithm in [5] contains a parameter called the rep-
etition index ; when , it reduces to the SGB algorithm.
When is equal to the size of a received block, the algorithm
reduces to the interesting method proposed by Manton, Neu-
mann, and Pham (the MNP algorithm [3], [4]), which allows
blind identification with as few as two received blocks. Even
though the performance with two blocks is usually not satis-
factory, with and the number of received blocks adjusted ap-
propriately, the performance of the generalized algorithm in [5]
is superior to those of the SGB and MNP algorithms, as docu-
mented in detail in [5].
The goal here is to quantify this performance improvement
theoretically. We study the channel estimation error (MSE) in
the algorithm of [5] and compare it with the corresponding
Cramer–Rao bound (CRB). This study is facilitated by the
fundamental work reported in [1], which analyzes the SGB
algorithm performance, derives appropriate CRB on the per-
formance, and compares them. The analysis of the generalized
algorithm of [5] is included in Section III and follows readily
from a modification of the analysis in [1]. We use the CRB
Manuscript received March 20, 2007; revised April 9, 2007. This work was
supported in part by the NSF Grant CCF-0428326, in part by ONR Grant
N00014-06-1-0011, and in part by the Moore Fellowship of the California
Institute of Technology. The associate editor coordinating the review of this
manuscript and approving it for publication was Dr. Xiang-Gen Xia.
The authors are with the California Institute of Technology, Pasadena, CA
91125 USA.
Digital Object Identifier 10.1109/LSP.2007.900033
expression given in [6] as a benchmark (this is a corrected
version of [1]).
The rest of this letter is organized as follows. Section II briefly
gives the problem statement and reviews the generalized algo-
rithm proposed in [5]. In Section III, we derive the theoretical
performance of the generalized algorithm and compare it with
the CRB. In Section IV, simulation results are given to compare
the theoretical performance, performance obtained by simula-
tion, and the CRB. Finally, the conclusion is given in Section V.
A. Notation
Boldfaced lowercase letters (e.g., ) represent column vec-
tors. Boldfaced uppercase letters and calligraphic uppercase let-
ters are reserved for matrices. Superscripts , and as in
, and denote the conjugate, transpose, and transpose-
conjugate operations, respectively. # represents the pseudo-
inverse of . denotes the th entry of vector . All the
vectors and matrices in this letter are complex-valued. Column
and row indices of all matrices and vectors begin at one.
is the identity matrix, and is the zero
matrix. is a -column full-banded Toeplitz matrix [7]
whose first column is and whose last column
is . The notation represents the column
vector constructed by concatenating columns of . de-
notes the Kronecker product [7] of the matrices and .
II. REVIEW OF THE GENERALIZED ALGORITHM
A. Problem Formulation
Consider a sequence of discrete-time information symbols
, which is blocked into vectors of size . Let
, where
for . Each block is precoded by a
linear transformation characterized by an nonsingular
matrix so that . Each precoded block is
appended at the end with a block of zeros, forming a vector
of size . The vector
signal is unblocked into scalar form before being
sent over the channel. The channel is characterized as a linear
time-invariant (LTI) finite impulse response (FIR) system whose
order is upper-bounded by as in the following:
Also define as the -vector
containing the channel coefficients. The channel output is cor-
rupted by an additive complex white Gaussian noise with
1070-9908/$25.00 © 2007 IEEE
790 IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 11, NOVEMBER 2007
variance . At the receiver side, the symbol stream is
blocked into vectors of size , which can be written as
, where for
. Assuming the block synchronization be-
tween the transmitter and the receiver is perfect, it can be shown
that [2]
where is a full-banded Toeplitz matrix and
is the blocked version of the additive noise .
Suppose we collect received blocks in a matrix
. Then it is clear that
noise (1)
where contains unknown
transmitted blocks.
The problem of blind channel estimation can be stated as fol-
lows. Given a matrix , how do we estimate the
channel coefficients blindly (i.e., when is unknown)?
This problem was first formulated and solved by Scaglione et
al. [2]. We will study here the performance of a generalization
of the SGB algorithm proposed in [5].
B. Generalized Algorithm
In this subsection, we review the generalized algorithm pro-
posed in [5]. We start with a subroutine that is used by both SGB
algorithm and the generalized algorithm.
Subroutine 1:
Input: Matrix of user-defined size .
Outputs: -vector matrix , and
matrix , where .
1) Take SVD on and denote this as
(2)
where has size and contains the smallest
singular values of . Columns of , denoted as
are the corresponding left singular
vectors. (Remark: In particular, if can be written as
, where the matrix has rank
, then it can be shown that , and
, i.e., columns of are
annihilators of .)
2) Construct the Hankel matrix
.
.
.
.
.
.
.
.
.
.
.
.
(3)
for , where represents the th element
of . Construct matrix . (Remark:
If , it is readily verified that .)
3) Let .
Subroutine 1 produces an output proportional to (i.e.,
for some ) if the input can be written as
, where has rank . When is corrupted with small
additive noise, then columns of are approximately annihila-
tors of , and an estimate of is outputted (with a scalar
ambiguity). These properties were first used by Scaglione et al.
in [2] when developing the SGB algorithm. In fact, the SGB al-
gorithm simply runs (under the
assumption that has full rank ) and takes as the esti-
mated channel coefficients.
Although the SGB algorithm uses Subroutine 1 as its kernel
routine, it does not take advantage of the flexibility on input ma-
trix size of Subroutine 1 (it always uses ). The generalized
algorithm in [5], on the other hand, fully exploits this flexibility
by using an extra parameter, namely, the repetition index , as
described below.
Algorithm 1:
Inputs: matrix and repetition index .
Output: channel estimate as an -vector .
1) Construct the matrix
2) Perform the subroutine
and output .
The generalized algorithm is based on the idea that (1) implies
noise (4)
where
(5)
Note that the noise autocorrelation in (4) is different from that
in (1). When , the generalized algorithm reduces to the
SGB algorithm. Also, when , the generalized algorithm
is equivalent to the MNP algorithm [4]. The matrix must
have full rank so that Algorithm 1 works, which implies
.
III. PERFORMANCE ANALYSIS AND THE CRAMER–RAO BOUND
A. Performance Analysis in Additive Noise
When evaluating the MSE performance of blind estimation
algorithms, it is natural to compare the estimated channel
and the true channel . However, due to an intrinsic scalar
ambiguity presented in all blind channel estimation algorithms,
the comparison should be done after normalizing this unknown
scalar. There are many options for doing this. Here we adopted
an option presented in [1], where the channel coefficient
with the largest magnitude is assumed known. That is, ,
where satisfies , is known.
After normalizing the estimated channel vector by letting
, the channel estimation error can be defined
as an -vector
(6)
where is an matrix obtained by removing the th
row of . We first review a result on estimating presented
SU AND VAIDYANATHAN: PERFORMANCE ANALYSIS OF GENERALIZED ZERO-PADDED BLIND CHANNEL ESTIMATION ALGORITHMS 791
in [1] using small perturbation analysis. In Lemma 1 presented
below, we assume the perturbation is small compared to .
That is, assume the first-order approximation
is accurate. We also assume is known to the receiver and
the output of has applied the scalar ambiguity
normalization based on the knowledge of .
Lemma 1: Let , where matrix has
rank . Let , where is a small perturba-
tion to . Perform subroutine ZPBLIND on and and de-
note them as and
, respectively. Consider error vector as de-
fined in (6). Then the first-order approximation of can be
expressed as
# #
where .
Proof: See [1, Eq. (28)].
Notice that and depend only on and and not on the
noise perturbation . Using Lemma 1, we can derive the MSE
performance of the generalized algorithm in [5] by computing
the autocorrelation matrix of , as described below.
Theorem 1: Consider as defined in (1) and
as the repetition index. Perform Algorithm 1:
. Then the autocorrelation matrix of
the channel estimation error vector [defined in (6)] can be
expressed as
# #
# # (7)
Here is defined as in (5), and is defined as
.
.
.
.
.
.
.
.
.
In (7), and are obtained by performing
and letting .
Proof: Using Lemma 1, the autocorrelation matrix of
can be written as
# # # #
(8)
where . The perturba-
tion matrix can be written as
One can verify that the
matrix can be written as , and the
proof of the theorem is complete.
Corollary 1: When , the channel estimation error au-
tocorrelation matrix can be expressed as
# # # #
which agrees with the analytical performance of the SGB algo-
rithm derived in [1].
Proof: Immediate from Theorem 1 using and
the fact that columns of are orthogonal to each other.
B. Cramer–Rao Bound
In [1], a CRB for the zero-padding blind channel estimation
problem was derived. We use the corrected version CRB pre-
sented in [6] as follows:
(9)
The Cramer–Rao bound presented in (9) is a lower bound for the
performance of all algorithms that attempt to solve the blind es-
timation problem described in Section II–A, including the SGB
algorithm [2], the MNP algorithm [4], and the generalized algo-
rithm [5].
IV. SIMULATIONS
In this section, we perform Monte Carlo simulation for the
generalized blind channel estimation algorithm [5] with dif-
ferent repetition indices and compare the performance obtained
by simulations and theory as well as the Cramer–Rao bound at
different SNR values.
In our simulations, the block size is chosen as and
the channel order is chosen as . The QPSK constellation
is used to generate i.i.d. symbols , and the linear precoder
is chosen as . The channel coefficients (elements of )
are chosen as i.i.d., zero-mean, unit variance complex Gaussian
random variables. The simulation is performed using 100 in-
dependent realizations of channel coefficients and ten indepen-
dent realizations of symbol streams (totally 1000 different
pairs of and ). Theoretical performances in (7) and the
CRB in (9) are computed accordingly and averaged over these
1000 pairs of and . Furthermore, to solve the scalar am-
biguity problem, the channel coefficient with the largest mag-
nitude, , is assumed known to the receiver. Two separate
simulation settings are considered: the first one uses 16 received
blocks and the second one uses .
Fig. 1 depicts the result of the first simulation setting, where
. We compare the MSE performances with
and . Both theoretical and simulation performances are
plotted for each case. The CRB is plotted as the benchmark.
We have the following observations. First of all, in both cases
792 IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 11, NOVEMBER 2007
Fig. 1. Channel estimation MSE versus SNR obtained by simulations, theoret-
ical values in (7), and CRB in (9) with 16 blocks.
Fig. 2. Channel estimation MSE versus SNR obtained by simulations, theoret-
ical values in (7), and CRB in (9) with five blocks.
of and , the simulation results are very close to
theory in the high SNR region. This validates the small pertur-
bation assumption given in Lemma 1. Second, performance of
is better than that of with a considerable margin.
However, the system with does not achieve the CRB yet.
Increasing might further improve the performance toward the
CRB, but we omit these curves here due to space limit.
In Fig. 2, simulation results are shown for the case when
. We choose and in this
simulation. Notice that represents the MNP algorithm
[4]. The simulation results approach the theoretical values when
SNR goes to infinity. In high SNR region, the performance for
is obviously better than that for , but it still
does not achieve the CRB. Notice that in this case, we need
in order to satisfy the full-rank assumption as described
in Section II-B.
V. CONCLUSION
In this letter, we derived the theoretical performance of
the generalized blind channel estimation algorithm [5] in the
high-SNR range. Simulation results and theory both suggest
that when the repetition index is larger, the performance is
usually better when SNR is large. A CRB presented in [1] and
corrected in [6] is used as a benchmark of the algorithm perfor-
mance. When the repetition index is large, the performance
curve tends to approach the CRB but does not appear to achieve
it.
In the future, a formal proof that the generalized algorithm
does not achieve the CRB for any is desirable. It also remains
an open question whether there exists another blind channel es-
timation algorithm that has a performance achieving the CRB.
REFERENCES
[1] S. Barbarossa, A. Scaglione, and G. B. Giannakis, “Performance anal-
ysis of a deterministic channel estimator for block transmission systems
with null guard intervals,” IEEE Trans. Signal Process., vol. 50, no. 3,
pp. 684–695, Mar. 2002.
[2] A. Scaglione, G. B. Giannakis, and S. Barbarossa, “Redundant
filter bank precoders and equalizers Part II: Synchronization and
direct equalization,” IEEE Trans. Signal Process., vol. 47, no. 7, pp.
2007–2002, Jul. 1999.
[3] J. H. Manton and W. D. Neumann, “Totally blind channel identification
by exploiting guard intervals,” Syst. Control Lett., vol. 48, no. 2, pp.
113–119, 2003.
[4] D. H. Pham and J. H. Manton, “A subspace algorithm for guard in-
terval based channel identification and source recovery requiring just
two received blocks,” in Proc. ICASSP, Hong Kong, China, 2003, pp.
317–320.
[5] B. Su and P. P. Vaidyanathan, “A generalized algorithm for blind
channel identification with linear redundant precoders,” EURASIP J.
Appl. Signal Process., vol. 2007, Article ID 25672, 13 pp.
[6] B. Su and P. P. Vaidyanathan, “Comments on ‘performance analysis
of a deterministic channel estimator for block transmission systems
with null guard intervals’,” IEEE Trans. Signal Process., submitted for
publication.
[7] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.:
Cambridge Univ. Press, 1996.


Performance Analysis of Generalized Zero-Padded Blind Channel Estimation Algorithms

IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 11, NOVEMBER 2007 789
Performance Analysis of Generalized Zero-Padded
Blind Channel Estimation Algorithms
Borching Su, Student Member, IEEE, and P. P. Vaidyanathan, Fellow, IEEE
Abstract—In this letter, we analyze the performance of a recently
reported generalized blind channel estimation algorithm. The al-
gorithm has a parameter called repetition index, and it reduces to
two previously reported special cases when the repetition index is
chosen as unity and as the size of received blocks, respectively. The
theoretical performance of the generalized algorithm is derived in
high-SNR region for any given repetition index. A recently derived
Cramer–Rao bound (CRB) is reviewed and used as a benchmark
for the performance of the generalized algorithm. Both theory and
simulation results suggest that the performance of the generalized
algorithm is usually closer to the CRB when the repetition index
is larger, but the performance does not achieve the CRB for any
repetition index.
Index Terms—Blind identification, Cramer–Rao bound, repeti-
tion index, zero padding.
I. INTRODUCTION
I N this letter, we analyze the performance of a recently re-ported blind channel estimation algorithm [5], which is a
generalization of the pioneering algorithm advanced in [2] by
Scaglione, Giannakis, and Barbarosa (the SGB algorithm). The
generalized algorithm in [5] contains a parameter called the rep-
etition index ; when , it reduces to the SGB algorithm.
When is equal to the size of a received block, the algorithm
reduces to the interesting method proposed by Manton, Neu-
mann, and Pham (the MNP algorithm [3], [4]), which allows
blind identification with as few as two received blocks. Even
though the performance with two blocks is usually not satis-
factory, with and the number of received blocks adjusted ap-
propriately, the performance of the generalized algorithm in [5]
is superior to those of the SGB and MNP algorithms, as docu-
mented in detail in [5].
The goal here is to quantify this performance improvement
theoretically. We study the channel estimation error (MSE) in
the algorithm of [5] and compare it with the corresponding
Cramer–Rao bound (CRB). This study is facilitated by the
fundamental work reported in [1], which analyzes the SGB
algorithm performance, derives appropriate CRB on the per-
formance, and compares them. The analysis of the generalized
algorithm of [5] is included in Section III and follows readily
from a modification of the analysis in [1]. We use the CRB
Manuscript received March 20, 2007; revised April 9, 2007. This work was
supported in part by the NSF Grant CCF-0428326, in part by ONR Grant
N00014-06-1-0011, and in part by the Moore Fellowship of the California
Institute of Technology. The associate editor coordinating the review of this
manuscript and approving it for publication was Dr. Xiang-Gen Xia.
The authors are with the California Institute of Technology, Pasadena, CA
91125 USA.
Digital Object Identifier 10.1109/LSP.2007.900033
expression given in [6] as a benchmark (this is a corrected
version of [1]).
The rest of this letter is organized as follows. Section II briefly
gives the problem statement and reviews the generalized algo-
rithm proposed in [5]. In Section III, we derive the theoretical
performance of the generalized algorithm and compare it with
the CRB. In Section IV, simulation results are given to compare
the theoretical performance, performance obtained by simula-
tion, and the CRB. Finally, the conclusion is given in Section V.
A. Notation
Boldfaced lowercase letters (e.g., ) represent column vec-
tors. Boldfaced uppercase letters and calligraphic uppercase let-
ters are reserved for matrices. Superscripts , and as in
, and denote the conjugate, transpose, and transpose-
conjugate operations, respectively. # represents the pseudo-
inverse of . denotes the th entry of vector . All the
vectors and matrices in this letter are complex-valued. Column
and row indices of all matrices and vectors begin at one.
is the identity matrix, and is the zero
matrix. is a -column full-banded Toeplitz matrix [7]
whose first column is and whose last column
is . The notation represents the column
vector constructed by concatenating columns of . de-
notes the Kronecker product [7] of the matrices and .
II. REVIEW OF THE GENERALIZED ALGORITHM
A. Problem Formulation
Consider a sequence of discrete-time information symbols
, which is blocked into vectors of size . Let
, where
for . Each block is precoded by a
linear transformation characterized by an nonsingular
matrix so that . Each precoded block is
appended at the end with a block of zeros, forming a vector
of size . The vector
signal is unblocked into scalar form before being
sent over the channel. The channel is characterized as a linear
time-invariant (LTI) finite impulse response (FIR) system whose
order is upper-bounded by as in the following:
Also define as the -vector
containing the channel coefficients. The channel output is cor-
rupted by an additive complex white Gaussian noise with
1070-9908/$25.00 © 2007 IEEE
790 IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 11, NOVEMBER 2007
variance . At the receiver side, the symbol stream is
blocked into vectors of size , which can be written as
, where for
. Assuming the block synchronization be-
tween the transmitter and the receiver is perfect, it can be shown
that [2]
where is a full-banded Toeplitz matrix and
is the blocked version of the additive noise .
Suppose we collect received blocks in a matrix
. Then it is clear that
noise (1)
where contains unknown
transmitted blocks.
The problem of blind channel estimation can be stated as fol-
lows. Given a matrix , how do we estimate the
channel coefficients blindly (i.e., when is unknown)?
This problem was first formulated and solved by Scaglione et
al. [2]. We will study here the performance of a generalization
of the SGB algorithm proposed in [5].
B. Generalized Algorithm
In this subsection, we review the generalized algorithm pro-
posed in [5]. We start with a subroutine that is used by both SGB
algorithm and the generalized algorithm.
Subroutine 1:
Input: Matrix of user-defined size .
Outputs: -vector matrix , and
matrix , where .
1) Take SVD on and denote this as
(2)
where has size and contains the smallest
singular values of . Columns of , denoted as
are the corresponding left singular
vectors. (Remark: In particular, if can be written as
, where the matrix has rank
, then it can be shown that , and
, i.e., columns of are
annihilators of .)
2) Construct the Hankel matrix
...
...
...
...
(3)
for , where represents the th element
of . Construct matrix . (Remark:
If , it is readily verified that .)
3) Let .
Subroutine 1 produces an output proportional to (i.e.,
for some ) if the input can be written as
, where has rank . When is corrupted with small
additive noise, then columns of are approximately annihila-
tors of , and an estimate of is outputted (with a scalar
ambiguity). These properties were first used by Scaglione et al.
in [2] when developing the SGB algorithm. In fact, the SGB al-
gorithm simply runs (under the
assumption that has full rank ) and takes as the esti-
mated channel coefficients.
Although the SGB algorithm uses Subroutine 1 as its kernel
routine, it does not take advantage of the flexibility on input ma-
trix size of Subroutine 1 (it always uses ). The generalized
algorithm in [5], on the other hand, fully exploits this flexibility
by using an extra parameter, namely, the repetition index , as
described below.
Algorithm 1:
Inputs: matrix and repetition index .
Output: channel estimate as an -vector .
1) Construct the matrix
2) Perform the subroutine
and output .
The generalized algorithm is based on the idea that (1) implies
noise (4)
where
(5)
Note that the noise autocorrelation in (4) is different from that
in (1). When , the generalized algorithm reduces to the
SGB algorithm. Also, when , the generalized algorithm
is equivalent to the MNP algorithm [4]. The matrix must
have full rank so that Algorithm 1 works, which implies
.
III. PERFORMANCE ANALYSIS AND THE CRAMER–RAO BOUND
A. Performance Analysis in Additive Noise
When evaluating the MSE performance of blind estimation
algorithms, it is natural to compare the estimated channel
and the true channel . However, due to an intrinsic scalar
ambiguity presented in all blind channel estimation algorithms,
the comparison should be done after normalizing this unknown
scalar. There are many options for doing this. Here we adopted
an option presented in [1], where the channel coefficient
with the largest magnitude is assumed known. That is, ,
where satisfies , is known.
After normalizing the estimated channel vector by letting
, the channel estimation error can be defined
as an -vector
(6)
where is an matrix obtained by removing the th
row of . We first review a result on estimating presented
SU AND VAIDYANATHAN: PERFORMANCE ANALYSIS OF GENERALIZED ZERO-PADDED BLIND CHANNEL ESTIMATION ALGORITHMS 791
in [1] using small perturbation analysis. In Lemma 1 presented
below, we assume the perturbation is small compared to .
That is, assume the first-order approximation
is accurate. We also assume is known to the receiver and
the output of has applied the scalar ambiguity
normalization based on the knowledge of .
Lemma 1: Let , where matrix has
rank . Let , where is a small perturba-
tion to . Perform subroutine ZPBLIND on and and de-
note them as and
, respectively. Consider error vector as de-
fined in (6). Then the first-order approximation of can be
expressed as
# #
where .
Proof: See [1, Eq. (28)].
Notice that and depend only on and and not on the
noise perturbation . Using Lemma 1, we can derive the MSE
performance of the generalized algorithm in [5] by computing
the autocorrelation matrix of , as described below.
Theorem 1: Consider as defined in (1) and
as the repetition index. Perform Algorithm 1:
. Then the autocorrelation matrix of
the channel estimation error vector [defined in (6)] can be
expressed as
# #
# # (7)
Here is defined as in (5), and is defined as
...
...
...
In (7), and are obtained by performing
and letting .
Proof: Using Lemma 1, the autocorrelation matrix of
can be written as
# # # #
(8)
where . The perturba-
tion matrix can be written as
One can verify that the
matrix can be written as , and the
proof of the theorem is complete.
Corollary 1: When , the channel estimation error au-
tocorrelation matrix can be expressed as
# # # #
which agrees with the analytical performance of the SGB algo-
rithm derived in [1].
Proof: Immediate from Theorem 1 using and
the fact that columns of are orthogonal to each other.
B. Cramer–Rao Bound
In [1], a CRB for the zero-padding blind channel estimation
problem was derived. We use the corrected version CRB pre-
sented in [6] as follows:
(9)
The Cramer–Rao bound presented in (9) is a lower bound for the
performance of all algorithms that attempt to solve the blind es-
timation problem described in Section II–A, including the SGB
algorithm [2], the MNP algorithm [4], and the generalized algo-
rithm [5].
IV. SIMULATIONS
In this section, we perform Monte Carlo simulation for the
generalized blind channel estimation algorithm [5] with dif-
ferent repetition indices and compare the performance obtained
by simulations and theory as well as the Cramer–Rao bound at
different SNR values.
In our simulations, the block size is chosen as and
the channel order is chosen as . The QPSK constellation
is used to generate i.i.d. symbols , and the linear precoder
is chosen as . The channel coefficients (elements of )
are chosen as i.i.d., zero-mean, unit variance complex Gaussian
random variables. The simulation is performed using 100 in-
dependent realizations of channel coefficients and ten indepen-
dent realizations of symbol streams (totally 1000 different
pairs of and ). Theoretical performances in (7) and the
CRB in (9) are computed accordingly and averaged over these
1000 pairs of and . Furthermore, to solve the scalar am-
biguity problem, the channel coefficient with the largest mag-
nitude, , is assumed known to the receiver. Two separate
simulation settings are considered: the first one uses 16 received
blocks and the second one uses .
Fig. 1 depicts the result of the first simulation setting, where
. We compare the MSE performances with
and . Both theoretical and simulation performances are
plotted for each case. The CRB is plotted as the benchmark.
We have the following observations. First of all, in both cases
792 IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 11, NOVEMBER 2007
Fig. 1. Channel estimation MSE versus SNR obtained by simulations, theoret-
ical values in (7), and CRB in (9) with 16 blocks.
Fig. 2. Channel estimation MSE versus SNR obtained by simulations, theoret-
ical values in (7), and CRB in (9) with five blocks.
of and , the simulation results are very close to
theory in the high SNR region. This validates the small pertur-
bation assumption given in Lemma 1. Second, performance of
is better than that of with a considerable margin.
However, the system with does not achieve the CRB yet.
Increasing might further improve the performance toward the
CRB, but we omit these curves here due to space limit.
In Fig. 2, simulation results are shown for the case when
. We choose and in this
simulation. Notice that represents the MNP algorithm
[4]. The simulation results approach the theoretical values when
SNR goes to infinity. In high SNR region, the performance for
is obviously better than that for , but it still
does not achieve the CRB. Notice that in this case, we need
in order to satisfy the full-rank assumption as described
in Section II-B.
V. CONCLUSION
In this letter, we derived the theoretical performance of
the generalized blind channel estimation algorithm [5] in the
high-SNR range. Simulation results and theory both suggest
that when the repetition index is larger, the performance is
usually better when SNR is large. A CRB presented in [1] and
corrected in [6] is used as a benchmark of the algorithm perfor-
mance. When the repetition index is large, the performance
curve tends to approach the CRB but does not appear to achieve
it.
In the future, a formal proof that the generalized algorithm
does not achieve the CRB for any is desirable. It also remains
an open question whether there exists another blind channel es-
timation algorithm that has a performance achieving the CRB.
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[5] B. Su and P. P. Vaidyanathan, “A generalized algorithm for blind
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A generalized algorithm for blind channel identification with linear redundant precoders,”

A subspace algorithm for guard interval based channel identification and source recovery requiring just two received blocks,” in

Comments on ‘performance analysis of a deterministic channel estimator for block transmission systems with null guard intervals’,” IEEE Trans. Signal Process., submitted for publication.

Matrix Analysis.

Performance analysis of a deterministic channel estimator for block transmission systems with null guard intervals,”

Redundant filter bank precoders and equalizers Part II: Synchronization and direct equalization,”

Totally blind channel identification by exploiting guard intervals,” Syst.

http://authors.library.caltech.edu/9223/1/SUBieeespl07.pdf

Performance Analysis of Generalized Zero-Padded Blind Channel Estimation Algorithms

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