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Blind Identification with Periodic Modulation
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We propose a method for blind identification of FIR channels with periodic modulation. The time-domain formulation in terms of block signals is simple compared with existing frequency-domain formulations. It is shown that the linear equations relating the products of channel coefficients and the autocorrelation matrix of the received signal can be further arranged into decoupled groups. The arrangement reduces computations and improves accuracy of the solution; it also leads to very sim-ple identifiability conditions and a very natural for-mulation of the optimal modulating sequence se-lection problem. The proposed optimal sese-lection minimizes the effects of channel noise and error in autocorrelation matrix estimation; it results in a consistent channel estimate when the channel noise
is white. Simulation results show the method yields good performance: it compares favorably with ex-isting subspace modulation- induced-cyclostationarity method and it is robust with respect to channel or-der overestimation.
Keywords: wireless communication, blind iden-tification, periodic modulation.
1 Introduction
To achieve high-speed reliable communication, chan-nel identification and equalization is necessary to reduce intersymbol interference (ISI) in many com-munication environments. Channel identification and equalization can be achieved either by send-ing trainsend-ing sequences, or by designsend-ing the equal-izer based on a priori knowledge of the channel. A
priori knowledge is often not available in a radio
(wireless) communication environment and send-ing trainsend-ing sequences reduced data transmission rate. Blind channel identification and equalization, which does not assume a priori channel knowl-edge or send training data, have traditionally re-lied on high-order statistics (HOS) of the station-ary received data, but this usually requires rela-tively long data which is needed to accurately es-timate the HOS [9]. This motivates approaches using only induced second-order cyclostationary statistics.
Blind identification and equalization of finite impulse response channel (FIR) channels which exploits cyclostationarity of second-order statistics of the received data is first proposed in [1]. Vari-ous schemes have since been proposed, e.g., [2, 3]. Cyclostationarity can be induced at the receiver or at the transmitter. While receiver induced cy-clostationarity is always through oversampling or multiple sensing, many different schemes have been proposed to induce cyclostationarity at the trans-mitter. They include periodic modulation [4, 5], repetition coding [6] and a combination of repeti-tion and modularepeti-tion[7] and filter bank precoding [8].
We study the problem of blind channel identi-fication with periodic modulation of source sym-bols. We formulate the problem in time-domain and in terms of block signals. The method exploits the linear relation between the products of chan-nel coefficients and the autocorrelation matrix of the received signal and computes the products first by solving a set of linear equations. The channel coefficients are then obtained (to within a scalar ambiguity) by computing the dominant eigenvec-tor of an associated Hermitian matrix. We show that the set of linear equations relating the products of coefficients and the autocorrelation matrix can be further arranged into decoupled groups. The arrangement reduces computations and improves accuracy of the solution; it also leads to very sim-ple identifiability conditions, which depend on the modulating sequence alone, and a very natural for-mulation of the optimal modulating sequence se-lection problem. The proposed optimal sese-lection minimizes the effects of channel noise and error in autocorrelation matrix estimation.
The report is organized as follows. Section 2 is the problem statement and preliminary. Section 3 establishes the identifiability conditions, proposes an identification algorithm and discusses numer-ical aspects associated with it. In Section 4, the problem of selecting the modulating sequence is formulated and solved. In Section 5, simulation examples are given to illustrate the performance of the proposed method. Section 6 is conclusions.
2 Problem statement and
pre-liminary
2.1 Problem statement
We consider the baseband transmission system. The source symbol sequence s(n) is modulated by a (real) periodic sequence p(n) with period N to ob-tain the modulated sequence w(n) = p(n)s(n) which is sent through the channel. The channel is modelled as an FIR filter, whose input-output relation is z(n) = L X l=0 h(l)w(n − l) (2.1)
where h(n) is the impulse response of the channel and L is the channel order.
The received signal sequence x(n) id the sum of the filtered signal z(n) and an additive noise,
i.e.,
x(n) = z(n) + v(n) (2.2)
We propose a method for identifying h(n) using second order statistics of x(n) and a method for optimal design of the modulation sequence p(n). The following assumptions are made throughout.
(A1) The source s(n) is zero mean, white, and
with unit variance.
(A2) The noise v(n) is stationary with zero mean
and is uncorrelated with s(n).
(A3) An upper bound ˆL on channel order L is
known and the period N > ˆL + 1.
2.2 Preliminary
Define the block received signal ¯
x(n) := h x(nN ) · · · x(nN + N − 1) iT and
let the block signals ¯w, ¯s, ¯z, and ¯v be similarly
defined. In terms of block signals the channel re-lation (2.1) and (2.2) can be written as
¯
x(n) = H0G¯s(n) + H1G¯s(n − 1) + ¯v(n) (2.3)
where H0 ∈ CN ×N is a lower triangular Toeplitz
matrix withh
h(0) · · · h(L) 0 · · · 0 iT as its
Toeplitz matrix with
0 · · · 0 h(L) · · · h(1)
as its first row, and G ∈ RN ×N is an diagonal
ma-trix whose jth diagonal entry is p(j −1). Equation (2.3) is a time-invariant description of the channel in terms of block signals.
3 Channel identification
3.1 Identification equation: noise free
case
We consider first the noise free case, i.e., x(n) = z(n). We assume for the moment that the chan-nel order is known. The autocorrelation matrix of ¯
x(n)can be computed from (2.3) as
Rx¯(0) = E ¯x(n)¯x(n)∗ = H0G 2 H∗ 0 + H1G 2 H∗ 1 (3.1)
The equation is quadratic in the channel coeffi-cients h(0), · · ·, h(L). If we consider the prod-ucts h(k)h(l) as unknowns, then (3.1) is a system of N(N + 1)/2 linear equations (we need only to consider the upper triangular part.) It can be fur-ther divided into L + 1 decoupled groups of equa-tions with smaller dimensions, by exploiting the
Toeplitz structure of H0 and H1. We describe
pre-cisely the equations below.
We define Γj[Q] ∈ CN −j as the vector consists
of the jth upper diagonal entries of the matrix Q ∈
CN ×N, and define the vector fj ∈ CL−j+1as
fj=
h(0)h(j)∗ h(1)h(j + 1) · · · h(L − j)h(L)
The L+1 decoupled groups of equations can then be expressed as
Γj[R¯x(0)] = Mjfj, 0 ≤ j ≤ L (3.2)
where the the klth entry of Mj ∈ R(N −j)×(L−j+1)
is (Mj)kl= p(0)2, if k = l; p(k − l)2, if k > l; p(N − l + k)2, if k < l.
We note that the matrix M0is a complex N ×(L+
1)circulant matrix with
p(0)2
· · · p(N − 1)2 T
as its first column. The matrix Mj can be
ob-tained from M0by deleting its last j rows and last j
columns. Solving these L + 1 sets of equations we
would get the products h(k)h(l)∗
, k = 0, · · · , L, l ≥ k.
3.2 Identifiability condition
Since we consider noise free case and choose N > L + 1, so every set of equations in (3.2) is
overde-termined and consistent. If each matrix Mj is full
column rank then the products of channel coeffi-cients can be solved uniquely as
fj = (MT
j Mj)
−1
MT
j Γj[R¯x(0)] (3.3)
Let Q be the Hermitian matrix whose ijth element
is h(i)h(j)∗
. The channel coefficientsh(0), · · · , h(L)
can be determined to within a scalar ambiguity by computing the eigenvector of Q associated with its largest eigenvalue. We thus have the follow-ing sufficient condition for identifiability.
Identifiability condition: The channel is
identifi-able if each Mj in (3.2) is full column rank.
Since Mj depends only on the modulating
se-quence p(n), by choosing p(n) properly we can
always make Mj full rank. We thus conclude that
every channel is identifiable.
3.3 Identification algorithm
Based on the discussions so far, we propose the following algorithm for the computation of chan-nel coefficients. Assume that the modulating
se-quence has been chosen so that each Mj is full
rank.
Identification algorithm
step1: Compute estimate of correlation matrixRx¯(0)
via time average ˆ R¯x(0) = 1 K K X i=1 ¯ x(i)¯x(i)∗ where K is the number of data block.
step2: Compute the product coefficients by (3.3). step3: Form the matrix Q defined previously and
compute its unit-norm eigenvector associated with the largest eigenvalue.
4 Optimal modulation sequence
We consider the general case, that is, the channel noise is present, and discuss the problem of select-ing the modulatselect-ing sequence p(n) . We first pro-pose an optimality criterion to select p(n) to re-duce the effect of noise. We will find a class of solutions which are optimal for noise attenuation. Among this class of solutions, we then choose the ”best” p(n), with which the channel coefficients can be most reliably computed.
4.1 Optimality criterion
Assume that the additive channel noise is white. Then Rx(0) = H¯ 0G2H ∗ 0 + H1G2H ∗ 1 + σ2IN (4.1)
where σ2 is the noise variance. From (4.1), noise
has contribution to only the diagonal entries of
R¯x(0). Thus the L + 1 groups of equations in (3.2)
remains the unchanged except that the j = 0 group becomes
Γ0[R¯x(0)] = M0f0+ σ2b (4.2)
where b = h
1 · · · 1 iT ∈ RN. Since σ2 is not
known, f0 can not be determined from (4.2).
In-stead the least squares solution ˆf0can be computed
as ˆ f0 = (MT 0 M0) −1 MT 0 Γj[R¯x(0)] = f0+ σ2(MT 0 M0) −1 MT 0 b
To eliminate the effect of noise, we should choose the modulating sequence so that the null space of
MT
0 contains b, or equivalently, we should make
R(M0) orthogonal to b. But this is impossible
since all entries of b and columns of M0 are
posi-tive. However, this suggests that we should choose
p(n)so that the angle between R(M0)and b is as
close to π/2 as possible. We note that each
col-umn of M0 makes the same angle with b since M0
is circulant. Let q be the first column of M0, i.e.,
q =h p(0)2 · · · p(N − 1)2 iT and define
γ = q
Tb
kqk2kbk2.
We formulate the problem of selecting optimal p(n) as the following optimization problem:
Minimize γ by choosing p(0), p(1), · · · , p(N − 1) Subject to 1 N N −1 X n=0
p(n)2 = 1 and p(n)2 ≥ δ > 0 for all n
4.2 Optimal solution
It turns out that for a given positive δ < 1, there are N optimal sequences:
|p(m)| =pN(1 − δ) + δ, |p(n)| =√δ for n 6= m
where m is any integer between 0 and N − 1. The optimal modulating sequences assume only two values with a single peak and the rest assum-ing the lower bound. In theory, where the peak occurs does not matter since each optimal choice gives the same γ and thus the same noise effect. In practice, it does matter since different choices
de-fine different M0 and thus all the submatrices Mj.
In particular, the condition numbers of the
matri-ces MT
j Mjwill be different. Making the condition
number small is crucial in obtaining reliable least squares solutions in (3.3). Thus the proposed op-timal choice of modulating sequence is one of the
N optimal sequences that results in the smallest
condition number of MT
j Mj.
5 Simulation results
To illustrate the performance of the proposed chan-nel identification method, we consider the five-tap channel used in [4]:
h(0) = 0.459 + 0.265j, h(1) = −0.2078 − 0.12j, h(2) =
−0.467 − 0.27j, h(3) = 0.095 + 0.055j, h(4) = −0.031 − 0.018j.
The input source symbols are drawn from an i.i.d QPSK constellation. The additive channel noise is white with normal distribution. The channel identification performance is measured by the nor-malized root-mean-squares error (NRMSE) defined as NRMSE := 1 khk2 v u u t 1 I I X i=1 kˆh(i)− hk2 2 where I is the number of Monte Carlo runs and
vector in the ith trial. For computation purpose, the scalar ambiguity is removed by a least squares fitting. The signal-to-noise ratio (SNR) is defined as SNR := v u u t 1 N PN −1 n=0 E|z(n)|2 E|v(n)|2
For all simulations, I = 100.
Figure 5.1 shows the dependence of NRMSE on the number of samples used in the computa-tion for different choice of peak value index m in the optimal modulating sequences. The choices
m = 0and m = 1 give the best condition
num-ber and thus yield smallest NRMSE. In this simu-lation, N = 6, δ = 0.5878, and SNR = 10 dB.
Figure 5.2 and Figure 5.3 compare the perfor-mance of the proposed method with those of the one cycle subspace method [5] and the structured subspace method [4]. In this simulation, N = 6, δ = 0.5878, and m = 0. In Figure 5.2, the SNR is fixed at 10 dB and in Figure 5.3 the num-ber of samples is fixed at 1000. The results show that the proposed method gives better performance than the two subspace methods do.
Figure 5.4 shows the effect of channel order over-estimation. For each channel order upper bound
ˆ
L, 4 ≤ ˆL ≤ 12, the length of modulation sequence
N = ˆL + 2. The SNR is fixed at 10 dB and the
number of samples is 1000. The result shows that as the channel upper bound increased from 4 to 12, the NRMSE increases only 5 dB.
100 200 300 400 500 600 700 800 900 1000 −25 −20 −15 −10 −5 0 Number of Samples S Channel NRMSE (dB) m=4 m=5 m=0 m=1
Figure 5.1: NRMSE for optimal p(n) with differ-ent peak value index m
100 200 300 400 500 600 700 800 900 1000 −24 −22 −20 −18 −16 −14 −12 −10 −8 Number of Samples S Channel NRMSE (dB)
one cycle subspace method
structured subspace method
proposed method
Figure 5.2: Comparison with subspace methods
−10 0 10 20 30 40 50 −25 −20 −15 −10 −5 0 SNR (dB) Channel NRMSE (dB)
one cycle subspace method
structured subspace method
proposed method
Figure 5.3: Comparison with subspace methods
6 Conclusions
We propose a method for blind identification of FIR channels with periodic modulation of source symbols. The time-domain formulation in terms of block signals is simple compared with exist-ing frequency-domain approaches. The method exploits the linear relation between the products of channel coefficients and the autocorrelation ma-trix of the received signal as well as the decoupled structure of the resulting linear system of equa-tions. The identifiability conditions so derived are particular simple: they depend on the modulat-ing sequence alone. Indeed, with the proposed method, any FIR channel is identifiable with an appropriate choice of the periodic modulating se-quence provided that the modulation period N ≥
4 5 6 7 8 9 10 11 12 −23 −22 −21 −20 −19 −18 −17 −16
Over−estimated Channel Order
Channel NRMSE (dB)
Figure 5.4: Robustness to channel order overesti-mation
L+2, where L is the channel order. In fact, almost
all periodic modulating sequences yield the chan-nel identifiable. The optimal modulating sequence selection problem formulated as one of minimiz-ing the effects of channel noise and error in es-timating the autocorrelation matrix is straightfor-ward and easy to solve. The proposed optimal solution also results in a consistent channel esti-mate when the channel noise is white. Simulation results show that the method yields good perfor-mance: it compares favorably with existing sub-space modulation-induced-cyclostationarity meth-ods and it is robust with respect to channel order overestimation.
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