# Block Iterative/Adaptive Frequency-Domain  Channel Estimation for Cyclic-Prefixed Single-Carrier  Broadband Wireless Systems

Authors: Jong-Seob Baek; Jong-Soo Seo
Journal: International Journal of Digital Multimedia Broadcasting
 (2008)

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## Abstract

This paper presents a new block iterative/adaptive frequency-domain channel estimation scheme, in which a channel frequency response (CFR) is estimated iteratively by the proposed weighted
element-wise block adaptive frequency-domain channel estimation (WEB-CE) scheme using the
soft information obtained by a soft-input soft-output (SISO) decoder. In the WEB-CE, an equalizer
coefficient is calculated by minimizing a weighted conditional squared-norm of the a posteriori error
vector with respect to its correction term. Simulation results verify the superiority of the WEB-CE
in a time-varying typical urban (TU)  channel.


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## Body

## 1. Introduction

Cyclic-prefixed single-carrier frequency-domain
equalization (SC-FDE) has received enormous attention in recent years because
of its efficient implementation and low peak-to-average power ratio (PAPR)
characteristics over broadband wireless channels. On the other hand, it is
noticed that most works have been performed with the assumption that a channel
frequency response (CFR) is perfectly known to the receiver [1, 2]. In practice, this assumption is not valid since a
real channel is unknown and time-varying. This, it highlights the need for the
precise estimation of CFR. In [3, 4], an adaptive channel estimation scheme using the hard
information from the decision was presented. In [5], an iterative (nonadaptive)
channel estimation scheme using the soft information obtained from a soft-input
soft-output (SISO) decoder was presented. Recently, an iterative/adaptive
scheme which performs a channel estimation iteratively by employing an adaptive
algorithm using the soft information was proposed in [6]. This work concludes that the iterative/adaptive
approach is adequate to support a good channel tracking performance over
time-varying channels. However, the scheme has not been applied to the SC-FDE.
In this correspondence, an iterative/adaptive channel estimation scheme for the
SC-FDE is studied. Moreover, a new block-type channel estimation scheme is
studied in order to provide a better channel tracking performance.In this paper, we propose a block iterative/adaptive
CFR estimation scheme, in which a weighted element-wise block adaptive
frequency-domain channel estimation (WEB-CE) using the soft information
obtained from the SISO decoder is presented. It is found that the WEB-CE has a
flexibility over the element-wise block length as compared to an approximated
recursive least square (RLS)-CE of [6] by applying a weighted conditional least-square (LS)
criterion formulated with the a posteriori error vector [7]. Moreover, mean square error
(MSE) of the WEB-CE with respect to the element-wise block length is analyzed.
Simulation results show that the WEB-CE yields good performance in a typical
urban (TU) channel as the iteration number and element-wise block length
increase.The paper is organized as follows. The next section
describes the system model. In Section3, the derivation procedure and
property of the proposed CFR estimation scheme are discussed. Simulation
results are discussed in Section 4, and conclusions are drawn in Section 5.Notation
E{⋅} and (⋅)T denote an expectation and a transpose
operator, respectively. (⋅)∗ denotes complex conjugates of a symbol,
vector, and matrix. IN, 0N,
and diag(⋅) denote the identity matrix of size N,
zero matrix of size N,
and diagonal matrix, respectively.


## 2. SC-FDE System Model

### 2.1. SC-FDE Transmitter

The transmitter
of SC-FDE is shown in Figure1. The blocks of data bits bm∈{0,1} are encoded by the convolutional encoder. The
code bits dm are interleaved by an interleaver Π(⋅), cm=Π(dm).
QPSK mapper maps a pair of
input bits to a symbol xm from the symbol alphabet 𝒮.
(In this paper, a constant-amplitude PSK
constellation is assumed.) Then, a block of signals xn=[xn,0⋯xn,N−2xn,N−1]T is structured with block length of N and a cyclic-prefix (CP) is inserted between
blocks to prevent an interblock interference (IBI) at block time n.
A training block (TB) that has block length of N is inserted every data frame that consists of
data blocks in order to initialize the channel coefficients and track the
channel variations.Figure 1
Block diagram of SC-FDE transmitter.


### 2.2. SC-FDE Receiver

Suppose that an
overall channel impulse response (CIR) containing the combined effects of
transmit and receive filters, multipath fading and sampling ishn=[hn,0⋯hn,L−2hn,L−1]T,
where L is the total length of the CIR, at a block
time n,
and it is constant within one block. It is also supposed that the CP length is
equal or larger than L−1.
At the receiver, discarding the CP, the time-domain received block yn can be expressed in a matrix form
as(1)yn=Hncxn+wn,where wn=[wn,0⋯wn,N−2wn,N−1]T is an additive white noise vector, and Hnc is a circular matrix. It is assumed that E{xnxn∗}=σx2IN, E{xnwn∗}=0N,
and E{wnwn∗}=σw2IN,
where σx2 and σw2 denote the signal power and noise variance,
respectively. Then, the discrete-time Fourier transform (DFT) matrix Q=(1/N)e−j2πil/N (for 0≤i,l≤N−1) yields the frequency-domain block as
follows:(2)Yn=HnXn+Wn,where Yn=Qyn=[Yn[0]⋯Yn[N−2]Yn[N−1]]T, Xn=Qxn=[Xn[0]⋯Xn[N−2]Xn[N−1]]T, Wn=Qwn=[Wn[0]⋯Wn[N−2]Wn[N−1]]T,
and Hn=QHncQ∗=diag(Hn[0],…,Hn[N−2],Hn[N−1]),
where Hn[i] represents the channel frequency response at
the ith frequency bin.From (2), the frequency-domain estimate of the
transmitted data block, X^n=[X^n[0]⋯X^n[N−2]X^n[N−1]]T,
is performed by employing FDE at a block time n.
In this paper, a zero-forcing FDE (ZF-FDE) is considered.
(Note that the minimum-mean-square error (MMSE)
equalization can be used in place of the ZF equalization.)
The received data at the lth frequency bin is described
as(3)X^n[l]=Yn[l]Hn[l],0≤l≤N−1.After the inverse DFT (IDFT)
operation, an extrinsic log-likelihood ratio (LLR) of a coded bit is computed
[8]. The extrinsic LLR
value is de-interleaved and fed to an SISO decoder. Then, the SISO decoder
outputs the a priori LLR, which is interleaved to compute the soft information,
that is, a priori mean vector E{xn}=[x¯n,0⋯x¯n,N−2x¯n,N−1]T,
in which x¯n,i is defined as [8](4)x¯n,i=∑s∈𝒮s⋅P(xn,i=s),where s denotes a symbol 𝒮.
Correspondingly, the DFT of the soft information such as the a priori mean
vector is given by(5)X¯n=QE{xn}=[X¯n[0]⋯X¯n[N−2]X¯n[N−1]]T.The soft information is used to
estimate Hn[l] in (3).


## 2.1. SC-FDE Transmitter

The transmitter
of SC-FDE is shown in Figure1. The blocks of data bits bm∈{0,1} are encoded by the convolutional encoder. The
code bits dm are interleaved by an interleaver Π(⋅), cm=Π(dm).
QPSK mapper maps a pair of
input bits to a symbol xm from the symbol alphabet 𝒮.
(In this paper, a constant-amplitude PSK
constellation is assumed.) Then, a block of signals xn=[xn,0⋯xn,N−2xn,N−1]T is structured with block length of N and a cyclic-prefix (CP) is inserted between
blocks to prevent an interblock interference (IBI) at block time n.
A training block (TB) that has block length of N is inserted every data frame that consists of
data blocks in order to initialize the channel coefficients and track the
channel variations.Figure 1
Block diagram of SC-FDE transmitter.


## 2.2. SC-FDE Receiver

Suppose that an
overall channel impulse response (CIR) containing the combined effects of
transmit and receive filters, multipath fading and sampling ishn=[hn,0⋯hn,L−2hn,L−1]T,
where L is the total length of the CIR, at a block
time n,
and it is constant within one block. It is also supposed that the CP length is
equal or larger than L−1.
At the receiver, discarding the CP, the time-domain received block yn can be expressed in a matrix form
as(1)yn=Hncxn+wn,where wn=[wn,0⋯wn,N−2wn,N−1]T is an additive white noise vector, and Hnc is a circular matrix. It is assumed that E{xnxn∗}=σx2IN, E{xnwn∗}=0N,
and E{wnwn∗}=σw2IN,
where σx2 and σw2 denote the signal power and noise variance,
respectively. Then, the discrete-time Fourier transform (DFT) matrix Q=(1/N)e−j2πil/N (for 0≤i,l≤N−1) yields the frequency-domain block as
follows:(2)Yn=HnXn+Wn,where Yn=Qyn=[Yn[0]⋯Yn[N−2]Yn[N−1]]T, Xn=Qxn=[Xn[0]⋯Xn[N−2]Xn[N−1]]T, Wn=Qwn=[Wn[0]⋯Wn[N−2]Wn[N−1]]T,
and Hn=QHncQ∗=diag(Hn[0],…,Hn[N−2],Hn[N−1]),
where Hn[i] represents the channel frequency response at
the ith frequency bin.From (2), the frequency-domain estimate of the
transmitted data block, X^n=[X^n[0]⋯X^n[N−2]X^n[N−1]]T,
is performed by employing FDE at a block time n.
In this paper, a zero-forcing FDE (ZF-FDE) is considered.
(Note that the minimum-mean-square error (MMSE)
equalization can be used in place of the ZF equalization.)
The received data at the lth frequency bin is described
as(3)X^n[l]=Yn[l]Hn[l],0≤l≤N−1.After the inverse DFT (IDFT)
operation, an extrinsic log-likelihood ratio (LLR) of a coded bit is computed
[8]. The extrinsic LLR
value is de-interleaved and fed to an SISO decoder. Then, the SISO decoder
outputs the a priori LLR, which is interleaved to compute the soft information,
that is, a priori mean vector E{xn}=[x¯n,0⋯x¯n,N−2x¯n,N−1]T,
in which x¯n,i is defined as [8](4)x¯n,i=∑s∈𝒮s⋅P(xn,i=s),where s denotes a symbol 𝒮.
Correspondingly, the DFT of the soft information such as the a priori mean
vector is given by(5)X¯n=QE{xn}=[X¯n[0]⋯X¯n[N−2]X¯n[N−1]]T.The soft information is used to
estimate Hn[l] in (3).


## 3. Iterative/Adaptive CFR Estimation Scheme

### 3.1. WEB-CE Using the Soft Information

The coefficient update algorithm of the WEB-CE atlth frequency bin
is expressed as [7](6)H^n[l]=H^n−1[l]+ΔH^n[l],where ΔH^n[l] is the correction term of H^n−1[l].
From (2), we define 1×B input vectors H^n[l] at a block time n as follows:(7)Yn[l]=[Yn[l]Yn−1[l]⋯Yn−B+1[l]],Xn[l]=[Xn[l]Xn−1[l]⋯Xn−B+1[l]],Wn[l]=[Wn[l]Wn−1[l]⋯Wn−B+1[l]],where B is the element-wise block length of the
WEB-CE. In (7), it is noticed that Xn[l] may actually not be known to the receiver,
thus Xn[l] is alternatively defined as [6](8)Xn[l]=X¯n[l]+Vn[l],where X¯n[l]=[X¯n[l]X¯n−1[l]⋯X¯n−B+1[l]]X¯n[l]X¯n−1[l]⋯X¯n−B+1[l] and Vn[l]=[Vn[l]Vn−1[l]⋯Vn−B+1[l]]Vn[l]Vn−1[l]⋯Vn−B+1[l],
whose element Vn−k[l] has zero mean and
variance given by(9)E{Vn−k[l]Vn−k∗[l]}=E{Xn−k[l]Xn−k∗[l]}−X¯n−k[l]X¯n−k∗[l].In addition, it is assumed that Vn−k[l] is independent of X¯n−k[l].
Furthermore, we define 1×Ba priori error vector en[l] and a posteriori error vector e¯n[l] as follows [7]:(10)en[l]=Yn[l]−H^n−1[l]Xn[l],e¯n[l]=en[l]−ΔH^n[l]Xn[l].From (7)–(10), a posteriori
error vector-based weighted conditional LS criterion is defined as
[6, 7](11)Jn[l]=E{e¯n[l]Fe¯n∗[l]∣Yn[l],H^n−1[l],Xn[l]},where a squared norm of e¯n[l] is attenuated geometrically by B×B diagonal matrix F=diag(1,…,βB−1) with the forgetting factor β that satisfies 0<β≤1.
Using (8) and (10), (11) can be rewritten as(12)Jn[l]=E{e¯n[l]Fe¯n∗[l]∣en[l],ΔH^n[l],Vn[l],X¯n[l]︸Gn}=E{en[l]Fen∗[l]∣Gn}+E{ΔH^n[l]Xn[l]FXn∗[l]ΔH^n∗[l]∣Gn}−E{ΔH^n[l]Xn[l]Fen∗[l]∣Gn}−E{en[l]FXn∗[l]ΔH^n∗[l]∣Gn}.In (12), using (8), the second
term is computed as(13)E{ΔH^n[l]Xn[l]FXn∗[l]ΔH^n∗[l]∣Gn}=ΔH^n[l]E{(X¯n[l]+Vn[l])F(X¯n∗[l]+Vn∗[l])}︸PnsΔH^n∗[l]=ΔH^n[l]PnsΔH^n∗[l].Using (9), Pns can be expressed as(14)Pns=X¯n[l]FX¯n∗[l]+E{Vn[l]FVn∗[l]}=∑k=0B−1βk(X¯n−k[l]X¯n−k∗[l]+E{Vn−k[l]Vn−k∗[l]})=∑k=0B−1βkqlE{xn−kxn−k∗}ql∗=σx2(1−βB1−β),where ql denotes the lth row vector of Q.
In (12), using (8) and (10), the third term is given by(15)E{ΔH^n[l]Xn[l]Fen∗[l]∣Gn}=ΔH^n[l](X¯n[l]FYn∗[l]+E{Vn[l]FYn∗[l]}−PnsH^n−1∗[l]).Furthermore, assuming Hn≃H^n−1 [6], E{Vn[l]FYn∗[l]} in (15) can be expressed as(16)E{Vn[l]FYn∗[l]}=E{Vn[l]F(Xn∗[l]Hn∗[l]+Wn∗[l])}=E{Vn[l]FVn∗[l]}H^n−1∗[l].Then, (15) can be rewritten
as(17)E{ΔH^n[l]Xn[l]Fen∗[l]∣Gn}=ΔH^n[l](X¯n[l]FYn∗[l]−X¯n[l]FX¯n∗[l]H^n−1∗[l]).The computation of the fourth
term in (12) can be done similarly as that of the third term as
follows:(18)E{en[l]FXn∗[l]ΔH^n∗[l]∣Gn}=(Yn[l]FX¯n∗[l]−H^n−1[l]X¯n[l]FX¯n∗[l])ΔH^n∗[l].Finally, substituting the
results of (13), (14), (17), and (18) into (12), and then minimizing Jn[l] with respect to ΔH^n∗[l], ΔH^n in (6) is computed:(19)∂Jn[l]∂ΔH^n∗[l]=ΔH^n[l]Pns−Yn[l]FX¯n∗[l]−H^n−1[l]X¯n[l]FX¯n∗[l]=ΔH^n[l]Pns−(Yn[l]−H^n−1[l]X¯n[l])FX¯n∗[l]=ΔH^n[l]Pns−ens[l]FX¯n∗[l]=0,where 1×Ba priori soft error vector ens[l] is defined as(20)ens[l]=Yn[l]−H^n−1[l]X¯n[l].From (19), ΔH^n[l] is given by(21)ΔH^n[l]=ens[l]FX¯n∗[l](Pns)−1.Consequently, substituting (21)
into (6), we can formulate WEB-CE algorithm as follows:(22)H^n[l]=H^n−1[l]+ens[l]FX¯n∗[l](Pns)−1.The overall iterative receive
structure employing the WEB-CE is depicted in Figure 2.Figure 2
Iterative receive structure employing the WEB-CE when the iteration number is greater than one. Dotted blocks correspond to registers (or buffers) preserving the past information.


### 3.2. Properties of WEB-CE

(1)Flexibility: In (14), (21),
and (22), assuming B=1,
it is found that the WEB-CE is similar to the approximated RLS-CE (or least
mean square (LMS)-CE) of [6] defined as(23)H^nRLS[l]=H^n−1RLS[l]+enRLS,s[l]X¯n∗[l](PnRLS,s)−1,where (PnRLS,s)−1≈(1−β)/σx2,
and enRLS,s[l]=Yn[l]−H^n−1RLS[l]X¯n[l].
As a result, we can see that the WEB-CE has flexibility over the block length B as compared to the approximated RLS-CE.(2)Computational complexity: The
computational complexity of the WEB-CE and the approximated RLS-CE are compared
with respect to a complex operator. We only focus on the required computation
number of the channel estimators, that is, (22) and (23). It is assumed that
the DFT operator requires (N/2)log2N complex multiplications (Mul.) and Nlog2N complex additions (Add.) per input symbols,
that is, X¯n and Yn.
It is noticed in Figure 2 that the dotted DFT blocks would not be considered in
computing the computational complexity because these correspond to the register
(or buffer). In the WEB-CE, the following steps are necessary for each block
length of B.(i)
ens[l] in (22) requires B complex multiplications and B complex additions, respectively, when Yn[l] and X¯n[l] are given. Considering the block length of N,
this step requires (Nlog2N+BN) complex multiplications and (2Nlog2N+BN) complex additions.(ii)
(22) requires 2B complex multiplications and B complex additions, respectively, where the
term (Pns)−1 is a constant from (14). Therefore, this step
requires 2BN complex multiplications and BN complex additions.
The
computational complexity of the approximated RLS-CE can be computed
identically. As a result, the overall computational complexities of the WEB-CE
and the approximated RLS-CE are shown in Table 1, which indicates that the
computational load involved in the WEB-CE is linearly proportional to B.Table 1
Computational complexity of approximated RLS-CE and WEB-CE algorithms.
AlgorithmOverall number of complex Mul./Add.ApproximatedNlog2N+3N (Mul.)RLS-CE2Nlog2N+2N (Add.)WEB-CENlog2N+3BN (Mul.)2Nlog2N+2BN (Add.)(3)MSE analysis: The MSE εn[l] of the WEB-CE with respect to B is analyzed in the Appendix. From (A.7), the
MSE is given by(24)εn[l]=σw2(1−β)(1+βB)(1+β)(1−βB).  When σw2=0.1,0.0316 and 0.01,
the MSE (dB) curves versus β and B is depicted in Figure 3, where it is found
that the MSE decreases exponentially as B increases for a given β.Figure 3
The MSE (dB) curves
versusβ and B when σw2=0.1,0.0316 and 0.01.


## 3.1. WEB-CE Using the Soft Information

The coefficient update algorithm of the WEB-CE atlth frequency bin
is expressed as [7](6)H^n[l]=H^n−1[l]+ΔH^n[l],where ΔH^n[l] is the correction term of H^n−1[l].
From (2), we define 1×B input vectors H^n[l] at a block time n as follows:(7)Yn[l]=[Yn[l]Yn−1[l]⋯Yn−B+1[l]],Xn[l]=[Xn[l]Xn−1[l]⋯Xn−B+1[l]],Wn[l]=[Wn[l]Wn−1[l]⋯Wn−B+1[l]],where B is the element-wise block length of the
WEB-CE. In (7), it is noticed that Xn[l] may actually not be known to the receiver,
thus Xn[l] is alternatively defined as [6](8)Xn[l]=X¯n[l]+Vn[l],where X¯n[l]=[X¯n[l]X¯n−1[l]⋯X¯n−B+1[l]]X¯n[l]X¯n−1[l]⋯X¯n−B+1[l] and Vn[l]=[Vn[l]Vn−1[l]⋯Vn−B+1[l]]Vn[l]Vn−1[l]⋯Vn−B+1[l],
whose element Vn−k[l] has zero mean and
variance given by(9)E{Vn−k[l]Vn−k∗[l]}=E{Xn−k[l]Xn−k∗[l]}−X¯n−k[l]X¯n−k∗[l].In addition, it is assumed that Vn−k[l] is independent of X¯n−k[l].
Furthermore, we define 1×Ba priori error vector en[l] and a posteriori error vector e¯n[l] as follows [7]:(10)en[l]=Yn[l]−H^n−1[l]Xn[l],e¯n[l]=en[l]−ΔH^n[l]Xn[l].From (7)–(10), a posteriori
error vector-based weighted conditional LS criterion is defined as
[6, 7](11)Jn[l]=E{e¯n[l]Fe¯n∗[l]∣Yn[l],H^n−1[l],Xn[l]},where a squared norm of e¯n[l] is attenuated geometrically by B×B diagonal matrix F=diag(1,…,βB−1) with the forgetting factor β that satisfies 0<β≤1.
Using (8) and (10), (11) can be rewritten as(12)Jn[l]=E{e¯n[l]Fe¯n∗[l]∣en[l],ΔH^n[l],Vn[l],X¯n[l]︸Gn}=E{en[l]Fen∗[l]∣Gn}+E{ΔH^n[l]Xn[l]FXn∗[l]ΔH^n∗[l]∣Gn}−E{ΔH^n[l]Xn[l]Fen∗[l]∣Gn}−E{en[l]FXn∗[l]ΔH^n∗[l]∣Gn}.In (12), using (8), the second
term is computed as(13)E{ΔH^n[l]Xn[l]FXn∗[l]ΔH^n∗[l]∣Gn}=ΔH^n[l]E{(X¯n[l]+Vn[l])F(X¯n∗[l]+Vn∗[l])}︸PnsΔH^n∗[l]=ΔH^n[l]PnsΔH^n∗[l].Using (9), Pns can be expressed as(14)Pns=X¯n[l]FX¯n∗[l]+E{Vn[l]FVn∗[l]}=∑k=0B−1βk(X¯n−k[l]X¯n−k∗[l]+E{Vn−k[l]Vn−k∗[l]})=∑k=0B−1βkqlE{xn−kxn−k∗}ql∗=σx2(1−βB1−β),where ql denotes the lth row vector of Q.
In (12), using (8) and (10), the third term is given by(15)E{ΔH^n[l]Xn[l]Fen∗[l]∣Gn}=ΔH^n[l](X¯n[l]FYn∗[l]+E{Vn[l]FYn∗[l]}−PnsH^n−1∗[l]).Furthermore, assuming Hn≃H^n−1 [6], E{Vn[l]FYn∗[l]} in (15) can be expressed as(16)E{Vn[l]FYn∗[l]}=E{Vn[l]F(Xn∗[l]Hn∗[l]+Wn∗[l])}=E{Vn[l]FVn∗[l]}H^n−1∗[l].Then, (15) can be rewritten
as(17)E{ΔH^n[l]Xn[l]Fen∗[l]∣Gn}=ΔH^n[l](X¯n[l]FYn∗[l]−X¯n[l]FX¯n∗[l]H^n−1∗[l]).The computation of the fourth
term in (12) can be done similarly as that of the third term as
follows:(18)E{en[l]FXn∗[l]ΔH^n∗[l]∣Gn}=(Yn[l]FX¯n∗[l]−H^n−1[l]X¯n[l]FX¯n∗[l])ΔH^n∗[l].Finally, substituting the
results of (13), (14), (17), and (18) into (12), and then minimizing Jn[l] with respect to ΔH^n∗[l], ΔH^n in (6) is computed:(19)∂Jn[l]∂ΔH^n∗[l]=ΔH^n[l]Pns−Yn[l]FX¯n∗[l]−H^n−1[l]X¯n[l]FX¯n∗[l]=ΔH^n[l]Pns−(Yn[l]−H^n−1[l]X¯n[l])FX¯n∗[l]=ΔH^n[l]Pns−ens[l]FX¯n∗[l]=0,where 1×Ba priori soft error vector ens[l] is defined as(20)ens[l]=Yn[l]−H^n−1[l]X¯n[l].From (19), ΔH^n[l] is given by(21)ΔH^n[l]=ens[l]FX¯n∗[l](Pns)−1.Consequently, substituting (21)
into (6), we can formulate WEB-CE algorithm as follows:(22)H^n[l]=H^n−1[l]+ens[l]FX¯n∗[l](Pns)−1.The overall iterative receive
structure employing the WEB-CE is depicted in Figure 2.Figure 2
Iterative receive structure employing the WEB-CE when the iteration number is greater than one. Dotted blocks correspond to registers (or buffers) preserving the past information.


## 3.2. Properties of WEB-CE

(1)Flexibility: In (14), (21),
and (22), assuming B=1,
it is found that the WEB-CE is similar to the approximated RLS-CE (or least
mean square (LMS)-CE) of [6] defined as(23)H^nRLS[l]=H^n−1RLS[l]+enRLS,s[l]X¯n∗[l](PnRLS,s)−1,where (PnRLS,s)−1≈(1−β)/σx2,
and enRLS,s[l]=Yn[l]−H^n−1RLS[l]X¯n[l].
As a result, we can see that the WEB-CE has flexibility over the block length B as compared to the approximated RLS-CE.(2)Computational complexity: The
computational complexity of the WEB-CE and the approximated RLS-CE are compared
with respect to a complex operator. We only focus on the required computation
number of the channel estimators, that is, (22) and (23). It is assumed that
the DFT operator requires (N/2)log2N complex multiplications (Mul.) and Nlog2N complex additions (Add.) per input symbols,
that is, X¯n and Yn.
It is noticed in Figure 2 that the dotted DFT blocks would not be considered in
computing the computational complexity because these correspond to the register
(or buffer). In the WEB-CE, the following steps are necessary for each block
length of B.(i)
ens[l] in (22) requires B complex multiplications and B complex additions, respectively, when Yn[l] and X¯n[l] are given. Considering the block length of N,
this step requires (Nlog2N+BN) complex multiplications and (2Nlog2N+BN) complex additions.(ii)
(22) requires 2B complex multiplications and B complex additions, respectively, where the
term (Pns)−1 is a constant from (14). Therefore, this step
requires 2BN complex multiplications and BN complex additions.
The
computational complexity of the approximated RLS-CE can be computed
identically. As a result, the overall computational complexities of the WEB-CE
and the approximated RLS-CE are shown in Table 1, which indicates that the
computational load involved in the WEB-CE is linearly proportional to B.Table 1
Computational complexity of approximated RLS-CE and WEB-CE algorithms.
AlgorithmOverall number of complex Mul./Add.ApproximatedNlog2N+3N (Mul.)RLS-CE2Nlog2N+2N (Add.)WEB-CENlog2N+3BN (Mul.)2Nlog2N+2BN (Add.)(3)MSE analysis: The MSE εn[l] of the WEB-CE with respect to B is analyzed in the Appendix. From (A.7), the
MSE is given by(24)εn[l]=σw2(1−β)(1+βB)(1+β)(1−βB).  When σw2=0.1,0.0316 and 0.01,
the MSE (dB) curves versus β and B is depicted in Figure 3, where it is found
that the MSE decreases exponentially as B increases for a given β.Figure 3
The MSE (dB) curves
versusβ and B when σw2=0.1,0.0316 and 0.01.


## 4. Simulation Results

We evaluate the
performance of SC-FDE employing WEB-CE in the slow time-varying channels
[3]. In the
simulation, a recursive systematic convolutional encoder with a generator G=[75] and BCJR SISO decoding algorithm were
employed. S-random interleaver with a construction S=0.50.5⋅I was used, where I=1024 is a block size of the interleaver [8]. One block consists of 64
symbols (N=64), where the symbol duration is Ts=3.69μs.
A normalized root mean square (rms) delay spread τ=0.2886 and radio channel memory of 3 (sample delay)
are assumed for the TU channel as in Jake's model [7]. The CP length of 3 was set
to eliminate the IBI. A data frame consists of 16 data blocks and one TS block. In the
simulations, the bit error rate (BER) was measured to evaluate the performance
of the ZF-FDE with respect to B of the WEB-CE. In addition, the MSE of channel
estimate, that is, MSE between the perfect CFR and the estimated CFR by the
WEB-CE, was also measured.Figure4 depicts the BER curve of the ZF-FDE employing
the WEB-CE (B=1,2,4), where the iteration number is 0 and 3. In this figure, the maximum Doppler
shift was set to fd=5 Hz and a parameter β=0.85 was chosen through a simulation in order to
allow a good tracking for the WEB-CE. In order to provide a reference BER, a
perfect channel condition was also considered. Here, the approximated RLS-CE
could be regarded equivalently as the WEB-CE (B=1). It is readily shown that the performance of
the ZF-FDE is improved as the iteration number increases. In particular, it is
noticed that the WEB-CE (B=2,4) yields a significant improvement of
performance as compared to the WEB-CE (B=1), which is caused by the gain obtained from B (see Figure 3). It is also noticed that the
error propagation due to the inaccurate past soft information [7] can be reduced as the
iteration number increases.Figure 4
BER
performance of ZF-FDE with the WEB-CE (B=1)/approx. RLS-CE, and WEB-CE (B=2,4) when fd=5 Hz.Figure5 depicts the corresponding MSE curve of Figure
4. It is shown that the WEB-CE (B=2,4) yields better MSE performance than the WEB-CE (B=1).Figure 5
MSE performance of the WEB-CE
(B=1)/approx. RLS-CE, and WEB-CE (B=2,4) when fd=5 Hz.Figures6 and 7 depict the BER and MSE curves when fd=10 Hz, where the simulation parameters are the
same as in Figures 4 and 5. From these figures, we can also see that the WEB-CE
yields superior performance as the iteration number and B increase in the presence of a relatively fast
fading. Furthermore, it is noticed from [9] that the TS blocks insertion, more often and using
smaller N,
can help improve the BER performance as the Doppler shift increases.Figure 6
BER performance of ZF-FDE with the
WEB-CE (B=1)/approx. RLS-CE, and WEB-CE (B=2,4) when fd=10 Hz.Figure 7
MSE performance of the WEB-CE
(B=1)/approx. RLS-CE, and WEB-CE (B=2,4) when fd=10 Hz.Figure8 depicts the performance comparison of the
ZF-FDE with WEB-CE (B=4) and the MMSE-FDE with WEB-CE (B=4),
where it is assumed that a noise variance is known to the receiver. Figure 8
shows that the MMSE-FDE provides the lower BER performance than the ZF-FDE.Figure 8
BER performance
comparison of ZF-FDE w/WEB-CE (B=4) and MMSE-FDE w/WEB-CE (B=4) when fd=5 and 10 Hz.


## 5. Conclusions

In this paper,
we proposed the novel block iterative/adaptive frequency-domain channel
estimation named as the WEB-CE that utilizes the soft information in the
time-varying channel. In the WEB-CE, the correction term of the coefficient was
calculated by minimizing the weighted conditionala posteriori error vector-based LS criterion at each block iteration. It was found that the MSE of
the WEB-CE decreases as B increases, which gives flexibility as compared
to the approximated RLS-CE. From the simulation results, it was shown that the
WEB-CE (B>1) would be a good choice for the channel
estimation scheme of the SC-FDE receiver in time-varying channels.



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*Source: 509171-2008-10-26.xml*