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

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.

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 Section 3, 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. 𝐈_𝑁, 𝟎_𝑁, 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 Figure 1. The blocks of data bits 𝑏_𝑚∈{0,1} are encoded by the convolutional encoder. The code bits 𝑑_𝑚 are interleaved by an interleaver Π(⋅), 𝑐_𝑚=Π(𝑑_𝑚). QPSK mapper maps a pair of input bits to a symbol 𝑥_𝑚 from the symbol alphabet 𝒮. (In this paper, a constant-amplitude PSK constellation is assumed.) Then, a block of signals 𝐱_𝑛=[𝑥_𝑛,0⋯𝑥_{𝑛,𝑁−2}𝑥_{𝑛,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 is 𝐡_𝑛=[ℎ_𝑛,0⋯ℎ_{𝑛,𝐿−2}ℎ_{𝑛,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 𝐲_𝑛 can be expressed in a matrix form as𝐲_𝑛=𝐇^𝑐_𝑛𝐱_𝑛+𝐰_𝑛, (1)where 𝐰_𝑛=[𝑤_𝑛,0⋯𝑤_{𝑛,𝑁−2}𝑤_{𝑛,𝑁−1}]^𝑇 is an additive white noise vector, and 𝐇^𝑐_𝑛 is a circular matrix. It is assumed that 𝐸{𝐱_𝑛𝐱^∗_𝑛}=𝜎²_𝑥𝐈_𝑁, 𝐸{𝐱_𝑛𝐰^∗_𝑛}=𝟎_𝑁, and 𝐸{𝐰_𝑛𝐰^∗_𝑛}=𝜎²_𝑤𝐈_𝑁, where 𝜎²_𝑥 and 𝜎²_𝑤 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:𝐘_𝑛=𝐇_𝑛𝐗_𝑛+𝐖_𝑛, (2)where 𝐘_𝑛=𝐐𝐲_𝑛=[𝑌_𝑛[0]⋯𝑌_𝑛[𝑁−2]𝑌_𝑛[𝑁−1]]^𝑇, 𝐗_𝑛=𝐐𝐱_𝑛=[𝑋_𝑛[0]⋯𝑋_𝑛[𝑁−2]𝑋_𝑛[𝑁−1]]^𝑇, 𝐖_𝑛=𝐐𝐰_𝑛=[𝑊_𝑛[0]⋯𝑊_𝑛[𝑁−2]𝑊_𝑛[𝑁−1]]^𝑇, and 𝐇_𝑛=𝐐𝐇^𝑐_𝑛𝐐^∗=diag(𝐻_𝑛[0],…,𝐻_𝑛[𝑁−2],𝐻_𝑛[𝑁−1]), where 𝐻_𝑛[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 as𝑋_𝑛𝑌[𝑙]=^𝑛[𝑙]𝐻_𝑛[𝑙],0≤𝑙≤𝑁−1. (3)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{𝐱_𝑛}=[𝑥_𝑛,0⋯𝑥_{𝑛,𝑁−2}𝑥_{𝑛,𝑁−1}]^𝑇, in which 𝑥_𝑛,𝑖 is defined as [8]𝑥_𝑛,𝑖=_𝑠∈𝒮𝑠⋅𝑃(𝑥_𝑛,𝑖=𝑠), (4)where 𝑠 denotes a symbol 𝒮. Correspondingly, the DFT of the soft information such as the a priori mean vector is given by𝐗_𝑛=𝐐𝐸{𝐱_𝑛}=𝑋_𝑛[0]⋯𝑋_𝑛[𝑁−2]𝑋_𝑛[𝑁−1]^𝑇. (5)The soft information is used to estimate 𝐻_𝑛[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 at 𝑙th frequency bin is expressed as [7]𝐻_𝑛[𝑙]=𝐻_𝑛−1[𝑙]+Δ𝐻_𝑛[𝑙], (6)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:𝐘_𝑛[𝑙]=𝑌_𝑛[𝑙]𝑌_𝑛−1[𝑙]⋯𝑌_{𝑛−𝐵+1},𝐗[𝑙]_𝑛[𝑙]=𝑋_𝑛[𝑙]𝑋_𝑛−1[𝑙]⋯𝑋_{𝑛−𝐵+1},𝐖[𝑙]_𝑛[𝑙]=𝑊_𝑛[𝑙]𝑊_𝑛−1[𝑙]⋯𝑊_{𝑛−𝐵+1}[𝑙], (7)where 𝐵 is the element-wise block length of the WEB-CE. In (7), it is noticed that 𝐗_𝑛[𝑙] may actually not be known to the receiver, thus 𝐗_𝑛[𝑙] is alternatively defined as [6]𝐗_𝑛[𝑙]=𝐗_𝑛[𝑙]+𝐕_𝑛[𝑙], (8)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 𝐕_𝑛[𝑙]=[𝑉_𝑛[𝑙]𝑉_𝑛−1[𝑙]⋯𝑉_{𝑛−𝐵+1}[𝑙]]𝑉_𝑛[𝑙]𝑉_𝑛−1[𝑙]⋯𝑉_{𝑛−𝐵+1}[𝑙], whose element 𝑉_𝑛−𝑘[𝑙] has zero mean and variance given by𝐸{𝑉_𝑛−𝑘[𝑙]𝑉^∗_𝑛−𝑘[𝑙]}=𝐸{𝑋_𝑛−𝑘[𝑙]𝑋^∗_𝑛−𝑘[𝑙]}−𝑋_𝑛−𝑘[𝑙]𝑋^∗_𝑛−𝑘[𝑙]. (9)In addition, it is assumed that 𝑉_𝑛−𝑘[𝑙] is independent of 𝑋_𝑛−𝑘[𝑙]. Furthermore, we define 1×𝐵a priori error vector 𝐞_𝑛[𝑙] and a posteriori error vector 𝐞_𝑛[𝑙] as follows [7]:𝐞_𝑛[𝑙]=𝐘_𝑛[𝑙]−𝐻_𝑛−1[𝑙]𝐗_𝑛[𝑙],𝐞_𝑛[𝑙]=𝐞_𝑛[𝑙]−Δ𝐻_𝑛[𝑙]𝐗_𝑛[𝑙]. (10)From (7)–(10), a posteriori error vector-based weighted conditional LS criterion is defined as [6, 7]𝐽_𝑛[𝑙]=𝐸{𝐞_𝑛[𝑙]𝐅𝐞^∗_𝑛[𝑙]∣𝐘_𝑛[𝑙],𝐻_𝑛−1[𝑙],𝐗_𝑛[𝑙]}, (11)where a squared norm of 𝐞_𝑛[𝑙] is attenuated geometrically by 𝐵×𝐵 diagonal matrix 𝐅=diag(1,…,𝛽^𝐵−1) with the forgetting factor 𝛽 that satisfies 0<𝛽≤1. Using (8) and (10), (11) can be rewritten as𝐽_𝑛[𝑙]=𝐸{𝐞_𝑛[𝑙]𝐅𝐞^∗_𝑛[𝑙]∣𝐞_𝑛[𝑙],Δ𝐻_𝑛[𝑙],𝐕_𝑛[𝑙],𝐗_𝑛[𝑙]_𝐺𝑛}=𝐸{𝐞_𝑛[𝑙]𝐅𝐞^∗_𝑛[𝑙]∣𝐺_𝑛}+𝐸{Δ𝐻_𝑛[𝑙]𝐗_𝑛[𝑙]𝐅𝐗^∗_𝑛[𝑙]Δ𝐻^∗_𝑛[𝑙]∣𝐺_𝑛}−𝐸{Δ𝐻_𝑛[𝑙]𝐗_𝑛[𝑙]𝐅𝐞^∗_𝑛[𝑙]∣𝐺_𝑛}−𝐸{𝐞_𝑛[𝑙]𝐅𝐗^∗_𝑛[𝑙]Δ𝐻^∗_𝑛[𝑙]∣𝐺_𝑛}. (12)In (12), using (8), the second term is computed as𝐸{Δ𝐻_𝑛[𝑙]𝐗_𝑛[𝑙]𝐅𝐗^∗_𝑛[𝑙]Δ𝐻^∗_𝑛[𝑙]∣𝐺_𝑛}=Δ𝐻_𝑛[𝑙]𝐸{(𝐗_𝑛[𝑙]+𝐕_𝑛[𝑙])𝐅(𝐗^∗_𝑛[𝑙]+𝐕^∗_𝑛[𝑙])}_𝑃𝑠𝑛Δ𝐻^∗_𝑛=Δ𝐻[𝑙]_𝑛[𝑙]𝑃^𝑠_𝑛Δ𝐻^∗_𝑛[𝑙]. (13)Using (9), 𝑃^𝑠_𝑛 can be expressed as𝑃^𝑠_𝑛=𝐗_𝑛[𝑙]𝐅𝐗^∗_𝑛[𝑙]+𝐸{𝐕_𝑛[𝑙]𝐅𝐕^∗_𝑛=[𝑙]}_{𝐵−1𝑘=0}𝛽^𝑘(𝑋_𝑛−𝑘[𝑙]𝑋^∗_𝑛−𝑘[𝑙]+𝐸{𝑉_𝑛−𝑘[𝑙]𝑉^∗_𝑛−𝑘=[𝑙]})_{𝐵−1𝑘=0}𝛽^𝑘𝐪_𝑙𝐸{𝐱_𝑛−𝑘𝐱^∗_𝑛−𝑘}𝐪^∗_𝑙=𝜎²_𝑥(1−𝛽^𝐵1−𝛽), (14)where 𝐪_𝑙 denotes the 𝑙th row vector of 𝐐. In (12), using (8) and (10), the third term is given by𝐸{Δ𝐻_𝑛[𝑙]𝐗_𝑛[𝑙]𝐅𝐞^∗_𝑛[𝑙]∣𝐺_𝑛}=Δ𝐻_𝑛[𝑙](𝐗_𝑛[𝑙]𝐅𝐘^∗_𝑛[𝑙]+𝐸{𝐕_𝑛[𝑙]𝐅𝐘^∗_𝑛[𝑙]}−𝑃^𝑠_𝑛𝐻^∗_𝑛−1[𝑙]). (15)Furthermore, assuming 𝐻_𝑛≃𝐻_𝑛−1 [6], 𝐸{𝐕_𝑛[𝑙]𝐅𝐘^∗_𝑛[𝑙]} in (15) can be expressed as𝐸{𝐕_𝑛[𝑙]𝐅𝐘^∗_𝑛[𝑙]}=𝐸{𝐕_𝑛[𝑙]𝐅(𝐗^∗_𝑛[𝑙]𝐻^∗_𝑛[𝑙]+𝐖^∗_𝑛[𝑙])}=𝐸{𝐕_𝑛[𝑙]𝐅𝐕^∗_𝑛[𝑙]}𝐻^∗_𝑛−1[𝑙]. (16)Then, (15) can be rewritten as𝐸{Δ𝐻_𝑛[𝑙]𝐗_𝑛[𝑙]𝐅𝐞^∗_𝑛[𝑙]∣𝐺_𝑛}=Δ𝐻_𝑛[𝑙](𝐗_𝑛[𝑙]𝐅𝐘^∗_𝑛[𝑙]−𝐗_𝑛[𝑙]𝐅𝐗^∗_𝑛[𝑙]𝐻^∗_𝑛−1[𝑙]). (17)The computation of the fourth term in (12) can be done similarly as that of the third term as follows:𝐸{𝐞_𝑛[𝑙]𝐅𝐗^∗_𝑛[𝑙]Δ𝐻^∗_𝑛[𝑙]∣𝐺_𝑛}=(𝐘_𝑛[𝑙]𝐅𝐗^∗_𝑛[𝑙]−𝐻_𝑛−1[𝑙]𝐗_𝑛[𝑙]𝐅𝐗^∗_𝑛[𝑙])Δ𝐻^∗_𝑛[𝑙]. (18)Finally, substituting the results of (13), (14), (17), and (18) into (12), and then minimizing 𝐽_𝑛[𝑙] with respect to Δ𝐻^∗_𝑛[𝑙], Δ𝐻_𝑛 in (6) is computed:𝜕𝐽_𝑛[𝑙]𝜕Δ𝐻^∗_𝑛𝐻[𝑙]=Δ_𝑛[𝑙]𝑃^𝑠_𝑛−𝐘_𝑛[𝑙]𝐅𝐗^∗_𝑛[𝑙]−𝐻_𝑛−1[𝑙]𝐗_𝑛[𝑙]𝐅𝐗^∗_𝑛𝐻[𝑙]=Δ_𝑛[𝑙]𝑃^𝑠_𝑛−(𝐘_𝑛[𝑙]−𝐻_𝑛−1[𝑙]𝐗_𝑛[𝑙])𝐅𝐗^∗_𝑛𝐻[𝑙]=Δ_𝑛[𝑙]𝑃^𝑠_𝑛−𝐞^𝑠_𝑛[𝑙]𝐅𝐗^∗_𝑛[𝑙]=0, (19)where 1×𝐵a priori soft error vector 𝐞^𝑠_𝑛[𝑙] is defined as𝐞^𝑠_𝑛[𝑙]=𝐘_𝑛[𝑙]−𝐻_𝑛−1[𝑙]𝐗_𝑛[𝑙]. (20)From (19), Δ𝐻_𝑛[𝑙] is given byΔ𝐻_𝑛[𝑙]=𝐞^𝑠_𝑛[𝑙]𝐅𝐗^∗_𝑛[𝑙](𝑃^𝑠_𝑛)⁻¹. (21)Consequently, substituting (21) into (6), we can formulate WEB-CE algorithm as follows:𝐻_𝑛[𝑙]=𝐻_𝑛−1[𝑙]+𝐞^𝑠_𝑛[𝑙]𝐅𝐗^∗_𝑛[𝑙](𝑃^𝑠_𝑛)⁻¹. (22)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 as𝐻^RLS_𝑛[𝑙]=𝐻^RLS_𝑛−1[𝑙]+𝑒^RLS_𝑛^,𝑠[𝑙]𝑋^∗_𝑛[𝑙](𝑃^RLS_𝑛^,𝑠)⁻¹, (23)where (𝑃^RLS_𝑛^,𝑠)⁻¹≈(1−𝛽)/𝜎²_𝑥, and 𝑒^RLS_𝑛^,𝑠[𝑙]=𝑌_𝑛[𝑙]−𝐻^RLS_𝑛−1[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 𝐘_𝑛. 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)𝐞^𝑠_𝑛[𝑙] in (22) requires 𝐵 complex multiplications and 𝐵 complex additions, respectively, when 𝐘_𝑛[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 (𝑃^𝑠_𝑛)⁻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.

(3) MSE analysis: The MSE 𝜀_𝑛[𝑙] of the WEB-CE with respect to 𝐵 is analyzed in the Appendix. From (A.7), the MSE is given by𝜀_𝑛[𝑙]=𝜎²_𝑤(1−𝛽)(1+𝛽^𝐵)(1+𝛽)(1−𝛽^𝐵). (24) When 𝜎²_𝑤=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 versus 𝛽 and 𝐵 when 𝜎2𝑤=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=0.5√0.5⋅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 𝑇_𝑠=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.

Figure 4 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 𝑓_𝑑=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.

Figure 5 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.

Figures 6 and 7 depict the BER and MSE curves when 𝑓_𝑑=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.

Figure 8 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 conditional a 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.

Appendix

MSE Analysis of the WEB-CE

In (22), assuming that the channel is stationary, 𝐗_𝑛 is perfectly known to the receiver and the WEB-CE is on the steady state, we analyze the MSE 𝜀_𝑛[𝑙] defined as𝜀_𝑛[𝑙]=𝐸{(𝐻^∗_𝑛[𝑙]−𝐻^∗_𝑛[𝑙])(𝐻_𝑛[𝑙]−𝐻_𝑛[𝑙])_Δ𝐻𝑛}=𝐸{Δ𝐻^∗_𝑛Δ𝐻_𝑛}, (A.1)where 𝐻_𝑛[𝑙] is defined as𝐻_𝑛[𝑙]=𝐻_𝑛−1[𝑙]+𝐞_𝑛[𝑙]𝐅𝐗^∗_𝑛[𝑙]𝑃_𝑛⁻¹ (A.2)with 𝑃_𝑛=𝐗_𝑛[𝑙]𝐅𝐗^∗_𝑛[𝑙]=∑_𝑘𝛽^𝑘𝐪_𝑙𝐱_𝑛−𝑘𝐱^∗_𝑛−𝑘𝐪^∗_𝑙, and 𝐞_𝑛[𝑙]=𝐘_𝑛[𝑙]−𝐻_𝑛−1[𝑙]𝐗_𝑛[𝑙]. Premultiplying both sides of (A.2) by 𝑃_𝑛 and using 𝐞_𝑛[𝑙], (A.2) can be rewritten as𝐻_𝑛[𝑙]=𝐘_𝑛[𝑙]𝐅𝐗^∗_𝑛[𝑙]𝑃_𝑛⁻¹. (A.3)Substituting 𝐘_𝑛[𝑙]=𝐻_𝑛[𝑙]𝐗_𝑛[𝑙]+𝐖_𝑛[𝑙] into (A.3), Δ𝐻_𝑛 is expressed asΔ𝐻_𝑛=𝐖_𝑛[𝑙]𝐅𝐗^∗_𝑛[𝑙]𝑃_𝑛⁻¹. (A.4)Correspondingly, the MSE 𝐸{Δ𝐻^∗_𝑛Δ𝐻_𝑛} is computed as𝐸{Δ𝐻^∗_𝑛Δ𝐻_𝑛}=𝜎²_𝑤𝑃_𝑛⁻¹𝐸{𝐗_𝑛[𝑙]𝐅²𝐗^∗_𝑛[𝑙]}_𝑃′𝑛𝑃_𝑛⁻¹. (A.5)Here, assuming 𝐸{𝐱_𝑛−𝑘𝐱^∗_𝑛−𝑘}=𝐈_𝑁 for all 𝑘 in (1), 𝑃^′_𝑛 and 𝑃_𝑛 are given by𝑃^′_𝑛=(1−𝛽^2𝐵)(1−𝛽²),𝑃_𝑛=(1−𝛽^𝐵).(1−𝛽) (A.6)Finally, substituting (A.6) into (A.5), the MSE of the WEB-CE is calculated as𝜀_𝑛[𝑙]=𝜎²_𝑤(1−𝛽)(1+𝛽^𝐵)(1+𝛽)(1−𝛽^𝐵). (A.7)

Acknowledgments

This research was supported by the MKE (Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute for Information Technology Advancement) (IITA-2008-(C1090-0801-0011)).