Multi-Tap Channel Estimation for Preamble-Based FBMC/OQAM Systems

The intrinsic imaginary inter-carrier/inter-symbol interference in filter bank multicarrier with offset quadrature amplitude modulation (FBMC/OQAM) systems makes accurate channel estimation remain an open issue, which motivates our work in this article. In order to improve the channel estimation performance for preamble-based FBMC/OQAM systems, a recursive least squares (RLS)-based multi-tap channel estimation (RLS-MTCE) scheme that can effectively reduce the influence of noise is proposed by adopting the widely used signal model of the analysis filter bank (AFB) output. On the other hand, by utilizing the new-built signal model of the AFB output, the authors develop least squares (LS)-based and minimum mean square error (MMSE)-based multi-tap channel estimation schemes, denoted by LS-MTCE and MMSE-MTCE, respectively, to accurately exploit the interference to improve the channel estimation performance. The complexity and performance analysis results show that, compared with the existing schemes, the proposed RLS-MTCE, LS-MTCE and MMSE-MTCE counterparts generally achieve a lower estimation error at the cost of an acceptable increase in computational complexity.

SECTION I. Introduction

Filter bank multicarrier with offset quadrature amplitude modulation (FBMC/OQAM) [1]–​[3] has been studied as a promising technique in future wireless communication, due to its enhanced robustness to synchronization requirements, higher spectrum efficiency, and better frequency localization over orthogonal frequency division multiplexing (OFDM) [4]. However, FBMC/OQAM suffers from imaginary inter-carrier/inter-symbol interference, which severely damages the pilot symbol, resulting in the wrong channel estimation. Therefore, the channel estimation is a challenge for FBMC/OQAM systems and has attracted numerous research efforts.

So far, many channel estimation methods for FBMC/ OQAM systems have been proposed to tackle this problem, including scattered pilots-based [5]–​[14] and preamble-based [15]–​[21] methods. For the former, the auxiliary pilot (AP) method was introduced in [5] to eliminate the undesired imaginary interference at pilot positions by inserting APs, so that the estimation could be performed in the same manner as OFDM systems. In [6]–​[8], the authors studied the coded AP (CAP) method that could achieve higher power efficiency than AP. For channel adaptive FBMC/OQAM systems, [9] derived the channel estimation error of AP and CAP and analyzed its effect on the bit error rate (BER) performance. By using the estimates of data to reconstruct the pseudo-pilot at the receiver side, [10] investigated the iterative scattered pilots-based channel estimation method to improve the estimation performance. The channel estimation method with continuous and burst pilot sequences was presented in [11] to obtain a good tracking of channels. In [12], a nonlinear complex support vector regression algorithm was developed for channel estimation of two auxiliary pilots-based FBMC/OQAM systems.

On the other hand, for preamble-based channel estimation, the interference approximation method (IAM) was proposed in [15], which constructs the complex pseudo-pilot by combining the real-valued pilot with the interference approximation from neighboring pilots and then uses the pseudo-pilot to estimate the channel frequency response (CFR). There are several variants of IAM [15]–​[18] such as IAM-R, IAM-I, IAM-C, E-IAM-C, and IAM-P, and all of these schemes aim at constructing approximate pseudo-pilots to improve the channel estimation performance. The interference cancellation method (ICM) proposed in [19] is another type of preamble-based channel estimation. However, different from IAM, it focuses on eliminating the interference by designing the symbols around the pilots. In [20], the preamble structure of ICM was used to develop a minimum mean square error (MMSE) channel estimator. [21] designed a preamble using comb-type pilots and it enables OFDM-like channel estimation in FBMC/OQAM systems. In addition, note that the pair of pilots (POP) method first analyzed in [15] is based on the preamble, which gets the estimated channel by eliminating the interference, while its variants [13], [14] utilizing the interference to improve the performance is of the scattered pilots-based method.

It is worth noting that most of the aforementioned methods [5]–​[12], [15]–​[19], [21] adopt the single-tap least squares (LS) algorithm to estimate the CFR, and very limited efforts have been made to improve the estimation performance by using multi-tap estimation methods [20]. Motivated by these, this article tries to develop multi-tap channel estimation schemes to achieve a more accurately estimated CFR for preamble-based FBMC/OQAM systems. Our main contributions can be summarized as follows.

1. By using the signal model of the analysis filter bank (AFB) output in [5]–​[21], we propose a recursive least squares (RLS)-based multi-tap channel estimation (RLS-MTCE) scheme that consists of forward and backward recursive operations. The proposed RLS-MTCE scheme can effectively use the pilots and the received signals of multiple subchannels to estimate the subchannel frequency response of interest, thus reducing the influence of noise on the channel estimation performance for preamble-based FBMC/OQAM systems.

2. A new frequency domain signal model of the AFB output is derived, which accurately includes all the inter-carrier/inter-symbol interference. Based on the new-built signal model, this article presents LS-based and MMSE-based multi-tap channel estimation schemes, denoted by LS-MTCE and MMSE-MTCE, respectively, and gives the theoretical mean square error (MSE) of the corresponding estimators. The proposed LS-MTCE and MMSE-MTCE schemes can accurately exploit the interference to improve the channel estimation performance, which are validated by simulations in terms of the normalized MSE (NMSE) of estimated channels and the BER.

Throughout this article, $diag(\mathbf {b})$ represents the diagonal matrix with the entries of the vector $\mathbf {b}$ on the main diagonal, $diag_{-1}(\mathbf {b})$ and $diag_{+1}(\mathbf {b})$ are defined at the bottom of this page,¹ $\mathbb {Z}^{+}$ denotes the set of positive integers, “$\thicksim$ ” is used to indicate “conforms to $\ldots$ distribution,” $(\cdot)^{\mathrm {T}}$ , $(\cdot)^{\mathrm {H}}$ and $(\cdot)^{\mathrm {\ast }}$ stand for transposition, conjugate transposition and complex conjugation, respectively, $\langle \cdot |\cdot \rangle$ symbolizes the inner product of two functions, $(\cdot)^{-1}$ denotes the inverse of the square matrix, $\|\cdot \|$ stands for the Euclidean norm, $|\cdot |$ represents the modulus operator, and $\mathbb {E}\{\cdot \}$ and $tr\{\cdot \}$ symbolize expectation and matrix trace, respectively.

SECTION II. FBMC/OQAM Signal Model

The discrete-time transmitted baseband signal, consisting of even $M$ subcarriers and $N$ FBMC/OQAM symbols, can be expressed as [1] $$\begin{equation*} s(l) = \sum \limits _{m=0}^{M-1}\sum \limits _{n=0}^{N-1} a_{m,n} \underbrace {g\left({l-n \frac {M}{2}}\right)e^{j \frac {2\pi }{M}m\left({l- \frac {L_{g} -1}{2}}\right)} e^{j\varphi _{m,n}}}_{g_{m,n}(l)} \tag{1}\end{equation*}$$ where $l=0,1,\ldots,(N-1)M/2 + L_{g}-1$ , $M=2^{z}$ , $z \in \mathbb {Z}^{+}$ , $z \geq 2$ , $N \in \mathbb {Z}^{+}$ , $a_{m,n}$ is the real-valued OQAM data carried on the $m$ th subcarrier during the $n$ th FBMC/OQAM symbol, and $\varphi _{m,n}=(m+n)\pi /2$ denotes an additional phase term. $g(l)$ stands for the discrete-time prototype filter impulse response with the length $L_{g}=KM+1$ , where $K$ is its overlapping factor. $g_{m,n}(l)$ represents the synthesis filter bank (SFB) impulse response for $a_{m,n}$ , which is derived by the time-frequency shifted version of $g(l)$ . This article considers the prototype filter of the PHYDYAS project in [22], with the impulse response coefficients expressed as $$\begin{equation*} g(i) = \frac {1}{K \sqrt {M}} \left [{G_{0} + 2\sum \limits _{k=1}^{K-1}(-1)^{k} G_{k} \cos \left ({\frac {2\pi ki}{KM} }\right) }\right] \tag{2}\end{equation*}$$ where $i=0,1,\ldots,L_{g} -1$ , $\sum _{i=0}^{L_{g}-1} g^{2}(i)=1$ , $K$ takes a typical value of 4, $G_{0}=1$ , $G_{1}=0.97196$ , $G_{2}=\sqrt {2}/2$ , and $G_{3}=0.235147$ .

Let $\mathbf {h}_{n}=[h(0,n),h(1,n),\ldots,h(L_{h}-1,n)]^{\mathrm {T}}$ be the impulse response of multipath fading channels during the $n$ th FBMC/OQAM symbol, where $L_{h} \leq M$ represents the maximum channel delay. At the receiver side, the received baseband signal can be written as $$\begin{equation*} y(l) = \sum \limits _{k=0}^{L_{h}-1}h(k,n)s(l-k)+w(l) \tag{3}\end{equation*}$$ where $w(l) \sim \mathcal {CN}(0,\sigma ^{2})$ denotes the complex additive white Gaussian noise. Then, the AFB output signal at the frequency-time (FT) point $(p,q)$ can be calculated by (4), as shown at the bottom of the page,

where $n_{p,q}$ is the noise term, and $\epsilon _{m,n}^{p,q}(k)$ represents the interference from which the FT point $(p,q)$ suffers, coming from $(m,n)$ of the $k$ th path. Since it is hard to use (4) as the signal model to estimate the CFR directly, most of the existing works [5]–​[21] tackle this problem by resorting to strict channel assumptions² to obtain a simplified approximate expression as follows: $$\begin{align*} y_{p,q} \!\approx \! H(p,q) \left({a_{p,q}\!+ \!\sum \limits _{(m,n)\in \Omega {_{p,q}^{F,T}}} a_{m,n} \langle g_{m,n}(l)|g_{p,q}(l) \rangle }\right) \!+\!n_{p,q}\!\! \\ \tag{5}\end{align*}$$ in which $H(p,q)=\sum _{k=0}^{M-1}h_{M}(k,q)e^{-j \frac {2\pi }{M}pk}$ is an $M$ -sized CFR of $h_{M}(k,q)$ , $h_{M}(k,q)$ is the zero-padding version of $h(k,q)$ , defined by $h_{M}(k,q)= \Big \{ {h(k,q),\,\,\,\,k=0,1,\ldots, L_{h}-1\quad 0,\qquad k=L_{h},L_{h}+1,\ldots,M-1}$ , $\langle g_{m,n}(l)|g_{p,q}(l) \rangle \triangleq \sum _{l=-\infty }^{+\infty }g_{m,n}(l)g_{p,q}^{*}(l)$ is a pure imaginary interference factor and determined only by the prototype filter, and $\Omega {_{p,q}^{F,T}}$ represents an FT neighborhood of $(p,q)$ (except for $(p,q)$ itself), with the definition given by $\Omega {_{p,q}^{F,T}} = \{(m,n)| |m-p| \leq F,|n-q| \leq T, (m,n) \neq (p,q)\}$ .

The approximation (5) has been used as the signal model for channel estimation in [5]–​[21]. Based on (5), we intend to develop a new channel estimation scheme, i.e., RLS-MTCE, to improve the channel estimation performance for preamble-based FBMC/OQAM systems in the next section.

SECTION III. RLS-Based Multi-Tap Estimation (RLS-MTCE) Scheme

The frame structure for preamble-based FBMC/OQAM systems is designed as shown in Fig. 1, where $\mathbf {a}_{q} =[a_{0,q}, a_{1,q},\ldots,a_{M-1,q}]^{\mathrm {T}}$ and $q=0,1,\ldots,N-1$ . $\mathbf {a}_{0}$ represents the preamble, and its entry $a_{m,0}$ is chosen as 1 or −1 (i.e., strictly OQAM) for $m=0,1,\ldots,M-1$ in a pseudo-random way. Three all-zero guard symbols are placed behind the preamble to avoid the imaginary interference from data symbols. In addition, note that the quasi-static multipath channels (i.e., the channel impulse response (CIR) is invariant over a frame, changing independently from one frame to another) are considered here, and thereby we only need to estimate the channels at the start of the frame. The proposed RLS-MTCE scheme consists of forward and backward recursive operations, with the detail described in the following.

FIGURE 1.

Frame structure for preamble-based FBMC/OQAM systems.

The forward recursive operation: Because the filter $g(l)$ is well localized in the FT domain, the intrinsic imaginary interference to the point ($p,0$ ) in Fig. 1 is mainly caused by the first-order frequency neighborhood $\Omega {_{p,0}^{1,0}} = \{(p \pm 1,0)\}$ . Define $c_{p,0} = a_{p,0}+ \sum _{(m,n)\in \Omega {_{p,0}^{1,0}}} a_{m,n} \langle g_{m,n}(l)|g_{p,0}(l) \rangle$ , and then, according to (5), the received pilot signal at the point ($p, 0$ ) can be further written as $y_{p,0} \approx c_{p,0}H(p,0) + n_{p,0}$ , where $p=0,1,\ldots,M-1$ , $c_{p,0}$ stands for the virtual transmitted pilot symbol and can be called pseudo-pilot. Forward recursive formulas using the RLS algorithm [23], [24] can be expressed as $$\begin{align*} X_{p,0}=&\lambda X_{p-1,0} + |c_{p,0}|^{2} \tag{6}\\ \widehat {H}_{F}(p,0)=&\widehat {H}_{F}(p-1,0) \\&+ \,\frac {c_{p,0}^{*}}{X_{p,0}} \{y_{p,0} - c_{p,0}\widehat {H}_{F}(p-1,0) \} \tag{7}\end{align*}$$ where $\widehat {H}_{F}(p,0)$ represents the estimate of $H(p,0)$ by the forward recursion, $0 < \lambda < 1$ denotes the forgetting factor, its optimal value is determined by the correlation of adjacent subchannels and the noise power, which is discussed in detail by simulations in Section VI, and the recursive initial conditions can be set to $\widehat {H}_{F}(0,0) = \frac {y_{0,0}}{c_{0,0}}$ and $X_{0,0} = |c_{0,0}|^{2}$ . The proof of forward recursive formulas (6) and (7) is given in Appendix.

The backward recursive operation: Assume that $\widehat {H}_{B}(p,0)$ is the estimate of $H(p,0)$ by the backward recursion. In contrast to the forward recursive operation, the initial value of estimated channels in the backward counterpart adopts the LS estimate of the ($M-1$ )th subchannel, i.e., $\widehat {H}_{B}(M-1,0) = \frac {y_{M-1,0}}{c_{M-1,0}}$ ; $\widehat {H}_{B}(p,0)$ is derived by updating $\widehat {H}_{B}(p+1,0)$ with the “new information” $\{y_{p,0} - c_{p,0}\widehat {H}_{B}(p+1,0) \}$ at the FT point ($p,0$ ). Backward recursive formulas can be given by $$\begin{align*} \mathcal {X}_{p,0}=&\lambda \mathcal {X}_{p+1,0} + |c_{p,0}|^{2} \tag{8}\\ \widehat {H}_{B}(p,0)=&\widehat {H}_{B}(p+1,0) \\&+ \,\frac {c_{p,0}^{*}}{\mathcal {X}_{p,0}} \{y_{p,0} - c_{p,0}\widehat {H}_{B}(p+1,0) \} \tag{9}\end{align*}$$ where the value of $\mathcal {X}_{M-1,0}$ is set to $|c_{M-1,0}|^{2}$ . Here, the proof of (8) and (9) is omitted due to the similar derivation process to (6) and (7) in Appendix.

Note that, in the above two recursive operations, $\widehat {H}_{F}(p,0)$ and $\widehat {H}_{B}(p,0)$ are derived only by making use of the received pilot signals [$y_{p,0},y_{p-1,0},y_{p-2,0},\ldots$ ] and [$y_{p,0},y_{p+1,0},y_{p+2,0},\ldots$ ], respectively. In view of this, the estimator of RLS-MTCE, which can utilize all the received pilot signals, is given as $$\begin{equation*} \widehat {H}_{RLS-MTCE}(p,0) = \frac {1}{2}(\widehat {H}_{F}(p,0) + \widehat {H}_{B}(p,0)) \tag{10}\end{equation*}$$ where $p=0,1,\ldots,M-1$ . Next, different from RLS-MTCE using the conventional signal model of the AFB output, we will study the multi-tap channel estimation schemes using the new-built signal model in the next section.

SECTION IV. LS-Based and MMSE-Based Multi-Tap Channel Estimation (LS-MTCE and MMSE-MTCE) Schemes

In Section II, (5) is an approximate expression of $y_{p,q}$ under the condition of strict channel assumptions, which will result in severe estimation error floors due to the residual interference. Motivated by these, in this section, we first derive a new frequency domain signal model of the AFB output that can accurately include all the inter-carrier/inter-symbol interference. Furthermore, two multi-tap channel estimation schemes, i.e., LS-MTCE and MMSE-MTCE, are proposed by utilizing the new-built signal model. The detail is described as follows.

A. New Frequency Domain Signal Model of the AFB Output

Let $l'=l-\frac {n}{2}M$ , $\Delta m=m-p$ , and $\Delta n=n-q$ , and then $\epsilon _{m,n}^{p,q}(k)$ in (4) can be written as $$\begin{align*} \epsilon _{m,n}^{p,q}(k)=&\underbrace {\left ({{\sum \limits _{l'=-\infty }^{+\infty }g(l' -k)g\left({l' + \frac {M}{2} \Delta n}\right)\quad \,\, \times e^{j \frac {2\pi }{M}\Delta m \left({l'- \frac {L_{g} -1}{2}}\right)} e^{j\frac {\pi }{2}(\Delta m+ \Delta n)}} }\right) }_{\epsilon ^{(\Delta m,\Delta n)}(k)} e^{j\pi \Delta mn} \\=&(-1)^{|\Delta m n|} \epsilon ^{(\Delta m,\Delta n)}(k) \tag{11}\end{align*}$$ where $\epsilon ^{(\Delta m,\Delta n)}(k)$ can be seen as an interference factor, which is determined only by $\Delta m$ , $\Delta n$ , and the prototype filter. Therefore, it can be precomputed once for all $k$ (i.e., $k=0,1,\ldots,L_{h}-1$ ) and be used subsequently in the system.

Let $E^{(\Delta m,\Delta n)}(v)=\sum _{k=0}^{M-1} \epsilon _{M}^{(\Delta m,\Delta n)}(k) e^{-j \frac {2\pi }{M}vk}$ be an $M$ -sized discrete Fourier transform (DFT) of $\epsilon _{M}^{(\Delta m,\Delta n)}(k)$ , which is the zero-padding version of $\epsilon ^{(\Delta m,\Delta n)}(k)$ and defined as $\epsilon _{M}^{(\Delta m,\Delta n)}(k)= \Big \{ {\epsilon ^{(\Delta m,\Delta n)}(k),\,\,\,\,k=0,1,\ldots,L_{h}-1\quad \,\, 0,\qquad \quad \,\,k=L_{h},L_{h}+1,\ldots,M-1}$ . Also let $\mathbf {H}_{n}^{(\Delta m,\Delta n)} = [H^{(\Delta m,\Delta n)}(0,n),H^{(\Delta m,\Delta n)}(1,n),\ldots,\,\,H^{(\Delta m,\Delta n)}(M-1,n)]^{\mathrm {T}}$ be an $M \times 1$ vector defined by $$\begin{equation*} \mathbf {H}_{n}^{(\Delta m,\Delta n)} = \mathbf {E}^{(\Delta m,\Delta n)}\mathbf {H}_{n} \tag{12}\end{equation*}$$ where $H^{(\Delta m,\Delta n)}(m,n) = \sum _{v=0}^{M-1} H(v,n) E^{(\Delta m,\Delta n)}((m-v) \mathrm {modulo}\,\,M)$ , $\mathbf {H}_{n}^{(\Delta m,\Delta n)}$ represents the CFR vector of the equivalent channel from the input of the OQAM modulator at the transmitter side to the output of the OQAM demodulator at the receiver side, $\mathbf {H}_{n} = [H(0,n), H(1,n),\ldots,H(M-1,n)]^{\mathrm {T}} \in \mathbb {C}^{M \times 1}$ denotes the CFR vector, and $\mathbf {E}^{(\Delta m,\Delta n)} \in \mathbb {C}^{M \times M}$ is the circulant matrix of $[E^{(\Delta m,\Delta n)}(0),E^{(\Delta m,\Delta n)}(1),\ldots,E^{(\Delta m,\Delta n)}(M-1)]^{\mathrm {T}}$ written as $$\begin{align*}&\hspace {-1pc} \mathbf {E}^{(\Delta m,\Delta n)} \\=&\left [{\! \begin{array}{cccc} E^{(\Delta m,\Delta n)}(0) &~E^{(\Delta m,\Delta n)}(M-1) &~\ldots &~E^{(\Delta m,\Delta n)}(1) \\ E^{(\Delta m,\Delta n)}(1) &~E^{(\Delta m,\Delta n)}(0) &~\ldots &~E^{(\Delta m,\Delta n)}(2) \\ \vdots &~\vdots &~\ddots &~\vdots \\ E^{(\Delta m,\Delta n)}(M-1) &~E^{(\Delta m,\Delta n)}(M-2) &~\ldots &~E^{(\Delta m,\Delta n)}(0) \end{array} \!}\right]\!. \\\tag{13}\end{align*}$$

Then, by substituting (11) into (4), with the help of (12) and frequency-domain convolution theorem [25], we can derive the expression of $y_{p,q}$ as (14), as shown at the bottom of the page.

Furthermore, only for the term of ($m,n$ )=($p,q$ ) in (14), we assume that the filter $g(l)$ has relatively low time variation over the interval $[{0,L_{h}-1}]$ , i.e., $g(l)\approx g(l-k)$ , $k \in [{0,L_{h}-1}]$ , and then one can attain $\epsilon ^{(0,0)}(k) = \sum _{l'=- \infty }^{+\infty }g(l'-k)g(l') \approx \sum _{l'=0}^{L_{g}-1}g^{2}(l')=1$ . With the aid of this formula, we can obtain $$\begin{align*} (1/M)H^{(0,0)}(p,q)=&\sum \limits _{k=0}^{L_{h}-1}h(k,q)\epsilon ^{(0,0)}(k) e^{-j \frac {2\pi }{M} pk} \\\approx&\sum \limits _{k=0}^{L_{h}-1}h(k,q) e^{-j \frac {2\pi }{M} pk} = H(p,q). \qquad \tag{15}\end{align*}$$ By substituting (15) into (14), a new frequency domain signal model of the AFB output at the FT point $(p,q)$ can be expressed as $$\begin{align*}&\hspace {-0.5pc}y_{p,q} \! \approx \! H(p,q)a_{p,q} \!+ \!\frac {1}{M}\sum \limits _{(\Delta m,\Delta n) \neq (0,0)} (-1)^{|\Delta m (q+\Delta n)|} \\&\qquad\qquad {\times \, a_{p+\Delta m,q\!+\!\Delta n}H^{(\Delta m,\Delta n)}(p\!+\!\Delta m,q\!+\!\Delta n) \!+ \!n_{p,q} }\tag{16}\end{align*}$$ where only the first term $H(p,q)a_{p,q}$ is approximated, and the second term $\frac {1}{M}\sum _{(\Delta m,\Delta n) \neq (0,0)} (-1)^{|\Delta m (q+\Delta n)|} \times a_{p+\Delta m,q+\Delta n} H^{(\Delta m,\Delta n)}(p+\Delta m,q+\Delta n)$ accurately includes all the inter-carrier/inter-symbol interference from the FT neighborhood. Therefore, (16) represents the AFB output signal more accurately than (5). Note that although $n_{p,q}$ in (16) follows Gaussian distribution with zero mean, it is correlated with the noise of its adjacent subcarriers. Based on [16, Appendix A], we can derive the covariance matrix of $\mathbf {n}_{q} = [n_{0,q},n_{1,q},\ldots,n_{M-1,q}]^{\mathrm {T}}$ as follows: $$\begin{align*}&\hspace {-1pc} \mathbf {C}_{q} \\[-2pt]=&\sigma ^{2} \\[-2pt]&\times \! \left [{ \!\!\begin{array}{cccccc} 1 & (-1)^{q}\beta j & 0 & \ldots & 0 & -(-1)^{q}\beta j \\[-2pt] -(-1)^{q}\beta j & 1 & (-1)^{q}\beta j & \ldots & 0 & 0 \\[-2pt] \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\[-2pt] (-1)^{q}\beta j & 0 & 0 & \ldots & -(-1)^{q}\beta j & 1 \end{array} \!\!}\right] \!\! \\[-2pt]\tag{17}\end{align*}$$ where $q=0,1,\ldots,N-1$ , $\beta =0.2393$ for the PHYDYAS prototype filter, and $\mathbf {C}_{q} \in \mathbb {C}^{M\times M}$ is a circulant Hermitian matrix.

B. LS-Based Multi-Tap Channel Estimation (LS-MTCE)

Using the frequency domain signal model (16) and the preamble $\mathbf {a}_{0}$ , this subsection studies the LS-MTCE scheme. Owing to the well FT localized property of the PHYDYAS prototype filter, the interference to $(p,0)$ in Fig. 1 mainly comes from the first-order frequency neighborhood $\Omega {_{p,0}^{1,0}}$ , and therefore the AFB output signal at the point $(p,0)$ can be approximated as $$\begin{align*}&\hspace {-0.5pc}y_{p,0} \approx a_{p,0}H(p,0) + \frac {1}{M} \Big [a_{p-1,0}H^{(-1,0)}(p-1,0) \\&\qquad\qquad\qquad\qquad {+\,a_{p+1,0}H^{(1,0)}(p+1,0) \Big] + n_{p,0}.}\tag{18}\end{align*}$$ Let $\mathbf {y}_{0} = [y_{0,0},y_{1,0},\ldots,y_{M-1,0}]^{\mathrm {T}}$ , and then the vector form of (18) can be written as follows: $$\begin{align*} \mathbf {y}_{0} \!\approx \! \underbrace {\left\{{ diag(\mathbf {a}_{0}) \!+\! \frac {1}{M} \bigg \{ {diag_{-1}(\mathbf {a}_{0})\mathbf {E}^{(-1,0)} \!+\! diag_{+1}(\mathbf {a}_{0})\mathbf {E}^{(1,0)}} \bigg \} }\right\}}_{\mathbf {A}_{0}} \mathbf {H}_{0} \! + \!\mathbf {n}_{0}\quad ~~ \tag{19}\end{align*}$$ where $\mathbf {A}_{0}$ is an $M\times M$ matrix and can be precomputed. By utilizing the LS algorithm [23], LS-MTCE for preamble-based FBMC/OQAM systems can be expressed as $$\begin{equation*} \widehat {\mathbf {H}}_{LS-MTCE}=\mathbf {A}_{0}^{-1} \mathbf {y}_{0}. \tag{20}\end{equation*}$$ Furthermore, the MSE matrix of $\widehat {\mathbf {H}}_{LS-MTCE}$ is given by $$\begin{align*}&\hspace {-2pc}\mathbf {M}_{LS-MTCE} \\=&\mathbb {E} \big \{(\mathbf {H}_{0}- \widehat {\mathbf {H}}_{LS-MTCE})(\mathbf {H}_{0}- \widehat {\mathbf {H}}_{LS-MTCE})^{\mathrm {H}} \big \} \\=&(\mathbf {A}_{0}^{\mathrm {H}}\mathbf {C}_{0}^{-1}\mathbf {A}_{0})^{-1} \tag{21}\end{align*}$$ where $\mathbf {C}_{0}$ is the covariance matrix of $\mathbf {n}_{0}$ provided by (17). Moreover, (21) implies that the MSE of $\widehat {\mathbf {H}}_{LS-MTCE}$ is equal to $tr\{\mathbf {M}_{LS-MTCE} \}$ .

C. MMSE-Based Multi-Tap Channel Estimation (MMSE-MTCE)

By utilizing the same vector signal model (19) as LS-MTCE, with the help of the MMSE algorithm [23], MMSE-MTCE for preamble-based FBMC/OQAM systems can be written as $$\begin{align*}&\hspace {-2pc} \widehat {\mathbf {H}}_{MMSE-MTCE} \\=&\mathbf {R}_{\mathbf {H}_{0} \mathbf {H}_{0}} (\mathbf {R}_{\mathbf {H}_{0} \mathbf {H}_{0}} + (\mathbf {A}_{0}^{\mathrm {H}}\mathbf {C}_{0}^{-1}\mathbf {A}_{0})^{-1})^{-1} \mathbf {A}_{0}^{-1} \mathbf {y}_{0} \\=&\mathbf {R}_{\mathbf {H}_{0} \mathbf {H}_{0}} (\mathbf {R}_{\mathbf {H}_{0} \mathbf {H}_{0}} + (\mathbf {A}_{0}^{\mathrm {H}}\mathbf {C}_{0}^{-1}\mathbf {A}_{0})^{-1})^{-1} \widehat {\mathbf {H}}_{LS-MTCE}\quad \tag{22}\end{align*}$$ where $\mathbf {R}_{\mathbf {H}_{0} \mathbf {H}_{0}} = \mathbb {E} \{ \mathbf {H}_{0} \mathbf {H}_{0}^{\mathrm {H}} \}$ represents the autocorrelation matrix of $\mathbf {H}_{0}$ . Note that in (22), $\widehat {\mathbf {H}}_{MMSE-MTCE}$ can be considered as the filtered output of $\widehat {\mathbf {H}}_{LS-MTCE}$ , and the filter coefficients are $\mathbf {R}_{\mathbf {H}_{0} \mathbf {H}_{0}} (\mathbf {R}_{\mathbf {H}_{0} \mathbf {H}_{0}} + (\mathbf {A}_{0}^{\mathrm {H}}\mathbf {C}_{0}^{-1}\mathbf {A}_{0})^{-1})^{-1}$ .

Moreover, the MSE matrix of $\widehat {\mathbf {H}}_{MMSE-MTCE}$ can be calculated as $$\begin{align*}&\hspace {-1.82pc}\mathbf {M}_{MMSE-MTCE} \\=&\mathbb {E} \big \{(\mathbf {H}_{0}- \widehat {\mathbf {H}}_{MMSE-MTCE})(\mathbf {H}_{0}- \widehat {\mathbf {H}}_{MMSE-MTCE})^{\mathrm {H}} \big \} \\=&\mathbf {R}_{\mathbf {H}_{0} \mathbf {H}_{0}} - \mathbf {R}_{\mathbf {H}_{0} \mathbf {H}_{0}} (\mathbf {R}_{\mathbf {H}_{0} \mathbf {H}_{0}} + (\mathbf {A}_{0}^{\mathrm {H}}\mathbf {C}_{0}^{-1}\mathbf {A}_{0})^{-1})^{-1} \mathbf {R}_{\mathbf {H}_{0} \mathbf {H}_{0}}. \\ \tag{23}\end{align*}$$ Then, we can obtain the MSE of $\widehat {\mathbf {H}}_{MMSE-MTCE}$ as $tr\{\mathbf {M}_{MMSE-MTCE} \}$ .

SECTION V. Complexity Analysis

In this work, the complexity analysis is based on the real-valued multiplication (RMUL) and the real-valued division (RDIV). Let $d_{1},~d_{2},~d_{3},~d_{4}$ and $d_{5}$ be pure real numbers, $v_{1} = d_{1} + d_{2}\,\,j$ and $v_{2} = d_{3} + d_{4}\,\,j$ , where $d_{1} \neq d_{2} \neq d_{3} \neq d_{4} \neq d_{5}$ . For a fair comparison, we assume that 4 RMULs, 2 RMULs and 2 RDIVs are required to compute $v_{1} \times v_{2}$ , $v_{1} \times d_{5}$ and $v_{1} \div d_{5}$ , respectively. For IAM-Random,³ since $1/c_{p,0}$ can be precomputed, we only need to consider the calculation of $y_{p,0} \times (1/c_{p,0})$ . Therefore, IAM-Random requires at most $4M$ RMULs for channel estimation of $M$ subchannels. Note that, for IAM-I, because the pseudo-pilots $c_{p,0}$ , $p = 0,1,\ldots,M-1$ are either imaginary or complex numbers and the imaginary pseudo-pilots account for one third, it takes an estimation complexity of $10M/3$ RMULs; for ICM, all the pseudo-pilots are real numbers, and therefore its calculation needs $2M$ RMULs.

Next, we analyze the complexities of the proposed RLS-MTCE, LS-MTCE and MMSE-MTCE schemes. In the RLS-MTCE scheme, since both $c_{p,0}$ and $|c_{p,0}|^{2}$ , $p = 0,1, \ldots,M-1$ can be precomputed, ($M-1$ ) recursive calculations of (6) and (7) require a total of 9($M-1$ ) RMULs and 2($M-1$ ) RDIVs. In addition, solving the initial condition $\widehat {H}_{F}(0,0) = y_{0,0} \times (1/c_{0,0})$ needs 4 RMULs. Consequently, the forward recursive operation requires ($9M-5$ ) RMULs and ($2M-2$ ) RDIVs. Because the backward recursive operation has the same computational complexity as the forward counterpart, the total complexity of RLS-MTCE is ($18M-10$ ) RMULs and ($4M-4$ ) RDIVs. For LS-MTCE and MMSE-MTCE, $\mathbf {A}_{0}^{-1}$ in (20) and $\mathbf {R}_{\mathbf {H}_{0} \mathbf {H}_{0}} (\mathbf {R}_{\mathbf {H}_{0} \mathbf {H}_{0}} + (\mathbf {A}_{0}^{\mathrm {H}}\mathbf {C}_{0}^{-1}\mathbf {A}_{0})^{-1})^{-1} \mathbf {A}_{0}^{-1}$ in (22) can also be precomputed similar to the pseudo-pilots $c_{p,0}$ , and therefore they have the same computational complexity of $4M^{2}$ RMULs, i.e., the complexity of multiplying one $M \times M$ complex matrix by one $M \times 1$ complex vector. Note that the noise variance is assumed to be known here. Then, the MMSE estimator [20, Eq. (10)] requires the same RMULs and RDIVs as MMSE-MTCE.

The computational complexities of the proposed and conventional schemes are summarized in Table 1. It can be seen that the complexities of the proposed RLS-MTCE, LS-MTCE and MMSE-MTCE schemes are higher than those of the conventional IAM-Random, IAM-I and ICM counterparts. However, the proposed schemes generally achieve excellent channel estimation performance, which is demonstrated by the following simulations.

TABLE 1 Complexity Comparison of the Proposed and Conventional Schemes

SECTION VI. Simulation Results

The channel estimation performance of the proposed RLS-MTCE, LS-MTCE and MMSE-MTCE schemes can be measured in terms of the NMSE and the BER. The NMSE is defined by $$\begin{equation*} NMSE = \frac {1}{Num} \sum \limits _{i=1}^{Num} \frac { \| \mathbf {H}(i) - \widehat {\mathbf {H}}(i) \|^{2}}{\| \mathbf {H}(i) \|^{2}} \tag{24}\end{equation*}$$ where $Num$ denotes the number of Monte Carlo experiments and is set to 1000 in this article, $\mathbf {H}(i)$ represents the CFR vector to be estimated in the $i$ th experiment, and $\widehat {\mathbf {H}}(i)$ stands for its estimate. Simulation parameters are set as follows:

• Number of Subcarriers: 128;

• Subcarrier spacing: 60KHz;

• Sampling frequency: 7.68MHz;

• Channel model: Extended Vehicular A Model (EVA), Extended Typical Urban Model (ETU) [26].

Figs. 2 and 3 depict the NMSE of RLS-MTCE with different forgetting factors over EVA and ETU channels, respectively. From the figures, the following remarks can be obtained.

• For a given SNR, the optimal value of forgetting factor (i.e., the value that can minimize the NMSE) over EVA channels is larger than that over ETU counterparts. The reason is that, compared with ETU, the adjacent subchannels in EVA are of stronger correlation, which means that the received pilots of adjacent subchannels in EVA play a more important role in the estimation of the subchannel of interest.

• Compared with high signal-to-noise ratio (SNR) regions, the forgetting factor takes a relatively large optimal value in the low counterparts. The reason can be explained as follows. In low SNR regions, the channel estimation performance is seriously affected by noise. Consequently, more received adjacent pilots (i.e., more taps) are required in the estimation of the subchannel of interest in order to effectively suppress the noise, implying that $\lambda$ should be set to a large value in (6) and (8). In addition, note that, with the increase of the gap between different subchannels, their correlation decreases. The optimal forgetting factor depends on the correlation of adjacent subchannels and the noise power.

FIGURE 2.

NMSE of RLS-MTCE with different forgetting factors over EVA channels.

FIGURE 3.

NMSE of RLS-MTCE with different forgetting factors over ETU channels.

Figs. 4 and 5 compare the NMSE of different schemes over EVA and ETU channels, respectively. It can be seen from the figures that RLS-MTCE with optimal forgetting factor and LS-MTCE significantly outperform the conventional IAM-Random, IAM-I⁴ and ICM schemes in low and high SNR regions, respectively. The MMSE estimator [20, Eq. (10)] can obtain a low estimation error only at low SNRs, because it uses the approximation (5) as the signal model of channel estimation and suffers from the inter-carrier/inter-symbol interference. In all the cases, the proposed MMSE-MTCE scheme achieves the best channel estimation performance. Moreover, the theoretical NMSE curves of LS-MTCE and MMSE-MTCE are also depicted in Figs. 4 and 5. Note that, due to the residual interference caused by the transmitted data $\mathbf {a}_{q}$ , $q=4,5,\ldots,N-1$ and the approximation (15), Monte Carlo simulations match with the theoretical curves only at low SNRs.

FIGURE 4.

NMSE comparison of the proposed and conventional channel estimation schemes over EVA channels.

FIGURE 5.

NMSE comparison of the proposed and conventional channel estimation schemes over ETU channels.

The BER results corresponding to different schemes over EVA and ETU channels are shown in Figs. 6 and 7, respectively. Herein, 4-QAM is used for data symbols. It can be observed that the BER performance of the proposed and conventional schemes is consistent with their NMSE in Figs. 4 and 5. RLS-MTCE and LS-MTCE obtain a low BER in low and high SNR regions, respectively. MMSE-MTCE achieves the best BER performance in all the cases.

FIGURE 6.

BER performance of single-tap zero-forcing (ZF) equalization with the estimated CFRs obtained from the proposed and conventional channel estimation schemes over EVA channels.

FIGURE 7.

BER performance of single-tap ZF equalization with the estimated CFRs obtained from the proposed and conventional channel estimation schemes over ETU channels.

SECTION VII. Conclusion

For preamble-based FBMC/OQAM systems, this article develops three multi-tap channel estimation schemes, i.e., RLS-MTCE, LS-MTCE and MMSE-MTCE. Simulation results demonstrate that the proposed schemes generally outperform the conventional counterparts. These new results can provide a valuable reference for the design and development of practical FBMC/OQAM systems. Extending the multi-tap channel estimation schemes to multi-antenna FBMC/OQAM systems will be left for our future work.

Appendix Proof of (6) and (7)

In order to obtain the forward recursive formulas, we first define the cost function of the RLS algorithm as follows: $$\begin{align*}&\hspace {-0.5pc}J(p) = \sum \limits _{m=0}^{p}\lambda ^{p-m}(y_{m,0} - c_{m,0}H(p,0))^{*} \\&\qquad\qquad\qquad\qquad\qquad\qquad {\times \,(y_{m,0} - c_{m,0}H(p,0)).}\tag{25}\end{align*}$$ Let $\mathbf {y}(p)=[y_{0,0},y_{1,0}, \ldots,y_{p,0}]^{\mathrm {T}}$ , $\mathbf {c}(p)=[c_{0,0},c_{1,0}, \ldots,\,\,c_{p,0}]^{\mathrm {T}}$ and $\boldsymbol {\lambda }(p) = diag([\lambda ^{p},\lambda ^{p-1}, \ldots,\lambda ^{0}]^{\mathrm {T}})$ , and then (25) can be rewritten as $$\begin{align*} J(p) = [\mathbf {y}(p)-\mathbf {c}(p) H(p,0)]^{\mathrm {H}} \boldsymbol {\lambda }(p) [\mathbf {y}(p) - \mathbf {c}(p) H(p,0)]. \\ \tag{26}\end{align*}$$ By utilizing the weighted least squares (WLS) algorithm [23], we can obtain the estimator as follows: $$\begin{equation*} \widehat {H}_{F}(p,0) =[\mathbf {c}^{\mathrm {H}}(p) \boldsymbol {\lambda }(p) \mathbf {c}(p)]^{-1} \mathbf {c}^{\mathrm {H}}(p) \boldsymbol {\lambda }(p) \mathbf {y}(p). \tag{27}\end{equation*}$$ Define $X_{p,0} = \mathbf {c}^{\mathrm {H}}(p) \boldsymbol {\lambda }(p) \mathbf {c}(p)$ , and its recursive expression is given by $$\begin{align*} X_{p,0}=&\left [{\mathbf {c}^{\mathrm {H}}(p-1),c_{p,0}^{\mathrm {*}} }\right] \left [{ \begin{array}{cc} \lambda \cdot \boldsymbol {\lambda }(p-1) &\quad \mathbf {0} \\ \mathbf {0} &\quad 1 \end{array} }\right] \left [{ \begin{array}{c} \mathbf {c}(p-1) \\ c_{p,0} \end{array} }\right] \\=&\lambda X_{p-1,0} + |c_{p,0}|^{2}. \tag{28}\end{align*}$$ With the help of (28), $\widehat {H}_{F}(p,0)$ can be further calculated by the recursive method as $$\begin{align*}&\hspace {-1.2pc} \widehat {H}_{F}(p,0) \\=&\frac {1}{X_{p,0}} \left \{{ \! \left [{\mathbf {c}^{\mathrm {H}}(p-1),c_{p,0}^{\mathrm {*}} }\right] \left [{ \! \begin{array}{cc} \lambda \cdot \boldsymbol {\lambda }(p-1) &\quad \mathbf {0} \\ \mathbf {0} &\quad 1 \end{array} \!}\right] \left [{\! \begin{array}{c} \mathbf {y}(p-1) \\ y_{p,0} \end{array} \!}\right] \!}\right \} \\=&\frac {1}{X_{p,0}} \left \{{\! \lambda X_{p-1,0}\widehat {H}_{F}(p-1,0) + c_{p,0}^{\mathrm {*}} y_{p,0} }\right \} \\=&\widehat {H}_{F}(p-1,0) + \frac {c_{p,0}^{\mathrm {*}}}{X_{p,0}} \left \{{y_{p,0} - c_{p,0} \widehat {H}_{F}(p-1,0) \!}\right \}. \tag{29}\end{align*}$$ According to the definition of $X_{p,0}$ and the expression (27) of $\widehat {H}_{F}(p,0)$ , we can attain the recursive initial conditions of (28) and (29), i.e., $X_{0,0} = |c_{0,0}|^{2}$ and $\widehat {H}_{F}(0,0) = \frac {y_{0,0}}{c_{0,0}}$ . Then, the proof of (6) and (7) is completed.