Iterative equalization for OFDM systems over wideband Multi-Scale Multi-Lag channels
- Tao Xu^{1}Email author,
- Zijian Tang^{1, 2},
- Rob Remis^{1} and
- Geert Leus^{1}
https://doi.org/10.1186/1687-1499-2012-280
© Xu et al.; licensee Springer. 2012
Received: 20 March 2012
Accepted: 10 August 2012
Published: 31 August 2012
Abstract
OFDM suffers from inter-carrier interference (ICI) when the channel is time varying. This article seeks to quantify the amount of interference resulting from wideband OFDM channels, which are assumed to follow the multi-scale multi-lag (MSML) model. The MSML channel model results in full channel matrices both in the frequency and time domains. However, banded approximations are possible, leading to a significant reduction in the equalization complexity. Measures for determining whether a time-domain or frequency-domain approach should be undertaken are provided based on the interference analysis, and we propose to use the conjugate gradient (CG) algorithm to equalize the channel iteratively. The suitability of a preconditioning technique, that often accompanies the CG method to accelerate the convergence, is also discussed. We show that in order for the diagonal preconditioner to function properly, optimal resampling is indispensable.
Introduction
With many desirable properties such as a high spectral efficiency and inherent resilience to the multipath dispersions of frequency-selective channels, the orthogonal frequency division multiplexing (OFDM) technology shows attractive features to wireless radio applications[1]. OFDM relies on the assumption that the channel stays constant within at least one OFDM symbol period. When Doppler effects due to temporal channel variation cannot be ignored, this assumption does not hold any more since the communication channel becomes time varying. The Doppler effects corrupt the orthogonality among OFDM subcarriers by inducing non-negligible inter-carrier interference (ICI)[2], and can therefore severely deteriorate the system performance. For traditional terrestrial radio systems, compensation of ICI in channel equalization has widely been researched for narrowband systems. Due to the small relative signal bandwidth (actual bandwidth divided by the center frequency) of narrowband systems, the Doppler effects can be modeled primarily by frequency shifts[3, 4], in which case it is reasonable to assume that each OFDM subcarrier experiences a statistically identical frequency offset[2]. Consequently, the effective channel matrix of a narrowband OFDM system in the presence of Doppler can be approximated as banded. Efficient equalization schemes for such a banded channel matrix have been studied in, e.g.,[5–7].
In a wideband system, where the relative signal bandwidth is large, the Doppler effects should be more appropriately modeled as scalings of the signal waveform[3, 4]. Wideband systems arise in, e.g., underwater acoustic (UWA) systems or wideband terrestrial radio frequency systems such as ultra wideband (UWB). Due to multipath, a wideband linear time-varying (LTV) channel can be more accurately described by a multi-scale multi-lag (MSML) model[3, 8]. Many signaling schemes have been studied for wideband systems. For instance,[9, 10] consider direct-sequence spread spectrum (DSSS). Recently, the use of OFDM for UWA or UWB has received considerable attention. To counteract the scaling effect due to Doppler,[11] proposes a multi-band OFDM system such that within each band, the narrowband assumption can still be valid. More commonly, many works assume a single-scale multi-lag (SSML) model for the wideband LTV channel. Based on the SSML assumption, after a resampling operation the channel can be approximated by a time-invariant channel but subject to a carrier frequency offset (CFO)[12, 13]. However, since the channel should be more accurately described by an MSML model, determining the optimal resampling rate is not trivial[14].
In this article, we consider OFDM transmission based on an MSML model. The resulting channel, which is a full matrix in the presence of Doppler, will be equalized by means of the conjugate gradient (CG) algorithm[15], whose performance is less sensitive to the condition of the channel matrix than, e.g., a least-squares approach. On the other hand, the convergence rate of CG is inversely proportional to the channel matrix condition number. This is especially of significance if a truncated CG is to be used in practice, which halts the algorithm after a limited number of iterations in order to reduce the overall complexity. Therefore, it is desired that the channel matrix is well-conditioned to ensure a fast convergence. To this end, preconditioning techniques can be invoked to enforce the eigenvalues of the channel matrix to cluster around one[16]. To achieve a balance between performance and complexity, we restrict the preconditioner to be a diagonal matrix, whose diagonal entries can be designed by following the steps given in[17]. We notice that a circulant preconditioner in the time domain was introduced in[18], which is equivalent to a diagonal preconditioner in the frequency domain. This preconditioner is introduced based on a basis expansion model (BEM), which is often used to approximate the channel’s time-variation for a narrowband system. For a wideband system as considered in this article, it can be shown that this preconditioner in the frequency domain is equal to the inverse of the diagonal entries of the frequency-domain channel matrix.
What is not considered in[17, 18] is the resampling operation at the receiver, which is an indispensable and crucial step for wideband LTV channels. Different from the trivial resampling scheme for SSML channel models, an optimum resampling method is proposed in[14] for MSML channels, which aims at minimizing the average error of approximating the MSML channel by an SSML model. This article studies the resampling from a preconditioning point of view. It is observed that if the major channel energy is located on the off-diagonals of the channel matrix, a diagonal preconditioner will deteriorate the channel matrix condition rather than improve it, thereby reducing the convergence rate of CG instead of increasing it as opposed to the claim of[17]. The energy distribution of the channel matrix is governed by the resampling. Different from[14], which only considers rescaling the received signal, and[19], which considers both rescaling and frequency synchronization, this article will show that for OFDM systems, all these three resampling parameters can have a significant impact on the system performance (i.e., rescaling, frequency synchronization and time synchronization). More specifically, we will extend the results of[19, 20] by jointly optimizing these three resampling parameters both in the frequency domain and the time domain.
Notation: Upper (lower) bold-face letters stand for matrices (vectors); superscripts T, H, and ∗ denote transpose, Hermitian transpose and conjugate transpose, respectively; we reserve j for the imaginary unit, <k> and ⌈k⌉ for integer rounding and ceiling of a number k, ∥x∥_{2} for the two norm of the vector x, ∥A∥_{Fro} for the Frobenius norm of the matrix A, [A]_{k,m} for the (k,m)th entry of the matrix A; diag(x) for a diagonal matrix with x on its main diagonal, and ⊙ for the Hadamard product of two matrices.
System model based on an MSML channel
Continuous data model
where the data symbol b_{ k } is modulated on the k th subcarrier f_{ k }=k Δf, for k = 0,1,…,K−1, with Δf being the OFDM subcarrier spacing. With T = 1/(k Δf), KT is the effective duration of an OFDM symbol. The cyclic prefix and postfix are given as T_{pre} and T_{post}, respectively. The cyclic prefix is assumed to be longer than the delay spread and the cyclic postfix is long enough to ensure signal completeness in case of scaling, which will be defined later on. The rectangular pulse u(t) is defined to be 1 within t∈[−T_{pre},KT + T_{post}] and 0 otherwise. Prior to transmission, s(t) is up-converted to passband, yielding$\left(\right)close="">\stackrel{\u0304}{s}(t)=\Re \{s(t){e}^{j2\Pi {f}_{c}t}\}$, where f_{ c }denotes the carrier frequency. With sufficient cyclic extensions, the interference form adjacent OFDM symbols can be neglected and hence we are allowed to consider an isolated OFDM symbol in this article without loss of generality. Although this article discusses the scenario when cyclic extensions are used, the analysis can be directly applied to zero padding OFDM (ZP-OFDM) with minor modifications.
The considered signal is transmitted over a wideband LTV channel, which is assumed to comprise multiple resolvable paths. The l th path can mathematically be characterized by the following three parameters:$\left(\right)close="">{\stackrel{\u0304}{h}}_{l}$, the path gain; v_{ l }, the radial velocity which is uniquely determined by the incident angle of this path; and τ_{ l }, the delay due to the propagation time. In compliance with the wideband assumption, the received signal resulting from the l th path is given by$\left(\right)close="">{\stackrel{\u0304}{h}}_{l}\sqrt{{\alpha}_{l}}\stackrel{\u0304}{s}\left({\alpha}_{l}\right(t-{\tau}_{l}\left)\right)$, where$\left(\right)close="">{\alpha}_{l}=\frac{c+{v}_{l}}{c-{v}_{l}}\approx 1+\frac{2{v}_{l}}{c}$ is the scaling factor with c the speed of the communication medium (normally c ≫ v_{ l }) and$\left(\right)close="">\sqrt{{\alpha}_{l}}$ is added as a normalization factor. Depending on the sign of v_{ l }, the received signal waveform via this path can be either dilated (a negative v_{ l }) or compressed (a positive v_{ l }).
where$\left(\right)close="">\stackrel{\u0304}{w}\left(t\right)$ stands for the passband noise. In the above, if there exist at least two paths l and l’, for which$\left(\right)close="">{\alpha}_{l}\ne {\alpha}_{{l}^{\prime}}$ and/or$\left(\right)close="">{\tau}_{l}\ne {\tau}_{{l}^{\prime}}$, the channel exhibits a multi-scale multi-lag (MSML) character. For a practical channel, it is realistic to assume that α_{ l }∈[1,α_{max} and τ_{ l }∈[0,τ_{max}^{a}, where α_{max} ≥ 1 and τ_{max} ≥ 0 determines the scale spread and delay spread, respectively. Note that in many prior works[12, 13], the approximation$\left(\right)close="">{\alpha}_{l}\approx {\alpha}_{{l}^{\prime}}$ for any l≠l^{ ′ } is adopted for the sake of analytical ease, which gives rise to an SSML model.
which stands for the time-varying channel frequency response seen by the k th subcarrier. From the definition of h_{ k }(t), we notice that the k th subcarrier experiences a frequency offset of (α_{ l }−1)(f_{ c } + f_{ k }) over the l th path.
Remark 1
When the above conditions are satisfied, we are allowed to drop the notation of the rectangular pulse u(t) embedded in h_{ k }(t) in the sequel for the sake of notational ease.
Discrete data model
- 1.
Which point should we consider as the starting point of the OFDM symbol (time synchronization)?
- 2.
What sampling rate should we adopt to discretize the received signal over MSML channels (rescaling)?
- 3.
What frequency shift should we apply to remove the residual carrier frequency offset (frequency synchronization)?
where β is a positive number within [1,α_{max}] and βT represents the sampling rate at the receiver; σ is the time shift factor, which is used to represent time synchronization; and likewise, ϕ is the phase shift factor used for frequency synchronization.$\left(\right)close="">\sqrt{\frac{1}{\beta}}$ is a normalization factor. Later on, we will show that a different choice of (β,ϕ,σ) can influence the energy distribution of the channel matrix significantly. For the moment, we leave the values of these parameters open to allow for a general treatment of the problems. It is clear that when (β,ϕ,σ) = (1,0,0), there is no resampling operation carried out.
In (9), the term$\left(\right)close="">{e}^{j2\Pi \omega \frac{({\alpha}_{l}-1+\varphi )}{\beta}\frac{n}{K}}$ corresponds to the residual CFO related with the l th path after resampling; the term$\left(\right)close="">{e}^{-j2\Pi f{\alpha}_{l}({\lambda}_{l}+\sigma )\frac{k}{K}}$ corresponds to the phase changes due to the time shift along the l th path; and the summation$\left(\right)close="">\sum _{k=0}^{K-1}{b}_{k}{e}^{j2\Pi \frac{{\alpha}_{l}}{\beta}\frac{\mathit{\text{nk}}}{K}}$ is the adapted version of the transmitted OFDM signal due to the channel time variation in the l th path.
where the superscript (β,ϕ) in$\left(\right)close="">{\mathbf{D}}_{l}^{(\beta ,\varphi )}$ and (σ) in$\left(\right)close="">{\mathbf{\bigwedge}}_{l}^{\left(\sigma \right)}$ reflects the dependence on the specific resampling parameters. This convention will hold throughout this article.
Interference analysis
where$\left(\right)close="">{\xi}_{l,\text{F1}}=\frac{{\alpha}_{l}-\beta}{\beta}$ and$\left(\right)close="">{\xi}_{l,\text{F2}}=\frac{{\alpha}_{l}-1+\varphi}{\beta}\omega $ with$\left(\right)close="">\text{sinc}(t)=\frac{sin\left(\Pi t\right)}{\Pi t}$.
Clearly, such an offset is not only dependent on the Doppler spread α and the carrier frequency f_{ c }, but also on the subcarrier frequency f_{ k }= k Δf. The dependence of the signal energy offset on the subcarrier index is unique to wideband channels, and is also referred to as nonuniform Doppler shifts in[13]. In contrast, the frequency offset for narrowband channels is statistically identical for all the subcarriers[2].
as the banded approximation of$\left(\right)close="">{\mathbf{H}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}$.
where$\left(\right)close="">{\stackrel{\u0304}{\mathbf{v}}}_{F}^{(\beta ,\varphi ,\sigma )}=\left({\mathbf{H}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}-{\stackrel{\u0304}{\mathbf{H}}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}\right)\mathbf{b}$.
The above analysis can also be applied in the time domain in an analogous manner. See Appendix 3 for the details. Here we only want to highlight that, different from the energy distribution in the FD channel matrix which is influenced by the rescaling factor β and the phase-shift factor ϕ[c.f. ξ_{l,F1} and ξ_{l,F2}in (16)], the energy distribution in the TD channel matrix is affected by the rescaling factor β and the time-shift factor σ[c.f. ξ_{l,T1} and ξ_{l,T2} in (39)]. However, similarly as the FD channel matrix, we can also understand from the right subplot of Figure2 that$\left(\right)close="">{\mathbf{H}}_{\text{T},l}^{(\beta ,\sigma )}$ is roughly banded along the l th path in the time domain, and so is the overall time-domain channel matrix$\left(\right)close="">{\mathbf{H}}_{\text{T}}^{(\beta ,\varphi ,\sigma )}$.
Channel equalization scheme
Additionally, we would like to highlight that just as a single-carrier channel can be equalized in the frequency domain, it is also possible to equalize an OFDM channel in the time domain. Due to the similarity, we again refer the reader to Appendix 3 for a detailed mathematical derivation of the time-domain method. The question in which domain the wideband channel should be equalized, shall be addressed in the following section.
Iterative equalization
where$\left(\right)close="">\widehat{\mathbf{b}}$ is the obtained estimate of b. Because the original channel matrix$\left(\right)close="">{\mathbf{H}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}$ is a full matrix, its inversion inflicts a complexity of$\left(\right)close="">\mathcal{O}\left({K}^{3}\right)$ and is thus not desired for a practical system. To lower the complexity,$\left(\right)close="">{\mathbf{H}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}$ has been replaced by the banded approximation$\left(\right)close="">{\stackrel{\u0304}{\mathbf{H}}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}$ in (22).
from which an estimate of$\left(\right)close="">{\mathbf{b}}_{\text{C}}={\mathbf{C}}_{\mathrm{F}}^{-1}\mathbf{b}$ is first obtained by applying CG on the preconditioned matrix$\left(\right)close="">{\stackrel{\u0304}{\mathbf{H}}}_{\text{FC}}^{(\beta ,\varphi ,\sigma )}={\stackrel{\u0304}{\mathbf{H}}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}{\mathbf{C}}_{\mathrm{F}}$. Afterwards,$\left(\right)close="">\widehat{\mathbf{b}}={\mathbf{C}}_{\mathrm{F}}{\widehat{\mathbf{b}}}_{\text{C}}$ is computed to obtain the final data estimates. For details about our CG equalization, see Appendix 4.
Diagonal preconditioning
where e_{ k } is the k th column of the identity matrix.
for k∈{0,…,K−1}.
which implies that all μ_{ k }’s at the same time lie inside a disk of$\left(\right)close="">\sqrt{K}{\epsilon}_{1}$ centered around one. It is clear that if ε_{0}<ε_{1}, then minimizing$\left(\right)close="">{\u2225{\stackrel{\u0304}{\mathbf{H}}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}{\mathbf{C}}_{\mathrm{F}}-{\mathbf{I}}_{K\times K}\u2225}_{\text{Fro}}^{2}$ will at the same time minimize the Frobenius norm$\left(\right)close="">{\u2225{\stackrel{\u0304}{\mathbf{H}}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}{\mathbf{C}}_{\mathrm{F}}\u2225}_{\text{Fro}}^{2}$ itself, making the eigenvalues more clustered around zero rather than one.
Channel I: a frequency-domain case
Channel I | path | scaleα_{ l } | delayλ_{ l } | path gain$\left(\right)close="">{\stackrel{\u0304}{h}}_{l}$ |
---|---|---|---|---|
(T=0.2 ms | l=0 | 1.0150 | 0.00 | 0 dB |
ω=256 | l=1 | 1.0154 | 10.15 | −3 dB |
K=128) | l=2 | 1.0201 | 20.40 | −5 dB |
Parameter | Original | (β,ϕ,σ)=(1,0,0) | ||
Resampled | (β_{F,⋆},ϕ_{F,⋆},σ_{F,⋆})=(1.0150,−0.0150,−15.00) | |||
Orig./no precond. | 4.26×10^{5} | |||
Cond. Num. | Orig./with precond. | 1.19×10^{6} | ||
for FD | Resampl./no precond. | 23.36 | ||
Resampl./with precond. | 7.17 | |||
FD Ratio | Original | $\left(\right)close="">{\rho}_{F}^{(1,0,0)}=0.0021$ | ||
Resampled | $\left(\right)close="">{\rho}_{\mathrm{F}}^{({\beta}_{\mathrm{F},\star},{\varphi}_{\mathrm{F},\star},{\sigma}_{\mathrm{F},\star})}=0.9279$ |
with$\left(\right)close="">{\widehat{\mathbf{b}}}^{\left(i\right)}$ being the result obtained at the i th iteration of our CG equalization as mentioned in Appendix 4. In the top-right plot of Figure5, it is clear that the CG convergence with such a diagonal preconditioner is even worse than without any preconditioning. This illustrates that the diagonal preconditioning defined in (25) may not always yield a better performance than without preconditioning, as opposed to what is claimed in[17, 18]. Using a more complex structured preconditioner can avoid this, which is, however, not desired due to complexity and implementation considerations.
In Section ‘Optimal resampling’, we will show how to enhance (32) with a higher probability by means of optimal resampling.
Optimal resampling
From the previous subsections, we understand that the effectiveness of a diagonal preconditioner depends on the energy distribution of the channel matrix. It is desired that the channel matrix should have most of its energy concentrated on the main diagonal. The analysis in Section ‘Discrete data model’ learns that the resampling operation (β ϕ σ) plays an important role in governing the energy distribution of the channel matrix, and so far we have left (β ϕ σ) open for choice. Recall that resampling is a standard step taken in many wideband LTV communication systems to compensate for the Doppler effect. For example, optimizing β is considered in[14], while β and ϕ are jointly optimized in[21]. In this sense, the optimal resampling proposed in this article can be considered as a generalization of[14, 21].
Next, we shall discuss how to jointly optimize the resampling parameters (β,ϕ,σ). Focusing on the FD matrix$\left(\right)close="">{\mathbf{H}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}$, we desire$\left(\right)close="">{\left|{\left[{\mathbf{H}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}\right]}_{k,k}\right|}^{2}\sum _{m\ne k}{\left|{\left[{\mathbf{H}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}\right]}_{m,k}\right|}^{2}$ for all k∈{0,1,…,K−1}. However, satisfying the above condition for each index k individually is expensive. As a relaxation, we practically seek$\left(\right)close="">\sum _{k}{\left|{\left[{\stackrel{\u0304}{\mathbf{H}}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}\right]}_{k,k}\right|}^{2}\sum _{k}\sum _{}^{m\ne k}{\left|{\left[{\stackrel{\u0304}{\mathbf{H}}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}\right]}_{m,k}\right|}^{2}.$
which leads to the maximal ratio$\left(\right)close="">{\rho}_{\mathrm{F}}^{({\beta}_{\mathrm{F},\star},{\varphi}_{\mathrm{F},\star},{\sigma}_{\mathrm{F},\star})}$. One can also explain this resampling as minimizing the total amount of ICI in the frequency domain.
where again$\left(\right)close="">{\xi}_{l,\text{F1}}=\frac{{\alpha}_{l}-\beta}{\beta}$ and$\left(\right)close="">{\xi}_{l,\text{F2}}=\frac{{\alpha}_{l}-1+\varphi}{\beta}\omega $. It is noteworthy that all three parameters, β, ϕ and σ, play a role in (35), indicating that separately considering one or two parameters as in[14, 21] might lead to a local maximum.
To illustrate our resampling approach in the frequency domain, we consider the channel example specified in Table1, where we also compare the properties of the resampled FD channel (i.e., the condition number and diagonal power ratio of the channel matrix) with the original MSML FD channel. A geometric interpretation may help to understand our resampling operation since β rotates the FD matrix through$\left(\right)close="">{\xi}_{l,\text{F1}}=\frac{{\alpha}_{l}-\beta}{\beta}$, ϕ shifts the FD matrix through$\left(\right)close="">{\xi}_{l,\text{F2}}=\frac{{\alpha}_{l}-1+\varphi}{\beta}\omega $ in (16), and σ influences the phase of each element in (35). The joint effect of these actions maximizes the matrix diagonal energy. The yielded resampling (β_{F,⋆}ϕ_{F,⋆}σ_{F,⋆})=(1.015,−0.015,−15.00) corresponds to a maximal diagonal power ratio$\left(\right)close="">{\rho}_{\mathrm{F}}^{({\beta}_{\mathrm{F},\star},{\varphi}_{\mathrm{F},\star},{\sigma}_{\mathrm{F},\star})}=0.9279$. We underscore that the condition number is already significantly reduced, solely by the optimum resampling, from 4.26×10^{5}to 23.36. In comparison, the resampling method proposed in[14] yields (β ϕ σ)=(1.016,0,0) and$\left(\right)close="">{\rho}_{\mathrm{F}}^{(1.016,0,0)}=0.3623$. Its corresponding condition number is 432.78, which is larger than our condition number after resampling. This is not surprising since the criterion adopted in[14] focuses only on minimizing the aggregate errors between the multi-scale channel and its single-scale approximation, which is different from our criterion.
In the lower plots of Figure5, we show the effectiveness of diagonal preconditioning applied to the resampled channel in Table1. It is clear that, after our resampling procedure, the diagonal preconditioner clusters the eigenvalues of the preconditioned FD channel matrix closer to one than without preconditioning, which further reduces the condition number from 23.36 to 7.17. In contrast, without optimal resampling, the preconditioner “wrongly” pushes the eigenvalues closer to zero. In this case, the matrix condition number increases from 4.26×10^{5} to 1.19×10^{6}, and hence the CG equalizer performs even worse than without preconditioning as shown in the top two plots of Figure5.
Channel II: a time-domain case
Channel II | path | scale α_{ l } | delayλ_{ l } | path gain$\left(\right)close="">{\stackrel{\mathbf{\u0304}}{\mathbf{h}}}_{\mathbf{l}}$ |
---|---|---|---|---|
(T=0.2 ms | l=0 | 1.0161 | 1.00 | 0 dB |
ω=640 | l=1 | 1.0180 | 0.80 | −3 dB |
K=128) | l=2 | 1.0244 | 3.00 | −5 dB |
Parameter | Original | (β,ϕ,σ)=(1,0,0) | ||
Resampled | (β_{T,⋆},ϕ_{T,⋆},σ_{T,⋆})=(1.0160,−0.0210,−1.00) | |||
Orig./no precond. | 2.54×10^{4} | |||
Cond. Num. | Orig./with precond. | 7.37×10^{4} | ||
for TD | Resampl./no precond. | 50.78 | ||
Resampl./with precond. | 15.03 | |||
TD Ratio | Original | $\left(\right)close="">{\rho}_{F}^{(1,0,0)}=0.0021$ | ||
Resampled | $\left(\right)close="">{\rho}_{\mathrm{F}}^{({\beta}_{\mathrm{F},\star},{\varphi}_{\mathrm{F},\star},{\sigma}_{\mathrm{F},\star})}=0.9168$ |
Frequency-domain or time-domain equalization?
In the previous sections, we showed that the equalization of an OFDM channel can be implemented in either the frequency or the time domain. With the CG algorithm specified in Appendix 4, it is clear that the cost of equalization in the frequency domain will be upper-bounded by$\left(\right)close="">\mathcal{O}\left({B}_{\mathrm{F}}^{(\beta ,\varphi )}K\right)$ with$\left(\right)close="">{B}_{\mathrm{F}}^{(\beta ,\varphi )}={\mathit{\text{max}}}_{k}{B}_{\mathrm{F}}^{(\beta ,\varphi )}\left(k\right)$ for each CG iteration. Likewise, the cost of equalization in the time domain will be upper-bounded by$\left(\right)close="">\mathcal{O}\left({B}_{\text{T}}^{(\beta ,\sigma )}K\right)$ with$\left(\right)close="">{B}_{\text{T}}^{(\beta ,\sigma )}={\mathit{\text{max}}}_{m}{B}_{\text{T}}^{(\beta ,\sigma )}\left(m\right)$. By assuming that the number of CG iterations is predetermined and identical in both domains, we can use the ratio$\left(\right)close="">{B}_{\mathrm{F}}^{(\beta ,\varphi )}/{B}_{\text{T}}^{(\beta ,\sigma )}$ as a criterion to choose in which domain the equalization will be realized in order to minimize the complexity.
which suggests that if the maximum difference between the Doppler shifts of each path (i.e.,$\left(\right)close="">\frac{{\alpha}_{l}-1}{\beta}\omega $) is smaller than the maximum difference between the time shifts of each path (i.e., α_{ l }λ_{ l }), then equalization should be realized in the frequency domain; otherwise, a time-domain approach will be preferred. A similar conclusion has been made for narrowband systems[24], though its extension to wideband systems is not straightforward as shown above.
To illustrate the above idea, we again use the channel examples specified in Tables1 and2, respectively. We use B_{rul}=5 to roughly capture γ=98% of the channel energy in both domains where γ is introduced in (18). In this way, we have ε≈0.10<1 for the channel in Table1, while for the channel in Table2, we have ε≈2.00>1.
For both channels, we compare the equalization performance in different domains. OFDM with K=128 subcarriers using QPSK is transmitted and the receiver is assumed to have perfect channel knowledge. We examine the bit error rate (BER) results of our CG equalization with a fixed CG iteration number (e.g., i_{F,max}=i_{T,max}=100). We use different bandwidths for the banded approximation$\left(\right)close="">{\stackrel{\u0304}{\mathbf{H}}}_{\mathrm{F}}^{({\beta}_{\mathrm{F},\star},{\varphi}_{\mathrm{F},\star},{\sigma}_{\mathrm{F},\star})}$ and$\left(\right)close="">{\stackrel{\u0304}{\mathbf{H}}}_{\text{T}}^{({\beta}_{\text{T},\star},{\varphi}_{\text{T},\star},{\sigma}_{\text{T},\star})}$ during the equalization and the values for$\left(\right)close="">({\beta}_{\mathrm{F},\star},{\varphi}_{\mathrm{F},\star},{\sigma}_{\mathrm{F},\star})$ and$\left(\right)close="">({\beta}_{\text{T},\star},{\varphi}_{\text{T},\star},{\sigma}_{\text{T},\star})$ have also been given in Tables1 and2, respectively. After our optimal resampling in either domain, the CG equalization is carried out using the appropriate preconditioner design.
The BER performance for Channel II is illustrated in the right subplot of Figure7, where the optimal resampling parameters are (β_{T,⋆},ϕ_{T,⋆},σ_{T,⋆})=(1.016,−0.021,−1) and (β_{F,⋆},ϕ_{F,⋆},σ_{F,⋆})=(1.016,−0.016,−3). In this case, it is evident that the TD equalizer is more appealing.
These observations made for the channels in Tables1 and2 confirm our metric ε for determining which domain is more suitable for channel equalization. Additionally, we like to point out that, in either domain, with a larger bandwidth the BER performance of our CG equalization will be increased.
Numerical results
Channel parameters
Case 1: ε<1 | Case 2: ε>1 | ||||||
---|---|---|---|---|---|---|---|
K=128,ω=256 | K=128,ω=640 | ||||||
L | α _{sp} | τ_{max}/T | L | α _{sp} | τ_{max}/T | ||
5 | 0.008 | 30.00 | 5 | 0.010 | 4.00 |
The proposed resampling and preconditioning method can also benefit from other Krylov-based algorithms. For instance, the LSQR algorithm exploiting a full channel matrix is studied in[18]. Note that[18] focuses on a narrowband LTV system where no resampling is required. Further, the preconditioner given in[18] is based on a truncated basis expansion model (BEM) which is usually used for the approximation of a narrowband time-varying channel. Because it is not clear whether such a truncated BEM is still suitable for a wideband LTV channel, in order to emulate a similar approach as in[18] for constructing the preconditioner, we utilize a (trivial) full-order critically-sampled complex exponential BEM (the CCE-BEM[25]) in the simulation. The preconditioner in[18] then boils down to the inverse of the diagonal of the frequency-domain channel matrix, which is obviously sub-optimal in the Frobenius norm sense. Consequently, it is no surprise that directly applying the equalizer of[18] to wideband LTV channels yields a bad performance as shown in Figure8. In comparison, the LSQR algorithm benefiting from the optimal resampling and our preconditioner renders the fastest convergence rate and lowest BER amongst all the equalization schemes. Of course, such an improved BER performance is achieved by leveraging the full channel matrix at the cost of a higher complexity, compared to our proposed method using banded matrices.
Conclusions
In this article, we have discussed iterative equalization of wideband channels using the conjugate gradient (CG) algorithm for OFDM systems. The channel follows a multi-scale multi-lag (MSML) model, and suffers therefore from interferences in both the frequency domain and time domain. To lower the equalization complexity, the channel matrices are approximated to be banded in both domains. A novel method of optimal resampling is proposed, which is indispensable for wideband communications. A diagonal preconditioning technique, that accompanies the CG method to accelerate the convergence, has also been adapted to enhance its suitability. Experimental results have shown that our equalization scheme allows for a superior performance to those schemes based on a single-scale resampling method, without any resampling operation, or using a traditional preconditioning procedure. In addition, we gave a simple criterion to determine whether to use a frequency-domain or time-domain equalizer, depending on the channel situation, to obtain the best BER performance with the same complexity. Such a criterion is also validated by experiments.
Appendix 1
Detailed derivation of the discrete data model
where h_{ k }(t) is defined in (5) and the embedded u(t) in h_{ k }(t) is considered to be one for the concerned observation window as clarified in Remark 1.
which gives (9).
Appendix 2
System model in the time domain and time-domain equalization
where$\left(\right)close="">{\xi}_{l,\text{T}1}=\frac{{\alpha}_{l}-\beta}{\beta}$ and ξ_{l,T2}=α_{ l }(λ_{ l } + σ).
which determines the index set of the data symbols that contribute the most to the m th received signal$\left(\right)close="">{\left[{\mathbf{r}}_{\text{T}}^{(\beta ,\varphi ,\sigma )}\right]}_{m}$ via the l th path. Note that$\left(\right)close="">{B}_{\mathrm{F},l}^{(\beta ,\varphi )}\left(k\right)$ in (18) depends on the resampling factor β and the frequency shift factor ϕ, whereas$\left(\right)close="">{B}_{\text{T},l}^{(\beta ,\sigma )}\left(m\right)$ in (18) depends on the resampling factor β and the time shift factor σ.
where$\left(\right)close="">{\stackrel{\u0304}{\mathbf{v}}}_{T}^{(\beta ,\varphi ,\sigma )}=\left({\mathbf{H}}_{\text{T}}^{(\beta ,\varphi ,\sigma )}-{\stackrel{\u0304}{\mathbf{H}}}_{\text{T}}^{(\beta ,\varphi ,\sigma )}\right)\mathbf{s}.$
where$\left(\right)close="">\mathbf{s}={\mathbf{F}}_{1}^{H}\mathbf{b}$, C_{T} is the preconditioner applied in the time domain and$\left(\right)close="">{\stackrel{\u0304}{\mathbf{H}}}_{\text{TC}}^{(\beta ,\varphi ,\sigma )}={\mathbf{C}}_{\text{T}}{\stackrel{\u0304}{\mathbf{H}}}_{\text{T}}^{(\beta ,\varphi ,\sigma )}$. We first estimate s by applying the CG algorithm on$\left(\right)close="">{\mathbf{r}}_{\text{TC}}^{(\beta ,\varphi ,\sigma )}$ to invert$\left(\right)close="">{\stackrel{\u0304}{\mathbf{H}}}_{\text{TC}}^{(\beta ,\varphi ,\sigma )}$ iteratively, and afterwards we obtain$\left(\right)close="">\widehat{\mathbf{b}}={\mathbf{F}}_{1}^{H}\widehat{\mathbf{s}}$.
Appendix 3
Equalization using the conjugate gradient algorithm
where$\left(\right)close="">{\stackrel{\u0304}{\mathbf{M}}}_{\text{FC}}^{(\beta ,\varphi ,\sigma )}={\stackrel{\u0304}{\mathbf{H}}}_{\text{FC}}^{{(\beta ,\varphi ,\sigma )}^{H}}{\stackrel{\u0304}{\mathbf{H}}}_{\text{FC}}^{(\beta ,\varphi ,\sigma )}$, and$\left(\right)close="">{\widehat{\mathbf{b}}}_{\text{C}}$ is the estimate of$\left(\right)close="">{\mathbf{b}}_{\text{C}}={\mathbf{C}}_{\mathrm{F}}^{-1}\mathbf{b}$.
- 1.
Define $\left(\right)close="">{\mathbf{d}}_{\mathrm{F}}={\stackrel{\u0304}{\mathbf{H}}}_{\mathrm{F}}^{{(\beta ,\varphi ,\sigma )}^{H}}{\mathbf{r}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}$ and i=0;
- 2.Perform the following iterations:$\begin{array}{l}\text{Loop}\phantom{\rule{2em}{0ex}}\\ {\mathbf{g}}^{\left(i\right)}={\mathbf{d}}_{\mathrm{F}}-{\stackrel{\u0304}{\mathbf{M}}}_{\text{FC}}^{(\beta ,\varphi ,\sigma )}{\widehat{\mathbf{b}}}_{\text{C}}^{\left(i\right)},\phantom{\rule{2em}{0ex}}\\ {\mathbf{a}}^{\left(i\right)}=\frac{\parallel {\mathbf{g}}^{\left(i\right)}{\parallel}_{2}^{2}}{\parallel {\mathbf{g}}^{(i-1)}{\parallel}_{2}^{2}}{\mathbf{a}}^{(i-1)}+{\mathbf{g}}^{\left(i\right)},\phantom{\rule{2em}{0ex}}\\ {u}^{\left(i\right)}=\frac{\parallel {\mathbf{g}}^{\left(i\right)}{\parallel}_{2}^{2}}{{\mathbf{a}}^{{\left(i\right)}^{H}}{\stackrel{\u0304}{\mathbf{M}}}_{\text{FC}}^{(\beta ,\varphi ,\sigma )}{\mathbf{a}}^{\left(i\right)}},\phantom{\rule{2em}{0ex}}\\ {\widehat{\mathbf{b}}}_{\text{C}}^{\left(i\right)}={\widehat{\mathbf{b}}}_{\text{C}}^{(i-1)}+{u}^{\left(i\right)}{\mathbf{a}}^{\left(i\right)}\phantom{\rule{2em}{0ex}}\\ \text{End Loop};\phantom{\rule{2em}{0ex}}\end{array}$(64)
where a^{(0)}=g^{(0)}=d_{F},$\left(\right)close="">{u}^{\left(0\right)}=\frac{\parallel {\mathbf{d}}_{\mathrm{F}}{\parallel}^{2}}{{\mathbf{d}}_{\mathrm{F}}^{H}{\mathbf{M}}_{\text{FC}}^{(\beta ,\varphi ,\sigma )}{\mathbf{d}}_{\mathrm{F}}}$ and$\left(\right)close="">{\widehat{\mathbf{b}}}^{\left(0\right)}={u}^{\left(0\right)}{\mathbf{d}}_{\mathrm{F}}$;
- 3.
Perform $\left(\right)close="">{\widehat{\mathbf{b}}}^{\left(i\right)}={\mathbf{C}}_{\mathrm{F}}{\mathbf{b}}_{\text{C}}^{\left(i\right)}$, which is the i th output of the equalization process, and the index i is incremental from 0 to i _{max} where i _{max} is the iteration number when the stopping criterion of the CG is satisfied.
Notably, the optimal stopping criterion for CG can be case dependent, e.g., as discussed in[23], and is not included in this article. When our CG iterations stop, we finally have$\left(\right)close="">\widehat{\mathbf{b}}={\widehat{\mathbf{b}}}^{\left({i}_{\mathit{\text{max}}}\right)}$, which is the data estimate.
It is worthy to note that the computational complexity of each CG iteration above is determined by the complex multiplication (CM) of$\left(\right)close="">{\stackrel{\u0304}{\mathbf{M}}}_{\text{FC}}^{(\beta ,\varphi ,\sigma )}$ with a vector (e.g.$\left(\right)close="">{\widehat{\mathbf{b}}}^{\left(i\right)}$ or a^{(i)}), e.g., as in (49). When$\left(\right)close="">{\mathbf{C}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}$ is a diagonal preconditioner as considered in this article, the bandwidth of the preconditioned$\left(\right)close="">{\stackrel{\u0304}{\mathbf{H}}}_{\text{FC}}^{(\beta ,\varphi ,\sigma )}$ equals that of$\left(\right)close="">{\stackrel{\u0304}{\mathbf{H}}}_{\text{C}}^{(\beta ,\varphi ,\sigma )}$, and consequently$\left(\right)close="">{\stackrel{\u0304}{\mathbf{M}}}_{\text{FC}}^{(\beta ,\varphi ,\sigma )}$ is banded with a bandwidth$\left(\right)close="">2{B}_{\mathrm{F}}^{(\beta ,\varphi )}$ where$\left(\right)close="">{B}_{\mathrm{F}}^{(\beta ,\varphi )}=\underset{k}{max}{B}_{\mathrm{F}}^{(\beta ,\varphi )}\left(k\right)$ with$\left(\right)close="">{B}_{\mathrm{F}}^{(\beta ,\varphi )}\left(k\right)$ defined in (19). In this case, the computational complexity of each iteration is upper-bounded by$\left(\right)close="">\mathcal{O}\left({B}_{\mathrm{F}}^{(\beta ,\varphi )}K\right)$ which is linear in the vector size K.
One can also repeat the above derivations using the TD notations for the TD CG equalization.
Appendix 4
Eigenvalue locations
We consider the diagonal matrix C_{F}=diag{[c_{F,0},c_{F,1},…,c_{F,K−1}]^{ T }}, and denote the eigenvalues of$\left(\right)close="">{\stackrel{\u0304}{\mathbf{H}}}_{\text{FC}}^{(\beta ,\varphi ,\sigma )}={\stackrel{\u0304}{\mathbf{H}}}_{\mathrm{F}}^{(\beta ,\varphi ,\sigma )}{\mathbf{C}}_{\mathrm{F}}$ as {μ_{1},μ_{2},…,μ_{K−1}}.
Similarly, we can also prove that$\left(\right)close="">\sum _{k=0}^{K-1}|{\mu}_{k}-1{|}^{2}\le K{\epsilon}_{1}^{2}$ associated with (26).
Endnote
^{a} As a matter of fact, the case where α_{ l }<1 or τ_{ l }<0 can be converted to the current situation by means of proper resampling and timing at the receiver. This justifies the assumption of a compressive and causal scenario without loss of generality.
Declarations
Acknowledgements
The first author wants to thank the National University of Defense Technology, China, and also the China Scholarship Council for the financial support. This work was supported in part by NWO-STW under the VICI program (project 10382). The work of Z. Tang is also supported in part by the European Defence Agency (EDA) project RACUN (Robust Acoustic Communication in Underwater Networks). In addition, we would like to thank Dr. Magnus Lundberg Nordenvaad from the Lulea University of Technology, Sweden, Prof. Urbashi Mitra from the University of Southern California, U.S., and Prof. Huihuang Chen from the Xiamen University, China, who participated in valuable discussions.
Authors’ Affiliations
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