Assume that one OFDM symbol contains K subcarriers, denoted as $\mathit{\boldsymbol{x}}={{[{{x}_{0}},{{x}_{1}},\cdots ,{{x}_{K-1}}]}^{\text{T}}}$. The symbol duration is T. The multipath channel model is $h(\tau ) = \sum\limits_{p = 1}^{{N_p}} {{h_p}\delta (\tau - {\tau _p})} $, where ${h_p}$ is the complex path gain of the p-th path, ${\tau _p}$ is the delay of p-th path, and ${N_p}$ is the total number of multipath. Assume that the maximum delay spread is ${\tau _{\max }}$. The multipath delay can be discretized with a sample rate ${T_s} = T/K$ to create the CCE model as follows

where $ L=\left\lceil {{{\tau }_{\max }}}/{{{T}_{s}}}\; \right\rceil +1$ is the total number of grids. If all the multipath delays lie exactly on these grids, i.e., ${\tau _p} \in \mathcal{A}$, the received symbol vector can be written as[14]:

where h=[h₀, h₁, …, h_L-1] is the channel coefficient vector to be estimated, X is a diagonal matrix with vector x being its diagonal elements, n is the complex Gaussian noise vector, and W is a partial FFT matrix by selecting the first L columns of the FFT matrix, i.e.,

where ${{w}^{kl}}\triangleq \exp ({-\text{j}2\pi kl}/{K}\;)$

The above system model is based on the assumption that the multipath delays lie exactly on the discrete grid set $\mathcal{A}$, which cannot be satisfied in practical applications. The actual multipath delays are distributed on the following perturbed delay grid set

Then the received symbol vector becomes

where matrix $\mathit{\boldsymbol{W}}(\mathit{\boldsymbol{ \theta}}\text{ })$ represents the basis mismatch on the delay grid, which is expressed as below:

where $w_{{{\theta }_{l}}}^{kl}=\exp ({-j2\pi k}/{T({lT}/{K}\;+{{\theta }_{l}})}\;)$, and $\theta = {[{\theta _0}, {\theta _1}, \cdots, {\theta _{L-1}}]^{\rm{T}}}$ is the basis mismatch vector. Assume the number of pilots is P. The received pilot vector can be expressed as

where the received pilot symbol vector ${{\mathit{\boldsymbol{y}}}_{p}}=\mathit{\boldsymbol{Sy}}$. The $P \times K$ matrix S is a selection matrix by selecting the rows indexed by the pilot positions in the eye matrix. ${{\mathit{\boldsymbol{X}}}_{p}}=\mathit{\boldsymbol{SX}}{{\mathit{\boldsymbol{S}}}^{\text{T}}}$, ${{\mathit{\boldsymbol{W}}}_{p}}(\mathit{\boldsymbol{ \theta}}\text{ })=\mathit{\boldsymbol{SW}}(\mathit{\boldsymbol{ \theta}}\text{ })$, and ${{\mathit{\boldsymbol{n}}}_{p}}=\mathit{\boldsymbol{Sn}}$. The compressed channel estimation model in (3) differs from (1) that (3) considers the basis mismatch. With the basis mismatch being taken into account, the sparse recovery problem becomes the following optimization problem^[9].

which will solve for the sparse model ${{\mathit{\boldsymbol{W}}}_{P}}(\mathit{\boldsymbol{ \theta}}\text{ })$ and the associated channel coefficient vector h. $\sigma $ indicates the noise energy.

Applying the Taylor expansion to the matrix function ${{\mathit{\boldsymbol{W}}}_{P}}(\mathit{\boldsymbol{ \theta}}\text{ })$ at θ=0 gives rise to:

where $\mathit{\boldsymbol{A}}(\mathit{\boldsymbol{ \theta}}\text{ })=\text{diag}(\mathit{\boldsymbol{\theta }}\text{ })$, $\mathit{\boldsymbol{W}}_{p}^{(i)}=\left[ \frac{{{\partial }^{i}}{{\mathit{\boldsymbol{w}}}_{0}}}{{{\partial }^{i}}{{\mathit{\boldsymbol{ \theta}}\text{ }}_{0}}^{i}},\frac{{{\partial }^{i}}{{\mathit{\boldsymbol{w}}}_{1}}}{{{\partial }^{i}}{{\mathit{\boldsymbol{ \theta}}\text{ }}_{1}}^{i}},\cdots ,\frac{{{\partial }^{i}}{{\mathit{\boldsymbol{w}}}_{L-1}}}{{{\partial }^{i}}{{\mathit{\boldsymbol{ \theta}}\text{ }}_{L-1}}^{i}} \right]$, and w_i is the (i-1)th column of matrix ${{\mathit{\boldsymbol{W}}}_{p}}(\mathit{\boldsymbol{ \theta}}\text{ })$. With the Taylor expansion, it follows from (4) that

where $\mathit{\boldsymbol{\varPhi}}(\mathit{\boldsymbol{\theta }})={{\mathit{\boldsymbol{W}}}_{p}}\mathit{\pmb{0}}+\mathit{\boldsymbol{W}}_{p}^{(1)}(\mathit{\pmb{0}})\mathit{\boldsymbol{ A}}(\mathit{\boldsymbol{\theta}})$ is a linear function of the basis mismatch θ, and the residual term $\mathit{\boldsymbol{E}}(\mathit{\boldsymbol{h}},\mathit{\boldsymbol{ \theta}}\text{ })={{\mathit{\boldsymbol{X}}}_{p}}\sum\limits_{i=2}^{\infty }{\mathit{\boldsymbol{W}}_{p}^{(i)}\mathit{\pmb{0}}\mathit{\boldsymbol{A}}{{(\mathit{\boldsymbol{\theta }}\text{ })}^{i}}\mathit{\boldsymbol{h}}}$ is a nonlinear function of θ. When the channel coefficient vector h and basis mismatch vector θ are given, $\mathit{\boldsymbol{E}}(\mathit{\boldsymbol{h}},\mathit{\boldsymbol{ \theta}}\text{ })$ can be computed as $\mathit{\boldsymbol{E}}(\mathit{\boldsymbol{h}},\mathit{\boldsymbol{\theta }}\text{ })={{\mathit{\boldsymbol{X}}}_{p}}({{\mathit{\boldsymbol{W}}}_{p}}(\mathit{\boldsymbol{ \theta}}\text{ })-\mathit{\boldsymbol{ \varPhi}}\text{ }(\mathit{\boldsymbol{ \theta}}\text{ }))\mathit{\boldsymbol{h}}$.

The optimization problem in (5) relates to two unknown vectors, i.e., h and θ. We propose the alternating iteration recovery method to estimate the two vectors. In the (i + 1)th iteration, we assume the basis mismatch vector is fixed as θ_(i), which is obtained from the previous iteration. Every iteration is comprised of two stages. The first stage of the (i + 1)th iteration is to solve for h through the following optimization problem.

where ${\varepsilon _{(i)}}$ represents the noise energy plus the residual basis mismatch in the ith iteration. The optimization problem of (6) is convex, which is also known as the basis pursuit denoising (BPDN) problem.

With the updated channel vector h_(i+1), the basis mismatch vector can be estimated in the second stage of the (i + 1)th iteration as follows

where $\beta $ is an upper bound of the basis mismatch. With the updated h_(i+1) and θ_(i+1), the residual noise energy can be calculated as (8)

Two scenarios will be discussed in the following. If ${\alpha _{(i + 1)}} \geqslant \alpha $, it means that there still exists residual basis mismatch after the (i + 1)th iteration. And the residual basis mismatch should be considered as an additional noise to the Gaussian noise. If ${\alpha _{(i + 1)}} < \alpha $, it indicates that the estimated h_(i+1) and θ_(i+1) are "excessive matching" to (5) after the (i + 1)th iteration, which leads the iteration to be derivation from the optimal solution. In order to guarantee the convergence of the proposed AIR algorithm, the noise control term ${\varepsilon _{(i + 1)}}$ after the (i +1)th iteration should be greater than the energy of the Gaussian noise. Therefore, we update the noise control item as follows

where function max(:;:) returns the greater parameter in the bracket. When the iteration number i becomes large enough, the frequency response between the ith and (i−1) th iteration would be smaller than a pre-defined iteration stopping threshold TOL. The AIR algorithm then stops. The proposed AIR algorithm is described in detail in Algorithm 1.

Algorithm 1 AIR Algorithm

Inputs y_p, X_p, W_p(0), W_p⁽¹⁾(0), σ, β

Initialize i=0, θ₍₀₎=0, ε₍₀₎=σ, count, TOL

Repeat

i=i+1

Update the channel coefficient vector h_(i) (6)

Update the basis mismatch vector θ_(i) as per (7)

Compute the residual noise α_(i)

Update ε_(i) as per (8)

If i > count then

  Break;

End if

Until $ \frac{{{\left\| {{\mathit{\boldsymbol{w}}}_{p}}({{\mathit{\boldsymbol{ \theta}}\text{ }}_{(i)}}){{\mathit{\boldsymbol{h}}}_{(i)}}-{{\mathit{\boldsymbol{w}}}_{p}}({{\mathit{\boldsymbol{\theta}}\text{ }}_{(i-1)}}){{\mathit{\boldsymbol{h}}}_{(i-1)}} \right\|}_{2}}}{{{\left\| {{\mathit{\boldsymbol{w}}}_{p}}({{\mathit{\boldsymbol{\theta }}\text{ }}_{(i-1)}}){{\mathit{\boldsymbol{h}}}_{(i-1)}} \right\|}_{2}}}≤\text{TOL}$

Output h, θ

In this section, we will provide two theorems to prove that the proposed AIR algorithm converges to the solution of (5).

Lemma 1: The output sequence triplet $\left\{ {{\mathit{\boldsymbol{h}}}_{(i)}},{{\mathit{\boldsymbol{\theta }}\text{ }}_{(i)}},{{\varepsilon }_{(i)}} \right\}$ of the proposed AIR algorithm converges to a stationary point.

Proof: As ${\alpha _{(i + 1)}} < {\varepsilon _{(i + 1)}}$, one can infer from (6) that ${{\left\| {{\mathit{\boldsymbol{h}}}_{(i+1)}} \right\|}_{1}}≤{{\left\| {{\mathit{\boldsymbol{h}}}_{(i)}} \right\|}_{1}}$, which means that ${{\left\| {{\mathit{\boldsymbol{h}}}_{(i)}} \right\|}_{1}}$, (i=0, 1, 2, …) is a monotonically decreasing sequence. $\{\mathit{\boldsymbol{\bar{h}}},\mathit{\boldsymbol{\bar{\theta}}},\bar{\varepsilon }\} $ is defined as the accumulation point of the bounded sequences $\{{{\mathit{\boldsymbol{h}}}_{(i)}},{{\mathit{\boldsymbol{\theta }}\text{ }}_{(i)}},{{\mathit{\boldsymbol{ }}\varepsilon\text{ }}_{(i)}}\} $. Then there exist sequences $\{{{\mathit{\boldsymbol{h}}}_{({{i}_{j}})}},{{\mathit{\boldsymbol{\theta }}\text{ }}_{({{i}_{j}})}},{{\mathit{\boldsymbol{ }}\varepsilon\text{ }}_{({{i}_{j}})}}\} $ (j=0, 1, 2, …), which converge to $\{\mathit{\boldsymbol{\bar{h}}},\mathit{\boldsymbol{\bar{\theta}}},\bar{\varepsilon }\} $ as subscript j tends to infinity, where h_(ij) denotes a subsequence of h_(i). It can be seen from (7) that ${{\left\| {{f}_{{{i}_{j}}}}(\mathit{\boldsymbol{\theta }}\text{ }) \right\|}_{2}}≥{{\left\| {{f}_{{{i}_{j}}}}({{\mathit{\boldsymbol{ \theta}}\text{ }}_{{{i}_{j}}}}) \right\|}_{2}}$, where ${{f}_{{{i}_{j}}}}({\mathit{\boldsymbol{\theta }}\text{ }})={{\mathit{\boldsymbol{y}}}_{p}}-{{\mathit{\boldsymbol{X}}}_{p}}\mathit{\boldsymbol{ \varPhi}}\text{ }({\mathit{\boldsymbol{\theta }}\text{ }}\text{ }){{\mathit{\boldsymbol{h}}}_{({{i}_{j}})}}-\mathit{\boldsymbol{E}}({{\mathit{\boldsymbol{h}}}_{({{i}_{j}})}},{{\mathit{\boldsymbol{ \theta}}\text{ }}_{({{i}_{j}}-1)}})$. The inequality is equivalent to

As discussed above, the sequence $\{{{\mathit{\boldsymbol{\theta }}\text{ }}_{({{i}_{j}})}}\} $(j=1, 2, …) converges to $\{\mathit{\boldsymbol{\bar{\theta}}}\} $ as j tends to infinity. For $\forall \delta > 0$, there exists a ${j_0}$ such that when $j > {j_0}$, the inequality $\left\| {{f}_{{{i}_{j}}}}(\mathit{\boldsymbol{ \theta}}\text{ })-{{u}_{{{i}_{j}}}}(\mathit{\boldsymbol{ \theta}}\text{ }) \right\|≤\delta $ holds, where ${{u}_{{{i}_{j}}}}(\mathit{\boldsymbol{\theta }}\text{ })={{\mathit{\boldsymbol{y}}}_{p}}-{{\mathit{\boldsymbol{X}}}_{p}}\mathit{\boldsymbol{ \varPhi}}\text{ }(\mathit{\boldsymbol{\theta }}\text{ }){{\mathit{\boldsymbol{h}}}_{({{i}_{j}})}}-\mathit{\boldsymbol{E}}({{\mathit{\boldsymbol{h}}}_{({{i}_{j}})}},{{\mathit{\boldsymbol{ \theta}}\text{ }}_{({{i}_{j}})}})$. This suggests that when j is large enough, ${{u}_{{{i}_{j}}}}({\mathit{\boldsymbol{\theta }}\text{ }}\text{ })$ can be used in lieu of ${{f}_{{{i}_{j}}}}({\mathit{\boldsymbol{\theta }}\text{ }}\text{ })$ for analytical purposes. When $j \to \infty $, then $\left\| {{u}_{{{i}_{j}}}}(\mathit{\boldsymbol{\bar{\theta}}}) \right\|\to \bar{\varepsilon } $.

Define two sets ${{D}_{({{i}_{j}})}}=\left\{ \mathit{\boldsymbol{h}}\left| {{\left\| {{g}_{{{i}_{j}}}}(\mathit{\boldsymbol{h}}) \right\|}_{2}}≤{{\varepsilon }_{({{i}_{j}})}} \right. \right\}$ and $\bar{D}=\{\mathit{\boldsymbol{h}}\left| {{\left\| \bar{g}(\mathit{\boldsymbol{h}}) \right\|}_{2}}≤\sigma \right.\} $, where ${{g}_{(i)}}(\mathit{\boldsymbol{h}})={{\mathit{\boldsymbol{y}}}_{p}}-{{\mathit{\boldsymbol{X}}}_{p}}\mathit{\boldsymbol{ \varPhi}}\text{ }({{\mathit{\boldsymbol{ \theta}}\text{ }}_{(i)}})\mathit{\boldsymbol{h}}-\mathit{\boldsymbol{E}}({{\mathit{\boldsymbol{h}}}_{(i)}},{{\mathit{\boldsymbol{\theta }}\text{ }}_{(i)}})$, and $\bar{g}(\mathit{\boldsymbol{h}})={{\mathit{\boldsymbol{y}}}_{p}}-{{\mathit{\boldsymbol{X}}}_{p}}\mathit{\boldsymbol{ \varPhi}}\text{ }(\mathit{\boldsymbol{\bar{\theta}}})\mathit{\boldsymbol{h}}-\mathit{\boldsymbol{E}}(\mathit{\boldsymbol{h}},\mathit{\boldsymbol{\bar{\theta}}})$. For $\forall \mathit{\boldsymbol{h}}\in \bar{D}$, the inequality $\left\| \bar{g}(\mathit{\boldsymbol{h}}) \right\|\le \sigma \le {{\varepsilon }_{({{i}_{j}})}}$ is satisfied. We can always create a subsequence h_(ij) that satisfies $\mathop {\lim }\limits_{j \to \infty }{{\mathit{\boldsymbol{h}}}_{({{i}_{j}})}}=\mathit{\boldsymbol{\bar{h}}}$. With norm inequality, the following inequality

holds. Based on the definition of limit, it is easy to show when j is large enough, the created sequence $\left\{ {{\mathit{\boldsymbol{h}}}_{\left( {{\text{i}}_{\text{j}}} \right)}} \right\} $, (j=1, 2, …, ) always belongs to set ${D_{({i_j})}}$. For $\forall {D_{({i_j})}}$, the solution to (6) achieves the minimum ${l_1}$ norm, denoted as ${{\mathit{\boldsymbol{h}}}_{(j)}}\in {{D}_{({{i}_{j}})}}$. Since $\left\| {{\mathit{\boldsymbol{h}}}_{({{i}_{j}})}} \right\|\le \left\| {{\mathit{\boldsymbol{h}}}_{(j)}} \right\|$, when $j \to \infty $,$\mathit{\boldsymbol{\bar{h}}} $ has the smallest ${l_1}$ norm in set $D$, i.e.,

Besides, ${{\left\| {{\mathit{\boldsymbol{h}}}_{(i)}} \right\|}_{1}}$is a monotonically decreasing sequence, which means the cost per iteration may either be reduced or maintained to its value. In summary, the generated sequence triplet $\{{{\mathit{\boldsymbol{h}}}_{(i)}},{{\mathit{\boldsymbol{\theta }}\text{ }}_{(i)}},{{\varepsilon }_{(i)}}\} $ converges to a stationary point of the AIR algorithm.

Lemma 2:When the iteration number is large enough, the generated sequence pair $\{{{\mathit{\boldsymbol{h}}}_{(i)}},{{\mathit{\boldsymbol{\theta }}\text{ }}_{(i)}}\} $ can be made arbitrarily close to the solution of (5).

Proof: $({{\mathit{\boldsymbol{h}}}^{*}},{{\mathit{\boldsymbol{\theta }}\text{ }}^{*}})$ is defined as the optimal solution to (5), which is also the solution to (11) with the energy of the residual noise being $\bar \varepsilon $. Assume that $({{\mathit{\boldsymbol{h}}}^{*}},{{\mathit{\boldsymbol{\theta }}\text{ }}^{*}})$ cannot satisfy (9), which means there exists a $\mathit{\boldsymbol{{\theta }\text{ }'}}$ that $\left\| q(\mathit{\boldsymbol{{ \theta}\text{ }'}}) \right\|\le \left\| q({{\mathit{\boldsymbol{\theta }}\text{ }}^{*}}) \right\|$, where $q(\mathit{\boldsymbol{ \theta}}\text{ })={{\mathit{\boldsymbol{y}}}_{p}}-{{\mathit{\boldsymbol{X}}}_{p}}\mathit{\boldsymbol{ \varPhi}}\text{ }(\mathit{\boldsymbol{\theta }}\text{ }){{\mathit{\boldsymbol{h}}}^{*}}-\mathit{\boldsymbol{E}}({{\mathit{\boldsymbol{h}}}^{*}},\mathit{\boldsymbol{\theta }}\text{ })$. Denote by h' as the solution to

where $p(\mathit{\boldsymbol{h}})={{\mathit{\boldsymbol{y}}}_{p}}-{{\mathit{\boldsymbol{X}}}_{p}}\mathit{\boldsymbol{ }}\varPhi\text{ }(\mathit{\boldsymbol{{ }\theta\text{ }'}})\mathit{\boldsymbol{h}}-\mathit{\boldsymbol{E}}(\mathit{\boldsymbol{h}},\mathit{\boldsymbol{{ }\theta\text{ }'}})$. Since $p(\mathit{\boldsymbol{{h}'}})\le \bar{\varepsilon } $, $(\mathit{\boldsymbol{{h}'}},\mathit{\boldsymbol{{ }\theta\text{ }'}})$ is also a feasible solution to (5). Because $\left\| {\mathit{\boldsymbol{{h}'}}} \right\|\le \left\| {{\mathit{\boldsymbol{h}}}^{*}} \right\|$, this contradicts the fact that $({{\mathit{\boldsymbol{h}}}^{*}},{{\mathit{\boldsymbol{ \theta}}\text{ }}^{*}})$is the optimal solution. Hence, the optimal solution $({{\mathit{\boldsymbol{h}}}^{*}},{{\mathit{\boldsymbol{ \theta}}\text{ }}^{*}})$ satisfies (7). In summary, the optimal solution to (5) is a stationary point of the proposed AIR algorithm. According to Lemma 1, it can be concluded that when the iteration number is large enough, the generated sequence pair $({{\mathit{\boldsymbol{h}}}_{(i)}},{{\mathit{\boldsymbol{\theta }}\text{ }}_{(i)}})$ can be made arbitrarily close to the solution of (5).

In this section, we will present simulation results to validate the proposed AIR algorithm. In our simulations, the multipath delay perturbations are assumed to be distributed uniformly in $[ - \beta ,\beta ]$. The maximum delay spread ${\tau _{\max }}$ is chosen to be T=4. The complex gains of the multipaths are assumed to obey a Rayleigh distribution with an average power decreasing exponentially with the delay. The total number of subcarriers in one OFDM is 512, and the number of pilots is assumed to be 40.

Two comparative algorithms are chosen from [11] and [13], which are suitable for the real-time channel estimation. All the algorithms are based on convex optimization. Fig. 1 plots the mean square error (MSE) versus the signal-to-noise (SNR) for these comparative algorithms. The sparsity of channel coefficients is set to be 10, and the off-grid bound is set to be 0.5. As can be seen from Fig. 1, the proposed AIR algorithm performs better in the sense of the MSE than the other two comparative algorithms. This is because the proposed AIR algorithm is based upon an accurate signal model than the other algorithms. The inaccurate model is equivalent to introducing additional noise. It also can be seen that the MSE performances of these algorithms are improved with an increase in SNR. However, their MSE performances saturate when the SNR is large enough, e.g., 25 dB. This is because the basis mismatch effect cannot be completely eliminated even at high SNRs. When the SNR is large enough, the residual noise energy is equivalent to the energy of residual basis mismatch.

Figure 1.  MSE performance comparisons among algorithms with different SNRs.

Fig.2 plots the MSE performances versus sparsity for the comparative algorithms. The SNR and the off-grid bound are set to be 20 dB and 0.5, respectively. The sparsity is defined as the number of non-zero elements in the channel coefficient vector $\mathit{\boldsymbol{h}}$, i.e., ${\left\| \mathit{\boldsymbol{h}} \right\|_0}$. It can be seen that the proposed AIR algorithm achieves better estimation performance compared with other algorithms. When the channel becomes more sparse, the MSE performances of the algorithms deteriorate.

Figure 2.  MSE performance comparisons among algorithms with different sparsity

Fig. 3 shows the MSE performance versus the off-grid bound. As can be seen from the figure, the estimation performances of the three comparative algorithms are identical, when the off-grid bound is zero. However, the proposed AIR algorithm performs the best when the off-grid bound is present, although the performances of all the algorithms deteriorate with the increase of the off-grid bound. This is because the energy of the residual noise becomes large with an increase in the off-grid bound.

Figure 3.  MSE performance comparisons among algorithms with basis mismatch.

The multipath delays in a wireless channel are actually sparse in the continuous space. The so-called basis mismatch phenomenon will cause the channel estimation performance to degrade. We propose a new convex optimization based alternating iteration recovery algorithm to reduce the effect of basis mismatch. The proposed AIR algorithm alternates to solve for the sparse channel coefficient vector and the basis mismatch vector. The algorithm uses an accurate optimization model while maintaining the convexity property at the same time. As a result, the AIR algorithm is able to obtain superior channel estimation performance.