Joint Channel Estimation and Signal Detection for MIMO-OFDM: A Novel Data-Aided Approach with Reduced Computational Overhead

An OFDM frame consisting of a pilot block and a data block is formed and transmitted at each antenna, under the duration of which the corresponding channel coefficients are assumed to be constant. $P$ out of $K$ subcarriers are chosen for inserting pilot sequences, while other subcarriers are used for data transmission. For more accurate estimation, the subcarriers taken up by pilot symbols are spaced evenly in $\lfloor{\frac{K}{P}}\rfloor$ intervals. Moreover, pilot sequences transmitted at different antennas are required to be orthogonal. The received signals corresponding to ${N_{\text{R}}}{N_{\text{T}}}$ sets of OFDM channels are demanded.

Specifically, the output at the $m$-th receiving antenna corresponding to the $n$-th transmitted pilot block ${\mathbf{y}}_{m,n}^{\text{p}}={[{y_{m,n}^{{p_{1}}},\ldots,y_{m,n}^{{p_{P}}}}]^{T}}\in{\mathbb{C}^{P\times 1}}$ can be expressed as

for $m=1,2,\dots,N_{\text{R}}$ and $n=1,2,\dots,N_{\text{T}}$, where $\{{{p_{k}}}\}_{k=1}^{P}$ denotes the index for pilot subcarriers. The transmitted pilot block is denoted as ${\mathbf{X}}_{n}^{\text{p}}={\text{diag}}({{\mathbf{x}}_{n}^{\text{p}}})={\text{diag}}({x_{n}^{{p_{1}}},\ldots,x_{n}^{{p_{P}}}})\in{\mathbb{C}^{P\times P}}$, and the corresponding channel coefficients are ${\mathbf{h}}_{m,n}^{\text{p}}={[{h_{m,n}^{{p_{1}}},\ldots,h_{m,n}^{{p_{P}}}}]^{T}}$. ${\mathbf{w}}_{m,n}^{\text{p}}={[{w_{m,n}^{{p_{1}}},\ldots,w_{m,n}^{{p_{P}}}}]^{T}}$ is an additive white Gaussian noise (AWGN) vector, i.e., ${\mathbf{w}}_{m,n}^{\text{p}}\sim\mathcal{N}({{\mathbf{0}},\sigma_{w}^{2}{\mathbf{I}}})$.

As for data transmission, the information sequences are modulated with a complex $M$-ary quadrature amplitude modulation (QAM) constellation $\mathcal{A}$ and assembled as data blocks, which are represented as ${\mathbf{X}}_{n}^{\text{d}}={\text{diag}}({{\mathbf{x}}_{n}^{\text{d}}})={\text{diag}}({x_{n}^{{d_{1}}},\ldots,x_{n}^{{d_{K-P}}}})\in{\mathbb{C}^{({K-P})\times({K-P})}}$ for $n=1,2,\dots,N_{\text{T}}$. The data blocks are then transmitted over the channel simultaneously. The received data block at the $m$-th receiving antenna ${\mathbf{y}}_{m}^{\text{d}}={[{y_{m}^{{d_{1}}},\ldots,y_{m}^{{d_{K-P}}}}]^{T}}\in{\mathbb{C}^{({K-P})\times 1}}$ is denoted as

for $m=1,2,\dots,N_{\text{R}}$. Similarly, $\{{{d_{k}}}\}_{k=1}^{K-P}$ denotes the index for data subcarriers, and the noise vector has independent components with zero-mean and $\sigma_{w}^{2}$-variance. The corresponding channel coefficients are ${\mathbf{h}}_{m,n}^{\text{d}}={[{h_{m,n}^{{d_{1}}},\ldots,h_{m,n}^{{d_{K-P}}}}]^{T}}$. Notably, for specific ${d_{k}}$, the corresponding vector ${{\mathbf{y}}^{{d_{k}}}}={[{y_{1}^{{d_{k}}},\ldots,y_{{N_{\text{R}}}}^{{d_{k}}}}]^{T}}\in{\mathbb{C}^{{N_{\text{R}}}\times 1}}$ consisting of ${N_{\text{R}}}$ components from different receiving antennas can also be represented as

where ${{\mathbf{x}}^{{d_{k}}}}={[{x_{1}^{{d_{k}}},\ldots,x_{{N_{\text{T}}}}^{{d_{k}}}}]^{T}}\in{\mathbb{C}^{{N_{\text{T}}}\times 1}}$ and ${{\mathbf{H}}^{{d_{k}}}}\in{\mathbb{C}^{{N_{\text{R}}}\times{N_{\text{T}}}}}$. Consequently, at any specific data subcarrier, the receiving representation can be treated as MIMO receiving signal equivalently. The equivalent MIMO system can be represented as

where the subcarrier index ${d_{k}}$ is omitted for simplicity.

Since only $P$ subcarriers are used for pilot transmission, channel estimation is expected to recover unknown channel coefficients at data subcarriers using acquirable information at pilot subcarriers. Based on least squares (LS) estimation at pilot subcarriers, LMMSE estimation can realize such a goal utilizing the frequency correlation in MIMO-OFDM channels. Specifically, the initial LS estimation at pilot subcarriers is

and the estimated channel coefficients at all subcarriers using LMMSE algorithm is

where the correlation among subcarriers ${{\mathbf{R}}_{{\mathbf{h}}{{\mathbf{h}}^{\text{p}}}}}\in{\mathbb{C}^{K\times P}}$ and ${{\mathbf{R}}_{{{\mathbf{h}}^{\text{p}}}{{\mathbf{h}}^{\text{p}}}}}\in{\mathbb{C}^{P\times P}}$ can be acquired from the channel correlation matrix in the frequency domain, i.e., $\mathbf{R}^{\text{Freq}}$.

After acquiring LMMSE estimation of ${N_{\text{R}}}{N_{\text{T}}}$ sets of OFDM channels, channel coefficients at data subcarriers are utilized during signal detection. According to (3), at any data subcarrier ${d_{k}}$, the ${N_{\text{R}}}{N_{\text{T}}}$ estimated components can be reassembled as a matrix such that ${\hat{\mathbf{H}}^{{d_{k}}}}\in{\mathbb{C}^{{N_{\text{R}}}\times{N_{T}}}}$, which can be further simplified as ${\hat{\mathbf{H}}}$ during the operation of signal detection.

EP approximates the posterior distribution with factorized Gaussian distributions as follows [8]

where ${p_{\text{a}}}({{x_{n}}})$ is the a priori pdf of $\mathbf{x}$, and the EP solution (9) approximates (8) by recursively updating $(\bm{\gamma},\bm{\Lambda})$. After $T$ iterations, data estimates $\hat{\mathbf{x}}$ are output.

When the actual channel coefficients are not perfectly known, EP detection can be performed according to Algorithm 1. However, under such circumstances, degradation in detection performance is exhibited [9]. This performance degradation necessitates an improved design that takes the imperfect CSI into account. Specifically, the data-aided method is considered, using the estimated symbols for a more accurate estimation of channel coefficients, which in turn mitigates the influences induced by channel estimation error. The detailed design is illustrated in the next section.

For the derivation of the estimation algorithm, we begin with the equivalent representation of MIMO-OFDM system. Specifically, consider MIMO-OFDM system as a “MIMO” system with large scale, where data blocks from ${N_{\text{R}}}$ receiving antennas $\{{{\mathbf{y}}_{m}^{\text{d}}}\}_{m=1}^{{N_{\text{R}}}}$ are concatenated as a vector ${\mathbf{y}}$, such that ${\mathbf{y}}={[{{{({{\mathbf{y}}_{1}^{\text{d}}})}^{T}},\ldots,{{({{\mathbf{y}}_{{N_{\text{R}}}}^{\text{d}}})}^{T}}}]^{T}}\in{\mathbb{C}^{{N_{\text{R}}}({K-P})\times 1}}$. Therefore, the relationship between received signals and transmitted symbols is represented as

where the transmitted block ${\mathbf{X}}={{\mathbf{I}}_{{N_{\text{R}}}}}\otimes[{({{{\mathbf{1}}_{1\times{N_{\text{T}}}}}\otimes{{\mathbf{I}}_{({K-P})}}}){\text{diag}}({\mathbf{x}})}]\in{\mathbb{C}^{{N_{\text{R}}}({K-P})\times{N_{\text{R}}}{N_{\text{T}}}({K-P})}}$ is derived by the concatenation of ${N_{\text{T}}}$ transmitted blocks ${\mathbf{x}}=[{{{({{\mathbf{x}}_{1}^{\text{d}}})}^{T}},\ldots,{{({{\mathbf{x}}_{{N_{\text{T}}}}^{\text{d}}})}^{T}}}]^{T}\in{\mathbb{C}^{{N_{\text{T}}}({K-P})\times 1}}$. Similarly, the corresponding channel vector ${\mathbf{h}}$ and noise vector ${\mathbf{w}}$ can be represented as ${\mathbf{h}}=[{{{({{\mathbf{h}}_{11}^{\text{d}}})}^{T}},\ldots,{{({{\mathbf{h}}_{1{N_{\text{T}}}}^{\text{d}}})}^{T}},\ldots,{{({{\mathbf{h}}_{{N_{\text{R}}}{N_{\text{T}}}}^{\text{d}}})}^{T}}}]^{T}\in{\mathbb{C}^{{N_{\text{R}}}{N_{\text{T}}}({K-P})\times 1}}$ and ${\mathbf{w}}=[{{{({{\mathbf{w}}_{1}^{\text{d}}})}^{T}},\ldots,{{({{\mathbf{w}}_{{N_{\text{R}}}}^{\text{d}}})}^{T}}}]^{T}\in{\mathbb{C}^{{N_{\text{R}}}({K-P})\times 1}}$, respectively.

Based on the rearrangement of the MIMO-OFDM system, the LMMSE principle can be utilized. Specifically, the following Wiener-Hopf equation is employed [39]

where ${{\mathbf{C}}_{{\mathbf{yh}}}}$ and ${{\mathbf{C}}_{{\mathbf{yy}}}}$ are defined as

The derivation involves three types of random variables: the channel coefficient $h$, the transmitted symbol $x$, and the noise variable $w$ according to (15). Therefore, we state the following assumptions before computing the second-order moments in (17):

1. 1)

   The concerned random variables $h$, $x$ and $w$ are mutually independent.

2. 2)

   The correlation of channel coefficients, i.e., $\mathbb{E}\{{{h_{i}}h_{j}^{*}}\}$, is derived from the second-order statistics of MIMO-OFDM channels.

3. 3)

   Different transmitted symbols are mutually independent, i.e., $\mathbb{E}\{{{x_{i}}x_{j}^{*}}\}=\mathbb{E}\{{{x_{i}}}\}\mathbb{E}\{{x_{j}^{*}}\}$ for any $i\neq j$.

4. 4)

   The signal detection error is defined as $\Delta e=x-\hat{x}$, where the estimated symbol is obtained by the expectation of symbol, i.e., $\mathbb{E}\{x\}=\hat{x}$. That is to say, the estimation of symbols is assumed to be unbiased.

Consequently, the following statistical properties are deduced:

1. (a)

   Statistical property of noise variable: $\mathbb{E}\{{{w_{i}}}\}=0$, $\mathbb{E}\{{{w_{i}}w_{i}^{*}}\}=\sigma_{w}^{2}$, and $\mathbb{E}\{{{w_{i}}w_{j}^{*}}\}=0$ for any $i\neq j$.

2. (b)

   Statistical property of detection error: $\mathbb{E}\{{\Delta{e_{i}}}\}=0$, $\mathbb{E}\{{\Delta{e_{i}}\Delta e_{i}^{*}}\}={v_{i}}$, and $\mathbb{E}\{{\Delta{e_{i}}\Delta e_{j}^{*}}\}=0$ for any $i\neq j$, where ${v_{i}}$ is the given information in ${\mathbf{V}}_{\text{p}}^{(T)}$ computed by (12).

3. (c)

   Statistical property of transmitted symbol: $\mathbb{E}\{{{x_{i}}}\}={\hat{x}_{i}}$, $\mathbb{E}\{{{x_{i}}x_{i}^{*}}\}={{\hat{x}}_{i}}\hat{x}_{i}^{*}+{v_{i}}$, and $\mathbb{E}\{{{x_{i}}x_{j}^{*}}\}={{{\hat{x}}_{i}}\hat{x}_{j}^{*}}$ for any $i\neq j$.

According to these properties, the detailed derivation is presented in Appendix A. In general, the deduced form of LMMSE estimation in channel coefficients corresponding to data subcarriers, using estimated symbols derived in the previous JCD layer, is given by (16). The formulation is as follows

with ${\mathbf{A}}={{\mathbf{I}}_{{N_{\text{R}}}}}\otimes{{\mathbf{1}}_{1\times{N_{\text{T}}}}}\otimes{{\mathbf{I}}_{K-P}}$ and ${\mathbf{B}}={{\mathbf{C}}_{{\mathbf{hh}}}}\odot({{{\mathbf{1}}_{{N_{\text{R}}}\times{N_{\text{R}}}}}\otimes{\mathbf{V}}})$. Here, ${{\mathbf{C}}_{{\mathbf{hh}}}}$ refers to the second-order statistics of MIMO-OFDM channels, which can be acquired through ${{\mathbf{C}}_{{\mathbf{hh}}}}=\mathbb{E}\{{{\mathbf{h}}{{\mathbf{h}}^{\mathrm{H}}}}\}\in{\mathbb{C}^{{N_{\text{R}}}{N_{\text{T}}}({K-P})\times{N_{\text{R}}}{N_{\text{T}}}({K-P})}}$. ${\mathbf{V}}={\text{diag}}({{v_{1}},\ldots,{v_{{N_{\text{T}}}({K-P})}}})$ refers to the autocorrelation of signal detection error acquired in the EP detector in the previous JCD layer. Specifically, ${v_{i}}=\mathbb{E}\{{\Delta{e_{i}}\Delta e_{i}^{*}}\}$ corresponds to the $i$-th diagonal elements in ${\mathbf{V}}_{\text{p}}^{(T)}$, which is computed according to (12).

Before conducting experimental realization, the number of layers in the proposed JCD structure, i.e., $I$, is worth discussing, to provide the most effective performance promotion with the least rounds of JCD iteration. It is empirically observed that $I=2$ is preferred, reaching the most efficient improvement compared to the original design, i.e., traditional LMMSE and EP. That is to say, after acquiring the original channel estimates and performing EP detection accordingly, only one extra JCD iteration is needed. The reason for the setting of $I=2$ is discussed in Section V.

Even though the least number of JCD layers has been adopted to reduce complexity as much as possible, the proposed MJCD-LMMSE still involves substantial computational cost. Specifically, the complexity of this method is $\mathcal{O}({N_{\text{R}}^{3}N_{\text{T}}^{2}{{({K-P})}^{3}}})$. For common MIMO-OFDM configuration, for example, $8\times 8$ MIMO, $K=256$ and $P=16$, the computational cost can be prohibitive.

The high complexity of the proposed MJCD-LMMSE can be intractable, where the cost is dominated by operations of matrix multiplication rather than matrix inversion. According to (18a), the computation of ${{\mathbf{C}}_{{\mathbf{yh}}}}$ and ${{\mathbf{C}}_{{\mathbf{yy}}}}$ involve the multiplications of two matrices with dimensions of ${{N_{\text{R}}}({K-P})\times{N_{\text{R}}}{N_{\text{T}}}({K-P})}$ and ${{N_{\text{R}}}{N_{\text{T}}}({K-P})\times{N_{\text{R}}}{N_{\text{T}}}({K-P})}$, respectively. Therefore, common algorithms that substitute iterative approximation for direct matrix inversion, for example, Gauss-Seidel method [40], are not suitable for the target of reducing complexity under such circumstances.

On the other hand, there is an opportunity for further exploration into methods for reducing complexity in the derivation process of the proposed approach. Given the notable computational overhead associated with matrix multiplications, careful attention to dimensionality reduction is warranted. Fundamentally, the utilization of the LMMSE principle within OFDM subsystems emerges as a potential solution. This approach eliminates the requirement for large-dimension equivalence and serves to mitigate concerns surrounding computational costs in matrix multiplications. A detailed discussion of this concept is provided in the next section.

The orthogonality principle for LMMSE-based channel estimation in OFDM systems has been applied in the traditional estimator in JCD-1 as shown in (5)-(6). Similarly, as for application in the second JCD layer, preliminary LS estimation at data subcarriers is

However, the computation in (19) analogous to (5) is intractable, not only because the transmitted data symbols are unknown for detection, but also because the output at the $m$-th receiving antenna corresponding to the $n$-th transmitted data block ${\mathbf{y}}_{m,n}^{\text{d}}$ is not available. Instead, the $m$-th received component corresponding to the $n$-th transmitted data block can be estimated [18] through

under which circumstance the data estimates error should be considered in the derivation of LMMSE channel estimates. A principle is presented in [35] for measuring the influence of symbol error through a weighting matrix. However, the decision of the matrix elements is made through a fixed threshold and is exclusively deduced for BEM-based methods under the Gaussian assumption of expansion coefficients. Therefore, this scheme does not apply to the general form of LMMSE estimates, let alone the potential inaccuracy induced by the nonuse of true data error information.

In our proposed method, the estimated received component is equivalently represented as

where ${{\mathbf{Z}}_{m,n}}$ includes all interference and is treated as the equivalent noise term

among which ${{\mathbf{\Delta h}}_{m,n^{\prime}}^{\text{d}}}$ and ${{\mathbf{\Delta}}{{\mathbf{E}}_{n^{\prime}}}}$ denote the channel estimation error and signal detection error, respectively

Consequently, the LS computation in (19) is now replaced by

and the LMMSE estimation is denoted by

which requires the derivation of weight ${\mathbf{W}}_{{\text{LMMSE}}}^{{\text{new}}}$ defined as follows

The proposed algorithm is summarized in Algorithm 2, and the detailed derivation process is written in Appendix B. Note that ${{\mathbf{V}}_{n}}={\text{diag}}({{v_{1}},\ldots,{v_{K-P}}})$ refers to the autocorrelation of signal detection error at the $n$-th transmitted data block acquired in EP detector in the first layer. Similarly, ${v_{i}}=\mathbb{E}\{{\Delta{e_{i}}\Delta e_{i}^{*}}\}$ corresponds to the $i$-th diagonal elements in ${\mathbf{V}}_{\text{p}}^{(T)}$, which is acquired through (12).

With the acquisition of two proposed equivalent representations of the LMMSE-based data-aided method, namely MJCD-LMMSE and OJCD-LMMSE, it is essential to conduct a comparative analysis of the computational complexity. The comparison using $\mathcal{O}(\cdot)$ notation is considered. Furthermore, for a more intuitive comparison, the floating point operations (FLOPs) ¹¹1We adopt the widely used definition of FLOPs as the number of multiply-add operations. involved in the realizations of both algorithms are considered. According to Table I, it is obvious that the realization of MJCD-LMMSE suffers from high computational cost at the $4\times 4$, $K=128$, and $P=8$ MIMO-OFDM configuration. In contrast, substituting the MJCD-LMMSE estimator with the OJCD-LMMSE estimator at JCD-2 effectively reduces the FLOPs by three orders of magnitude.

A MIMO-OFDM system configured with $4\times 4$ or $8\times 8$ antennas and $K=128$ or $256$ subcarriers is considered. The modulation type for transmitted data streams is set as quadrature phase-shift keying (QPSK) or 16-QAM. Both block-fading and time-varying scenarios are implemented, utilizing different frame structures as illustrated in Fig. 3. For evaluations of performance improvements based on the proposed algorithms, the frame design depicted in Fig. 3(a), consistent with the assumption in Section II, is utilized in Sections V-B through V-D. Conversely, for the performance assessment of the proposed receiver under various time-selective fading environments, the pilot pattern shown in Fig. 3(b) is employed. Note that the figure takes the $4\times 4$ with $K=128$, $P=8$ case as an example, and uses the corresponding pilot interval to represent the resource block. The number of JCD layers, as mentioned above, is $I=2$. Besides, in the detection module, the number of EP iterations is fixed at $T=5$, and the damping factor value is empirically set as $\beta=0.2$.

The tapped delay line (TDL) channel model is adopted during the generation of MIMO-OFDM channel datasets, where TDL-C, the non-line-of-sight (NLOS) case profile with 24 taps, is utilized. The correlation between antennas is also taken into consideration. Specifically, the Kronecker model is adopted for the representation of MIMO characteristic, i.e., ${{\mathbf{H}}_{{\text{corr}}}}={\mathbf{R}}_{\text{r}}^{1/2}{{\mathbf{H}}_{{\text{iid}}}}{\mathbf{R}}_{\text{t}}^{1/2}$, where ${\mathbf{R}}_{\text{t}}$ and ${\mathbf{R}}_{\text{r}}$ denotes the spatial correlation of transmitter and receiver, which is illustrated by a correlation coefficient $\rho$ using exponential correlation model such that the element of matrices ${r_{ij}}$ fulfills

The corresponding spatial matrices are utilized for assignment in nrTDLChannel ²²2The MATLAB 5G Toolbox function.. The configurations concerned for MIMO-OFDM system and channel generation are summarized in Table II. Note that $\rho=0$ is used for channel generation unless noted otherwise.

The high complexity of the proposed MJCD-LMMSE makes it necessary to avoid extra JCD iterations, such that the computational cost is reduced as much as possible. For discussions on the convergence of the MJCD-LMMSE-based JCD design, $4\times 4$ MIMO with $K=128$ and $P=16$ is configured. Both estimation accuracy of ${\hat{\mathbf{h}}_{{\text{LMMSE}}}}$ and detection error of ${\hat{\mathbf{x}}}$ are evaluated under the JCD setting at $I=2$ and $I=5$, respectively. The case of the traditional method, i.e., $I=1$, is utilized as the benchmark under different signal-to-noise ratios (SNRs) so that the extent of performance promotion after different JCD iterations is observed intuitively. Notably, the channel estimation accuracy is measured by the mean square error (MSE)

where ${\mathbf{h}}_{i}$ denotes the MIMO-OFDM channels corresponding to data transmission during the $i$-th test, and ${{N_{{\text{test}}}}}$ is the number of testing realizations. The signal detection error is evaluated by BER.

Fig. 4 shows the performance comparison under QPSK modulation. From the figure, it is obvious that the gains in performance occur in JCD-2, while from JCD-3 to JCD-5, the performance tends to saturate. Moreover, according to Fig. 4(a), under $\text{SNR}\in[{4,\,8}]\ \text{dB}$, the use of JCD design may lead to negative effects on estimation accuracy. This is reasonable, for the detection accuracy is poor in low SNRs according to Fig. 4(b), thus the estimated symbol ${\hat{\mathbf{X}}}$ used in MJCD-LMMSE is inaccurate, which may deviate from the assumption of unbiased estimation according to Section III-A, 4). Despite the lower estimation accuracy compared to $I=1$ in low SNRs, the corresponding detection performance does not visibly degrade, as shown in Fig. 4(b). As for $\text{SNR}\in[{12,\,28}]\ \text{dB}$, the improvement of both estimation and detection accuracy can be observed in JCD-2. The comparisons under 16-QAM modulation showcase similar results, which are not exhibited. In general, it is empirically concluded that $I=2$ provides the most efficient improvement, which is implemented in the rest of the simulation.

In this subsection, we evaluate the performance comparison between the proposed OJCD-LMMSE and MJCD-LMMSE. Theoretically, equivalence in performance is expected, as OJCD-LMMSE is derived equivalently using the LMMSE principle in OFDM subsystems. To further validate this, the following simulation is organized: Similar to the previous subsection, a MIMO-OFDM system configured with $4\times 4$ antennas and $K=128$, $P=16$ is considered, and the detection error at the output of JCD-2 under different SNRs is evaluated, where the proposed algorithms are utilized at JCD-2 for performance comparison. Besides, the case of $I=1$, i.e., traditional LMMSE and EP, is utilized as the baseline.

Fig. 5 shows the comparison of the proposed methods under QPSK modulation. On the one hand, both of the JCD designs adopting the proposed algorithms show significant improvement compared to the benchmark with traditional algorithms. On the other hand, OJCD-LMMSE provides comparable performance to that of MJCD-LMMSE based JCD structure. In summary, simulation results in Fig. 5 further validate the performance equivalence of our proposed designs, yet OJCD-LMMSE based JCD structure significantly reduces the computational cost. Therefore, OJCD-LMMSE-based JCD structure is adopted in the remainder of the performance comparisons.

Another factor that influences the performance of JCD remains for discussion, which is the length of inserted pilot sequences for preliminary estimation in JCD-1. According to (5)-(7), using more subcarriers for inserting pilots naturally leads to more accurate channel estimation with traditional LMMSE, as more information becomes available. Consequently, it is predictable that if the number of inserted pilots is sufficient, utilizing the JCD structure to facilitate more reliable EP detection becomes unnecessary since traditional LMMSE can already provide accurate estimation results. Conversely, the fewer the pilots inserted, the more significant the performance improvement offered by the proposed JCD structure compared to the baseline scenario where $I=1$. To provide empirical evidence for the aforementioned analysis, the following simulation is conducted.

The introduction of basis functions in [35] provides another baseline for conducting LMMSE-based data-aided estimation, using $M_{t}$ B-Spline functions [37, 38] and $M_{f}$ DPS sequences [36] to represent the time-frequency characteristics of fading channels. A weighting principle (i.e., WLMMSE) is designed under BEM to measure the influence of data detection error, which is also deployed in our proposed system. In this section, simulations are conducted under different time-varying scenarios to evaluate whether our proposed JCD receiver reaches comparable accuracy to that of BEM-based receiver, and complexity analysis is investigated.

The $4\times 4$ MIMO system with $K=128$ subcarriers is configured, and the spatial correlation coefficient is set to $\rho=0.5$. The maximum Doppler shift for generating time-varying TDL channels is determined based on the velocity $v\in\{100,300\}$ km/h. The transmission frame shown in Fig. 3(b) is adopted, where ${N_{\text{p}}}$ out of 14 OFDM symbols are allocated for pilot transmission. Traditional LMMSE and EP methods are employed in the first layer, consistent with the previous evaluations. For BEM-based receivers,³³3BEM-WLMMSE integrates DPS and B-Spline basis functions to model fading across both time and frequency domains, utilizing time-frequency concatenation for data-aided computations. Conversely, the DPS-WLMMSE method applies DPS basis functions to model frequency-domain autocorrelation at each time instant, offering a significant reduction in computational complexity. the JCD loop requires $I=11$ iterations,⁴⁴4Note that $I=11$ is selected for BEM-based methods to achieve optimal performance. However, performance at $I=2$ is also included in Fig. 10 to ensure a fair comparison of computational complexity. A total of $M_{f}=9$ DPS sequences are chosen based on delay spread characteristics, while $M_{t}=4$ is empirically selected to optimize the tradeoff between computational complexity and accuracy [38].