In my research, I focus on the reliability enhancement of wireless communication links involving an unmanned aerial vehicle (UAV) in multiple-input multiple-output (MIMO) systems. With the widespread deployment of fifth-generation (5G) networks, the unmanned aerial vehicle has become an essential component in both civil and military applications, ranging from temporary base stations to aerial relays and remote sensing platforms. However, the high mobility of the unmanned aerial vehicle introduces severe time-varying channel conditions, which impose significant challenges on channel estimation and beam tracking. In this thesis, I explore two key techniques: a Kalman filter (KF) based channel estimation scheme using a sparse pilot pattern, and an extended Kalman filter (EKF) beam tracking scheme assisted by a reconfigurable intelligent surface (RIS). I systematically analyze the system models, derive the estimation algorithms, and evaluate the performance through computer simulations.
The wireless channel between an unmanned aerial vehicle and a ground terminal is characterized by both large-scale fading and small-scale fading. The large-scale fading includes path loss and shadowing, while the small-scale fading arises from multipath propagation and Doppler effects. In UAV communications, the Doppler effect is particularly pronounced because of the rapid relative movement between the airborne platform and the ground node. The Doppler shift can be expressed as:
$$f_d = \frac{v}{\lambda} \cos\theta$$
where \(v\) is the relative velocity, \(\lambda\) is the wavelength, and \(\theta\) is the angle between the direction of motion and the signal arrival direction. The coherence time \(T_c\) is inversely proportional to the maximum Doppler frequency \(f_m\):
$$T_c \approx \frac{1}{2 f_m}$$
For an unmanned aerial vehicle flying at a moderate speed, the Doppler shift can reach several hundred hertz at carrier frequencies around 5 GHz, which makes the channel vary rapidly within a single transmission frame. Therefore, conventional channel estimators that assume a quasi-static channel may suffer from significant performance degradation.
1. System Model and Channel Representation
I consider a downlink MIMO communication system where a UAV base station equipped with \(N_t\) transmit antennas communicates with a ground terminal equipped with \(N_r\) receive antennas. The received signal vector at time slot \(t\) is given by:
$$\mathbf{y}[t] = \mathbf{H}[t] \mathbf{x}[t] + \mathbf{n}[t]$$
where \(\mathbf{H}[t] \in \mathbb{C}^{N_r \times N_t}\) is the channel matrix, \(\mathbf{x}[t] \in \mathbb{C}^{N_t \times 1}\) is the transmitted signal vector, and \(\mathbf{n}[t] \sim \mathcal{CN}(0, \sigma_n^2 \mathbf{I})\) is the additive white Gaussian noise vector. For the time-varying channel, I adopt an autoregressive (AR) model to capture the temporal correlation:
$$\mathbf{H}[t] = \sum_{i=1}^{p} \mathbf{A}_i \mathbf{H}[t-i] + \mathbf{u}[t]$$
where \(p\) is the AR model order, \(\mathbf{A}_i\) are the AR coefficient matrices, and \(\mathbf{u}[t]\) is the process noise with covariance matrix \(\mathbf{Q}\). In my work, I use a first-order AR model (\(p=1\)) for simplicity and efficiency.
To estimate the channel effectively, I design a sparse pilot pattern that balances pilot overhead and estimation accuracy. The pilot symbols are arranged in a comb-type pattern within each coherence block, as shown in the following schematic:
| Parameter | Symbol | Typical Value |
|---|---|---|
| Number of subcarriers per symbol | \(N\) | 256 |
| Pilot spacing in frequency | \(N_d\) | 256 |
| Pilot length per symbol | \(N_p\) | 2 or 4 |
| Number of OFDM symbols per coherence time | \(N_s\) | 512 or 1024 |
| Coherence time | \(T_c\) | 0.0385 s or 0.077 s |
| Carrier frequency | \(f_c\) | 5 GHz |
| Subcarrier spacing | \(\Delta f\) | 15 kHz |
The sparse pilot pattern ensures that only a small fraction of resources is dedicated to channel estimation, while the Kalman filter can exploit the temporal correlation of the channel to improve the estimation accuracy.
2. Traditional Channel Estimation Methods
Before introducing the Kalman filter based approach, I briefly review three classical channel estimation methods: least squares (LS), minimum mean square error (MMSE), and linear MMSE (LMMSE).
2.1 LS Channel Estimation
The LS estimator minimizes the squared error between the received signal and the pilot-based signal model:
$$\hat{\mathbf{H}}_{LS} = \arg\min_{\mathbf{H}} \|\mathbf{Y} – \mathbf{X}\mathbf{H}\|^2$$
which yields the closed-form solution:
$$\hat{\mathbf{H}}_{LS} = \mathbf{X}^{-1} \mathbf{Y}$$
Assuming perfect knowledge of the transmitted pilot symbols, the LS estimate is simple to implement. However, its mean square error (MSE) is inversely proportional to the signal-to-noise ratio (SNR), and in deep fading scenarios its performance becomes poor.
2.2 MMSE Channel Estimation
The MMSE estimator incorporates the statistical knowledge of the channel. It minimizes:
$$\hat{\mathbf{H}}_{MMSE} = \mathbf{R}_{HY} \mathbf{R}_{YY}^{-1} \mathbf{Y}$$
where \(\mathbf{R}_{HY} = E\{\mathbf{H} \mathbf{Y}^H\}\) and \(\mathbf{R}_{YY} = E\{\mathbf{Y} \mathbf{Y}^H\}\). Using the relation with the LS estimate, the MMSE estimate can be expressed as:
$$\hat{\mathbf{H}}_{MMSE} = \mathbf{R}_{HH} \left( \mathbf{R}_{HH} + \frac{\sigma_z^2}{\sigma_x^2} \mathbf{I} \right)^{-1} \hat{\mathbf{H}}_{LS}$$
where \(\mathbf{R}_{HH}\) is the channel autocorrelation matrix.
2.3 LMMSE Channel Estimation
The LMMSE estimator reduces the complexity of MMSE by replacing the matrix inverse term \( (\mathbf{X}\mathbf{X}^H)^{-1} \) with its average value. The resulting expression is:
$$\hat{\mathbf{H}}_{LMMSE} = \mathbf{R}_{HH} \left( \mathbf{R}_{HH} + \frac{\beta}{SNR} \mathbf{I} \right)^{-1} \hat{\mathbf{H}}_{LS}$$
where \(\beta\) depends on the modulation constellation. The performance of LMMSE is very close to that of MMSE, while requiring significantly less computation.
In the context of UAV communications, these traditional methods may not provide sufficient accuracy because the channel changes rapidly. To overcome this limitation, I propose a Kalman filter based channel estimation scheme that dynamically tracks the channel state evolution.
3. Kalman Filter Based MIMO Channel Estimation
Kalman filtering is a recursive state estimation technique that is well suited for tracking time-varying channels. The state-space model for the MIMO channel is defined as:
State equation:
$$\mathbf{H}[t] = \mathbf{A} \mathbf{H}[t-1] + \mathbf{w}[t]$$
Measurement equation:
$$\mathbf{y}[t] = \mathbf{X}[t] \mathbf{H}[t] + \mathbf{v}[t]$$
where \(\mathbf{A}\) is the state transition matrix obtained from the AR model, \(\mathbf{w}[t]\) is the process noise with covariance \(\mathbf{Q}\), and \(\mathbf{v}[t]\) is the measurement noise with covariance \(\mathbf{R}\).
The Kalman filter operates in two phases: prediction and update.
3.1 Prediction Phase
In the prediction phase, I compute the prior estimate of the channel state and its error covariance:
$$\hat{\mathbf{H}}[t|t-1] = \mathbf{A} \hat{\mathbf{H}}[t-1|t-1]$$
$$\mathbf{P}[t|t-1] = \mathbf{A} \mathbf{P}[t-1|t-1] \mathbf{A}^H + \mathbf{Q}$$
where \(\mathbf{P}\) is the error covariance matrix.
3.2 Update Phase
In the update phase, the Kalman gain is calculated as:
$$\mathbf{K}[t] = \mathbf{P}[t|t-1] \mathbf{X}^H[t] \left( \mathbf{X}[t] \mathbf{P}[t|t-1] \mathbf{X}^H[t] + \mathbf{R} \right)^{-1}$$
The posterior state estimate and covariance are then updated:
$$\hat{\mathbf{H}}[t|t] = \hat{\mathbf{H}}[t|t-1] + \mathbf{K}[t] \left( \mathbf{y}[t] – \mathbf{X}[t] \hat{\mathbf{H}}[t|t-1] \right)$$
$$\mathbf{P}[t|t] = \left( \mathbf{I} – \mathbf{K}[t] \mathbf{X}[t] \right) \mathbf{P}[t|t-1]$$
The state transition matrix \(\mathbf{A}\) is obtained from the Yule-Walker equations. For the first-order AR model, the autocorrelation matrices are given by:
$$\mathbf{R}[l] = E\{ \mathbf{H}[t] \mathbf{H}^H[t-l] \}$$
and the AR matrix is:
$$\mathbf{A} = \mathbf{R}[1] \mathbf{R}^{-1}[0]$$
The process noise covariance is:
$$\mathbf{Q} = \mathbf{R}[0] – \mathbf{A} \mathbf{R}[1]^H$$
In my implementation, the initial channel estimate is obtained from a small number of pilot symbols using the LS method. The Kalman filter then iteratively refines the estimate as more pilot and data symbols are received.
3.3 Pilot Design and Feedback
A critical contribution of my work is the design of a sparse pilot pattern that is specifically tailored for Kalman filter based channel estimation in UAV MIMO systems. The pilot symbols are placed in a comb-type structure with a careful selection of the pilot spacing and pilot length. The design criteria include:
- Minimizing the pilot overhead while ensuring that the Kalman filter has sufficient measurements to track the channel variations.
- Maintaining orthogonality among pilot sequences transmitted from different antennas.
- Enabling reliable interpolation for data symbols between pilot positions.
After the channel estimation is performed, the ground terminal feeds back the CSI to the UAV base station. The UAV base station then uses the feedback information to design a precoder for the subsequent data transmission. In my work, I adopt the MMSE precoder given by:
$$\mathbf{F} = \alpha \mathbf{G}^H \left( \mathbf{G} \mathbf{G}^H + \frac{\sigma_n^2}{P} \mathbf{I}_{N_r} \right)^{-1}$$
where \(\alpha\) is a normalization factor, and \(\mathbf{G} = \mathbf{H}\) is the estimated channel matrix.
3.4 Simulation Results
I conducted extensive simulations to compare the proposed Kalman filter based channel estimation with the conventional LS estimator. The simulation parameters are listed in the following tables:
| Parameter | Value |
|---|---|
| Number of transmit antennas | 1 |
| Number of receive antennas | 1 |
| Doppler frequency | 11 Hz |
| Number of subcarriers per OFDM symbol | 256 |
| Number of OFDM symbols per coherence time | 512 |
| Carrier frequency | 5 GHz |
| Modulation | 4QAM |
| Channel model | Rayleigh flat fading |
Figure 1 shows the MSE performance comparison between the Kalman filter estimator and the LS estimator for the single-antenna case with 11 Hz Doppler frequency. The Kalman filter achieves a lower MSE than the LS estimator across all SNR values, which confirms the advantage of exploiting temporal correlation.
Table below summarizes the simulation parameters for the case with a lower Doppler frequency of 5.5 Hz. As expected, the performance of both estimators improves with lower Doppler, but the Kalman filter still outperforms the LS method by a clear margin.
| Parameter | Value |
|---|---|
| Number of transmit antennas | 1 |
| Number of receive antennas | 1 |
| Doppler frequency | 5.5 Hz |
| Number of subcarriers per OFDM symbol | 256 |
| Number of OFDM symbols per coherence time | 1024 |
| Carrier frequency | 5 GHz |
| Modulation | 4QAM |
| Channel model | Rayleigh flat fading |
For the MIMO case with \(N_t = N_r = 4\), the pilot length was increased to 4 to accommodate the additional channel coefficients. The performance comparison is shown in the following table:
| Parameter | Value |
|---|---|
| Number of transmit antennas | 4 |
| Number of receive antennas | 4 |
| Doppler frequency | 11 Hz |
| Number of subcarriers per OFDM symbol | 256 |
| Number of OFDM symbols per coherence time | 512 |
| Pilot length | 4 |
| Carrier frequency | 5 GHz |
| Modulation | 4QAM |
From the simulations, I observed that the Kalman filter based estimator provides a consistent MSE gain over the LS estimator in all tested scenarios. Furthermore, the performance gap becomes more pronounced in the low SNR regime, which is particularly important for UAV communication links that often operate under limited power constraints.
4. Extended Kalman Filter Beam Tracking with RIS Assistance
In the second part of my research, I investigate beam tracking in a millimeter-wave (mmWave) MIMO system assisted by a reconfigurable intelligent surface (RIS). The use of mmWave bands is attractive for UAV communications due to the abundant spectrum and high data rates, but the high path loss and sensitivity to blockage demand intelligent beam management. RIS provides an additional degree of freedom by controlling the phase shifts of the reflected signals, which can create virtual line-of-sight links when the direct path is obstructed.
4.1 System Model
Consider a downlink system where a base station (BS) with \(N_{BS}\) antennas communicates with an unmanned aerial vehicle equipped with \(N_{UAV}\) antennas through the assistance of an RIS with \(N_{RIS}\) reflecting elements. The direct path between the BS and the UAV is blocked, and the communication is established via the BS-RIS-UAV link. The received signal at the UAV is:
$$\mathbf{y}[t] = \mathbf{G} \boldsymbol{\Theta}[t] \mathbf{H}[t] \mathbf{f} s[t] + \mathbf{z}[t]$$
where \(\mathbf{G} \in \mathbb{C}^{N_{RIS} \times N_{BS}}\) is the BS-RIS channel matrix, \(\mathbf{H}[t] \in \mathbb{C}^{N_{UAV} \times N_{RIS}}\) is the RIS-UAV channel matrix at time slot \(t\), \(\boldsymbol{\Theta}[t]\) is the diagonal phase-shift matrix of the RIS, \(\mathbf{f}\) is the transmit beamforming vector, \(s[t]\) is the pilot symbol, and \(\mathbf{z}[t]\) is the noise.
For the mmWave channel, I adopt a geometric channel model with a single dominant path:
$$\mathbf{G} = \alpha_{br} \mathbf{a}_{ris}(\phi_{br}) \mathbf{a}_{bs}^H(\theta_{br})$$
$$\mathbf{H}[t] = \alpha_{ru}[t] \mathbf{a}_{uav}(\phi_{ru}[t]) \mathbf{a}_{ris}^H(\theta_{ru}[t])$$
where \(\alpha_{br}\) and \(\alpha_{ru}[t]\) are the path gains, and \(\mathbf{a}_{bs}\), \(\mathbf{a}_{ris}\), \(\mathbf{a}_{uav}\) are the array response vectors for a uniform linear array (ULA). The array response is given by:
$$\mathbf{a}( \phi ) = \frac{1}{\sqrt{N}} \left[ 1, e^{j \frac{2\pi d}{\lambda} \sin(\phi)}, \ldots, e^{j \frac{2\pi d (N-1)}{\lambda} \sin(\phi)} \right]^T$$
4.2 State and Measurement Models
To track the beam direction, I define the state vector as:
$$\mathbf{x}[t] = \left[ \operatorname{Re}(\alpha[t]), \operatorname{Im}(\alpha[t]), \theta_{ru}[t], \phi_{ru}[t] \right]^T$$
where \(\alpha[t] = \alpha_{br} \alpha_{ru}[t]\) is the composite path gain. The state evolution is modeled as a first-order Gaussian-Markov process:
$$\mathbf{x}[t] = \mathbf{F} \mathbf{x}[t-1] + \mathbf{u}[t]$$
with
$$\mathbf{F} = \operatorname{diag}(\rho, \rho, 1, 1)$$
where \(\rho\) is the correlation coefficient, and \(\mathbf{u}[t] \sim \mathcal{N}(0, \mathbf{Q}_u)\). The measurement model is:
$$\mathbf{y}[t] = \mathbf{h}(\mathbf{x}[t]) + \mathbf{z}[t]$$
where \(\mathbf{h}(\mathbf{x}[t])\) is a nonlinear function of the state because the array responses involve sinusoidal functions of the angles.
4.3 Extended Kalman Filter Algorithm
Since the measurement model is nonlinear, I use the extended Kalman filter (EKF) which linearizes the nonlinearity around the current state estimate. The EKF recursions are summarized as follows:
Prediction:
$$\hat{\mathbf{x}}[t|t-1] = \mathbf{F} \hat{\mathbf{x}}[t-1|t-1]$$
$$\mathbf{P}[t|t-1] = \mathbf{F} \mathbf{P}[t-1|t-1] \mathbf{F}^T + \mathbf{Q}$$
Update:
$$\mathbf{K}[t] = \mathbf{P}[t|t-1] \mathbf{J}^H[t] \left( \mathbf{J}[t] \mathbf{P}[t|t-1] \mathbf{J}^H[t] + \mathbf{R} \right)^{-1}$$
where \(\mathbf{J}[t]\) is the Jacobian matrix of \(\mathbf{h}\) evaluated at \(\hat{\mathbf{x}}[t|t-1]\). The Jacobian can be expressed as:
$$\mathbf{J}[t] = \begin{bmatrix} \frac{\partial \mathbf{h}}{\partial \operatorname{Re}(\alpha)} & \frac{\partial \mathbf{h}}{\partial \operatorname{Im}(\alpha)} & \frac{\partial \mathbf{h}}{\partial \theta_{ru}} & \frac{\partial \mathbf{h}}{\partial \phi_{ru}} \end{bmatrix}$$
The posterior state and covariance are updated as:
$$\hat{\mathbf{x}}[t|t] = \hat{\mathbf{x}}[t|t-1] + \mathbf{K}[t] \left( \mathbf{y}[t] – \mathbf{h}(\hat{\mathbf{x}}[t|t-1]) \right)$$
$$\mathbf{P}[t|t] = \left( \mathbf{I} – \mathbf{K}[t] \mathbf{J}[t] \right) \mathbf{P}[t|t-1]$$
To compute the Jacobian, I need the derivatives of the array response vectors with respect to the angles. For an ULA, the derivative is:
$$\frac{\partial \mathbf{a}(\phi)}{\partial \phi} = j \frac{2\pi d}{\lambda} \cos(\phi) \mathbf{D} \mathbf{a}(\phi)$$
where \(\mathbf{D} = \operatorname{diag}(0, 1, \ldots, N-1)\). This enables a straightforward calculation of the Jacobian.
4.4 Simulation Results
I simulated the EKF beam tracking algorithm with and without RIS assistance. In the RIS-assisted case, the BS transmits through the RIS, while in the no-RIS case I assumed a direct link with the same total path loss for a fair comparison. The simulation parameters are listed below:
| Parameter | Value |
|---|---|
| Number of tracking time slots | 100 |
| Correlation coefficient \(\rho\) | 0.995 |
| Number of BS antennas | 32 |
| Number of RIS elements | 32 |
| Number of UAV antennas | 1 |
| Number of multipath components | 1 |
| SNR | 10 dB or 20 dB |
Figure 2 and Figure 3 (not shown here) display the MSE performance of the EKF for different SNR values. The results demonstrate that the RIS-assisted configuration consistently yields a lower MSE compared to the no-RIS configuration. This improvement is due to the additional signal path provided by the RIS, which enhances the received signal strength and the observability of the angles.
The following table outlines the performance gain in terms of steady-state MSE for the two SNR levels:
| SNR | Without RIS | With RIS | Gain |
|---|---|---|---|
| 10 dB | 1.2e-3 | 4.5e-4 | 0.75e-3 |
| 20 dB | 3.8e-4 | 1.1e-4 | 0.27e-3 |
I also investigated the tracking behavior of the estimated angles versus the time slot index. The EKF was able to converge quickly and accurately track the varying AOD/AOA, which is essential for maintaining a reliable link between the UAV and the ground infrastructure. The improved beam tracking accuracy directly translates to higher data rates and lower outage probability for the unmanned aerial vehicle communication system.
5. Conclusion and Future Work
In this thesis, I have presented two main contributions aimed at improving the reliability of unmanned aerial vehicle MIMO communications. First, I proposed a Kalman filter based channel estimation scheme that uses a sparse pilot pattern to balance overhead and performance. The simulation results showed that the proposed estimator outperforms the conventional LS estimator in terms of MSE for various Doppler frequencies and antenna configurations. Second, I investigated an extended Kalman filter beam tracking algorithm assisted by a reconfigurable intelligent surface. The numerical results demonstrated that the RIS-assisted EKF achieves lower tracking error than the configuration without RIS, confirming the benefit of deploying RIS in UAV mmWave systems.
There are several directions for future investigation. In the channel estimation area, I plan to extend the proposed method to multi-user scenarios and to massive MIMO systems where the channel dimension is much larger. In the beam tracking area, I aim to design adaptive tracking algorithms for different types of UAV platforms, such as fixed-wing and rotary-wing drones, and to incorporate the UAV navigation data into the filtering process. Additionally, the integration of deep learning techniques with Kalman filtering could further enhance the performance in highly dynamic environments.
Overall, my work provides practical solutions for enhancing the performance of unmanned aerial vehicle communication systems through advanced signal processing algorithms. The combination of Kalman filtering, sparse pilot design, and intelligent reflecting surfaces offers a promising framework for future wireless networks that must support highly mobile aerial users.