# Improved Low-Complexity, Pilot-Based Channel Estimation for Large Intelligent Surface Systems
## 3 Receiver Design Considerations for LIS: Proposed Improvements in Channel Estimation
### Optimized Pilot Design and Enhancing the Quality of Pilot Data
#### 3.1.2 Hadamard as Orthogonal Pilot Sequences  
Orthogonal pilot sequences are employed to minimize interference and pilot contamination. Orthogonal sequences such as Hadamard matrices ensure minimal cross-correlation between pilots transmitted by different terminals. For a system with \(T\) terminals and \(N\) pilot symbols, the pilot matrix \(S_{k}\) is constructed as \(Hadamard(N)\), where rows are assigned to terminals in a mutually orthogonal manner. This design ensures that the channel estimates for different users are independent, even under multi-user interference [26]. To generate Hadamard sequences, one should create a Hadamard matrix, where each different row represents a complete orthogonal Hadamard sequence. A Hadamard matrix is an \(n\times n\) matrix \(S\), with entries of either \(+1\) or \(-1\), and its rows are mutually orthogonal. Let \(\mathbf{H}\) be a Hadamard matrix of order \(n\). The transpose of \(\mathbf{H}\) is closely related to its inverse  
\[\mathbf{H}\,\mathbf{H}^{T}=n\mathbf{I}_{n} \tag{11}\]  
where \(\mathbf{I}_{n}\) is the \(n\times n\) identity matrix, and \(\mathbf{H}^{T}\) is the transpose of \(\mathbf{H}\). Algorithm 1 provides the initial LS estimation of the channel response matrix based on received pilot symbols. ```
Input:
1. Define system parameters: - Hadamard matrix \(\mathbf{H}\) of size \(M\), used for pilot sequences; - Received signal matrix \(\mathbf{Y}\) of size \(N\); - Noise variance \(\sigma^{2}\); - LIS configuration parameters: number of LIS antennas. Preprocessing:
2. Extract the pilot sequence matrix \(\mathbf{S}\), which corresponds to the first \(M\) rows of the Hadamard matrix \(\mathbf{H}\). 3. Initialize the channel estimate \(\hat{\mathbf{H}}\) to zero. Repeat: For each subcarrier or time slot \(t\): 4. _Step 1_: extract the received signal \(\mathbf{Y}_{t}\) for the current time slot \(t\) from the received matrix \(Y\). 5. _Step 2_: separate the pilot signals by applying the Hadamard inverse transform \(\mathbf{S}^{-1}\) to \(\mathbf{Y}_{t}\). 6. _Step 3_: estimate the channel matrix \(\hat{\mathbf{H}}_{t}\) using the LS method: \[\hat{\mathbf{H}}_{t}=\mathbf{S}^{H}\Big{(}\mathbf{S}\,\mathbf{S}^{H}\Big{)}^{-1}\mathbf{Y}_{t}\] where \(\mathbf{S}^{H}\) is the Hermitian transpose of the pilot matrix \(\mathbf{S}\). 7. _Step 4_: update the overall channel estimate \(\hat{\mathbf{H}}\) by averaging over time slots or subcarriers: \[\hat{\mathbf{H}}\rightarrow\hat{\mathbf{H}}+\frac{\hat{\mathbf{H}}_{t}}{T}\] where \(T\) is the total number of time slots or subcarriers. until:
8. The convergence criteria are met, or all subcarriers/time slots are processed. Output:
9. The estimated channel matrix \(\hat{\mathbf{H}}\). B. _Properties of Hadamard Pilots_: * _Orthogonality_: each pilot sequence is orthogonal to others, allowing easy separation at the receiver [27]. * _Low complexity_: Hadamard sequences are computationally efficient to generate and process. * _Binary nature_: They are simple to implement in hardware or software with \(+1/-1\) entries. #### 3.1.2 Pilot Power Optimization [19]

The pilot power is scaled to improve the SNR of the received pilot signals, resulting in more accurate channel estimation. When the pilot signals are stronger relative to noise, the effect of noise on the estimation process is reduced, leading to more reliable estimates of the channel coefficients [28]. Higher pilot power helps in mitigating the high-dimensional complexity of estimating channels for a massive number of antennas or surface elements. Improved estimation leads to better beamforming and higher energy efficiency, which are crucial for LIS systems. The power allocated to pilot symbols is scaled to enhance the SNR of the received pilot signals. The channel estimation process becomes more robust to noise, at the cost of slightly reduced data throughput [29]. Increasing pilot power improves estimation accuracy and the SNR, but it also raises the computational complexity, energy consumption, and potential interference.