# 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
Balancing pilot power is essential to maintain spectral efficiency and avoid excessive overhead [30]. Algorithm 2 improves LS estimation by using more pilot symbols power than Algorithm 1. This helps to make channel estimation more accurate, especially at a low SNR. According to the algorithm, after estimating the channel at pilot subcarriers, the next step involves interpolating these estimates to obtain the full channel response across all subcarriers. This step is essential for accurate signal reconstruction and optimal receiver performance [31]. * _Interpolation process:_ the estimated pilot-based channel matrix is interpolated across all subcarriers using a suitable interpolation function, which represents the chosen interpolation technique. * _Interpolation methods:_ commonly used interpolation techniques include the following: 1. _Linear interpolation:_ a simple method that connects pilot estimates with straight-line approximations; 2. _Spline interpolation:_ a more accurate approach that ensures smooth transitions between estimated points. * _Low-pass filtering:_ this mitigates noise effects and provides a smoother channel response by eliminating high-frequency artifacts. * _Defining the Optimization Problem_

To optimize pilot power, we need to define a function to minimize channel estimation error or reduce the BER. The proposed model is formulated as [24]:

\[\underset{P}{min}\ E\left[\left\|\hat{\textbf{H}}-\textbf{H}\right\|^{2}\right] \tag{12}\]

where \(H\) is the actual channel matrix, \(\hat{H}\) defines the estimated channel matrix using LSand, and \(P=[P_{1},\ P_{2},\ \ldots,P_{N}]\) shows the pilot power allocation vector. The sum of all allocated pilot powers must not exceed the available power budget \(P_{max}\)[32]:

\[\sum_{i=1}^{N}\textbf{P}_{i}\leq\textbf{P}_{max} \tag{13}\]

where \(N\) is the number of pilot subcarriers. Each pilot subcarrier must achieve a minimum SNR for reliable estimation [33]:

\[\frac{\textbf{P}_{i}|\textbf{H}_{i}|^{2}}{\sigma^{2}}\geq\gamma_{min},\forall_ {i}\in\{1,\ 2,\ \ldots,N\} \tag{14}\]

where \(\textbf{H}_{i}\) is **the** channel coefficient for the \(i\)th pilot subcarrier. \(\sigma^{2}\) is the noise variance (as defined in Algorithm 2). \(\gamma_{min}\) shows the minimum required SNR. A simple adaptive pilot power allocation strategy based on the estimated channel quality can be:

\[\textbf{P}_{i}\leq\frac{\textbf{P}_{max}}{N}\cdot\frac{|\textbf{H}_{i}|^{2}}{ \sum_{j=1}^{N}|\textbf{H}_{i}|^{2}} \tag{15}\]Pilots with stronger channel conditions receive higher power. The total allocated power remains within \(\mathbf{P}_{max}\). ```
Input:
1. Define system parameters: -  Define \(N\) subcarriers, \(P\) pilot subcarriers, \(\rho\) used for pilot power scaling; - \(S_{p}\) as pilots; -  Received signal matrix \(\mathbf{Y}_{p}\); -  Noise variance \(\sigma^{2}\); -  Scale pilot power : \(S_{p}^{scaled}=\rho.S_{p}\). Pilot symbol insertion:
2. Step 1: spread pilots over subcarriers. 3. Step 2: define new pilot symbols. LS estimation:
4. Step 1: compute received scaled pilots. 5. Step 2: estimate channel on pilots : \(\hat{\mathbf{H}}_{p}=\mathsf{S}^{H}\Big{(}\mathsf{S}\,\mathsf{S}^{H}\Big{)}^{-1} \mathbf{Y}_{p}\). Interpolation:
6. Interpolate over all subcarriers : \(\hat{\mathbf{H}}_{Total}=Interp\big{(}\hat{\mathbf{H}}_{p}\big{)}\). 7. Apply low-pass filtering in frequency domain. Output:
8. Estimated channel response\(\hat{\mathbf{H}}=\big{\{}\hat{\mathbf{H}}_{Total}\big{\}}\). ```
**Algorithm 2** Improved LS Channel Estimation with Increased Pilot Power

#### 3.1.3 Increasing the Number of Pilots [19]

Increasing the number of pilots in channel estimation involves transmitting a greater quantity of known pilot signals during the training phase. This approach enhances the accuracy of channel estimation by providing more reference data, reducing the impact of noise and interference. However, it comes with trade-offs such as increased overhead and reduced spectral efficiency, which should be balanced to optimize system performance [34].