# 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
Algorithm 3 shows LS Channel Estimation with increased Pilot Number. **A**. _Optimization Problem for Number of Pilots \(N_{p}\)_

To optimize the number of pilot symbols, we formulate a minimization problem that balances channel estimation accuracy and pilot overhead while ensuring SNR constraints are met, as given below. Subject to:

\[N_{min}\leq N_{p}\leq N_{max} \tag{16}\]

\[\frac{1}{N_{p}}\sum_{i=1}^{N_{p}}\frac{\mathbf{P}_{i}|\mathbf{H}_{i}|^{2}}{\sigma^{2}} \geq\gamma_{min} \tag{17}\]

Optimal solution [24]:

\[N_{p}^{*}=\underset{N_{p}}{\arg}_{N_{p}}\Bigg{[}\frac{1}{N_{p}}\sum_{i=1}^{N_{ p}}\frac{\mathbf{P}_{i}|\mathbf{H}_{i}|^{2}}{\sigma^{2}}\Bigg{]} \tag{18}\]

\(N_{p}\) represents the number of pilot symbols, constrained between \(N_{min}\) and \(N_{max}\). The objective is to minimize channel estimation error while ensuring that the received pilot SNR meets a required threshold \(\gamma_{min}\). The optimal number of pilots, \(N_{p}^{*}\), is determined by balancing pilot overhead and estimation accuracy. ```
Input:
1. Define system parameters: - \(N\) subcarriers and \(P\) initial pilot subcarriers; - \(S_{p}\) pilots; - \(Y_{p}\) received signal matrix; - Select new increased number of pilots \(\mathbf{H}_{p}^{(i)}\). Pilot symbol insertion:
2. Step 1: spread pilots over \(P_{n}\) subcarriers. 3. Step 2: define new pilot symbols \(\mathbf{S}_{p}^{n}\). LS estimation:
4. Step 1: compute received pilots \(Y_{p}^{n}\). 5. Step 2: estimate channel on pilots : \(\hat{\mathbf{H}}_{p}^{n}=\frac{\mathbf{Y}_{p}^{n}}{\mathbf{S}_{p}^{n}}=\ \mathbf{S}_{p}^{nH}\left( \mathbf{S}_{p}^{n}\ \mathbf{S}_{p}^{nH}\right)^{-1}\mathbf{Y}_{p}^{n}\). Improved interpolation:
6. Interpolate over all subcarriers : \(\hat{\mathbf{H}}_{Total}=Interp\left(\hat{\mathbf{H}}_{p}^{n}\right)\). 7. Low-pass filtering in frequency domain. Output:
8. Estimated channel response \(\hat{\mathbf{H}}=\left\{\hat{\mathbf{H}}_{Total}\right\}\). ```
**Algorithm 3**Improved LS Channel Estimation with Increased Pilot Number

#### 3.1.4 Adding Pilot Iteration to Channel Estimation [10]

In LIS systems, iterative pilot-based channel estimation techniques can significantly enhance estimation accuracy, especially in complex or interference-prone environments. We can use the identified data to improve channel estimation in subsequent iterations. In iterative channel estimation, the channel estimate is updated over several iterations using the estimated channel from the previous iteration. This technique is particularly useful when the channel is subject to noise, interference, and fading, which makes single-pass estimation insufficient. After the first pass, we can use the output of the initial estimation to refine the channel estimate. This can be achieved by applying the data symbols in the next iteration to further improve the estimation [35]. Applying pilot iteration improves the estimation accuracy and interference mitigation but increases the computational complexity and energy consumption. Proper tuning ensures spectral efficiency gains while minimizing processing overhead. Algorithm 4 improves LS estimation by using an iterative process. It repeats the estimation steps multiple times, making each iteration's channel estimation more accurate. ### Optimization Problem for Number of Pilot Iterations \(N_{iter}\)

To optimize the number of pilot symbols, we formulate a minimization problem that balances channel estimation accuracy and pilot overhead while ensuring SNR constraints are met, as given below. Subject to:

The number of iterations does not exceed a predefined limit. \[\hat{\mathbf{H}}_{t}=\mathbf{S}^{H}\Big{(}\mathbf{S}\,\mathbf{S}^{H}\Big{)}^{-1}\mathbf{Y}_{t}\]0\[\hat{\mathbf{H}}_{t}=\mathbf{S}^{H}\Big{(}\mathbf{S}\,\mathbf{S}^{H}\Big{)}^{-1}\mathbf{Y}_{t}\]1

The iterative refinements stop when the channel estimate improvement is below the threshold \(\mathbf{\epsilon}\).