## 6 Frequency Domain Optimization
### Detailed Optimization Example: Circular Scan Pattern
Furthermore, the optimal scan parameters presented in this section are only valid for measurements of rotor effective wind speed; when measuring other wind variables such as shear or wind direction, the optimal parameters are typically different. For example, as discussed by [PERSON] (2015), measurements of horizontal and vertical wind shear are more accurate with larger scan radii. Additionally, as presented by [PERSON] et al. (2015), measurements of wind direction tend to improve with larger cone angles. Finally, the presence of yaw misalignment or vertical inflow may change the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{6}{c}{**Optimal Parameters for Maximizing Coherence Bandwidth**} \\ \hline \multirow{2}{*}{**Scan Pattern**} & **Coherence** & \multirow{2}{*}{\(r\)} & \multirow{2}{*}{\(d\)} & \multirow{2}{*}{\(\theta\) (deg.)} & \multirow{2}{*}{\(F\) (m)} \\  & **Bandwidth (Hz)** & & & & \\ \hline Point Measurement & 0.025 & N/A & 50 m (0.4 \(D\)) & N/A & N/A \\
2-Beam CW & 0.037 & 22.1 m (0.35 \(R\)) & 88.2 m (0.7 \(D\)) & 14.0 & 90.9 \\
2-Beam Pulsed & 0.046 & 44.1 m (0.7 \(R\)) & 163.8 m (1.3 \(D\)) & 15.1 & 169.6 \\
4-Beam CW & 0.056 & 31.5 m (0.5 \(R\)) & 75.6 m (0.6 \(D\)) & 22.6 & 81.9 \\
4-Beam Pulsed & 0.073 & 53.6 m (0.85 \(R\)) & 138.6 m (1.1 \(D\)) & 21.1 & 148.6 \\ Circular Scan CW & 0.091 & 37.8 m (0.6 \(R\)) & 75.6 m (0.6 \(D\)) & 26.6 & 84.5 \\ Rotor Disk Avg. & 0.27 & N/A & 50 m (0.4 \(D\)) & N/A & N/A \\ \hline \multicolumn{6}{c}{**Optimal Parameters for Minimizing Rotor Effective Wind Speed Measurement MSE**} \\ \hline \multirow{2}{*}{**Scan Pattern**} & **Normalized** & \multirow{2}{*}{\(r\)} & \multirow{2}{*}{\(d\)} & \multirow{2}{*}{\(\theta\) (deg.)} & \multirow{2}{*}{\(F\) (m)} \\  & \(u_{\text{\emph{eff}}}\) **MSE** & & & & \\ \hline Point Measurement & 0.242 & N/A & 50 m (0.4 \(D\)) & N/A & N/A \\
2-Beam CW & 0.151 & 25.2 m (0.4 \(R\)) & 100.8 m (0.8 \(D\)) & 14.0 & 103.9 \\
2-Beam Pulsed & 0.118 & 50.4 m (0.8 \(R\)) & 201.6 m (1.6 \(D\)) & 14.0 & 207.8 \\
4-Beam CW & 0.079 & 37.8 m (0.6 \(R\)) & 100.8 m (0.8 \(D\)) & 20.6 & 107.7 \\
4-Beam Pulsed & 0.054 & 59.9 m (0.95 \(R\)) & 151.2 m (1.2 \(D\)) & 21.6 & 162.6 \\ Circular Scan CW & 0.045 & 41 m (0.65 \(R\)) & 88.2 m (0.7 \(D\)) & 24.9 & 97.2 \\ Rotor Disk Avg. & 0.007 & N/A & 50 m (0.4 \(D\)) & N/A & N/A \\ \hline \multicolumn{6}{c}{**Optimal Parameters for Minimizing Generator Speed MSE**} \\ \hline \hline \multirow{2}{*}{**Scan Pattern**} & **Normalized** & \multirow{2}{*}{\(r\)} & \multirow{2}{*}{\(d\)} & \multirow{2}{*}{\(\theta\) (deg.)} & \multirow{2}{*}{\(F\) (m)} \\  & \(\omega_{\text{\emph{gean}}}\) **MSE** & & & & \\ \hline Point Measurement & 0.63 & N/A & 50 m (0.4 \(D\)) & N/A & N/A \\
2-Beam CW & 0.52 & 22.1 m (0.35 \(R\)) & 75.6 m (0.6 \(D\)) & 16.3 & 78.8 \\
2-Beam Pulsed & 0.45 & 44.1 m (0.7 \(R\)) & 163.8 m (1.3 \(D\)) & 15.1 & 169.6 \\
4-Beam CW & 0.38 & 31.5 m (0.5 \(R\)) & 75.6 m (0.6 \(D\)) & 22.6 & 81.9 \\
4-Beam Pulsed & 0.29 & 53.6 m (0.85 \(R\)) & 138.6 m (1.1 \(D\)) & 21.1 & 148.6 \\ Circular Scan CW & 0.23 & 37.8 m (0.6 \(R\)) & 75.6 m (0.6 \(D\)) & 26.6 & 84.5 \\ Rotor Disk Avg. & 0.046 & N/A & 50 m (0.4 \(D\)) & N/A & N/A \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimal coherence bandwidth, rotor effective wind speed measurement mean square error (MSE) (normalized by rotor effective wind speed variance), and linearized generator speed MSE (normalized by generator speed MSE with baseline feedback control) along with optimal scan parameters for each scan pattern. Optimal scan radii and preview distances are expressed in meters and as fractions of the rotor radius and preview distance, respectively. optimal scan parameters, likely favoring shorter preview distances so that the measured wind is more representative of what hits the rotor.