## 6 Frequency Domain Optimization
### Frequency Domain Calculations for Assessing Measurement Quality  
Because lidar measurements and the rotor effective wind speed can be modeled as linear combinations of the wind speeds at different points in space, it is relatively straightforward to compute the measurement quality metrics in Section 5.1 using a frequency domain wind field model containing the power spectral densities (PSDs) of the three wind speed components and the spatial coherence between wind speeds at different points. As discussed in Section 2.4.3, direct frequency domain calculations of the metrics describing measurement quality are much more computationally efficient than calculations based on simulated lidar measurements and wind speeds. The wind field model used here consists of the Kaimal turbulence spectrum and the spatial coherence model described in the International Standard IEC 61400-1 Ed. 3 [56]. Whereas the IEC standard only describes spatial coherence for the transverse \(y\) and vertical \(z\) directions, wind evolution is modeled here using the empirical longitudinal coherence formula based on LES data developed by [PERSON] and [PERSON] [46]. Note that the theoretical longitudinal spatial coherence model described by [PERSON] [33] is often used as well. The spatial coherence model defined in the IEC standard [56] is modified further by extending the transverse and vertical spatial coherence formula used for the \(u\) component of the wind to the \(v\) and \(w\) components as well (the IEC standard defines the \(v\) and \(w\) components as having zero spatial correlation, but this could lead to unrealistically low error from LOS limitations). Although Section 5.1 describes the possibility of assessing measurement quality using the raw lidar measurement MSE (Equation (1)), it is more appropriate to filter the lidar measurements to remove unwanted noise before using them in the controller (Equation (2)) [42]. Therefore, the following three metrics introduced in Section 5.1 will be used for scan pattern optimization in this section: (1) mean square measurement error with optimal filtering; (2) generator speed variance resulting from combined feedback-feedforward control with optimally filtered lidar measurements; and (3) the 0.5 measurement coherence bandwidth. As explained in [PERSON] et al. [39] and [PERSON] [42], the optimal filter that minimizes the MSE between the lidar measurement and the true rotor effective wind speed is given by the transfer function  
\[H_{opt}\left(f\right)=\frac{S_{u_{off}u_{off}}\left(f\right)}{S_{d_{off}u_{ off}}\left(f\right)}, \tag{9}\]  
where \(S_{u_{off}u_{off}}\left(f\right)\) is the CPSD between the rotor effective wind speed and its lidar-based estimate and \(S_{d_{off}u_{off}}\left(f\right)\) is the PSD of the lidar measurement. The simplest measurement quality metric to calculate is the correlation bandwidth, based on the measurement coherence: \(\gamma_{u_{off}u_{off}}^{2}\left(f\right)\). As derived by [PERSON] and [PERSON] [42], the measurement MSE using the optimal filter from Equation (9) can be calculated by integrating over the PSD of the measurement error:  
\[\mathbf{E}\left[\left|u_{off}-H_{opt}\hat{u}_{off}\right|^{2}\right]=\int_{0} ^{\infty}S_{u_{off}u_{off}}\left(f\right)\left(1-\gamma_{u_{off}u_{off}}^{2} \left(f\right)\right)df, \tag{10}\]  
where \(S_{u_{off}u_{off}}\left(f\right)\) is the PSD of the true rotor effective wind speed. Additionally, as shown by [PERSON] and [PERSON] [42], the generator speed MSE (letting \(\omega_{gen}\) simply represent the deviation from the generator speed setpoint) resulting from the use of combined feedback-feedforward control with optimally-filtered lidar measurements can be expressed as  
\[\mathbf{E}\left[\left|\omega_{gen}\right|^{2}\right]=\int_{0}^{\infty}\left|T _{u_{gen}u_{off}}\left(f\right)\right|^{2}S_{u_{off}u_{off}}\left(f\right) \left(1-\gamma_{u_{off}u_{off}}^{2}\left(f\right)\right)df, \tag{11}\]  
where \(T_{u_{gen}u_{off}}\) is the closed-loop transfer function from rotor effective wind speed to generator speed error.