## 2 Background
### Lidar System Optimization for Control Applications
#### 2.4.3 Lidar Optimization Using Frequency Domain Techniques
Furthermore, the preview time provided by the measurement configuration must be large enough to overcome the time delay resulting from the measurement filter required to remove the uncorrelated high frequencies (discussed in [PERSON] et al. (2019); [PERSON] and [PERSON] (2018)). Adhering to the constraints, the optimal scan pattern is found to consist of six beam directions along the scan circle (requiring a scan time of 4.8 s) with a cone angle of 21.8\({}^{\circ}\), with the first range gate at 68 m (0.625 \(D\)) and a range gate spacing of 13.6 m.  
This configuration allows the range gates to radially cover almost all of the rotor area. These parameters result in a measurement transfer function cutoff frequency of 0.03 rad/m (which can be converted to a temporal frequency in Hz by multiplying by \(U/2\pi\), where \(U\) is the mean wind speed). More details about the semianalytic frequency domain model for computing measurement coherence used by [PERSON] et al. [40] can be found in the dissertation of [PERSON] [43]. [PERSON] and [PERSON] [44] use a frequency domain wind field model similar to that of [PERSON] et al. [40], combining the [PERSON] wind evolution model [33] with a standard wind field definition to semianalytically calculate measurement coherence and other metrics. However, instead of modeling the rotor disk average wind speed, the authors calculate measurement coherence for rotating \"blade effective wind speeds\", defined as the average of the longitudinal wind speeds along the length of the blade weighted by the local contribution to aerodynamic torque production. To determine measurement quality, the authors calculate the mean square error (MSE) between the lidar measurements and the blade effective wind speed, assuming an optimal minimum-MSE measurement filter is used (see [42]). Using the NREL 5 MW reference turbine model, [PERSON] and [PERSON] [44] optimize the scan parameters of a circularly-scanning hub-mounted CW lidar, finding that the measurement MSE is minimized using a scan radius of 44 m (70% R) and preview distance of 170 m (1.35 \(D\)). In the dissertation of [PERSON] [45], the blade effective wind speed model from [PERSON] and [PERSON] [44] is extended to calculate the \"rotor effective\" wind speed as well as linear horizontal and vertical shear that result from three rotating blades. The lidar scenario modeled in [45] consists of a CW lidar with three rotating beams, one for each blade. The measurements from the three beams are used to estimate the rotor effective wind speed and shear components. Wind evolution is included using an empirical exponential coherence formula based on the coherence measured in a variety of LES wind fields [46]. As described by [PERSON] [45], for a mean wind speed of 13 m/s and a turbulence intensity (TI) of 10%, the optimal scan radius and preview distance for measuring the rotor effective wind speed for the NREL 5 MW reference turbine are found to be 38 m (60% R) and 100 m (0.8 \(D\)), respectively. The same parameters for measuring horizontal and vertical shear are found to be 44 m (70% R) and 113 m (0.9 \(D\)), respectively. By exploring different wind conditions, [PERSON] [45] finds that the optimal lidar configuration changes very little as the mean wind speed varies. However, as TI increases, wind evolution becomes more severe, causing the optimal preview distances to become shorter.