## 2 Background
### Lidar System Optimization for Control Applications
#### 2.4.3 Lidar Optimization Using Frequency Domain Techniques  
As an alternative to performing lidar simulations in stochastic or CFD-based wind fields, metrics of lidar measurement quality can often be calculated directly in the frequency domain. This approach is possible because (a) stochastic wind fields are defined by their frequency domain statistics (turbulence power spectra and spatial coherence) and (b) many useful metrics for measurement quality are either directly based on frequency domain definitions, such as the coherence bandwidth, or can be easily calculated using frequency domain information, such as mean square measurement error. Measurement coherence (the correlation between the lidar measurements and the true wind speed at the rotor plane as a function of frequency) can be calculated using a frequency domain wind field model as long as the lidar measurements and true wind speed variables are defined as linear combinations of wind speeds. Because lidar measurements and rotor effective wind speeds can be modeled as weighted averages of the wind speeds along the lidar beam and along the blades or across the rotor plane, respectively, they are well-suited for frequency domain analysis. There are two main advantages to determining measurement quality directly in the frequency domain rather than based on time domain simulations. First, generating stochastic or CFD-based wind fields and performing time domain lidar simulations is computationally expensive. Second, to arrive at statistically significant measurement quality statistics, results from many different time domain simulations must be averaged together. Frequency domain calculations, on the other hand, only need to be performed once for each lidar configuration and set of wind field parameters, and produce the exact measurement coherence. [PERSON] et al. (2019); [PERSON] (2018) combine a simple exponential longitudinal coherence model of wind evolution, introduced by [PERSON] and [PERSON] [PERSON] and [PERSON] (2018), with a standard wind field model based on frozen turbulence to form an evolving wind field model. The model is used to calculate the measurement coherence between pulsed lidar measurements and the rotor effective wind speed, defined as the spatial average of the longitudinal wind speeds across the rotor disk. Specifically, the authors calculate the \(-3\) dB cutoff frequency of the transfer function from the lidar measurements to the rotor effective wind speed, describing the bandwidth where the measurements are correlated with the true wind disturbance. In addition to lidar range weighting and LOS velocity limitations, the frequency domain measurement coherence model used in [PERSON] and [PERSON] (2018) includes the effects of sequential scanning (i.e., the use of a single laser to scan across all beam directions in a finite amount of time). In the pulsed lidar measurement model used, the velocity measurements at different range gates are delayed by the amount of time it takes them to reach the first range gate (determined by the mean wind speed according to [PERSON]'s frozen turbulence hypothesis [PERSON] (1958)). The delayed velocities, which should be roughly in phase with each other, are then averaged together to estimate the rotor effective wind speed. [PERSON] et al. (2019) describe the direct frequency domain calculation of lidar measurement coherence as a \"semianalytic\" approach because spatial averages are approximated using discretization along the lidar beams and across the rotor disk. Building upon the theory described in [PERSON] et al. (2019), [PERSON] et al. (2019) optimize the scan trajectory of a nacelle-mounted circularly-scanning pulsed lidar with five range gates for a wind turbine with a 109 m rotor diameter by maximizing the \(-3\) dB cutoff frequency of the rotor effective wind speed measurement transfer function. Subject to realistic lidar constraints, the cone angle, number of beam directions, the distance of the first range gate, and the spacing between range gates are optimized.