## 2 Material and Methods
### Power Spectral Density Features from Fast Fourier Transform  
The power spectral density (PSD) of the fast Fourier-transform (FFT) was used to analyze the time series of the drag, codend motions, and the flow field inside and behind different codend based Welch's method and using a Hann window. An examination of the power distribution throughout the whole frequency range is provided by the PSD of a signal. The primary objective of this approach is to estimate the spectral density from the provided data. It is calculated by performing an autocorrelation function FFT on the signals. It determines the signal's strength after interpreting it as a stochastic process [28; 29]. The method has been implemented using MATLAB R2019B software. During the implementation of the Fourier spectral analysis, the Welch method is used to compute spectra in order to improve statistical convergence. This is because of the complexity of the interaction between the flow and the codend structure linked to the catches that allow the need for the much longer signals in order to evaluate the predominant frequency peaks with sufficient precision. Indeed, the PSD describes how power is distributed over the frequency content of the random process. It complements the probability density function in the definition of a specific random process. An intuitive and historically important way of obtaining a PSD was by filtering and averaging random process. Abundant processing capabilities of modern computers make it possible to obtain the PSD by means of the Fourier-transform, commonly using the FFT algorithm of the Discrete Fourier-Transform (DFT) [30; 31; 32; 33; 34]. The transformed vector \(X(k)\) is generally a complex value, and the spectral content of the signal is expressed by the real value of the PSD function:  
\[S(f_{k})=S(k)=\frac{1}{N}|X(f_{k})|^{2} \tag{7}\]  
where  
\[f_{k}=\frac{k}{N} \tag{8}\]  
As the PSD is based on a finite set of samples, it can be calculated even for periodic signals. In the case of noncoherent sampling, the estimation suffers from the phenomena of picket fence and leakage. To suppress these effects, windowing techniques have been  
Figure 4: Schematic of the experimental setup of flow velocity measurement inside and behind codend: The two grey solid dot indicates the measurement points.  
developed. Windowing means that the signal \(x_{n}\) is multiplied by the so-called window function \(w_{n}\) prior to the transform:  
\[X_{w}(k)=\sum_{n=0}^{N-1}x_{n}w_{n}e^{-j\frac{2\alpha}{N}}\ n,\ k=0\ldots\ldots N-1 \tag{9}\]  
A huge set of window functions have been developed in the last decades. All of them can improve the result of the estimation, and many of them are optimal in a certain sense. A significant application of PSD calculation is the analysis of periodic or quasi-periodic signals corrupted by measurement noise. Unfortunately, the measurement noise can hinder the detection of all important harmonic components of the signal. In this case, one finite set of \(N\) samples is insufficient, a long series of samples are recorded, and many consecutive blocks of \(N\) samples are transformed, and the estimator is obtained by averaging the individual PSDs. The blocks can overlap, according to the Welch method. The mean of the individual estimates can be calculated by linear averaging:  
\[\overline{S}(k)=\frac{1}{M}\sum_{m=0}^{M-1}S_{m}(k) \tag{10}\]  
where \(\overline{S}(k)\) denotes the averaged PSD, and \(S_{m}(k)\) is the PSD of block \(m\). Exponential averaging is also commonly used, when the averaged PSD is calculated in the following way:  
\[\overline{S}(k)=\overline{S}(k)+\alpha\big{(}S_{m}(k)-\overline{S}(k)\big{)} \tag{11}\]  
where \(\alpha\) is the so-called smoothing constant, \(\overline{S}(k)\) and \(S_{m}(k)\) denote the averaged and the individual PSD, respectively. For high-precision measurements, the bias caused by the noise can be eliminated by subtracting the PSD of the noise from \(\overline{S}(k)\).