# Optimization of Parametric Equalizer Filters in In-Vehicle Audio Systems with a Genetic Algorithm
## 3 Materials and Methods
### Modeling and Optimization of System Output  
In this study, to ensure an ideal audio experience in vehicle multimedia systems, the processes of signal processing, filtering, integration of amplifier, and speaker responses were detailed, and the combined effect of these components was optimized. While creating the system model, a B-weighting white-noise signal was used as the input signal, followed by the integration of parametric equalizer filters, amplifier, and speaker components, sequentially. This integration is based on the cascading method, where the output of each system component serves as the input for the next component. This method allows for the step-by-step processing of the signal and the sequential application of each component's effect. In this cascading process, the role of convolution, one of the fundamental concepts of signal processing theory, is of great importance. Convolution, as a mathematical operation, involves applying a system response (e.g., the frequency response of an equalizer filter or an amplifier) to an input signal. This process is essentially performed by \"folding\" each point of the input signal with the system response and summing the results. The convolution theorem explains the frequency domain representation of this process: the convolution of two signals (input and system response) in the time domain is equivalent to the multiplication of these signals in the frequency domain. This transformation is shown in (2). \[y(t)=x(t)*h(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau\leftrightarrow Y( f)=X(f)H(f) \tag{2}\]  
Here, \(y(t)\) represents the output signal in the time domain, \(x(t)\) denotes the input signal in the time domain, and \(h(t)\) represents the system's impulse response in the time domain. The variable \(\tau\) signifies the time at which the \(x(t)\) signal is shifted over the \(h(t)\) signal. In the frequency domain, \(Y(f)\) represents the output signal, \(X(f)\) denotes the input signal, and \(H(f)\) is the system's frequency response. This property allows engineers and designers in the field of signal processing, particularly in filter design and audio processing applications, to perform complex signal processing operations more efficiently. In this study, since the system components were modeled in the frequency domain, the convolution operation was also performed in this domain [17]. To model the resulting signal at the system output, the amplitude responses of the input signal, amplifier, and speaker components described in Figures 4-6 were first imported into Matlab. Since the signals were defined in the frequency domain, the frequency responses of the parametric equalizer filters were calculated using the \"freqz\" function,  
Figure 6: Full-range speaker frequency response. and their amplitude responses were obtained. Here, a sampling frequency of 96 kHz, which is also used in the actual setup, was employed. The convolution of these obtained signals was performed by multiplication, as shown in Equation (2), and the speaker output was modeled. In this study, statistical analyses were used to measure the performance by comparing the experimental results obtained from the speaker output. Pearson correlation analysis and RMSE methods were utilized for this purpose. Pearson correlation analysis is a statistical method used to measure the linear relationship between two data sets. This analysis helps determine how two variables change together, producing a correlation coefficient (r) between \(-1\) and \(1\). A correlation coefficient close to \(1\) indicates a strong positive linear relationship, while a value close to \(-1\) indicates a strong negative linear relationship. A value near \(0\) suggests no relationship between the two data sets. In this study, Pearson correlation analysis was used to examine the linear relationship between the obtained results. The Pearson correlation coefficient is calculated using (3).