update Mastersthesis.pdf and main.tex for enhanced clarity and detail in...

update Mastersthesis.pdf and main.tex for enhanced clarity and detail in bandpass filtering explanation

Thesis_Docs/main.tex

@@ -660,7 +660,15 @@ A multi-stage processing pipeline has been developed to isolate genuine periodic
 
 \begin{enumerate}
     \item \textbf{Bandpass Filtering:} \\
-    The initial stage involves preprocessing the raw time-series data to remove extraneous noise while preserving the essential periodic components. A bandpass filter is applied to the data, ensuring that unwanted frequency components are suppressed without introducing phase distortions. This filtering step is critical as it isolates the frequency range where the periodic signals are expected to reside, thereby laying the foundation for subsequent analysis.
+    In the opening act of the analysis, the raw time-series data is prepared by removing unwanted noise while preserving its hidden rhythmic patterns. The process begins with a calculation: determining the Nyquist frequency, which is defined as half the sampling rate. This frequency sets the ultimate limit for the representation of signal components.
+
+    Next, the desired temporal constraints—specified in seconds—are transformed into the language of frequencies. This is achieved by taking the reciprocal of the low and high periods, thus converting them into lower and upper frequency bounds. These bounds pinpoint the interval in which the periodic signals are expected to emerge.
+
+    With these frequency limits in hand, they are normalized against the Nyquist frequency, establishing a precise passband. A Butterworth filter is then crafted to operate within this band, celebrated for its smooth, flat response that faithfully preserves the integrity of the true periodic components.
+
+    Finally, the filtering is applied in a zero-phase manner, meaning that the data is processed in both forward and reverse directions. This dual-pass approach ensures that no phase distortion creeps into the signal, thereby maintaining the original timing of the periodic events.
+
+    Thus, through this thoughtful orchestration of calculations and filtering, the bandpass stage effectively isolates the frequency region where the periodic signals reside, setting the stage for subsequent, more refined analyses.
 
     \item \textbf{Permutation-Based FFT Thresholding:} \\
     Following filtering, the signal is transformed into the frequency domain using a Fast Fourier Transform (FFT). In order to distinguish significant periodic components from random noise, a dynamic threshold is computed. This threshold is derived by repeatedly randomizing the filtered data and analyzing the resulting spectral amplitudes. The underlying idea is that random permutations will destroy any inherent periodicity; therefore, frequency components in the original signal that exceed the threshold—determined based on a high confidence level—are likely to represent true periodic behavior.