Update gitignore, readme and explanations directory

.gitignore

@@ -1 +1,6 @@
+# Directory provided by Mike X Cohen from Udemy Course
+Fourier_intro/
+FourierTransform_course/
+# Extensions from VSCode
+.vscode/
 

README.md

@@ -1,3 +1,46 @@
-Welcome to my **Deep Learning** repository! 🐒
+# **Fourier Transform Learning Journey** 📊  
 
-This repository documents my personal journey in learning **Fourier Transform**. It includes practical tools, resources, and notes that I use in my personal projects. The goal of this repository is to provide a collection of useful materials, experiments, and insights for anyone diving into the exciting field of Deep Learning.
+Welcome to my **Deep Learning** repository! 🐒  
+
+This repository serves as a documentation of my personal journey in mastering the **Fourier Transform**, a foundational mathematical tool widely used in engineering, data science, and signal processing.
+---
+
+## **Overview** 🌟  
+
+The **Fourier Transform** is a powerful technique that allows us to analyze signals in the frequency domain, revealing how complex signals can be decomposed into simpler sinusoidal components. It has a wide range of applications in fields such as:
+
+- **Signal Processing**: Filtering, compression, and noise reduction.  
+- **Data Analysis**: Spectral analysis of time series data.  
+- **Image Processing**: Compression techniques (e.g., JPEG) and feature extraction.  
+- **Control Systems**: Stability and frequency response analysis.  
+- **Audio and Speech**: Equalization, synthesis, and noise cancellation.  
+
+This repository contains structured **notes, tools, and experiments** that I develop as I progress in my studies.  
+
+---
+
+## **What You'll Find in This Repository** 📁  
+
+1. **Learning Notes**: Summaries and explanations of key Fourier Transform concepts.  
+2. **Practical Tools**: Scripts and utilities for performing Fourier analysis (Matlab/Python-based).  
+3. **Experiments**: Step-by-step projects that demonstrate the use of Fourier Transform in real-world problems. 
+
+---
+
+## **Goals of the Repository** 🎯  
+
+- To **document my learning process** as I explore the theory and applications of the Fourier Transform.  
+- To create a **reference hub** for anyone interested in signal and frequency analysis.  
+- To share **practical implementations** and insights gained through experimentation and personal projects.  
+
+---
+
+## **Current Status** 🚧  
+
+I have started my learning journey with resources like a course on **Udemy**. The focus is currently on building a solid theoretical foundation and gradually moving into practical applications using tools like **Python** and libraries such as:  
+
+- **NumPy**: For numerical analysis.  
+- **Matplotlib**: For visualizing signals and frequency components.  
+- **SciPy**: For advanced signal processing tasks.  
+
+## 🐒🍌🐒🍌🐒🍌🐒🍌🐒🍌🐒🍌

explanations/complexnumber.md

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+# **The Importance of Complex Numbers in the Fourier Transform** 📈  
+
+Complex numbers play a **fundamental role** in the Fourier Transform because they allow us to represent oscillations (sinusoids) in a concise and elegant mathematical form. This simplifies both the theory and practical calculations, particularly when analyzing signals in the frequency domain.  
+
+---
+
+## **1. Representing Sinusoids Using Complex Numbers**  
+
+The Fourier Transform decomposes a signal into its sinusoidal components. Instead of dealing with separate sine and cosine functions, complex numbers unify these components using the **complex exponential form** of Euler's formula:
+
+\[
+e^{j\omega t} = \cos(\omega t) + j\sin(\omega t)
+\]
+
+- \( e^{j\omega t} \): A complex exponential representing an oscillation with angular frequency \( \omega \).  
+- \( j \): The imaginary unit (\( j^2 = -1 \)).  
+
+This compact representation allows us to describe both **cosine** (real part) and **sine** (imaginary part) components simultaneously.  
+
+---
+
+## **2. The Fourier Transform and Complex Numbers**  
+
+The **Fourier Transform** of a signal \( f(t) \) is defined as:  
+
+\[
+F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} \, dt
+\]
+
+Here:  
+- \( e^{-j\omega t} \) serves as a **basis function** for decomposing the signal into frequencies.  
+- \( F(\omega) \) is the resulting **complex spectrum**, which contains:  
+   - **Amplitude**: The magnitude of each frequency component.  
+   - **Phase**: The shift (angle) of each frequency component.  
+
+The use of complex numbers allows the spectrum \( F(\omega) \) to carry **both amplitude and phase information**, which is essential for accurately reconstructing the original signal.  
+
+---
+
+## **3. Why Phase Information Matters**  
+
+The phase tells us how much a sinusoid is **shifted in time** relative to its origin. If we ignored phase, we would lose critical information about the signal's structure.  
+
+For example, a sinusoidal wave with phase shift \( \phi \) can be written as:  
+
+\[
+A \cdot e^{j\phi} = A (\cos(\phi) + j\sin(\phi))
+\]
+
+Where:  
+- \( A \): Amplitude.  
+- \( \phi \): Phase shift.  
+
+The combination of amplitude and phase uniquely describes each frequency component.
+
+---
+
+## **4. Geometric Interpretation of Complex Numbers**  
+
+In the **complex plane**:  
+- The **real axis** represents the cosine (real) components.  
+- The **imaginary axis** represents the sine (imaginary) components.  
+
+A complex exponential \( e^{j\omega t} \) can be visualized as a **rotation** around the complex plane. This rotation simplifies the analysis of oscillations in signals.  
+
+---
+
+## **5. An Illustrative Example**  
+
+Let’s consider a simple signal:
+
+\[
+f(t) = \cos(2\pi t)
+\]
+
+Using Euler's formula, the cosine can be rewritten as:  
+
+\[
+\cos(2\pi t) = \frac{e^{j2\pi t} + e^{-j2\pi t}}{2}
+\]
+
+Here:  
+- \( e^{j2\pi t} \): Represents a **positive frequency** oscillation.  
+- \( e^{-j2\pi t} \): Represents a **negative frequency** oscillation.  
+
+Thus, the cosine wave is composed of two rotating phasors (complex exponentials) in the frequency domain, one rotating **counterclockwise** (positive frequency) and the other **clockwise** (negative frequency).  
+
+By expressing signals in terms of complex exponentials, the Fourier Transform can efficiently analyze these components, extracting both amplitude and phase.
+
+---
+
+## **6. Why Use Complex Numbers in Fourier Analysis?**  
+
+1. **Mathematical Simplicity**: Complex exponentials unify sine and cosine terms into a single, elegant form.  
+2. **Compact Representation**: The complex spectrum \( F(\omega) \) captures both amplitude and phase information.  
+3. **Efficient Calculations**: Many mathematical operations, such as differentiation and integration, are simplified using complex exponentials.  
+4. **Geometric Insight**: Signals can be visualized as rotations in the complex plane.  
+
+---
+
+## **Summary**  
+
+Complex numbers are crucial in the Fourier Transform because they:  
+- Represent sinusoidal oscillations using a single exponential term \( e^{j\omega t} \).  
+- Combine both amplitude and phase into a **compact form**.  
+- Provide a mathematically efficient and geometrically intuitive framework for signal analysis.  
+
+By leveraging complex numbers, the Fourier Transform becomes a powerful tool for understanding the frequency content of signals and reconstructing them accurately.
+

explanations/euler_formula.md

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+# **Euler's Formula and Its Importance** 🍌  
+
+Euler's formula is a fundamental relationship in mathematics that connects complex exponentials with trigonometric functions. It provides an elegant way to represent oscillations using complex numbers, which is essential for applications like the **Fourier Transform**.
+
+---
+
+## **Euler's Formula**  
+
+Euler's formula states:
+
+\[
+e^{j\theta} = \cos(\theta) + j\sin(\theta)
+\]
+
+Where:  
+- \( e \): The base of the natural logarithm (\( e \approx 2.718 \)).  
+- \( j \): The imaginary unit, where \( j^2 = -1 \).  
+- \( \theta \): An angle in radians. ⚠️ 
+- \( \cos(\theta) \): The real part of the expression.  
+- \( \sin(\theta) \): The imaginary part of the expression.  
+
+This formula shows that a **complex exponential** combines both cosine and sine terms, which are the building blocks of oscillations.
+
+---
+
+## **Geometric Interpretation**  
+
+Euler's formula can be interpreted geometrically in the **complex plane**:  
+- The complex number \( e^{j\theta} \) represents a point on the unit circle (radius \( 1 \)).  
+- \( \theta \) is the angle between the point and the positive real axis, measured counterclockwise.
+
+### **Key Rotations on the Complex Plane**  
+
+| Angle \( \theta \)       | \( e^{j\theta} \) | Coordinates in Complex Plane |
+|--------------------------|------------------|------------------------------|
+| \( 0 \)                  | \( 1 \)          | \( (1, 0) \)                 |
+| \( \frac{\pi}{2} \)      | \( j \)          | \( (0, 1) \)                 |
+| \( \pi \)                | \( -1 \)         | \( (-1, 0) \)                |
+| \( \frac{3\pi}{2} \)     | \( -j \)         | \( (0, -1) \)                |
+| \( 2\pi \)               | \( 1 \)          | \( (1, 0) \)                 |
+
+Thus, \( e^{j\theta} \) describes a **rotation** by angle \( \theta \) around the origin in the complex plane.
+
+---
+
+## **Decomposing Cosine and Sine**  
+
+Euler's formula can also be used to derive expressions for cosine and sine in terms of exponentials:
+
+1. **Cosine**:
+   \[
+   \cos(\theta) = \frac{e^{j\theta} + e^{-j\theta}}{2}
+   \]
+
+2. **Sine**:
+   \[
+   \sin(\theta) = \frac{e^{j\theta} - e^{-j\theta}}{2j}
+   \]
+
+These formulas are particularly useful in the **Fourier Transform**, where signals are represented as combinations of complex exponentials.
+
+---
+
+## **Example 1: Representing a Cosine Wave**  
+
+Suppose we have the cosine function:
+
+\[
+f(t) = \cos(2\pi t)
+\]
+
+Using Euler's formula, we can rewrite it as:
+
+\[
+\cos(2\pi t) = \frac{e^{j2\pi t} + e^{-j2\pi t}}{2}
+\]
+
+This means that a **cosine wave** can be expressed as the sum of two complex exponentials:  
+- \( e^{j2\pi t} \): A rotating vector in the counterclockwise direction (positive frequency).  
+- \( e^{-j2\pi t} \): A rotating vector in the clockwise direction (negative frequency).  
+
+---
+
+## **Example 2: Rotating a Point on the Unit Circle**  
+
+Let \( \theta = \pi/4 \) (45 degrees). Using Euler's formula:
+
+\[
+e^{j\frac{\pi}{4}} = \cos\left(\frac{\pi}{4}\right) + j\sin\left(\frac{\pi}{4}\right)
+\]
+
+From trigonometric values:
+
+\[
+\cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
+\]
+
+Thus:
+
+\[
+e^{j\frac{\pi}{4}} = \frac{\sqrt{2}}{2} + j\frac{\sqrt{2}}{2}
+\]
+
+This represents a point on the unit circle at a 45° angle in the complex plane.
+
+---
+
+## **Why Is Euler's Formula Important?**  
+
+1. **Compact Representation**: It unifies cosine and sine into a single exponential function.  
+2. **Signal Analysis**: Used in the Fourier Transform to represent signals as sums of complex exponentials.  
+3. **Simplified Math**: Operations like differentiation, integration, and multiplication become much easier with exponentials.  
+4. **Geometric Insight**: Provides a clear understanding of rotations in the complex plane.
+
+---
+
+## **Summary**  
+
+Euler's formula:
+
+\[
+e^{j\theta} = \cos(\theta) + j\sin(\theta)
+\]
+
+- Connects trigonometric functions and complex exponentials.  
+- Represents rotations in the complex plane.  
+- Allows us to decompose and analyze signals efficiently, particularly in **Fourier analysis**.