My example note

Structure

Structuring goal per chapter:

• Intuition / motivation (why it’s useful)
• Mental model, diagrams, analogies (no equation)
• Method (Equations, algorithms, procedures)
• How to use the method, step by step
• Key takeaways / summary / typical pitfalls
• Applications or examples
• Related content

Use question driven titles :
Bad:
“Kalman Filter”

Good:
“How do we optimally estimate state under noise?”

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1. [Question-Driven Title: e.g. How do we filter noise?]

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1.1 Intuition & Mental Model

CONTEXT

Goal: [One sentence: What physical problem are we solving?]
Analogy: [A non-technical comparison, e.g., “Think of voltage like water pressure”]

[Concept Name]:
Explanation of the mechanism without heavy math. Because the image is floated right, this text wraps around it.

• Key Insight: The system reacts to change, not steady state.
• Visual Cue: Note the curve in the diagram.

1.2 The Method (Theory)

Def

[Formal Definition]: Precise mathematical or logical definition of the concept.

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Governing Equation:
Description of the variables.

• $x(t)$: Input signal.
• $h(t)$: Impulse response.

Form

$y(t)={\int }_{-\infty }^{\infty }x(\tau )h(t-\tau )d\tau$

Thm

[Important Theorem Name]:
If the system is LTI, the output is the convolution of input and impulse response.

[!warning]

Pay attention to this

1.3 Deep Dive / Derivation

Critical Assumption:
This only holds true if $\text{Linearity}$ is preserved.

Geometric Interpretation:
As shown in the sketch:

1. The vector rotates counter-clockwise.
2. Magnitude remains constant at $|A|$.

Feature	Behavior	Description
Direction	Counter-clockwise	The vector follows a positive angular rotation.
Magnitude	Constant	The value remains fixed at $
Trajectory	Circular	The path traces a circle in the complex plane.

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2. Application & Execution

2.1 How to Use It (Algorithm)

Standard Procedure:
3. Identify: Check if the system is linear.
4. Transform: Convert to Frequency domain ($\mathcal{F}$).
5. Solve: Multiply $Y(\omega )=X(\omega )H(\omega )$.
6. Invert: Convert back to Time domain.

2.2 Implementation (Code/Logic)

[Algorithm Name] Implementation:

def compute_signal(x, h):
    # Convolution operation
    N = len(x) + len(h) - 1
    y = zeros(N)
    for i in range(N):
        # ... math logic ...
    return y

2.3 System Behavior (Diagrams)

Flow Logic:

graph LR
    A[Input Signal] --> B[Filter System]
    B -->|Noise Removed| C[Clean Output]
    B -->|High Freq| D[Ground]

2.4 Summary & Traps

KEY TAKEAWAYS

• Main Idea: Convolution in time is multiplication in frequency.
• Utility: Simplifies solving differential equations.

EXAM TRAPS

• ⚠️ Trap 1: Forgetting the minus sign in the exponent.
• ⚠️ Trap 2: Confusing $H(s)$ (Laplace) with $H(j\omega )$ (Fourier).

Related: Next Chapter Title, Prerequisite Topic

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