Master Note-Making Roadmap: ECE 2107 & ECE 2108

This document serves as the absolute architectural blueprint for your academic vault. It reorganizes the entire signals, systems, and network coursework into a highly modular, exam-optimized set of standalone note-making templates.

To ensure absolute clarity, the curriculum is strictly divided by examiner:

Part I: Instructor 2 (Time-Domain Fundamentals, Systems & Network Applications)
Part II: Instructor 1 (Transform-Domain Analysis & Fast Algorithms)

Every single note-making blueprint below is engineered for A+ optimization, specifying the exact definitions, step-by-step derivations, graphical assets, numerical types, and common exam traps required to build complete, standalone study guides.

Part I: Instructor 2 (Time-Domain Fundamentals, Systems & Network Applications)

Chapter 1: Signal Fundamentals & Operations (Notes 1.00 - 1.06)
Chapter 2: LTI System Properties & Time-Domain Convolution (Notes 2.00 - 2.05)
Chapter 3: State-Space Representation of CT Systems (Notes 3.00 - 3.04)
Chapter 4: The Sampling Theorem & Multi-Rate Processing (Notes 4.00 - 4.04)
Chapter 5: Analog Filter Design (Notes 5.00 - 5.04)
Chapter 6: Two-Port Network Theory (Notes 6.00 - 6.04)

Chapter 1: Signal Fundamentals & Operations

Directory Path

01 UNI/2-1/ece 2107 signals/01 signal operations/

1.00 Chapter Map - Signal Fundamentals & Operations.md
1.01 Signals, Systems & Singularity Functions.md
1.02 Elementary Operations on Signals.md
1.03 Signal Classification I — Theory & Periodicity.md
1.04 Signal Classification II — Symmetry, Energy & Power.md
1.05 Typical Signal Processing Operations.md

`1.00 Chapter Map - Signal Fundamentals & Operations.md`

Type: Map of Content (MOC)
Syllabus & Core Objectives: Visualizes the foundational pathway of continuous-time and discrete-time signals, organizing basic mathematical expressions, operations, classifications, and system block interfaces.

Vault Frontmatter:

title: "1.00 Chapter Map - Signal Fundamentals & Operations"
aliases: [Signal Map, Chapter 1 MOC]
tags: [signals-and-systems/moc, ece-2107, term-2-1]
type: MOC
course: "[[ECE 2107 - Signals and Systems]]"
chapter: 1

Note Sequence Matrix:
- [[1.01 Signals, Systems & Singularity Functions]] — Elementary physical math definitions and singularity vectors.
- [[1.02 Elementary Operations on Signals]] — Shifting, scaling, folding, and order-of-operation execution rules.
- [[1.03 Signal Classification I — Theory & Periodicity]] — Structural classifications and multi-tone period combinations.
- [[1.04 Signal Classification II — Symmetry, Energy & Power]] — Even/odd mathematical decomposition and finite energy/power bounds.
- [[1.05 Typical Signal Processing Operations]] — Functional block-diagram behaviors of systems (modulation, filtering, correlation, multiplexing).

`1.01 Signals, Systems & Singularity Functions.md`

Type: Modular Concept Note
Syllabus Boundaries: Conceptual definitions of signals and systems, functional independent variables ($t$ vs. $n$), real-world communications applications, and fundamental singularity functions.
Definitions to Cover:
- Signal — Physical representation of an informational variable $x(t)$ or $x[n]$.
- System — Operator $\mathbb{H}[\cdot]$ mapping an independent input $x(t)$ to a dependent output response $y(t)$.
- Unit Step Function — $u(t)$ and $u[n]$ switching boundaries.
- Unit Impulse Function (Dirac Delta) — $\delta(t)$ distribution and Kronecker delta $\delta[n]$.
- Ramp Function — $r(t)$ and $r[n]$ linear growth profiles.
- Signum Function — $\text{sgn}(t)$ polarity indicator.
- Rectangular Pulse (Gate) — $\Pi(t/\tau)$ windowing function.
Mathematical Derivations:
- Prove the derivative/integral relationship between unit impulse and unit step functions: $u (t) = \int_{- \infty}^{t} δ (τ) d τ ⟺ δ (t) = \frac{d u ( t )}{d t}$
Key Diagrams / Graphics:
- [GRAPH: u(t) discontinuous transition at t=0 with height 1]
- [GRAPH: \delta(t) pointing vertical arrow at t=0, labeled with area weight (1)]
- [GRAPH: r(t) diagonal line with slope = 1 starting at t=0]
- [GRAPH: \text{sgn}(t) switching at t=0 between amplitude -1 and +1]
Common Mistakes Callout: Real-world physical impossibility of the infinite amplitude of $\delta(t)$ at $t=0$; integration boundary limits at $t=0^-$ versus $t=0^+$.

`1.02 Elementary Operations on Signals.md`

Type: Computational & Sketching Note
Syllabus Boundaries: Time shifting, time scaling, and time reversal (folding) operations on independent variables for both continuous-time and discrete-time signals.
Definitions to Cover:
- Time Shift — Right shift (delay $t - t_0$) and left shift (advance $t + t_0$).
- Time Scale — Horizontal compression ($|a| > 1$) and expansion ($0 < |a| < 1$).
- Time Reversal — Mirrored reflection across the vertical axis ($x(-t)$ or $x[-n]$).
Rigid Step-by-Step Sketching Workflow:
- Rule: To sketch $x(at - b)$, always apply operations in this order: $Step 1: Shift First x (t - b) \to Step 2: Scale Second x (a t - b) \to Step 3: Reverse Last (if a < 0)$
Core Solved Numericals:
- Step-by-step piecewise sketching of $x(t) = 4r(3t - 1)$ — setting $3t - 1 = 0 \implies t = 1/3$ as the shifted ramp origin, with a transformed slope of 12.
- Piecewise sketching of $x(t) = r(-0.5t + 2)$ — factoring to $r(-0.5(t - 4))$, starting at $t=4$ and expanding to the left.
- Continuous step gate formulation: $x(t) = u(t-5) - u(t-7)$ representing a window of width 2.
Key Diagrams / Graphics:
- [GRAPH: Step-by-step construction phases of x(3t - 1) starting from the raw ramp r(t)]
- [GRAPH: Discrete stem profile of delayed sequence x[n-2] compared to original x[n]]
Common Mistakes Callout: Applying scaling before shifting, which scales the shift factor itself (the “phase shift trap”); failing to reverse discrete boundaries properly.

`1.03 Signal Classification I — Theory & Periodicity.md`

Type: Classification & Analytical Note
Syllabus Boundaries: Continuous-time vs. discrete-time, analog vs. digital, deterministic vs. random, and periodic vs. aperiodic signals.
Definitions to Cover:
- Analog Signal — Continuous amplitude range over a continuous domain.
- Digital Signal — Quantized discrete amplitude values over discrete indexes.
- Deterministic Signal — Perfectly modeled by a deterministic mathematical equation.
- Random Signal — Governed by probabilistic density profiles (e.g., Gaussian noise).
- Periodic Signal — Satisfies $x(t + T) = x(t)$ or $x[n + N] = x[n]$.
Mathematical Derivations:
- Derive the rational ratio criterion for multi-tone periodic combination signals: $\frac{T _{1}}{T _{2}} = \frac{m}{k} \in Q ⟹ T_{0} = k \cdot T_{1} = m \cdot T_{2}$
Core Solved Numericals:
- Test the periodicity and find the fundamental period of $x(t) = 2\cos(4\pi t) + 3\sin(3\pi t)$ using the fraction LCM formula: $LCM (\frac{a}{b}, \frac{c}{d}) = \frac{LCM ( a , c )}{HCF ( b , d )}$
- Prove the aperiodicity of $x(t) = \sin(\sqrt{2}\pi t) + \sin(15\pi t)$ due to the irrational ratio of the periods: $\frac{T _{1}}{T _{2}} = \frac{2}{2/15} \in / Q$
Key Diagrams / Graphics:
- [GRAPH: Periodic composite wave highlighting the envelope repetition boundaries]
Common Mistakes Callout: Assuming the sum of two periodic discrete-time signals is always periodic (unlike continuous-time, discrete sinusoids are periodic only if their frequency is a rational multiple of $2\pi$).

`1.04 Signal Classification II — Symmetry, Energy & Power.md`

Type: Derivation & Calculation Note
Syllabus Boundaries: Causal, non-causal, and anticausal systems; even and odd symmetries; signal energy and average power bounds in both continuous-time (CT) and discrete-time (DT) domains.
Definitions to Cover:
- Causal Signal — $x(t) = 0$ for $t < 0$.
- Even Signal — Symmetrical across the vertical axis: $x(t) = x(-t)$.
- Odd Signal — Anti-symmetrical across the origin: $x(t) = -x(-t)$.
- Energy Signal — Finite energy ($0 < E < \infty$) and zero power ($P = 0$).
- Power Signal — Finite power ($0 < P < \infty$) and infinite energy ($E = \infty$).
Mathematical Derivations:
- Even-Odd Decomposition — Prove that any signal can be written as $x(t) = x_e(t) + x_o(t)$, where: $x_{e} (t) = \frac{x ( t ) + x ( - t )}{2}, x_{o} (t) = \frac{x ( t ) - x ( - t )}{2}$
- Even-Odd Energy Orthogonality Proof — Prove that the total energy of a signal is the sum of the energies of its even and odd parts: $\int_{- \infty}^{\infty} x^{2} (t) d t = \int_{- \infty}^{\infty} x_{e}^{2} (t) d t + \int_{- \infty}^{\infty} x_{o}^{2} (t) d t$
Core Solved Numericals:
- Calculate the power of $x(t) = u(t)$ over infinite boundaries to prove $P = 0.5 \text{ W}$.
- Find the energy and power of the discrete signal $x[n] = \cos(\pi n)u[n]$ to show it represents a power signal with $P = 0.5 \text{ W}$.
- Calculate the energy of the finite-duration, bounded sequence $x[n] = {0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0}$ ($E = 85 \text{ J}$).
Key Diagrams / Graphics:
- [GRAPH: Graphical decomposition of an asymmetrical pulse into symmetric even and anti-symmetric odd components]
Common Mistakes Callout: Forgetting the time-reversal sign when computing the odd component ($x_o(t)$); evaluating the average power of a periodic signal over an incorrect period boundary (always integrate over exactly one fundamental period $T_0$).

`1.05 Typical Signal Processing Operations.md`

Type: Architectural Block Note
Syllabus Boundaries: High-signal physical overviews, functional math definitions, and block-diagram behaviors of correlation, filtering, modulation/demodulation, transformation, and multiplexing.
Definitions to Cover:
- Correlation — Measure of similarity between signals: Auto-correlation ($R_{xx}$) and Cross-correlation ($R_{xy}$).
- Filtering — Selective frequency suppression: Low-pass (LPF), High-pass (HPF), Band-pass (BPF), Band-stop (BSF), and Notch.
- Modulation — Translation of a low-frequency message signal onto a high-frequency carrier wave for transmission: $y (t) = m (t) cos (ω_{c} t)$
- Transformation — Dynamic coordinate domain switching (Fourier, Laplace, s-plane, z-plane).
- Multiplexing — Interleaving multiple signals over a shared transmission medium: Time Division (TDM), Frequency Division (FDM), and Code Division (CDM).
Key Diagrams / Graphics:
- [DIAGRAM: Flow pipeline of a superheterodyne transmitter modulator block - source: Senior Notes]
- [DIAGRAM: FDM spectrum layout showing guard bands separating adjacent signal channels]
Common Mistakes Callout: Convolving instead of correlating (correlation does not involve folding the signal: $R_{xy}(\tau) = \int x(t)y(t+\tau) , dt$, whereas convolution does).

Chapter 2: LTI System Properties & Time-Domain Convolution

Directory Path

01 UNI/2-1/ece 2107 signals/02 time domain lti/

2.00 Chapter Map - LTI System Properties & Convolution.md
2.01 Systems Classification, LTI Properties & Stability.md
2.02 The Continuous Convolution Integral.md
2.03 The Discrete Convolution Sum & Realization.md
2.04 Systems Described by Differential & Difference Equations.md
2.05 Block-Diagram Representations & System Interconnections.md

`2.00 Chapter Map - LTI System Properties & Convolution.md`

Type: Map of Content (MOC)
Syllabus & Core Objectives: Links the raw input signal configurations from Chapter 1 to system operators, focusing on the impulse response ($h(t)$ or $h[n]$), time-domain continuous/discrete convolution, and classic linear system equations.

Vault Frontmatter:

title: "2.00 Chapter Map - LTI System Properties & Convolution"
aliases: [LTI Map, Chapter 2 MOC]
tags: [signals-and-systems/moc, ece-2107, term-2-1]
type: MOC
course: "[[ECE 2107 - Signals and Systems]]"
chapter: 2

Note Sequence Matrix:
- [[2.01 Systems Classification, LTI Properties & Stability]] — System tests for linearity, time-invariance, and BIBO stability.
- [[2.02 The Continuous Convolution Integral]] — Graphical and analytical evaluation of continuous convolution.
- [[2.03 The Discrete Convolution Sum & Realization]] — Slant-sum, matrix, and tabular methods for discrete convolution.
- [[2.04 Systems Described by Differential & Difference Equations]] — Classic homogenous and particular solutions for system dynamics.
- [[2.05 Block-Diagram Representations & System Interconnections]] — Realization of systems using Direct Form I and II block structures.

`2.01 Systems Classification, LTI Properties & Stability.md`

Type: Proof & Verification Note
Syllabus Boundaries: Testing continuous-time and discrete-time systems for linearity, time-invariance, causality, memory, and BIBO stability.
Definitions to Cover:
- Linearity — Perfect adherence to superposition (additivity + homogeneity).
- Time-Invariance — System parameters do not change with time. A delay in the input produces an identical delay in the output.
- Causality — Output depends strictly on present and past inputs.
- Memoryless — Output at any instant $t_0$ depends strictly on the input at $t_0$.
- BIBO Stability — Every bounded input produces a bounded output.
Mathematical Derivations:
- BIBO Stability Integrability Derivation — Prove that a continuous-time LTI system is BIBO stable if and only if its impulse response is absolutely integrable: $\int_{- \infty}^{\infty} ∣ h (τ) ∣ d τ < \infty$
Core Solved Numericals:
- Test the system $y(t) = 3x(-t) + 2x(t)$ to prove it is linear but time-variant (due to the time-reversal operator).
- Analyze the non-linear differential equation: $\frac{d y ( t )}{d t} + 3 y (t) = x (t) \frac{d x ( t )}{d t}$
- Test the discrete-time difference equation $y[n] = 3y[n-1] - n^2 x[n]$ for linearity, time-invariance, and causality.
Common Mistakes Callout: Confusing time-variance with amplitude scaling; assuming a system is causal when it depends on future states (e.g., $x[n+1]$).

`2.02 The Continuous Convolution Integral.md`

Type: Derivation & Analytical Note
Syllabus Boundaries: Continuous-time convolution derivation, and step-by-step graphical/analytical convolution of piecewise and exponential signals.
Definitions to Cover:
- Impulse Response ($h(t)$) — System output when the input is a unit impulse ($\delta(t)$).
- Convolution Integral — Time-domain representation of an LTI system output: $y (t) = x (t) * h (t) = \int_{- \infty}^{\infty} x (τ) h (t - τ) d τ$
Mathematical Derivations:
- Convolution Integral Derivation — Derive the convolution integral by representing input signal $x(t)$ as an infinite sum of weighted shifted impulse functions.
- Scaling Property Proof — Prove that if $y(t) = x(t) * h(t)$, then: $y (2 t) = 2 x (2 t) * h (2 t)$
Core Solved Numericals:
- Convolve input signal $x(t) = e^{-t/RC}u(t)$ with the impulse response of a low-pass RC network $h(t) = \frac{1}{RC} e^{-t/RC} u(t)$ to find: $y (t) = \frac{t}{RC} e^{- t / RC} u (t)$
Key Diagrams / Graphics:
- [GRAPH: Graphical convolution steps showing: (1) original functions, (2) folding of h(\tau) to h(-\tau), (3) shifting to h(t-\tau), and (4) shaded overlap integration regions over piecewise boundaries]
- [CIRCUIT: Simple low-pass RC network - source: Senior Notes]
Common Mistakes Callout: Failing to adjust the integration limits when working with piecewise signals (always verify the active overlap interval for each range of $t$).

`2.03 The Discrete Convolution Sum & Realization.md`

Type: Computational Note
Syllabus Boundaries: Discrete-time convolution sum, the length of convolved sequences, and tabular, matrix, and graphical solution methods.
Definitions to Cover:
- Convolution Sum — Time-domain output of a discrete LTI system: $y [n] = x [n] * h [n] = \sum_{k = - \infty}^{\infty} x [k] h [n - k]$
- Sequence Length Theorem — If $x[n]$ has length $N_1$ and $h[n]$ has length $N_2$, convolving them produces an output sequence of length: $L = N_{1} + N_{2} - 1$
Core Solved Numericals:
- Convolve $x_1[n] = {3, \underline{2}, 1, 2}$ and $x_2[n] = {\underline{1}, 2, 1, 2}$ using the tabular (slant-sum) method.
- Find the response of an FIR filter with impulse response $h[n] = {1, \underline{2}, -2, -1, 0}$ and input $x[n] = {\underline{1}, 2, 3, 0}$ using linear convolution graphically.
Key Diagrams / Graphics:
- [DIAGRAM: Tabular layout matrix of x[k] versus h[n-k] showing slant-line summation pathways - source: Textbook Ch 2]
Common Mistakes Callout: Misaligning the index origin ($n=0$, typically underlined or marked with an arrow) when setting up convolved arrays.

`2.04 Systems Described by Differential & Difference Equations.md`

Type: Differential/Difference Equation Solving Note
Syllabus Boundaries: Linear constant-coefficient differential equations (LCCDE) and difference equations, complementary (homogeneous) solutions, particular solutions, and classical solving techniques.
Definitions to Cover:
- Natural (Homogeneous) Response — System output when input is zero, determined by initial conditions.
- Forced (Particular) Response — System output when input is non-zero, determined by the input signal.
Core Solved Numericals:
- Solve the differential equation $\frac{dy(t)}{dt} + y(t) = x(t)$ with $y(0)=0$ to find its step and impulse responses.
- Solve the second-order recursive system $y[n] - \frac{1}{9}y[n-2] = x[n-1]$ with input $x[n] = u[n]$, and initial conditions $y[-1]=1, y[-2]=0$.
Common Mistakes Callout: Applying initial conditions to the homogeneous solution before finding the total response ($y(t) = y_h(t) + y_p(t)$); using the wrong form for the particular solution when the input matches a root of the characteristic equation (requires multiplying by $t$ or $n$).

`2.05 Block-Diagram Representations & System Interconnections.md`

Type: Implementation & Realization Note
Syllabus Boundaries: Realizing LTI systems using integrators, adders, and multipliers in Direct Form I and Direct Form II structures; cascade and parallel system interconnections.
Definitions to Cover:
- Direct Form I Realization — Separate realization of the input zeros and output poles (uses more delay elements).
- Direct Form II Realization — Merges delay elements to minimize memory requirements (canonic form).
- Cascade Interconnection — Subsystems connected in series: $h_{total} (t) = h_{1} (t) * h_{2} (t)$
- Parallel Interconnection — Subsystems connected in parallel: $h_{total} (t) = h_{1} (t) + h_{2} (t)$
Core Solved Numericals:
- Draw the Direct Form II block diagram for the second-order differential equation: $\frac{d ^{2} y ( t )}{d t ^{2}} + 2 \frac{d y ( t )}{d t} + y (t) = \frac{d ^{2} x ( t )}{d t ^{2}} - \frac{d x ( t )}{d t} - 6 x (t)$
- Draw the Direct Form II block diagram for the difference equation: $y [n] + \frac{1}{3} y [n - 1] + \frac{1}{5} y [n - 3] = x [n - 1]$
Key Diagrams / Graphics:
- [DIAGRAM: Direct Form I and Direct Form II block diagram structures showing delay buffers and feedback paths - source: Textbook Ch 2]
Common Mistakes Callout: Mixing up the signs of feedback coefficients in Direct Form II realizations (always negate the coefficients of the feedback terms when mapping to the diagram multipliers).

Chapter 3: State-Space Representation of Continuous-Time Systems

Directory Path

01 UNI/2-1/ece 2107 signals/03 state space/

3.00 Chapter Map - State-Space Analysis of CT Systems.md
3.01 Introduction to State-Space & Matrix Formulations.md
3.02 State-Space Modeling of Electrical Circuits.md
3.03 State-Space Modeling of Block Diagrams.md
3.04 State Transition Matrix: Properties & Solution Methods.md

`3.00 Chapter Map - State-Space Analysis of CT Systems.md`

Type: Map of Content (MOC)
Syllabus & Core Objectives: Translates higher-order differential equations and complex electrical networks into a unified system of first-order matrix equations.

Vault Frontmatter:

title: "3.00 Chapter Map - State-Space Analysis of CT Systems"
aliases: [State-Space Map, Chapter 3 MOC]
tags: [signals-and-systems/moc, ece-2107, term-2-1]
type: MOC
course: "[[ECE 2107 - Signals and Systems]]"
chapter: 3

Note Sequence Matrix:
- [[3.01 Introduction to State-Space & Matrix Formulations]] — Mathematical phase-variable formulations for higher-order differential equations.
- [[3.02 State-Space Modeling of Electrical Circuits]] — Mapping physical circuits to state-space systems using capacitor voltages and inductor currents as state variables.
- [[3.03 State-Space Modeling of Block Diagrams]] — Realizing state equations directly from operational block diagrams.
- [[3.04 State Transition Matrix: Properties & Solution Methods]] — Deriving the state transition matrix ($e^{\mathbf{A}t}$) using Laplace and infinite series expansion.

`3.01 Introduction to State-Space & Matrix Formulations.md`

Type: Proof & Formulation Note
Syllabus Boundaries: Defining the state of a system, state vector, state variables, and state equations; converting higher-order differential equations into state matrices using the phase-variable method.
Definitions to Cover:
- State of a System — The minimum set of variables ${x_1(t), x_2(t), \dots}$ required to uniquely determine the system’s future response, given its current state and input.
- Continuous-Time State Equation — First-order matrix equation tracking internal system states: $\dot{X} (t) = AX (t) + BU (t)$
- Continuous-Time Output Equation — Matrix equation mapping internal states to external outputs: $Y (t) = CX (t) + DU (t)$
Core Solved Numericals:
- Obtain the state-space model $(\mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D})$ of the third-order differential equation: $\frac{d ^{3} y ( t )}{d t ^{3}} + \frac{d ^{2} y ( t )}{d t ^{2}} + 6 \frac{d y ( t )}{d t} + 7 y (t) = 2 u (t)$
Common Mistakes Callout: Incorrectly writing the size dimensions of the matrices (for an $n$-th order system with $m$ inputs and $p$ outputs: $\mathbf{A}$ is $n \times n$, $\mathbf{B}$ is $n \times m$, $\mathbf{C}$ is $p \times n$, and $\mathbf{D}$ is $p \times m$).

`3.02 State-Space Modeling of Electrical Circuits.md`

Type: Matrix Formulation Note
Syllabus Boundaries: Writing state-space models for complex electrical networks; choosing state variables based on the energy storage elements in the circuit.
Formulation Rule:
- Rule: Always select the voltages across capacitors ($v_C$) and the currents through inductors ($i_L$) as the state variables, representing the physical energy stored in the circuit: $x_{1} (t) = v_{C} (t), x_{2} (t) = i_{L} (t)$
Core Solved Numericals:
- Derive the state equations for a two-stage cascaded RC network to find matrices $\mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D}$.
- Derive the state-space representation for a second-order series RLC network.
Key Diagrams / Graphics:
- [CIRCUIT: Two-stage passive RC cascade circuit with labeled node voltages - source: Textbook Ch 3]
- [CIRCUIT: Series RLC circuit with loop current and component voltages - source: Textbook Ch 3]
Common Mistakes Callout: Writing dependent loops in KVL/KCL; violating polarity conventions when setting up capacitor current equations ($i_C = C \frac{dv_C}{dt}$) and inductor voltage equations ($v_L = L \frac{di_L}{dt}$).

`3.03 State-Space Modeling of Block Diagrams.md`

Type: Block Diagram Note
Syllabus Boundaries: Assigning state variables to operational block diagrams; formulating state matrices directly from the outputs of integrators or unit delays.
Formulation Rule:
- Rule: Assign a state variable to the output of each integrator (in continuous-time systems) or each unit delay block (in discrete-time systems): $Output of Integrator k = x_{k} (t) ⟹ Input of Integrator k = \overset{x}{˙}_{k} (t)$
Core Solved Numericals:
- Determine the state-space representation for a discrete-time system from its delay-block diagram (Direct Form II canonic structure).
Key Diagrams / Graphics:
- [DIAGRAM: Integrator-based block diagram highlighting the location of state variables and feedback loops - source: Slide State Space]
Common Mistakes Callout: Mismapping feedback coefficients to matrix rows; failing to include feedforward paths ($\mathbf{D}$ matrix) when input terms bypass the integrators.

`3.04 State Transition Matrix: Properties & Solution Methods.md`

Type: Derivation & Proof Note
Syllabus Boundaries: Defining the state transition matrix ($\phi(t) = e^{\mathbf{A}t}$), its mathematical properties, and analytical evaluation using the infinite series method and s-domain inverse Laplace method.
Definitions to Cover:
- State Transition Matrix ($\phi(t)$) — Matrix exponential $e^{\mathbf{A}t}$ that determines the unforced natural response of the system’s states over time.
Mathematical Derivations:
- Laplace Resolvent Derivation — Derive the Laplace-domain formula for the state transition matrix: $ϕ (t) = e^{A t} = L^{- 1} {(s I - A)^{- 1}}$
- State-Space to Transfer Function Conversion — Prove that the continuous-time transfer function of a state-space model is: $H (s) = C (s I - A)^{- 1} B + D$
Core Solved Numericals:
- Evaluate the state transition matrix for system matrix $A = [1011]$ using the infinite series method: $e^{A t} = I + A t + \frac{A ^{2} t ^{2}}{2 !} + \dots = [e^{t} 0 t e^{t} e^{t}]$
- Obtain the s-domain transfer function matrix for the system: $A = [- 1 1 0 - 2], B = [10], C = [11]$
Common Mistakes Callout: Forgetting to subtract $A$ from $s I$ before taking the inverse; incorrectly calculating the adjoint and determinant of $s I - A$ for higher-order systems.

Chapter 4: The Sampling Theorem & Multi-Rate Processing

Directory Path

01 UNI/2-1/ece 2107 signals/04 sampling/

4.00 Chapter Map - Sampling & Multi-Rate Processing.md
4.01 The Shannon-Nyquist Sampling Theorem.md
4.02 Spectral Replications & Aliasing Dynamics.md
4.03 Ideal, Natural & Flat-Top Sampling Techniques.md
4.04 Multi-Rate Signal Processing: Decimation & Interpolation.md

`4.00 Chapter Map - Sampling & Multi-Rate Processing.md`

Type: Map of Content (MOC)
Syllabus & Core Objectives: This chapter is the critical link between continuous-time (analog) and discrete-time (digital) domains. It covers uniform sampling, spectral replication, aliasing, and discrete interpolation/decimation.

Vault Frontmatter:

title: "4.00 Chapter Map - Sampling & Multi-Rate Processing"
aliases: [Sampling Map, Chapter 4 MOC]
tags: [signals-and-systems/moc, ece-2107, term-2-1]
type: MOC
course: "[[ECE 2107 - Signals and Systems]]"
chapter: 4

Note Sequence Matrix:
- [[4.01 The Shannon-Nyquist Sampling Theorem]] — Statement, proof, and Nyquist rate calculations.
- [[4.02 Spectral Replications & Aliasing Dynamics]] — Replications of frequency spectra, aliasing, and anti-aliasing filtering.
- [[4.03 Ideal, Natural & Flat-Top Sampling Techniques]] — Practical physical sampling techniques and reconstruction filters.
- [[4.04 Multi-Rate Signal Processing: Decimation & Interpolation]] — Up-sampling, down-sampling, and discrete array manipulation.

`4.01 The Shannon-Nyquist Sampling Theorem.md`

Type: Proof & Calculation Note
Syllabus Boundaries: Formal statement of the sampling theorem, mathematical proof of impulse train sampling, and calculating the Nyquist rate for single and multi-tone signals.
Definitions to Cover:
- Shannon-Nyquist Sampling Theorem — An analog signal can be perfectly reconstructed from its samples if the sampling frequency is at least twice the highest frequency present in the signal: $f_{s} \geq 2 f_{max}$
- Nyquist Rate — The absolute minimum sampling rate required for perfect signal reconstruction: $f_{Nyquist} = 2 f_{max}$
Mathematical Derivations:
- Sampling Theorem Derivation — Mathematically prove the sampling theorem by showing how multiplying an analog signal $x_{a} (t)$ by an impulse train $p (t) = \sum δ (t - n T_{s})$ replicates its spectrum in the frequency domain: $X_{p} (j Ω) = \frac{1}{T _{s}} \sum_{k = - \infty}^{\infty} X_{a} (j (Ω - k Ω_{s}))$
Core Solved Numericals:
- Calculate the Nyquist rate and Nyquist interval for $x (t) = 5 cos (200 π t) + 10 sin (500 π t)$ .
- Calculate the Nyquist rate for $x (t) = sinc^{2} (100 t)$ .
Common Mistakes Callout: Misidentifying the maximum frequency component when working with composite signals; confusing angular frequency ( $ω = 2 π f$ in rad/s) with cyclic frequency ( $f$ in Hz).

`4.02 Spectral Replications & Aliasing Dynamics.md`

Type: Spectral Analysis Note
Syllabus Boundaries: Replicated frequency spectra under over-sampling, Nyquist-rate sampling, and under-sampling; aliasing distortion; anti-aliasing low-pass filtering.
Definitions to Cover:
- Aliasing — High-frequency components fold back into the passband and masquerade as false low frequencies when the sampling rate is below the Nyquist limit ( $f_{s} < 2 f_{max}$ ).
- Anti-Aliasing Filter — A low-pass filter placed before the sampler to strip away frequencies above $f_{s} /2$ and prevent aliasing.
Key Diagrams / Graphics:
- [GRAPH: Replicated frequency spectra showing: (1) Over-sampling (gaps between bands), (2) Nyquist sampling (bands exactly touching), and (3) Under-sampling (overlapping bands creating shaded aliasing regions)]
Common Mistakes Callout: Thinking that digital filtering can remove aliasing after a signal has been sampled (aliasing permanently overlaps frequency bands during sampling and must be prevented before sampling with an analog filter).

`4.03 Ideal, Natural & Flat-Top Sampling Techniques.md`

Type: Circuit Modeling Note
Syllabus Boundaries: Ideal (impulse) sampling, natural sampling (chopping), flat-top sampling (sample-and-hold), and aperture effect distortion.
Definitions to Cover:
- Ideal Sampling — Multiplying a signal by a train of zero-width Dirac impulses.
- Natural Sampling — Multiplying a signal by a train of finite-width rectangular pulses, preserving the shape of the analog signal at the top of each pulse.
- Flat-Top Sampling — Sampling a signal’s amplitude and holding it constant for the duration of the pulse, producing a flat-topped staircase waveform.
- Aperture Effect — High-frequency amplitude attenuation caused by the flat-top pulse-width ( $s in c$ function envelope distortion).
Key Diagrams / Graphics:
- [GRAPH: Waveforms comparing natural sampling (curved pulse tops) and flat-top sampling (staircase flat pulse tops) - source: Textbook Ch 6]
Common Mistakes Callout: Forgetting the need for equalizer filtering ( $sinc^{- 1}$ compensation) to correct the high-frequency droop caused by the aperture effect in flat-top sampling.

`4.04 Multi-Rate Signal Processing: Decimation & Interpolation.md`

Type: Multi-Rate Calculation Note
Syllabus Boundaries: Decimation (down-sampling) and interpolation (up-sampling) in the discrete-time domain, and algebraic array manipulation.
Definitions to Cover:
- Down-Sampling (Decimation by $M$ ) — Keeping only every $M$ -th sample of a sequence and discarding the rest: $x_{d} [n] = x [n M]$
- Up-Sampling (Interpolation by $L$ ) — Inserting $L - 1$ zeros between each consecutive pair of samples: $x_{e} [n] = {x [n / L], 0, n = 0, \pm L, \pm 2 L, \dots otherwise$
Core Solved Numericals:
- Given sequence $x [n] = [2, 6, \underline{0}, 3, 1, 2]$ , find the down-sampled sequence $x_{d} [n]$ by factor $M = 3$ , and the up-sampled sequence $x_{e} [n]$ by factor $L = 3$ .
Common Mistakes Callout: Mismapping index values around the $n = 0$ origin when down-sampling; failing to include anti-imaging low-pass filters after up-sampling to remove high-frequency spectral replicas.

Chapter 5: Analog Filter Design

Directory Path

01 UNI/2-1/ece 2107 signals/05 analog filters/

5.00 Chapter Map - Analog Filter Design.md
5.01 Filter Specifications & Tolerance Curves.md
5.02 Butterworth Filter Approximation.md
5.03 Chebyshev & Elliptic Filter Approximations.md
5.04 Active Filter Realization & Sallen-Key RC Networks.md

`5.00 Chapter Map - Analog Filter Design.md`

Type: Map of Content (MOC)
Syllabus & Core Objectives: This chapter covers the mathematical design of continuous-time low-pass filters (Butterworth, Chebyshev, and Elliptic) and their active RC op-amp realizations.

Vault Frontmatter:

title: "5.00 Chapter Map - Analog Filter Design"
aliases: [Filter Map, Chapter 5 MOC]
tags: [signals-and-systems/moc, ece-2107, term-2-1]
type: MOC
course: "[[ECE 2107 - Signals and Systems]]"
chapter: 5

Note Sequence Matrix:
- [[5.01 Filter Specifications & Tolerance Curves]] — Continuous low-pass specifications and passband/stopband tolerance bands.
- [[5.02 Butterworth Filter Approximation]] — Maximally flat design equations, pole placements, and filter order derivation.
- [[5.03 Chebyshev & Elliptic Filter Approximations]] — Chebyshev polynomials, passband ripples, and Elliptic equal-ripple behavior.
- [[5.04 Active Filter Realization & Sallen-Key RC Networks]] — Realizing transfer functions using active Sallen-Key op-amp circuits.

`5.01 Filter Specifications & Tolerance Curves.md`

Type: Spec Design Note
Syllabus Boundaries: Ideal filter brick-wall limits, practical low-pass specifications, and passband/stopband tolerance regions.
Definitions to Cover:
- Passband Edge Frequency ( $Ω_{p}$ ) — Boundary defining the end of the passband.
- Stopband Edge Frequency ( $Ω_{s}$ ) — Boundary defining the start of the stopband.
- Passband Attenuation Ripple ( $A_{p}$ ) — Maximum allowed gain variation in the passband (in dB).
- Stopband Attenuation ( $A_{s}$ ) — Minimum required attenuation in the stopband (in dB).
Key Diagrams / Graphics:
- [GRAPH: Practical low-pass filter tolerance curve showing \Omega_p, \Omega_s, passband ripple width, and stopband attenuation floor - source: Textbook Ch 5]

`5.02 Butterworth Filter Approximation.md`

Type: Derivation & Design Note
Syllabus Boundaries: Maximally flat magnitude response, deriving filter order ( $N$ ) and cutoff frequency ( $Ω_{c}$ ), and locating stable poles on the left-half s-plane.
Definitions to Cover:
- Butterworth Filter — Analog filter with a maximally flat magnitude response in the passband, decaying monotonically in the transition and stopbands: $∣ H (j Ω) ∣ = \frac{1}{1 + ( \frac{Ω}{Ω _{c}} ) ^{2 N}}$
Mathematical Derivations:
- Filter Order Derivation — Derive the formula to calculate the minimum Butterworth filter order $N$ given passband and stopband specifications: $N = \frac{l o g ( \frac{1 0 ^{A_{s} /10} - 1}{1 0 ^{A_{p} /10} - 1} )}{2 l o g ( \frac{Ω _{s}}{Ω _{p}} )}$
- Stable Pole Location Derivation — Derive the equation for locating the stable poles $p_{k}$ of a Butterworth filter, which lie symmetrically on a semicircle of radius $Ω_{c}$ in the left-half s-plane: $p_{k} = Ω_{c} e^{j (\frac{π}{2} + \frac{( 2 k - 1 ) π}{2 N})}, k = 1, 2, \dots, N$
Core Solved Numericals:
- Design a Butterworth low-pass filter with $A_{p} = 1 dB$ , $Ω_{p} = 2000 π rad/s$ , $Ω_{s} = 10000 π rad/s$ , and $A_{s} \geq 40 dB$ . Find its order $N$ , cutoff frequency $Ω_{c}$ , and s-domain transfer function $H (s)$ .
Key Diagrams / Graphics:
- [GRAPH: Butterworth pole locations on a left-half s-plane circle for order N=3 - source: Textbook Ch 5]
Common Mistakes Callout: Selecting poles in the right-half s-plane (which makes the filter unstable); failing to round up the calculated filter order $N$ to the next highest integer.

`5.03 Chebyshev & Elliptic Filter Approximations.md`

Type: Comparative Design Note
Syllabus Boundaries: Chebyshev Type I and Type II characteristics, Chebyshev polynomial equations, Elliptic equal-ripple behavior, and comparing transition band roll-off.
Definitions to Cover:
- Chebyshev Type I Filter — Exhibits equal ripple behavior in the passband and a monotonic roll-off in the stopband.
- Chebyshev Type II (Inverse Chebyshev) — Exhibits monotonic behavior in the passband and equal ripples in the stopband.
- Elliptic Filter — Exhibits equal ripple behavior in both the passband and stopband, offering the sharpest transition band roll-off for a given filter order.
Mathematical Formulations:
- Chebyshev magnitude approximation: $∣ H (j Ω) ∣ = \frac{1}{1 + ϵ ^{2} C _{N}^{2} ( \frac{Ω}{Ω _{p}} )}$
- Chebyshev polynomial recursive definition: $C_{N} (x) = 2 x C_{N - 1} (x) - C_{N - 2} (x) with C_{0} (x) = 1, C_{1} (x) = x$
Key Diagrams / Graphics:
- [GRAPH: Comparative magnitude response curves of Butterworth, Chebyshev, and Elliptic filters showing transition steepness differences - source: Textbook Ch 5]

`5.04 Active Filter Realization & Sallen-Key RC Networks.md`

Type: Circuit Analysis Note
Syllabus Boundaries: Active filter topologies, op-amp realization, and deriving the transfer function of a Sallen-Key second-order low-pass filter.
Definitions to Cover:
- Sallen-Key Filter — Active filter topology used to realize second-order filter stages using resistors, capacitors, and an operational amplifier.
Mathematical Derivations:
- Sallen-Key Low-Pass Transfer Function — Use nodal analysis to derive the transfer function $H (s) = V_{o} (s) / V_{i} (s)$ of a Sallen-Key second-order low-pass filter: $H (s) = \frac{K}{s ^{2} ( R _{1} R _{2} C _{1} C _{2} ) + s ( R _{1} C _{2} + R _{2} C _{2} + R _{1} C _{1} ( 1 - K ) ) + 1}$
Key Diagrams / Graphics:
- [CIRCUIT: Active second-order low-pass Sallen-Key circuit with op-amp feedback paths - source: Lab Manual]
ECE 2108 Laboratory Connection:
- MATLAB script for plotting the frequency response of a Sallen-Key low-pass filter using the step, pzmap, and ss commands.

Chapter 6: Two-Port Network Theory

Directory Path

01 UNI/2-1/ece 2107 signals/06 two port networks/

6.00 Chapter Map - Two-Port Networks.md
6.01 Open-Circuit Impedance (Z) Parameters.md
6.02 Short-Circuit Admittance (Y) Parameters.md
6.03 Transmission (ABCD) Parameters & Cascaded Networks.md
6.04 Hybrid (h) & Inverse Hybrid (g) Parameter Formulations.md

`6.00 Chapter Map - Two-Port Networks.md`

Type: Map of Content (MOC)
Syllabus & Core Objectives: This chapter treats linear networks as “black boxes,” modeling their terminal behaviors using two-port matrix parameters (Z, Y, ABCD, and hybrid parameters).

Vault Frontmatter:

title: "6.00 Chapter Map - Two-Port Networks"
aliases: [Two-Port Map, Chapter 6 MOC]
tags: [signals-and-systems/moc, ece-2107, term-2-1]
type: MOC
course: "[[ECE 2107 - Signals and Systems]]"
chapter: 6

Note Sequence Matrix:
- [[6.01 Open-Circuit Impedance (Z) Parameters]] — Defining and calculating impedance matrices for T-networks.
- [[6.02 Short-Circuit Admittance (Y) Parameters]] — Defining and calculating admittance matrices for Pi-networks.
- [[6.03 Transmission (ABCD) Parameters & Cascaded Networks]] — Matrix multiplication for cascaded networks.
- [[6.04 Hybrid (h) & Inverse Hybrid (g) Parameter Formulations]] — Parameter definitions and conversions.

`6.01 Open-Circuit Impedance (Z) Parameters.md`

Type: Formulation & Calculation Note
Syllabus Boundaries: Defining two-port networks, Z-parameter equations, calculating parameters by selectively open-circuiting ports ( $I_{k} = 0$ ), and testing for reciprocity and symmetry.
Definitions to Cover:
- Z-Parameters (Open-Circuit Impedance) — Relates terminal voltages to currents: $V_{1} = Z_{11} I_{1} + Z_{12} I_{2}, V_{2} = Z_{21} I_{1} + Z_{22} I_{2}$
- Reciprocal Network — Satisfies $Z_{12} = Z_{21}$ .
- Symmetrical Network — Satisfies $Z_{11} = Z_{22}$ .
Core Solved Numericals:
- Calculate the Z-parameters for a resistive T-network with branch resistor values of $R_{a} = 24 Ω$ , $R_{b} = 8 Ω$ , and $R_{c} = 8 Ω$ .
Key Diagrams / Graphics:
- [CIRCUIT: Resistive T-network configuration showing terminal voltages V_1, V_2 and current directions - source: Textbook Ch 4]
Common Mistakes Callout: Misidentifying current directions (two-port parameter convention defines both currents $I_{1}$ and $I_{2}$ as flowing into their respective ports).

`6.02 Short-Circuit Admittance (Y) Parameters.md`

Type: Formulation & Calculation Note
Syllabus Boundaries: Y-parameter equations, calculating parameters by short-circuiting ports ( $V_{k} = 0$ ), mapping to Pi-networks, and reciprocity/symmetry conditions.
Definitions to Cover:
- Y-Parameters (Short-Circuit Admittance) — Relates terminal currents to voltages: $I_{1} = Y_{11} V_{1} + Y_{12} V_{2}, I_{2} = Y_{21} V_{1} + Y_{22} V_{2}$
- Y-Z Inverse Matrix Identity — Admittance matrix is the exact inverse of the impedance matrix: $[Y] = [Z]^{- 1}$
Core Solved Numericals:
- Calculate the Y-parameters for a resistive Pi-network with node resistors $R_{1}$ , $R_{2}$ , and feedback resistor $R_{3}$ .
Key Diagrams / Graphics:
- [CIRCUIT: Resistive Pi-network configuration with terminal voltages V_1, V_2 - source: Textbook Ch 4]
Common Mistakes Callout: Forgetting the negative sign on transfer admittance terms ( $Y_{12}$ and $Y_{21}$ ) in resistive Pi-networks.

`6.03 Transmission (ABCD) Parameters & Cascaded Networks.md`

Type: Matrix Modeling Note
Syllabus Boundaries: ABCD parameter equations, negative current convention for $I_{2}$ , cascading networks using matrix multiplication, and reciprocity/symmetry checks.
Definitions to Cover:
- Transmission (ABCD) Parameters — Relates sending-end variables to receiving-end variables: $V_{1} = A V_{2} - B I_{2}, I_{1} = C V_{2} - D I_{2}$
- Transmission Reciprocity Condition — Satisfies: $A D - BC = 1$
- Transmission Symmetry Condition — Satisfies: $A = D$
Core Solved Numericals:
- Calculate the ABCD parameters for a resistive T-network ( $R_{1} = 1 Ω$ , $R_{2} = 2 Ω$ , $R_{3} = 5 Ω$ ).
- Calculate the overall ABCD transmission matrix for a cascade of two identical T-networks by multiplying their individual ABCD matrices.
Key Diagrams / Graphics:
- [DIAGRAM: Block diagram of two cascaded networks showing: [ABCD_1] \cdot [ABCD_2] - source: Textbook Ch 4]
Common Mistakes Callout: Forgetting the negative sign on current $I_{2}$ in the transmission equations (the ABCD matrix is modeled with $I_{2}$ flowing out of port 2 to simplify cascade calculations).

`6.04 Hybrid (h) & Inverse Hybrid (g) Parameter Formulations.md`

Type: Formulation Note
Syllabus Boundaries: Hybrid (h) and inverse hybrid (g) parameter equations, applications in transistor modeling, and parameter conversion matrices.
Definitions to Cover:
- h-Parameters (Hybrid) — Relates input voltage and output current to input current and output voltage: $V_{1} = h_{11} I_{1} + h_{12} V_{2}, I_{2} = h_{21} I_{1} + h_{22} V_{2}$
- g-Parameters (Inverse Hybrid) — Inverse hybrid formulations: $I_{1} = g_{11} V_{1} + g_{12} I_{2}, V_{2} = g_{21} V_{1} + g_{22} I_{2}$
Conversion Table: Reference matrix mapping Z, Y, ABCD, and hybrid parameters to one another.

Part II: Instructor 1 (Transform-Domain Analysis & Fast Algorithms)

Chapter 7: Continuous-Time Fourier Series (CTFS) (Notes 7.00 - 7.04)
Chapter 8: Continuous-Time Fourier Transform (CTFT) (Notes 8.00 - 8.07)
Chapter 9: Laplace Transform & s-Domain Circuit Applications (Notes 9.00 - 9.08)
Chapter 10: Z-Transform & Discrete-Time Analysis (Notes 10.00 - 10.05)
Chapter 11: DFT, FFT, and Frequency-Domain Discrete Analysis (Notes 11.00 - 11.04)

Chapter 7: Continuous-Time Fourier Series (CTFS)

Directory Path

01 UNI/2-1/ece 2107 signals/07 fourier series/

7.00 Chapter Map - Continuous-Time Fourier Series.md
7.01 Trigonometric Fourier Series Representation.md
7.02 Symmetry Conditions & Waveform Analysis.md
7.03 Exponential Fourier Series & Complex Spectra.md
7.04 Parseval’s Theorem & Convergence Conditions.md

`7.00 Chapter Map - Continuous-Time Fourier Series.md`

Type: Map of Content (MOC)
Syllabus & Core Objectives: This chapter covers decomposing continuous, periodic signals into sinusoidal and exponential harmonics, setting up the foundation for frequency-domain system analysis.

Vault Frontmatter:

title: "7.00 Chapter Map - Continuous-Time Fourier Series"
aliases: [Fourier Series Map, Chapter 7 MOC]
tags: [signals-and-systems/moc, ece-2107, term-2-1]
type: MOC
course: "[[ECE 2107 - Signals and Systems]]"
chapter: 7

Note Sequence Matrix:
- [[7.01 Trigonometric Fourier Series Representation]] — Integrating trigonometric coefficients and orthogonal bases.
- [[7.02 Symmetry Conditions & Waveform Analysis]] — Waveform symmetry shortcuts (even, odd, half-wave).
- [[7.03 Exponential Fourier Series & Complex Spectra]] — Complex coefficients ( $C_{n}$ ) and single/double-sided line spectra.
- [[7.04 Parseval’s Theorem & Convergence Conditions]] — Power conservation and Dirichlet convergence criteria.

`7.01 Trigonometric Fourier Series Representation.md`

Type: Analytical & Derivation Note
Syllabus Boundaries: Trigonometric Fourier Series expansion, orthogonality of sines and cosines, and calculating coefficients $a_{0}, a_{n}, b_{n}$ .
Definitions to Cover:
- Trigonometric Fourier Series — Representation of a periodic signal as a sum of harmonically related sines and cosines: $f (t) = \frac{a _{0}}{2} + \sum_{n = 1}^{\infty} a_{n} cos (n ω_{0} t) + \sum_{n = 1}^{\infty} b_{n} sin (n ω_{0} t)$
Mathematical Derivations:
- Trigonometric Coefficient Derivation — Derive the Euler-Fourier formulas for calculating coefficients $a_{0}, a_{n}, b_{n}$ using the orthogonality properties of sinusoidal functions over a period $T_{0}$ : $a_{0} = \frac{2}{T _{0}} \int_{T_{0}} f (t) d t, a_{n} = \frac{2}{T _{0}} \int_{T_{0}} f (t) cos (n ω_{0} t) d t, b_{n} = \frac{2}{T _{0}} \int_{T_{0}} f (t) sin (n ω_{0} t) d t$
Core Solved Numericals:
- Calculate the trigonometric Fourier series of a periodic rectangular pulse train.
- Calculate the trigonometric Fourier series for a triangular waveform ( $a_{0} = 0, a_{n} = 0$ for all $n$ , $b_{n} = \frac{8 A}{π ^{2} n ^{2}} sin (nπ /2)$ ).
Key Diagrams / Graphics:
- [GRAPH: 3D frequency visualization showing periodic time-domain square wave decomposing into separate sinusoidal harmonics - source: Senior Notes]

`7.02 Symmetry Conditions & Waveform Analysis.md`

Type: Shortcut & Verification Note
Syllabus Boundaries: Simplifying Fourier series calculations using waveform symmetry: even, odd, and half-wave symmetries.
Definitions to Cover:
- Even Symmetry — Symmetrical across the vertical axis ( $f (t) = f (- t)$ ) $⟹ b_{n} = 0$ (contains only cosine terms).
- Odd Symmetry — Symmetrical across the origin ( $f (t) = - f (- t)$ ) $⟹ a_{0} = 0, a_{n} = 0$ (contains only sine terms).
- Half-Wave Symmetry — Symmetrical under a half-period shift and inversion ( $f (t - T_{0} /2) = - f (t)$ ) $⟹ a_{0} = 0$ , and coefficients $a_{n} = b_{n} = 0$ for all even values of $n$ (contains only odd harmonics).
Mathematical Derivations:
- Odd Symmetry Coefficient Proof — Prove that odd periodic functions contain only sine terms ( $b_{n}$ ) in their Fourier series representation.
Key Diagrams / Graphics:
- [GRAPH: Waveforms illustrating even, odd, and half-wave symmetries - source: Textbook Ch 5]
Common Mistakes Callout: Forgetting that half-wave symmetry must satisfy both the shift ( $T_{0} /2$ ) and the amplitude inversion (multiplying by $- 1$ ).

`7.03 Exponential Fourier Series & Complex Spectra.md`

Type: Formulation & Spectral Note
Syllabus Boundaries: Complex exponential Fourier series, complex coefficients ( $C_{n}$ ), converting between complex and trigonometric forms, and plotting single and double-sided amplitude and phase line spectra.
Definitions to Cover:
- Complex Exponential Fourier Series — Compact representation of a periodic signal using complex exponentials: $f (t) = \sum_{n = - \infty}^{\infty} C_{n} e^{jn ω_{0} t} where C_{n} = \frac{1}{T _{0}} \int_{T_{0}} f (t) e^{- jn ω_{0} t} d t$
Mathematical Derivations:
- Trigonometric-to-Exponential Conversion Derivation — Use Euler’s formula to derive the relationship between trigonometric and complex exponential Fourier coefficients: $C_{n} = \frac{a _{n} - j b _{n}}{2}, C_{- n} = C_{n}^{*} = \frac{a _{n} + j b _{n}}{2}, C_{0} = \frac{a _{0}}{2}$
Core Solved Numericals:
- Determine the complex exponential Fourier series for a half-wave rectified cosine signal.
- Determine the period $T$ and plot the double-sided amplitude and phase spectrum for the signal: $x (t) = \sum_{n = - \infty}^{\infty} \frac{3}{4 + ( nπ ) ^{2}} e^{j 3 nπ t}$
Key Diagrams / Graphics:
- [GRAPH: Double-sided discrete line spectrum showing amplitude impulses (|C_n|) and corresponding phase angles (\angle C_n) versus index n - source: Senior Notes]

`7.04 Parseval’s Theorem & Convergence Conditions.md`

Type: Proof & Analytical Note
Syllabus Boundaries: Dirichlet conditions for Fourier series convergence, Parseval’s power theorem, and harmonic power allocation.
Definitions to Cover:
- Parseval’s Identity (Power) — States that the total average power of a periodic signal is equal to the sum of the average powers of its individual harmonic components: $P = \frac{1}{T _{0}} \int_{T_{0}} ∣ f (t) ∣^{2} d t = \sum_{n = - \infty}^{\infty} ∣ C_{n} ∣^{2} = (\frac{a _{0}}{2})^{2} + \frac{1}{2} \sum_{n = 1}^{\infty} (a_{n}^{2} + b_{n}^{2})$
- Dirichlet Conditions — Mathematical prerequisites a periodic signal must satisfy to have a valid Fourier series representation (must be absolutely integrable over a period, have a finite number of maxima and minima, and have a finite number of discontinuities).
Mathematical Derivations:
- Parseval’s Power Theorem Proof — Mathematically prove Parseval’s identity for a periodic signal using the complex exponential Fourier series.
Core Solved Numericals:
- Find the percentage of power contained in the first five harmonic terms of the complex Fourier series: $x (t) = \sum_{n = - \infty}^{\infty} \frac{3}{4 + ( nπ ) ^{2}} e^{j 3 nπ t}$ given a maximum power of $0.7 W$ .
Common Mistakes Callout: Forgetting the factor of $1/2$ when calculating power from trigonometric coefficients ( $a_{n}^{2} /2 + b_{n}^{2} /2$ ); violating the absolute integrability condition of Dirichlet.

Chapter 8: Continuous-Time Fourier Transform (CTFT)

Directory Path

01 UNI/2-1/ece 2107 signals/08 fourier transform/

8.00 Chapter Map - Continuous-Time Fourier Transform.md
8.01 Foundation of the Continuous-Time Fourier Transform (CTFT).md
8.02 Properties of the Continuous-Time Fourier Transform.md
8.03 Rayleigh's Energy Theorem & Spectral Density.md
8.04 CTFT Pairs for Singularity & Common Functions.md
8.05 Time-Domain Convolution & Multiplication Properties.md
8.06 Frequency Spectra, Phase Spectra & Modulation.md
8.07 CTFT System Analysis of Continuous Networks.md

`8.00 Chapter Map - Continuous-Time Fourier Transform.md`

Type: Map of Content (MOC)
Syllabus & Core Objectives: This chapter covers continuous-time Fourier transforms (CTFT) and their properties, energy spectral density, Fourier pairs, and frequency-domain LTI system analysis.

Vault Frontmatter:

title: "8.00 Chapter Map - Continuous-Time Fourier Transform"
aliases: [CTFT Map, Chapter 8 MOC]
tags: [signals-and-systems/moc, ece-2107, term-2-1]
type: MOC
course: "[[ECE 2107 - Signals and Systems]]"
chapter: 8

Note Sequence Matrix:
- [[8.01 Foundation of the Continuous-Time Fourier Transform (CTFT)]] — Deriving the CTFT as a limiting case of the Fourier series.
- [[8.02 Properties of the Continuous-Time Fourier Transform]] — Linearity, scaling, time shift, modulation, and differentiation.
- [[8.03 Rayleigh's Energy Theorem & Spectral Density]] — Energy spectral density (ESD) and Rayleigh’s energy theorem.
- [[8.04 CTFT Pairs for Singularity & Common Functions]] — Common Fourier pairs (impulses, steps, exponentials).
- [[8.05 Time-Domain Convolution & Multiplication Properties]] — Equivalence of time convolution to spectral multiplication.
- [[8.06 Frequency Spectra, Phase Spectra & Modulation]] — Plotting spectra and analyzing amplitude modulation.
- [[8.07 CTFT System Analysis of Continuous Networks]] — Analyzing low-pass RC filter circuits in the frequency domain.

`8.01 Foundation of the Continuous-Time Fourier Transform (CTFT).md`

Type: Derivation & Foundation Note
Syllabus Boundaries: Deriving the Continuous-Time Fourier Transform (CTFT) as the period of a Fourier series approaches infinity ( $T_{0} \to \infty$ ), existence conditions, and forward and inverse CTFT integrals.
Definitions to Cover:
- Continuous-Time Fourier Transform (CTFT) — Converts a continuous time-domain signal into its continuous frequency spectrum: $F (jω) = \int_{- \infty}^{\infty} f (t) e^{- jω t} d t$
- Inverse CTFT — Reconstructs a time-domain signal from its frequency spectrum: $f (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} F (jω) e^{jω t} d ω$
Mathematical Derivations:
- Fourier Transform Derivation — Mathematically derive the continuous Fourier transform by taking the limit of the exponential Fourier series as $T_{0} \to \infty$ (forcing the fundamental frequency spacing $ω_{0}$ to become an infinitesimal frequency step $d ω$ ).
Common Mistakes Callout: Confusing cyclic frequency ( $f$ in Hz) with angular frequency ( $ω$ in rad/s), which changes the scaling factor of the inverse Fourier integral ( $1/2 π$ versus 1).

`8.02 Properties of the Continuous-Time Fourier Transform.md`

Type: Proof & Formulation Note
Syllabus Boundaries: Linearity, scaling, time shifting, frequency shifting (modulation), time differentiation, time integration, and duality properties of the CTFT.
Mathematical Derivations:
- Time Shifting Property Proof — Prove that delaying a signal in time shifts its phase in frequency: $f (t - t_{0}) \leftrightarrow F (jω) e^{- jω t_{0}}$
- Time Scaling Property Proof — Prove the time scaling property: $f (a t) \leftrightarrow \frac{1}{∣ a ∣} F (j \frac{ω}{a})$
- Time Differentiation Property Proof — Prove the differentiation property: $\frac{d ^{n} f ( t )}{d t ^{n}} \leftrightarrow (jω)^{n} F (jω)$
Core Solved Numericals:
- Use Fourier properties to find the Fourier transform of $f (2 t)$ , $f (t - 2) e^{j t}$ , and $3 \frac{df ( t )}{d t}$ given the source spectrum: $F (jω) = \frac{1}{ω ^{2} + 1} e^{- j ω^{3} + 1}$
Common Mistakes Callout: Forgetting the scaling factor $1/∣ a ∣$ when applying the scaling property; applying a time delay before scaling, which yields an incorrect phase shift.

`8.03 Rayleigh's Energy Theorem & Spectral Density.md`

Type: Proof & Calculation Note
Syllabus Boundaries: Energy conservation in the time and frequency domains, Rayleigh’s energy theorem, Energy Spectral Density (ESD), and Power Spectral Density (PSD).
Definitions to Cover:
- Rayleigh’s Energy Theorem — States that the total energy of a signal calculated in the time domain is equal to the integral of its energy spectral density across the frequency spectrum: $E = \int_{- \infty}^{\infty} ∣ f (t) ∣^{2} d t = \frac{1}{2 π} \int_{- \infty}^{\infty} ∣ F (jω) ∣^{2} d ω$
- Energy Spectral Density (ESD) — Distribution of a signal’s energy across the frequency spectrum, defined as $∣ F (jω) ∣^{2}$ .
Mathematical Derivations:
- Rayleigh’s Energy Theorem Proof — Mathematically prove Rayleigh’s theorem using the properties of the Fourier transform and its conjugate.
Core Solved Numericals:
- Find the total energy of the signal $f (t) = e^{- a t} u (t)$ .
- Determine the frequency limit $ω_{1}$ below which 95% of the total energy of $f (t) = e^{- a t} u (t)$ is contained.
Common Mistakes Callout: Integrating the magnitude $F (jω)$ instead of the squared magnitude $∣ F (jω) ∣^{2}$ when calculating energy in the frequency domain.

`8.04 CTFT Pairs for Singularity & Common Functions.md`

Type: Derivation & Reference Note
Syllabus Boundaries: Deriving and compiling standard CTFT pairs for impulse, constant, step, exponential, and Gaussian pulse signals.
Mathematical Derivations:
- Gaussian Pulse Self-Transform Proof — Mathematically prove that the normalized Gaussian pulse is its own Fourier transform: $F {e^{- π t^{2}}} = e^{- π f^{2}} (or e^{- ω^{2} /4 π})$
- Signum Function Transform Derivation — Derive the Fourier transform of the Signum function: $F {sgn (t)} = \frac{2}{jω}$
- Dirac Delta Transform Derivation — Derive the Fourier transform of the Dirac delta function: $F {δ (t)} = 1$
Core Solved Numericals:
- Calculate the Fourier transform of the piecewise step pulse: $f (t) = 5 [u (t + 3) + u (t + 2) - u (t - 2) - u (t - 3)]$
- Calculate the Fourier transform of the double exponential pulse $f (t) = e^{- a ∣ t ∣}$ ( $F (jω) = \frac{2 a}{a ^{2} + ω ^{2}}$ ).
- Calculate the Fourier transform of the discrete impulse train: $f (t) = \frac{1}{2} [δ (t + 1) + δ (t + \frac{1}{2}) + δ (t - \frac{1}{2}) + δ (t - 1)]$
Key Diagrams / Graphics:
- [GRAPH: Bell curve shape of the Gaussian pulse in both time and frequency domains - source: Slide Fourier]
Common Mistakes Callout: Writing the Fourier transform of a unit step $u (t)$ as simply $1/ jω$ (it must include the DC impulse component $π δ (ω)$ due to its non-zero average value).

`8.05 Time-Domain Convolution & Multiplication Properties.md`

Type: Proof Note
Syllabus Boundaries: Continuous-time convolution property (time-domain convolution corresponds to spectral multiplication) and multiplication property (time-domain multiplication corresponds to spectral convolution).
Mathematical Derivations:
- Convolution-Multiplication Equivalence Proof — Mathematically prove that time-domain convolution simplifies to direct algebraic multiplication in the frequency domain: $F {x (t) * h (t)} = X (jω) H (jω)$
Common Mistakes Callout: Forgetting the scaling factor $1/2 π$ when performing frequency-domain convolution: $F {x_{1} (t) \cdot x_{2} (t)} = \frac{1}{2 π} [X_{1} (jω) * X_{2} (jω)]$

`8.06 Frequency Spectra, Phase Spectra & Modulation.md`

Type: Plotting & Analytical Note
Syllabus Boundaries: Plotting amplitude and phase spectra, double-sided spectral representation, and the modulation theorem.
Mathematical Formulations:
- Modulation Theorem — Multiplying a signal by a carrier wave shifts its spectrum to the carrier frequency: $m (t) cos (ω_{c} t) \leftrightarrow \frac{1}{2} [M (j (ω - ω_{c})) + M (j (ω + ω_{c}))]$
Core Solved Numericals:
- Plot the double-sided frequency spectrum of the signal $x (t) = 10 cos (20 π t - π /6)$ .
- Plot the single and double-sided amplitude and phase spectrum of $x (t) = 25 cos (5 π t - π /2)$ .
Key Diagrams / Graphics:
- [GRAPH: Double-sided amplitude line impulses and phase angle shifts for a cosine wave on separate axes - source: Senior Notes]

`8.07 CTFT System Analysis of Continuous Networks.md`

Type: Circuit Analysis Note
Syllabus Boundaries: Frequency-domain analysis of analog circuits, system transfer functions $H (jω)$ , and calculating steady-state responses.
Core Solved Numericals:
- Analyze a passive series low-pass RC network with input $x (t) = e^{- t / RC} u (t)$ . Find its frequency response $H (jω)$ , output spectrum $Y (jω)$ , and time-domain output $y (t)$ using the inverse Fourier transform: $y (t) = \frac{t}{RC} e^{- t / RC} u (t)$
- Analyze the same low-pass RC network with input $x (t) = t e^{- t / RC} u (t)$ to find its time-domain output $y (t)$ .
Key Diagrams / Graphics:
- [CIRCUIT: Series RC circuit with input x(t) and output voltage y(t) across the capacitor - source: Senior Notes]
Common Mistakes Callout: Incorrectly writing the capacitor impedance in the frequency domain (it should be $1/ jω C$ , not $jω C$ or $1/ s C$ in Fourier analysis).

Chapter 9: Laplace Transform & s-Domain Circuit Applications

Directory Path

01 UNI/2-1/ece 2107 signals/09 laplace transform/

9.00 Chapter Map - Laplace Transform & s-Domain Analysis.md
9.01 Foundation of Laplace Transform & s-Plane Representation.md
9.02 Laplace Transforms of Singularity & Elementary Functions.md
9.03 Mathematical Properties of the Laplace Transform.md
9.04 Initial and Final Value Theorems: Statements, Proofs, and Bounds.md
9.05 Pole-Zero Analysis, Transfer Functions & s-Domain Stability.md
9.06 s-Domain Modeling of Passive Circuit Elements with Initial Conditions.md
9.07 Transient Response of RL & RC Circuits using Laplace Transform.md
9.08 Transient Response of Series & Parallel RLC Networks.md

`9.00 Chapter Map - Laplace Transform & s-Domain Analysis.md`

Type: Map of Content (MOC)
Syllabus & Core Objectives: This chapter covers s-domain system modeling and transient circuit analysis, linking the properties of continuous-time LTI systems to the s-plane.

Vault Frontmatter:

title: "9.00 Chapter Map - Laplace Transform & s-Domain Analysis"
aliases: [Laplace Map, Chapter 9 MOC]
tags: [signals-and-systems/moc, ece-2107, term-2-1]
type: MOC
course: "[[ECE 2107 - Signals and Systems]]"
chapter: 9

Note Sequence Matrix:
- [[9.01 Foundation of Laplace Transform & s-Plane Representation]] — Unilateral and Bilateral Laplace transforms and their relationship to the Fourier transform.
- [[9.02 Laplace Transforms of Singularity & Elementary Functions]] — Common Laplace pairs (step, ramp, impulses).
- [[9.03 Mathematical Properties of the Laplace Transform]] — Laplace properties and s-domain differentiation.
- [[9.04 Initial and Final Value Theorems: Statements, Proofs, and Bounds]] — Value theorems and their system stability limits.
- [[9.05 Pole-Zero Analysis, Transfer Functions & s-Domain Stability]] — Poles and zeros, Routh-Hurwitz stability, and s-plane pole mapping.
- [[9.06 s-Domain Modeling of Passive Circuit Elements with Initial Conditions]] — modeling components with initial conditions.
- [[9.07 Transient Response of RL & RC Circuits using Laplace Transform]] — s-domain transient solving workflows.
- [[9.08 Transient Response of Series & Parallel RLC Networks]] — Second-order transient analysis.

`9.01 Foundation of Laplace Transform & s-Plane Representation.md`

Type: Theoretical & Boundary Note
Syllabus Boundaries: Unilateral and Bilateral Laplace transform definitions, the complex s-plane ( $s = σ + jω$ ), the relationship between the Laplace and Fourier transforms, the existence conditions of the Laplace transform, and the Region of Convergence (ROC).
Definitions to Cover:
- Unilateral Laplace Transform — Maps a causal time-domain signal to the complex s-domain: $F (s) = \int_{0^{-}}^{\infty} f (t) e^{- s t} d t$
- Inverse Laplace Transform — Reconstructs a time-domain signal from its s-domain representation: $f (t) = \frac{1}{2 πj} \int_{σ - j \infty}^{σ + j \infty} F (s) e^{s t} d s$
- Region of Convergence (ROC) — The region in the complex s-plane where the Laplace integral converges to a finite value.
Mathematical Derivations: Laplace-Fourier Equivalence — Derive the mathematical relationship between the Laplace and Fourier transforms by substituting $s = σ + jω$ into the unilateral Laplace integral, showing that the Laplace transform is the Fourier transform of a damped signal: $L {f (t)} = F {f (t) e^{- σ t}}$
Key Diagrams / Graphics:
- [GRAPH: Complex s-plane diagram showing a shaded vertical region of convergence (ROC) for F(s) = 1/(s-3) with boundary \sigma > 3 - source: Senior Notes]
Common Mistakes Callout: Attempting to evaluate an inverse Laplace transform without considering its region of convergence, which can lead to non-unique solutions.

`9.02 Laplace Transforms of Singularity & Elementary Functions.md`

Type: Analytical Note
Syllabus Boundaries: Standard Laplace transform pairs for unit impulse $δ (t)$ , unit step $u (t)$ , ramp $r (t)$ , real exponential $e^{- a t}$ , sin/cos, hyperbolic sinh/cosh, and $t^{n}$ .
Mathematical Derivations:
- Power Function Laplace Derivation — Derive the Laplace transform of the power function $t^{n}$ using mathematical induction and integration by parts: $L {t^{n}} = \frac{n !}{s ^{n + 1}}$
Summary Table: A clean, 2-column reference table of standard Laplace transform pairs.

`9.03 Mathematical Properties of the Laplace Transform.md`

Type: Proof & Calculation Note
Syllabus Boundaries: Linearity, scaling, time shift, frequency shift (modulation), time differentiation, time integration, and s-domain differentiation properties of the Laplace transform.
Mathematical Derivations:
- s-Domain Differentiation Proof — Prove the s-domain differentiation property: $\frac{d F ( s )}{d s} = L {- t f (t)}$
Core Solved Numericals:
- Find the Laplace transform of a sawtooth pulse wave.
- Find the Laplace transform of composite functions using properties (e.g., $t sin (ω t)$ using s-domain differentiation, or $cos^{3} (3 t)$ using trigonometric identities).
Common Mistakes Callout: Applying frequency scaling before time-shifting, which scales the delay factor itself and leads to incorrect results.

`9.04 Initial and Final Value Theorems: Statements, Proofs, and Bounds.md`

Type: Proof & Analytical Note
Syllabus Boundaries: Initial Value Theorem (IVT) and Final Value Theorem (FVT) statements, proofs, and system stability bounds.
Definitions to Cover:
- Initial Value Theorem (IVT) — Finds the value of a signal immediately after a transition: $lim_{t \to 0^{+}} f (t) = lim_{s \to \infty} s F (s)$
- Final Value Theorem (FVT) — Finds the steady-state final value of a stable signal as $t \to \infty$ : $lim_{t \to \infty} f (t) = lim_{s \to 0} s F (s)$
Mathematical Derivations:
- Value Theorems Proof — Mathematically prove the Initial and Final Value Theorems starting from the Laplace transform of a derivative: $L {\frac{df ( t )}{d t}} = s F (s) - f (0^{-})$
Core Solved Numericals:
- Find the initial and final values of the function: $F (s) = \frac{17 s ^{3} + 7 s ^{2} + s + 6}{s ^{5} + 3 s ^{4} + 5 s ^{3} + 4 s ^{2} + 2 s}$
Common Mistakes Callout: Applying the Final Value Theorem to an unstable system or a system with poles on the imaginary $jω$ -axis (e.g., $sin (ω t)$ does not settle to a single final value as $t \to \infty$ ).

`9.05 Pole-Zero Analysis, Transfer Functions & s-Domain Stability.md`

Type: Spectral Mapping Note
Syllabus Boundaries: Poles and zeros in the s-domain, system transfer functions $H (s)$ , system stability classifications, s-plane pole mapping, Routh-Hurwitz stability criterion, and inverse Laplace transform using partial fraction expansion.
Definitions to Cover:
- Poles — The values of $s$ where the transfer function approaches infinity (denominator roots).
- Zeros — The values of $s$ where the transfer function equals zero (numerator roots).
- Absolute Stability — All poles of the transfer function lie strictly in the left-half of the complex s-plane.
- Marginal Stability — Simple, non-repeated poles lie on the imaginary $jω$ -axis, with no poles in the right-half plane (produces sustained oscillations).
- Unstable System — Any poles of the transfer function lie in the right-half of the s-plane.
Core Solved Numericals:
- Plot the poles and zeros for current $I (s) = \frac{3 s}{( s + 2 ) ( s + 4 )}$ and find its time-domain output $i (t)$ .
- Plot the poles and zeros for $V (s) = \frac{( s + 1 ) ( s + 3 )}{( s + 2 ) ( s + 4 )}$ and find $v (t)$ .
- Plot the poles and zeros for current $I (s) = \frac{3 ( s - 1 )}{( s + 1 ) ( s + 3 )}$ and find $i (t)$ .
- Find the impulse response $h (t)$ for the transfer function $H (s) = \frac{9 s}{s ^{2} + 2 s + 2}$ and plot its pole-zero diagram.
- Determine the inverse Laplace transform of the higher-order function: $F (s) = \frac{2 s ^{2} + 3 s + 3}{( s + 1 ) ( s + 3 ) ^{2}}$
Key Diagrams / Graphics:
- [GRAPH: s-plane pole-zero map showing stable (LHP), unstable (RHP), and marginally stable (imaginary axis) pole locations - source: Senior Notes]

`9.06 s-Domain Modeling of Passive Circuit Elements with Initial Conditions.md`

Type: Circuit Modeling Note
Syllabus Boundaries: Time-domain to s-domain modeling of passive resistors, inductors, and capacitors; modeling initial inductor current $i (0^{-})$ and initial capacitor voltage $v_{C} (0^{-})$ using series voltage sources or parallel current sources.
Formulation Equations:
- S-domain capacitor voltage: $V_{C} (s) = \frac{1}{C s} I (s) + \frac{v _{C} ( 0 ^{-} )}{s}$
- S-domain inductor voltage: $V_{L} (s) = L s I (s) - L i (0^{-})$
Key Diagrams / Graphics:
- [CIRCUIT: Reference table showing passive elements (R, L, C) in the time domain alongside their corresponding s-domain series voltage and parallel current equivalent models with initial conditions - source: Textbook Ch 4]
Common Mistakes Callout: Mixing up the polarities of initial condition voltage sources (the inductor initial current source acts as a voltage source opposing KVL, whereas the capacitor initial voltage source acts as a step source supporting KVL).

`9.07 Transient Response of RL & RC Circuits using Laplace Transform.md`

Type: Circuit Solving Note
Syllabus Boundaries: Transient response of first-order series RL and RC circuits, formulating s-domain loop equations, step response, and impulse response.
Core Solved Numericals:
- Find the current $i (t)$ of a first-order circuit when a switch moves from position 1 to position 2 at $t = 0$ (with initial condition $i (0^{-})$ considered).
- Draw the s-domain equivalent circuit and solve for the transient current $i (t)$ of a series RL circuit when the switch is closed.
- Formulate the loop equations and solve for the output voltage $v_{o} (t)$ of a passive RC low-pass filter circuit.

`9.08 Transient Response of Series & Parallel RLC Networks.md`

Type: Circuit Solving Note
Syllabus Boundaries: Transient analysis of series and parallel second-order RLC circuits, second-order s-domain equations, over-damped, under-damped, and critically-damped response conditions, and Laplace convolution.
Core Solved Numericals:
- Formulate loop equations and solve for the currents $i_{1} (t)$ and $i_{2} (t)$ , output voltage, and initial/final current values of a multi-loop second-order RLC network when a switch is closed.
- Obtain the inductor current expression in a second-order circuit after switch $S$ has been closed for a long time and is opened at $t = 0$ .
- Find the transient current $i (t)$ of a series RLC circuit with no initial charge when a switch is closed at $t = 0$ .
- Formulate the differential equation relating $I_{s}$ and $I_{L}$ in a parallel RLC circuit ( $R = 1 Ω$ , $L = 1 H$ ). Find the zero-state response for $I_{L} (t)$ when the input is $I_{s} (t) = e^{- 3 t} u (t)$ .

Chapter 10: Z-Transform & Discrete-Time Analysis

Directory Path

01 UNI/2-1/ece 2107 signals/10 z transform/

10.00 Chapter Map - Z-Transform & Discrete Analysis.md
10.01 Z-Transform: Definitions & Region of Convergence (ROC).md
10.02 Z-Transform Properties & Standard Pairs.md
10.03 Initial & Final Value Theorems in the Z-Domain.md
10.04 Inverse Z-Transform Methods.md
10.05 Convolution, Correlation & Realization in Z-Domain.md

`10.00 Chapter Map - Z-Transform & Discrete Analysis.md`

Type: Map of Content (MOC)
Syllabus & Core Objectives: The Z-transform is the discrete-time equivalent of the Laplace transform. This chapter covers Z-domain mapping, properties, region of convergence (ROC), value theorems, and solving discrete-time LTI difference equations.

Vault Frontmatter:

title: "10.00 Chapter Map - Z-Transform & Discrete Analysis"
aliases: [Z-Transform Map, Chapter 10 MOC]
tags: [signals-and-systems/moc, ece-2107, term-2-1]
type: MOC
course: "[[ECE 2107 - Signals and Systems]]"
chapter: 10

Note Sequence Matrix:
- [[10.01 Z-Transform: Definitions & Region of Convergence (ROC)]] — Unilateral and bilateral Z-transforms and ROC boundaries.
- [[10.02 Z-Transform Properties & Standard Pairs]] — Z-transform properties and standard Z-pairs.
- [[10.03 Initial & Final Value Theorems in the Z-Domain]] — Value theorems and their discrete unit-circle stability bounds.
- [[10.04 Inverse Z-Transform Methods]] — Inverse Z-transforms using long division, partial fractions, and residue calculus.
- [[10.05 Convolution, Correlation & Realization in Z-Domain]] — Convolution, discrete correlation, and system realization block diagrams.

`10.01 Z-Transform: Definitions & Region of Convergence (ROC).md`

Type: Theoretical & Boundary Note
Syllabus Boundaries: Bilateral and unilateral Z-transforms, mapping to the complex z-plane, the relationship between the Z-transform and the Laplace transform, and Z-domain Region of Convergence (ROC) properties.
Definitions to Cover:
- Bilateral Z-Transform — Maps a discrete-time sequence to the complex z-domain: $X (z) = \sum_{n = - \infty}^{\infty} x [n] z^{- n}$
- Unilateral Z-Transform — Maps causal discrete-time sequences: $X (z) = \sum_{n = 0}^{\infty} x [n] z^{- n}$
Mathematical Derivations:
- Laplace-to-Z Mapping Derivation — Derive the mapping from the s-plane to the z-plane by sampling a continuous signal ( $t = n T_{s}$ ), showing that $z = e^{s T_{s}}$ maps the LHP of the s-plane to the inside of the unit circle in the z-plane.
Core Solved Numericals:
- Calculate the Z-transform and ROC of a finite-duration window sequence: $x [n] = [u [n] - u [n - 5]]$
- Calculate the Z-transform of a sum of shifted impulses: $x [n] = δ [n + 1] + 3 δ [n] + 7 δ [n - 3] - 3 δ [n - 4]$
- Calculate the Z-transform of the shifted steps $u [n - k]$ and the time-reversed step $u [- n]$ .
- Calculate the Z-transform of exponential sequences: $(- \frac{1}{3})^{n} u [- n] and e^{j \frac{nπ}{4}} u [n]$
- Calculate the Z-transform of $x [n] = e^{- 4 n} u [n - 1]$ and plot its ROC.
Key Diagrams / Graphics:
- [GRAPH: Complex z-plane pole-zero map highlighting the unit circle and ROC shaded bounds for causal, anticausal, and double-sided sequences - source: Senior Notes]

`10.02 Z-Transform Properties & Standard Pairs.md`

Type: Proof & Reference Note
Syllabus Boundaries: Linearity, scaling, time shifting, time reversal, differentiation, and convolution properties of the Z-transform.
Mathematical Derivations:
- Z-Domain Differentiation Proof — Prove the Z-domain differentiation property (multiplication by $n$ ): $Z {n x [n]} = - z \frac{d X ( z )}{d z}$
Summary Table: Master table matching standard discrete sequences to their transforms and exact ROCs.

`10.03 Initial & Final Value Theorems in the Z-Domain.md`

Type: Proof & Calculation Note
Syllabus Boundaries: Z-domain Initial and Final Value Theorems, statements, proofs, and pole stability boundaries.
Definitions to Cover:
- Initial Value Theorem — Finds the starting value of a causal sequence: $x = lim_{z \to \infty} X (z)$
- Final Value Theorem — Finds the steady-state final value of a stable sequence: $x [\infty] = lim_{z \to 1} (1 - z^{- 1}) X (z)$
Core Solved Numericals:
- Find the initial and final values of the sequence $x [n]$ given the Z-domain function: $X (z) = 2 + 3 z^{- 1} + 4 z^{- 2}$
- Prove that the final value of $x [n]$ is 1.25 and its initial value is unity for the system function: $X (z) = \frac{z ^{2}}{( z - 1 ) ( z - 0.2 )}$
Common Mistakes Callout: Applying the Final Value Theorem to systems with poles on or outside the unit circle ( $∣ z ∣ \geq 1$ ), which causes the theorem to fail.

`10.04 Inverse Z-Transform Methods.md`

Type: Computational Note
Syllabus Boundaries: Inverse Z-transform techniques, including power series expansion (long division), partial fraction expansion, and Cauchy’s residue contour integration.
Mathematical Derivations:
- Cauchy’s Residue Integration Proof — Mathematically derive Cauchy’s residue integration formula for the inverse Z-transform: $x [n] = \frac{1}{2 πj} \oint_{C} X (z) z^{n - 1} d z = \sum [Residues of X (z) z^{n - 1} at internal poles]$
Core Solved Numericals:
- Calculate the inverse Z-transform of $H (z) = \frac{z ^{2} - 3 z + 8}{( z - 2 ) ( z + 2 ) ( z + 3 )}$ using partial fractions.
- Calculate the inverse Z-transform of $X (z) = \frac{1}{z ( z - 0.8 ) ( z + 0.4 )}$ assuming a causal signal.
- Calculate the inverse Z-transform of $X (z) = \frac{z ^{- 1}}{1 - 3 z ^{- 1}}$ using the time-shifting property.
- Calculate the inverse Z-transform of $X (z) = \frac{z}{( z - a ) ( z - b )}$ assuming a right-sided sequence.
- Calculate the inverse Z-transform of $X (z) = \frac{1}{( z - 0.2 ) ( z - 0.9 )}$ using the residue method.
- Calculate the inverse Z-transform of $H (z) = \frac{- 4 + 8 z ^{- 1}}{1 + 6 z ^{- 1} + 8 z ^{- 2}}$ .
- Calculate the inverse Z-transform of $X (z) = \frac{z}{3 z ^{2} - 4 z + 1}$ under three different ROC conditions.
- Calculate the inverse Z-transform of $X (z) = \frac{1}{( 1 + z ^{- 1} ) ( 1 - z ^{- 1} ) ^{2}}$ for a causal signal.
- Calculate the inverse Z-transform using long division for a causal sequence ( $∣ z ∣ > 2$ ) and an anticausal sequence ( $∣ z ∣ < 0.5$ ).

`10.05 Convolution, Correlation & Realization in Z-Domain.md`

Type: System realization Note
Syllabus Boundaries: Discrete convolution in the Z-domain, cross-correlation and auto-correlation in the Z-domain, discrete transfer functions, system stability, and impulse response.
Core Solved Numericals:
- Calculate the convolution of the sequences $x_{1} [n] = {4, \underline{- 2}, 1}$ and $x_{2} [n] = {\underline{1}, 1, 1, 1, 1, 1}$ using Z-transforms.
- Find the cross-correlation sequence $r_{x y} [l]$ of the sequences $x [n] = {\underline{1}, 2, 3, 4}$ and $y [n] = {4, 3, 2, \underline{1}}$ using Z-transforms.
- Find the impulse response $h [n]$ of the system described by the difference equation: $y [n] - \frac{5}{6} y [n - 1] + \frac{1}{6} y [n - 2] = \frac{1}{3} x [n - 1]$
- Determine the transfer function, stability, impulse response $h [n]$ , and step response of the system $y [n] + y [n - 1] = x [n]$ using Z-transforms.
- Find the input sequence $x [n]$ given the system’s impulse response $h [n] = {\underline{1}, 2, 3}$ and output response $y [n] = {\underline{1}, 1, 2, - 1, 3}$ using Z-transforms.

Chapter 11: DFT, FFT, and Frequency-Domain Discrete Analysis

Directory Path

01 UNI/2-1/ece 2107 signals/11 dft fft/

11.00 Chapter Map - Discrete Fourier Transforms.md
11.01 The Discrete Fourier Transform (DFT).md
11.02 Properties of the DFT & Twiddle Factors.md
11.03 Radix-2 Decimation-In-Time FFT Algorithm (DIT-FFT).md
11.04 Radix-2 Decimation-In-Frequency FFT Algorithm (DIF-FFT).md

`11.00 Chapter Map - Discrete Fourier Transforms.md`

Type: Map of Content (MOC)
Syllabus & Core Objectives: This chapter covers the Discrete Fourier Transform (DFT), twiddle factors, circular convolution, and the computationally efficient Radix-2 FFT algorithms (DIT and DIF).

Vault Frontmatter:

title: "11.00 Chapter Map - Discrete Fourier Transforms"
aliases: [FFT Map, Chapter 11 MOC]
tags: [signals-and-systems/moc, ece-2107, term-2-1]
type: MOC
course: "[[ECE 2107 - Signals and Systems]]"
chapter: 11

Note Sequence Matrix:
- [[11.01 The Discrete Fourier Transform (DFT)]] — N-point DFT equations and computational complexity.
- [[11.02 Properties of the DFT & Twiddle Factors]] — Twiddle factor properties, circular convolution, and circular shifting.
- [[11.03 Radix-2 Decimation-In-Time FFT Algorithm (DIT-FFT)]] — Bit-reversal and DIT butterfly diagrams.
- [[11.04 Radix-2 Decimation-In-Frequency FFT Algorithm (DIF-FFT)]] — DIF division and DIF butterfly diagrams.

`11.01 The Discrete Fourier Transform (DFT).md`

Type: Analytical Note
Syllabus Boundaries: N-point DFT and IDFT equations, and comparing the computational complexity of the DFT ( $O (N^{2})$ ) and the FFT ( $O (N lo g_{2} N)$ ).
Definitions to Cover:
- Discrete Fourier Transform (DFT) — Converts a finite-length discrete sequence into its discrete frequency spectrum: $X [k] = \sum_{n = 0}^{N - 1} x [n] e^{- j \frac{2 π}{N} nk} = \sum_{n = 0}^{N - 1} x [n] W_{N}^{nk}, k = 0, 1, \dots, N - 1$
- Inverse DFT (IDFT) — Reconstructs a time sequence from its DFT bins: $x [n] = \frac{1}{N} \sum_{k = 0}^{N - 1} X [k] e^{j \frac{2 π}{N} nk} = \frac{1}{N} \sum_{k = 0}^{N - 1} X [k] W_{N}^{- nk}, n = 0, 1, \dots, N - 1$
Core Solved Numericals:
- Evaluate the DFT of sequence $x [n] = {1, 2, 3, 4}$ using direct matrix multiplication.
Common Mistakes Callout: Forgetting the scaling factor $1/ N$ in the IDFT equation.

`11.02 Properties of the DFT & Twiddle Factors.md`

Type: Formulation & Circular Math Note
Syllabus Boundaries: Twiddle factor ( $W_{N}$ ) definition, its symmetry and periodicity properties, circular shifting, circular convolution, and circular-to-linear convolution mapping.
Definitions to Cover:
- Twiddle Factor ( $W_{N}$ ) — Complex exponential coefficient: $W_{N} = e^{- j \frac{2 π}{N}}$
- Twiddle Periodicity — Satisfies $W_{N}^{k + N} = W_{N}^{k}$ .
- Twiddle Symmetry — Satisfies $W_{N}^{k + N /2} = - W_{N}^{k}$ .
Core Solved Numericals:
- Determine the circular convolution of sequences $x_{1} [n] = {0, 1, 2, 3}$ and $x_{2} [n] = {1, 1, 2, 2}$ .
- Compare the circular convolution of the sequences above to their linear convolution, showing that linear convolution can be obtained using circular convolution by padding both sequences with zeros to length $L \geq N_{1} + N_{2} - 1$ .
Common Mistakes Callout: Performing linear convolution instead of circular convolution (circular convolution wraps indices around the sequence length modulo $N$ ).

`11.03 Radix-2 Decimation-In-Time FFT Algorithm (DIT-FFT).md`

Type: Butterfly Realization Note
Syllabus Boundaries: Radix-2 DIT-FFT algorithm, bit-reversal input sorting, and drawing 8-point DIT butterfly signal-flow diagrams.
Definitions to Cover:
- Decimation-In-Time (DIT) — Recursively splits the input sequence $x [n]$ into even and odd indices to compute the DFT: $X [k] = G [k] + W_{N}^{k} H [k]$
- Bit-Reversal — Reversing the binary representation of an index to sort the input sequence before applying the DIT-FFT algorithm.
Core Solved Numericals:
- Calculate the 8-point DFT of sequence $x [n] = {0, 1, 2, 3, 0, 0, 0, 0}$ using the Radix-2 DIT-FFT algorithm. Draw its complete butterfly diagram.
Key Diagrams / Graphics:
- [DIAGRAM: Complete 8-point Radix-2 DIT-FFT butterfly signal-flow graph showing bit-reversed inputs, twiddle factor multipliers, and normally ordered outputs - source: Senior Notes]
Common Mistakes Callout: Mismapping twiddle factor exponents across the butterfly stages; applying bit-reversal sorting to the outputs instead of the inputs in the DIT algorithm.

`11.04 Radix-2 Decimation-In-Frequency FFT Algorithm (DIF-FFT).md`

Type: Butterfly Realization Note
Syllabus Boundaries: Radix-2 DIF-FFT algorithm, dividing the output sequence into even and odd frequency bins, and drawing 8-point DIF butterfly signal-flow diagrams.
Definitions to Cover:
- Decimation-In-Frequency (DIF) — Recursively splits the output frequency bins into even and odd frequencies to compute the DFT: $X [2 k] = \sum_{n = 0}^{N /2 - 1} (x [n] + x [n + N /2]) W_{N /2}^{nk}$
Core Solved Numericals:
- Calculate the 8-point DFT of sequence $x [n] = n + 1$ using the Radix-2 DIF-FFT algorithm.
- Calculate the 8-point DFT of sequence $x [n] = n^{2}$ using the Radix-2 DIF-FFT algorithm.
Key Diagrams / Graphics:
- [DIAGRAM: Complete 8-point Radix-2 DIF-FFT butterfly signal-flow graph showing normally ordered inputs, subtraction stages, twiddle multipliers, and bit-reversed outputs - source: Senior Notes]
Common Mistakes Callout: Applying bit-reversal sorting to the inputs instead of the outputs in the DIF algorithm.

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note making roadmap

Master Note-Making Roadmap: ECE 2107 & ECE 2108

Part I: Instructor 2 (Time-Domain Fundamentals, Systems & Network Applications)

Chapter 1: Signal Fundamentals & Operations

Directory Path

1.00 Chapter Map - Signal Fundamentals & Operations.md

1.01 Signals, Systems & Singularity Functions.md

1.02 Elementary Operations on Signals.md

1.03 Signal Classification I — Theory & Periodicity.md

1.04 Signal Classification II — Symmetry, Energy & Power.md

1.05 Typical Signal Processing Operations.md

Chapter 2: LTI System Properties & Time-Domain Convolution

Directory Path

2.00 Chapter Map - LTI System Properties & Convolution.md

2.01 Systems Classification, LTI Properties & Stability.md

2.02 The Continuous Convolution Integral.md

2.03 The Discrete Convolution Sum & Realization.md

2.04 Systems Described by Differential & Difference Equations.md

2.05 Block-Diagram Representations & System Interconnections.md

Chapter 3: State-Space Representation of Continuous-Time Systems

Directory Path

3.00 Chapter Map - State-Space Analysis of CT Systems.md

3.01 Introduction to State-Space & Matrix Formulations.md

3.02 State-Space Modeling of Electrical Circuits.md

3.03 State-Space Modeling of Block Diagrams.md

3.04 State Transition Matrix: Properties & Solution Methods.md

Chapter 4: The Sampling Theorem & Multi-Rate Processing

Directory Path

4.00 Chapter Map - Sampling & Multi-Rate Processing.md

4.01 The Shannon-Nyquist Sampling Theorem.md

4.02 Spectral Replications & Aliasing Dynamics.md

4.03 Ideal, Natural & Flat-Top Sampling Techniques.md

4.04 Multi-Rate Signal Processing: Decimation & Interpolation.md

Chapter 5: Analog Filter Design

Directory Path

5.00 Chapter Map - Analog Filter Design.md

5.01 Filter Specifications & Tolerance Curves.md

5.02 Butterworth Filter Approximation.md

5.03 Chebyshev & Elliptic Filter Approximations.md

5.04 Active Filter Realization & Sallen-Key RC Networks.md

Chapter 6: Two-Port Network Theory

Directory Path

6.00 Chapter Map - Two-Port Networks.md

6.01 Open-Circuit Impedance (Z) Parameters.md

6.02 Short-Circuit Admittance (Y) Parameters.md

6.03 Transmission (ABCD) Parameters & Cascaded Networks.md

6.04 Hybrid (h) & Inverse Hybrid (g) Parameter Formulations.md

Part II: Instructor 1 (Transform-Domain Analysis & Fast Algorithms)

Chapter 7: Continuous-Time Fourier Series (CTFS)

Directory Path

7.00 Chapter Map - Continuous-Time Fourier Series.md

7.01 Trigonometric Fourier Series Representation.md

7.02 Symmetry Conditions & Waveform Analysis.md

7.03 Exponential Fourier Series & Complex Spectra.md

7.04 Parseval’s Theorem & Convergence Conditions.md

Chapter 8: Continuous-Time Fourier Transform (CTFT)

Directory Path

8.00 Chapter Map - Continuous-Time Fourier Transform.md

8.01 Foundation of the Continuous-Time Fourier Transform (CTFT).md

8.02 Properties of the Continuous-Time Fourier Transform.md

8.03 Rayleigh's Energy Theorem & Spectral Density.md

8.04 CTFT Pairs for Singularity & Common Functions.md

8.05 Time-Domain Convolution & Multiplication Properties.md

8.06 Frequency Spectra, Phase Spectra & Modulation.md

8.07 CTFT System Analysis of Continuous Networks.md

Chapter 9: Laplace Transform & s-Domain Circuit Applications

Directory Path

9.00 Chapter Map - Laplace Transform & s-Domain Analysis.md

9.01 Foundation of Laplace Transform & s-Plane Representation.md

9.02 Laplace Transforms of Singularity & Elementary Functions.md

9.03 Mathematical Properties of the Laplace Transform.md

9.04 Initial and Final Value Theorems: Statements, Proofs, and Bounds.md

9.05 Pole-Zero Analysis, Transfer Functions & s-Domain Stability.md

9.06 s-Domain Modeling of Passive Circuit Elements with Initial Conditions.md

9.07 Transient Response of RL & RC Circuits using Laplace Transform.md

9.08 Transient Response of Series & Parallel RLC Networks.md

Chapter 10: Z-Transform & Discrete-Time Analysis

Directory Path

10.00 Chapter Map - Z-Transform & Discrete Analysis.md

`1.00 Chapter Map - Signal Fundamentals & Operations.md`

`1.01 Signals, Systems & Singularity Functions.md`

`1.02 Elementary Operations on Signals.md`

`1.03 Signal Classification I — Theory & Periodicity.md`

`1.04 Signal Classification II — Symmetry, Energy & Power.md`

`1.05 Typical Signal Processing Operations.md`

`2.00 Chapter Map - LTI System Properties & Convolution.md`

`2.01 Systems Classification, LTI Properties & Stability.md`

`2.02 The Continuous Convolution Integral.md`

`2.03 The Discrete Convolution Sum & Realization.md`

`2.04 Systems Described by Differential & Difference Equations.md`

`2.05 Block-Diagram Representations & System Interconnections.md`

`3.00 Chapter Map - State-Space Analysis of CT Systems.md`

`3.01 Introduction to State-Space & Matrix Formulations.md`

`3.02 State-Space Modeling of Electrical Circuits.md`

`3.03 State-Space Modeling of Block Diagrams.md`

`3.04 State Transition Matrix: Properties & Solution Methods.md`

`4.00 Chapter Map - Sampling & Multi-Rate Processing.md`

`4.01 The Shannon-Nyquist Sampling Theorem.md`

`4.02 Spectral Replications & Aliasing Dynamics.md`

`4.03 Ideal, Natural & Flat-Top Sampling Techniques.md`

`4.04 Multi-Rate Signal Processing: Decimation & Interpolation.md`

`5.00 Chapter Map - Analog Filter Design.md`

`5.01 Filter Specifications & Tolerance Curves.md`

`5.02 Butterworth Filter Approximation.md`

`5.03 Chebyshev & Elliptic Filter Approximations.md`

`5.04 Active Filter Realization & Sallen-Key RC Networks.md`

`6.00 Chapter Map - Two-Port Networks.md`

`6.01 Open-Circuit Impedance (Z) Parameters.md`

`6.02 Short-Circuit Admittance (Y) Parameters.md`

`6.03 Transmission (ABCD) Parameters & Cascaded Networks.md`

`6.04 Hybrid (h) & Inverse Hybrid (g) Parameter Formulations.md`

`7.00 Chapter Map - Continuous-Time Fourier Series.md`

`7.01 Trigonometric Fourier Series Representation.md`

`7.02 Symmetry Conditions & Waveform Analysis.md`

`7.03 Exponential Fourier Series & Complex Spectra.md`

`7.04 Parseval’s Theorem & Convergence Conditions.md`

`8.00 Chapter Map - Continuous-Time Fourier Transform.md`

`8.01 Foundation of the Continuous-Time Fourier Transform (CTFT).md`

`8.02 Properties of the Continuous-Time Fourier Transform.md`

`8.03 Rayleigh's Energy Theorem & Spectral Density.md`

`8.04 CTFT Pairs for Singularity & Common Functions.md`

`8.05 Time-Domain Convolution & Multiplication Properties.md`

`8.06 Frequency Spectra, Phase Spectra & Modulation.md`

`8.07 CTFT System Analysis of Continuous Networks.md`

`9.00 Chapter Map - Laplace Transform & s-Domain Analysis.md`

`9.01 Foundation of Laplace Transform & s-Plane Representation.md`

`9.02 Laplace Transforms of Singularity & Elementary Functions.md`

`9.03 Mathematical Properties of the Laplace Transform.md`

`9.04 Initial and Final Value Theorems: Statements, Proofs, and Bounds.md`

`9.05 Pole-Zero Analysis, Transfer Functions & s-Domain Stability.md`

`9.06 s-Domain Modeling of Passive Circuit Elements with Initial Conditions.md`

`9.07 Transient Response of RL & RC Circuits using Laplace Transform.md`

`9.08 Transient Response of Series & Parallel RLC Networks.md`

`10.00 Chapter Map - Z-Transform & Discrete Analysis.md`

`10.01 Z-Transform: Definitions & Region of Convergence (ROC).md`

`10.02 Z-Transform Properties & Standard Pairs.md`

`10.03 Initial & Final Value Theorems in the Z-Domain.md`

`10.04 Inverse Z-Transform Methods.md`

`10.05 Convolution, Correlation & Realization in Z-Domain.md`

`11.00 Chapter Map - Discrete Fourier Transforms.md`

`11.01 The Discrete Fourier Transform (DFT).md`

`11.02 Properties of the DFT & Twiddle Factors.md`

`11.03 Radix-2 Decimation-In-Time FFT Algorithm (DIT-FFT).md`

`11.04 Radix-2 Decimation-In-Frequency FFT Algorithm (DIF-FFT).md`