Laplace Transform Properties

Laplace Transform Definition

• The Laplace transform of a function $f(t)$ is defined as $F(s) = \mathcal{L} {f(t)} = \int_{0}^{\infty} e^{-st} f(t) dt$ for all real numbers $t \geq 0$.
• $s$ represents a complex number frequency parameter, $s = \sigma + j\omega$.

Region of Convergence (ROC)

• The Laplace transform exists if the defining integral converges.
• Convergence occurs when $Re(s) > a$ for some real number $a$, so the ROC is $Re(s) > a$.

Linearity Property

• The Laplace transform exhibits linearity: $\mathcal{L} {af(t) + bg(t)} = a\mathcal{L} {f(t)} + b\mathcal{L} {g(t)}$.

Time Scaling Property

• Time scaling affects the Laplace transform as follows: $\mathcal{L} {f(at)} = \frac{1}{|a|} F(\frac{s}{a})$.

Time Shifting Property

• Shifting in time is represented by $\mathcal{L} {f(t - a)u(t - a)} = e^{-as}F(s)$, where $u(t)$ is the Heaviside step function.

Shifting in the s-Domain Property

• Shifting in the s-domain is represented by $\mathcal{L} {e^{at}f(t)} = F(s - a)$.

Differentiation in the Time Domain Property

• First derivative: $\mathcal{L} {\frac{d}{dt}f(t)} = sF(s) - f(0)$.
• Second derivative: $\mathcal{L} {\frac{d^2}{dt^2}f(t)} = s^2F(s) - sf(0) - f'(0)$.
• nth derivative: $\mathcal{L} {\frac{d^n}{dt^n}f(t)} = s^nF(s) - s^{n-1}f(0) - s^{n-2}f'(0) -... - f^{(n-1)}(0)$.

Integration in the Time Domain Property

• The Laplace transform of an integral is given by: $\mathcal{L} {\int_{0}^{t} f(\tau) d\tau} = \frac{F(s)}{s}$.

Differentiation in the s-Domain Property

• $\mathcal{L} {tf(t)} = -\frac{d}{ds}F(s)$.
• $\mathcal{L} {t^nf(t)} = (-1)^n \frac{d^n}{ds^n}F(s)$.

Convolution Property

• $\mathcal{L} {(f * g)(t)} = F(s)G(s)$, where $(f * g)(t) = \int_{0}^{t} f(\tau)g(t - \tau) d\tau$.

Initial Value Theorem

• The initial value of $f(t)$ can be found by: $\lim_{t \to 0} f(t) = \lim_{s \to \infty} sF(s)$.

Final Value Theorem

• The final value of $f(t)$ can be found by: $\lim_{t \to \infty} f(t) = \lim_{s \to 0} sF(s)$.

Common Laplace Transforms

• Unit Impulse: $f(t) = \delta(t)$, $F(s) = 1$, ROC: All s.
• Unit Step: $f(t) = u(t)$, $F(s) = \frac{1}{s}$, ROC: $Re(s) > 0$.
• Ramp: $f(t) = t$, $F(s) = \frac{1}{s^2}$, ROC: $Re(s) > 0$.
• Exponential: $f(t) = e^{at}$, $F(s) = \frac{1}{s - a}$, ROC: $Re(s) > Re(a)$.
• Sine: $f(t) = \sin(\omega t)$, $F(s) = \frac{\omega}{s^2 + \omega^2}$, ROC: $Re(s) > 0$.
• Cosine: $f(t) = \cos(\omega t)$, $F(s) = \frac{s}{s^2 + \omega^2}$, ROC: $Re(s) > 0$.
• Hyperbolic Sine: $f(t) = \sinh(at)$, $F(s) = \frac{a}{s^2 - a^2}$, ROC: $Re(s) > |a|$.
• Hyperbolic Cosine: $f(t) = \cosh(at)$, $F(s) = \frac{s}{s^2 - a^2}$, ROC: $Re(s) > |a|$.
• Damped Sine: $f(t) = e^{-at}\sin(\omega t)$, $F(s) = \frac{\omega}{(s + a)^2 + \omega^2}$, ROC: $Re(s) > -a$.
• Damped Cosine: $f(t) = e^{-at}\cos(\omega t)$, $F(s) = \frac{s + a}{(s + a)^2 + \omega^2}$, ROC: $Re(s) > -a$.
• t to the power of n: $f(t) = t^n$, $F(s) = \frac{n!}{s^{n+1}}$, ROC: $Re(s) > 0$.

Sum of Vectors - Graphical Method

• Vectors A and B are placed sequentially, maintaining their magnitudes, directions, and senses.
• The resultant vector R connects the origin of the first vector to the endpoint of the last vector.

Sum of Vectors - Analytical Method

Rectangular Components

• $A_x = A \cos \theta$.
• $A_y = A \sin \theta$.

Summation by Components

• $R_x = A_x + B_x + ...$.
• $R_y = A_y + B_y + ...$.
• Magnitude: $R = \sqrt{R_x^2 + R_y^2}$.
• Direction: $\theta = \arctan \frac{R_y}{R_x}$.

Dot Product (Scalar Product)

• $\vec{A} \cdot \vec{B} = AB \cos \theta = A_x B_x + A_y B_y + A_z B_z$.
• The dot product results in a scalar value.

Cross Product (Vector Product)

• $\vec{A} \times \vec{B} = AB \sin \theta \hat{n}$.
• $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix} = (A_y B_z - A_z B_y)\hat{i} + (A_z B_x - A_x B_z)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$.
• The cross product results in a vector.
• The resulting vector is perpendicular to the plane formed by vectors A and B.
• Direction determined by the right-hand rule.

Energy Bands

• E vs. k Diagram: solutions of Schrödinger's equation for an electron in a crystal lattice.

Metals

• The Fermi level lies within an allowed band, resulting in a partially filled band.

Semiconductors

• The Fermi level lies within a band gap separating a completely filled band (valence band) from an empty band (conduction band).

Insulators

• The Fermi level also lies within a band gap, but the gap is much larger than in semiconductors.

Intrinsic Semiconductor Definition

• A perfect crystal with no impurities or lattice defects.

Examples of Intrinsic Semiconductors

• Silicon (Si) and Germanium (Ge) are Group IV elements.
• These elements have a diamond crystal structure where each atom is covalently bonded to four neighbors.

Energy Band Diagram at 0K

• All valence band states are full.
• All conduction band states are empty.

Intrinsic Carrier Concentration Definition

• At $T > 0K$, some electrons are thermally excited from the valence band to the conduction band.
• This excitation creates electron-hole pairs.
• Variable relationship: The concentration of electrons in the conduction band ($n$) is equal to the concentration of holes in the valence band ($p$).
• $n = p = n_i$, where $n_i$ is the intrinsic carrier concentration.

Temperature Dependence Characteristic

• The intrinsic carrier concentration ($n_i$) increases exponentially with temperature.

Fermi Level Location in Intrinsic Semiconductor Characteristic

• The Fermi level ($E_F$) lies near the middle of the band gap ($E_g$).

Mathematical Expression for Intrinsic Carrier Concentration

• $n_i = \sqrt{N_c N_v} e^{-E_g / 2kT}$, where:
  • $N_c$ is the effective density of states in the conduction band.
  • $N_v$ is the effective density of states in the valence band.
  • $E_g$ is the band gap energy.
  • $k$ is Boltzmann's constant.
  • $T$ is the temperature in Kelvin.

Doping Definition

• Doping is the process of intentionally adding impurities to a semiconductor to control its electrical properties.

n-type Semiconductor details

• Doped with donor impurities.
• Donor impurities have more valence electrons than the host atoms (e.g., Phosphorus in Silicon).
• Donors contribute electrons to the conduction band.
• $n > n_i$.
• $E_F$ closer to $E_c$

p-type Semiconductor details

• Doped with acceptor impurities.
• Acceptor impurities have fewer valence electrons than the host atoms (e.g., Boron in Silicon).
• Acceptors create holes in the valence band.
• $p > n_i$.
• $E_F$ closer to $E_v$

Compensation Definition

• Occurs when both donor and acceptor impurities are present in the same semiconductor material.
• The conductivity type (n-type or p-type) is determined by the impurity with the higher concentration.

Drift, Carrier Transport

• Motion of charge carriers in response to an electric field.
• $J_{drift} = \sigma E = q(n\mu_n + p\mu_p)E$, where:
  • $\sigma$ is the conductivity.
  • $E$ is the electric field.
  • $q$ is the elementary charge.
  • $n$ and $p$ are the electron and hole concentrations, respectively.
  • $\mu_n$ and $\mu_p$ are the electron and hole mobilities, respectively.

Diffusion, Carrier Transport

• Movement of charge carriers from a region of high concentration to a region of low concentration.
• $J_{diffusion} = qD_n \frac{dn}{dx} - qD_p \frac{dp}{dx}$, where:
  • $D_n$ and $D_p$ are the electron and hole diffusion coefficients, respectively.
  • $\frac{dn}{dx}$ and $\frac{dp}{dx}$ are the electron and hole concentration gradients, respectively.

Einstein Relation

• Relates the diffusion coefficient and mobility of charge carriers.
• $\frac{D_n}{\mu_n} = \frac{D_p}{\mu_p} = \frac{kT}{q}$, where:
  • $k$ is Boltzmann's constant.
  • $T$ is the temperature in Kelvin.
  • $q$ is the elementary charge.

Algorithmic Complexity - Purpose

• Quantify the amount of resources an algorithm requires.
• Facilitate the comparison of algorithm efficiency.
• Predict scalability relative to input size.

Algorithmic Complexity - Use

• Choose suitable algorithms.
• Optimize code performance.
• Understand computational limits.

Algorithmic Complexity - Process

1. Choose a computation model.
2. Define input size, $n$.
3. Count operations as a function of $n$.
4. Express complexity using Big-O.

Computation Models

• Turing Machine: abstract model with tape manipulation based on rules.
• Random Access Machine (RAM): practical model allowing random memory access.
• Word RAM: RAM model with fixed-size word memory division.
• Real RAM: RAM model that allows storage of real numbers.

Big-O Notation - Definition

• $f(n) = O(g(n))$ if constants exist such that $f(n) \le cg(n)$ for large $n$.

Big-O Notation - Intuition

• $f(n)$ grows no faster than $g(n)$ asymptotically.

Using Big-O Notation - Example

• $n^2 + n = O(n^2)$.
• $100n = O(n)$.
• $\log(n) = O(n)$.

Common Complexities

Simplifying Big-O Expressions Tips

• Drop lower order terms: $n^2 + n \rightarrow n^2$.
• Ignore constant factors: $100n \rightarrow n$.
• Use simplest expression: $O(n^2 + n) \rightarrow O(n^2)$.
• Different logarithmic bases are equivalent: $O(log_a n) = O(log_b n)$.

Searching Algorithms

• Linear Search: $O(n)$.
• Binary Search: $O(log n)$.

Sorting Algorithms

• Bubble Sort: $O(n^2)$.
• Merge Sort: $O(n log n)$.

Big-O Notation - Limitations

• Asymptotic behavior only.
• Constant factors can matter practically.
• Doesn't reflect actual running time.
• Differentiates between worst-case and average-case complexity.

Game Theory Defined

• Models strategic interactions among rational agents in various disciplines.

Game Theory Sub Disciplines

• Cooperative: Focus on coalitions.
• Non-Cooperative: Focus on strategies of individual players.

Normal-Form Games Defined

• Games described via a payoff matrix.

Normal-Form Games, key components

• A tuple $(N, A, u)$, where:
  • $N$ is a finite set of $n$ players, indexed by $i$.
  • $A = A_1 \times \dots \times A_n$, where $A_i$ is a finite set of actions for player $i$.
  • $u = (u_1, \dots, u_n)$, where $u_i : A \mapsto \mathbb{R}$ is a utility function for player $i$.

Prisoner's Dilemma - Scenario

• Two suspects, insufficient evidence.
• Confess and other doesn't: free vs. 3 years.
• Both confess: 2 years each.
• Neither confess: 1 year each.

Prisoner's Dilemma - Game Representation

• $N = {1, 2}$.
• $A_i = {\text{Cooperate}, \text{Defect}}$.

Prisoner's Dilemma - Utility Table

Pure Strategy Definition

• An action $a_i \in A_i$.

Mixed Strategy Definition

• A probability distribution over $A_i$. $S_i$ is the set of all possible mixed strategies for player $i$. Let $s_i(a_i)$ be the probability of playing action $a_i$ under strategy $s_i$.

Strategy Profile Definition

• A tuple $s = (s_1, \dots, s_n)$, where $s_i \in S_i$ is a mixed strategy for player $i$. Let $S = S_1 \times \dots \times S_n$.

Expected Utility Definition

• $u_i(s) = \sum_{a \in A} u_i(a) \prod_{j \in N} s_j(a_j)$.

Nash Equilibrium Definition

• A strategy profile $s^* \in S$ with $u_i(s_i^, s_{-i}^) \ge u_i(s_i, s_{-i}^*)$ for every player $i$.

Nash Equilibrium - Interpretation

• No incentive for unilateral deviation.

Prisoner's Dilemma Nash Equilibrium

• (Defect, Defect).

Key Game Theory Theorem

• Every normal-form game with finite players and actions has at least one Nash Equilibrium, possibly in mixed strategies (Nash, 1950).