Transmitting Station Basics

Transmitting Station Basics

• A transmitting station is an electronic facility that generates and radiates radio frequency (RF) signals into space via an antenna.
• These signals carry info like audio, video, or data over long distances, enabling broadcasting and telecommunications.

Transmitting Station Components

• Signal Source: Generates the information for transmission (microphone, camera, data stream).
• Modulator: Combines the signal with a carrier wave using techniques like AM, FM, or PM.
• Oscillator: Generates the high-frequency carrier wave for transmission, determining the radio frequency.
• Amplifier: Increases the power of the modulated signal while minimizing distortion.
• Antenna: Radiates the amplified signal as electromagnetic waves, either omnidirectionally or directionally.
• Transmission Line: Connects the amplifier to the antenna, efficiently transferring the signal.
• Power Supply: Provides the electrical power to operate the station components.
• Control and Monitoring Equipment: Monitors station performance and allows operators to make adjustments.
• Protection Circuitry: Protects the station from damage due to abnormal conditions.

Types of Transmitting Stations

• Broadcast Transmitters: High power for radio and TV over wide areas.
• Telecommunications Transmitters: Moderate power for point-to-point communication.
• Mobile Transmitters: Low power for mobile devices to conserve battery.
• Radar Transmitters: Pulse transmissions for object detection.
• Satellite Transmitters: For communication between satellites and ground stations.

Factors Affecting Transmission Range

• Transmitter Power: Higher power equates to greater range.
• Antenna Height: Higher antennas increase line-of-sight and reduce blockage.
• Antenna Gain: Higher gain focuses the signal, increasing effective power.
• Frequency: Lower frequencies have greater range due to better propagation.
• Atmospheric Conditions: Rain, fog, and ducting can affect range.
• Terrain: Hills, mountains, and buildings can block or reflect signals.

Regulations and Standards

• Government agencies like the FCC (US) and ITU (global) regulate frequency allocation, power limits, and emission standards.
• These regulations prevent interference and ensure efficient use of the radio spectrum.

Applications of Transmitting Stations

• Used in radio and TV broadcasting, mobile communications, satellite communications, radar systems, and more.

Algorithmic Game Theory Basics

• Game Theory entails the study of mathematical models of strategic interactions among rational agents.
• Algorithmic Game Theory merges computer science with game theory, focusing on computationally feasible solutions to game-theoretic problems.
• Computer Science contributes algorithms, data structures, and computational complexity.
• Game Theory provides insights into economic incentives and strategic behavior.

Core Concepts

• Strategic Interaction: Actions of different players mutually affect payoffs.
• Rational Agent: An agent optimizes based on its payoff (values such as money or utility).
• Model: A simplified representation of reality.

Examples of Alorithmic Game Theory

Selfish Routing

• Road network with latency dependent on traffic.
• Drivers choose routes to minimize travel time.
• Selfish behavior impacts overall traffic flow.

Mechanism Design

• Seller wants to sell an item to a buyer.
• Seller aims to maximize revenue without knowing the buyer's valuation.
• Buyer seeks to maximize utility (valuation minus price).

Braess's Paradox

• Adding a new road ironically increases overall latency in a network due to drivers' selfish routing decisions.
• 2000 drivers travel from A to B selecting between two routes.
• Nash equilibrium is both routes are equally good and 1000 drivers choose both routes .
• Latency for each driver is 55.
• Adding a new road can make all drivers choose the new route with a latency of 40.
• Total latency has increased to 40 from 55 in the previous equilibrium.

Mechanism Design Example: VCG Auction

• Government wants to build a new road, with $n$ nearby residents.
• Each agent $i$ values the road at $v_i$, and the cost of building is $C$.
• The road is built if and only if ∑i​vi​≥C.
• The goal is to maximize social welfare ∑i​vi​−C.
• Use a VCG auction to ask every agent to bid $b_i$.
• If ∑i=1n​bi​≥C, build the road and Charge each agent a price pi​=∑j≠i​bj​−C

Properties of a VCG Auction

• Incentive Compatible: Truthful bidding is a dominant strategy.
• Socially Efficient: It maximizes social welfare.
• Individual Rational: Agent's utility is non-negative.

Maxwell Equations: Displacement Current

• Ampere's law is not valid for non-static situations.
• Corrected using displacement current Id, which is given as $$I_d = \epsilon_0 \frac{d\phi_E}{dt}$$
• Ampere-Maxwell law: $$\oint \overrightarrow{B} \cdot \overrightarrow{dl} = \mu_0 I_{enc} = \mu_0 I + \mu_0 \epsilon_0 \frac{d\phi_E}{dt}$$

Complex Number Basics

• A complex number is composed of a real and an imaginary number
• Real Number: A number on the number line
• Imaginary Number: A real number × i (√-1 )
• A Complex Number: a+bi

Operations with Complex Numbers

• Adding and Subtracting: like terms

(a+bi)+(c+di)=(a+c)+(b+d)i
(a+bi)-(c+di)=(a-c)+(b-d)i

• Multiplying: distribution

(a+bi)(c+di)=(ac−bd)+(ad+bc)i

• Dividing: $\frac{a + bi}{c + di} =$ $\frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$

Calculus Notes

• Function Definition: Assigns each element x in domain D exactly one element f(x) in range E.
• A function can be represented verbally, numerically, visually, or algebraically.
• Vertical Line Test: Determines if a curve is a function by allowing no vertical line to intersect it more than once.

Function Types

• Linear Function: f(x) = mx + b
• Polynomial Function: f(x) = anxn + an-1xn-1 +...+ a1x + a0
• Power Function: f(x) = xa
• Rational Function: f(x) = P(x)/Q(x)
• Algebraic Function: Constructed with algebraic operations.
• Trigonometric Function: sin x, cos x, etc.
• Exponential Function: f(x) = ax
• Logarithmic Function: f(x) = loga x

Symmetry

• Even Function: f(-x) = f(x) —symmetric about the y-axis.
• Odd Function: f(-x) = -f(x) —symmetric about the origin.

Function Transformations

• Transformations include vertical/horizontal shifts and stretching/compression; reflections about axes.

Derivative Concept

• Definition: Derivative of f(x) at x=a is defined as: $f'(a) = $$\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$
• Represents the slope of the tangent line at a point on the function's graph.
• Yields a new function, ${f}'(x), which gives slope of original function at each point.

Derivative Interpretation

Geometric

• Slope of the line tangent to the graph of the function at that point.

Physical

• Velocity (if function is position over time) OR acceleration (if function is velocity over time)

Side Derivatives

• Useful in the following scenarios:

f'(a+) = $\lim_{h \to 0+} \frac{f(a + h) - f(a)}{h}$

f'(a-) = $\lim_{h \to 0-} \frac{f(a + h) - f(a)}{h}$

Differentiability and Continuity

• If a function is differentiable at any given point, it MUST be continuous at the point BUT, a function can be continuous at point BUT NOT differentiable at the SAME point.

Algorithmic Trading Notes

Order Book

• Consists of Limit Order Book.

Limitation Book Order

• It has outstanding orders by order and/or price
• New orders can either match with existing orders OR will be placed if it can't be matched.

Market Order

• Executed immediately upon arrival of price
• Buyers should execute at lowest asking price
• Sellers should execute at best bidding price

Limitation Order

• Executed specifically at best price better than order
• Buyers will always pay highest willing price.
• Seller should always receive lowest willing amount

Market Impact

• Large orders WILL move price
• Market orders will walk the book.
• Limit orders have different parameters

Techniques

• VWAP, TWAP, Implementation Shortfall

Others

• Slippage, Adverse Selection, Market Manipulation

Kinematics Coordinate System (Physics Notes)

Position and Frame of Reference

Carteisan Coordinates:

$$\overrightarrow{OM} = x(t) \vec{i} + y(t) \vec{j} + z(t) \vec{k}$$.

Cylidrical Coordinates:

$$\overrightarrow{OM} = r(t) \vec{u_r} + z(t) \vec{k}$$.

Spherical Coordinates:

$$\overrightarrow{OM} = r(t) \vec{u_r}$$

Velocity:

• The speed at which position changes $$\overrightarrow{v}(t) = \frac{d\overrightarrow{OM}}{dt}$$.

Carteisan:

$$\overrightarrow{v}(t) = \frac{dx}{dt} \vec{i} + \frac{dy}{dt} \vec{j} + \frac{dz}{dt} \vec{k} = \dot{x}(t) \vec{i} + \dot{y}(t) \vec{j} + \dot{z}(t) \vec{k}$$.

Cylindrical:

$$\overrightarrow{v}(t) = \dot{r}(t) \vec{u_r} + r(t)\dot{\theta}(t) \vec{u_{\theta}} + \dot{z}(t) \vec{k}$$.

Spherical

$$\overrightarrow{v}(t) = \dot{r}(t) \vec{u_r} + r(t)\dot{\theta}(t) \vec{u_{\theta}} + r(t)sin\theta(t)\dot{\phi}(t) \vec{u_{\phi}}$$.

Acceleration:

• The speed at which the velocity changes $$\overrightarrow{a}(t) = \frac{d\overrightarrow{v}}{dt} = \frac{d^2\overrightarrow{OM}}{dt^2}$$.

Carterisian:

$$\overrightarrow{a}(t) = \frac{d^2x}{dt^2} \vec{i} + \frac{d^2y}{dt^2} \vec{j} + \frac{d^2z}{dt^2} \vec{k} = \ddot{x}(t) \vec{i} + \ddot{y}(t) \vec{j} + \ddot{z}(t) \vec{k}$$.

Cylindrical Coordinates :

$$\overrightarrow{a}(t) = (\ddot{r} - r\dot{\theta}^2) \vec{u_r} + (r\ddot{\theta} + 2\dot{r}\dot{\theta}) \vec{u_{\theta}} + \ddot{z} \vec{k}$$.

Spherical:

$$\overrightarrow{a}(t) = (\ddot{r} - r\dot{\theta}^2 - r\dot{\phi}^2 sin^2\theta) \vec{u_r} + (r\ddot{\theta} + 2\dot{r}\dot{\theta} - r\dot{\phi}^2 sin\theta cos\theta) \vec{u_{\theta}} + (r\ddot{\phi}sin\theta + 2\dot{r}\dot{\phi}sin\theta + 2r\dot{\theta}\dot{\phi}cos\theta) \vec{u_{\phi}}$$.

Numerical Intergration: Gaussian Quadrature Notes:

Newton-Cotes Formula Recap:

• Integrate by $f(x)$ by polynomial, either closing or opening w nodes.

Gaussian Qualrdature: Finding Optimal Evaluation POints

• All coefficients and nodes are unknown- determined to polynomials w exact degrees.
• The system is as follows: $$\begin{aligned} \int_{-1}^{1} 1 dx &= c_1 \cdot 1 + c_2 \cdot 1 = 2 \ \int_{-1}^{1} x dx &= c_1 \cdot x_1 + c_2 \cdot x_2 = 0 \ \int_{-1}^{1} x^2 dx &= c_1 \cdot x_1^2 + c_2 \cdot x_2^2 = \frac{2}{3} \ \int_{-1}^{1} x^3 dx &= c_1 \cdot x_1^3 + c_2 \cdot x_2^3 = 0 \end{aligned}$$

The General Formula on -1,1 :

• Gaussian Quadrature: $$\int_{-1}^{1} f(x) dx \approx \sum_{i=1}^{n} c_i f(x_i)$$.
• Xi is Legendre Poly Root N $ci = c_i = \int_{-1}^{1} \prod_{\substack{j=1 \ j \neq i}}^{n} \frac{x - x_j}{x_i - x_j} dx$

Legendre Polynimials (IMPORTANT):

Notes

The Definition is: $$(n+1)P_{n+1}(x) = (2n+1)xP_n(x) - nP_{n-1}(x)$$. WHERE: $$P_0(x) = 1, \quad P_1(x) = x$$.

CHANGE of Intervals

Formula to change Intervals for easy solving! :$$x = \frac{b - a}{2} t + \frac{a + b}{2}$$. Integral: $$\int_{a}^{b} f(x) dx = \frac{b - a}{2} \int_{-1}^{1} f\left(\frac{b - a}{2} t + \frac{a + b}{2}\right) dt$$-.

Skteching Graphs Notes:

Domain:

The set of all inputs

Intercepts:

The cross by with x/y axis

Symmetry:

Even functions yield symmetry

Asymptotes:

Behavior at limits are defined

Intervals of Indreasing or Decreasing:

• Where the first derivative is, can show a great deal

Local Mininum and Maximum Values:

• First derivative test can show local Mins and maxs

Concavity and Points of Inflection:

• Second Derivaives:
• Can see curves at point
• Can show inflection changes