Fourier Analysis and Signal Representation

Fourier Analysis

• A powerful tool for analyzing signals and linear time-invariant (LTI) systems.
• Applications include solving differential equations, analyzing frequency responses, designing filters, and compressing signals.

Fourier Series

• Represents a periodic signal as a weighted sum of sines and cosines.

Periodic Signals

• A signal $x(t)$ is periodic with period $T$ if $x(t + T) = x(t)$ for all $t$.
• The fundamental frequency of a periodic signal is $f_0 = \frac{1}{T}$.

Trigonometric Fourier Series

• Represents a periodic signal $x(t)$ with period $T$ as $x(t) = a_0 + \sum_{n=1}^{\infty} a_n \cos(2\pi nf_0t) + \sum_{n=1}^{\infty} b_n \sin(2\pi nf_0t)$.
• $a_0 = \frac{1}{T} \int_{0}^{T} x(t) dt$ represents the DC component.
• $a_n = \frac{2}{T} \int_{0}^{T} x(t) \cos(2\pi nf_0t) dt$ are the cosine coefficients.
• $b_n = \frac{2}{T} \int_{0}^{T} x(t) \sin(2\pi nf_0t) dt$ are the sine coefficients.

Exponential Fourier Series

• Represents a periodic signal $x(t)$ with period $T$ as $x(t) = \sum_{n=-\infty}^{\infty} c_n e^{j2\pi nf_0t}$.
• $c_n = \frac{1}{T} \int_{0}^{T} x(t) e^{-j2\pi nf_0t} dt$ are the complex exponential coefficients.

Relationship Between Trigonometric and Exponential Fourier Series

• $a_0 = c_0$
• $a_n = c_n + c_{-n}$
• $b_n = j(c_n - c_{-n})$

Fourier Transform

• Represents an aperiodic signal as an integral of complex exponentials.

Fourier Transform Definition

• The Fourier Transform of a signal $x(t)$ is $X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt$.

Inverse Fourier Transform Definition

• The Inverse Fourier Transform of $X(f)$ is $x(t) = \int_{-\infty}^{\infty} X(f) e^{j2\pi ft} df$.

Properties of the Fourier Transform

• Linearity, time shifting, time scaling, convolution, multiplication, duality.

Energy Spectral Density

• Defined as $S_x(f) = |X(f)|^2$.
• Represents the distribution of signal energy as a function of frequency.

Applications of Fourier Analysis

• Signal processing, communications, control systems, image processing, acoustics, and medicine.

Example: Frequency Response Analysis

• Frequency response of a filter is a function describing how it affects different signal frequencies.
• An input signal to the filter and the output is measured.
• The Fourier Transform of the output signal is divided by the Fourier Transform of the input signal, yielding the filter's frequency response.

Mathematical Analysis

• A curriculum from the academic year 2005-2006.

Exercise 1

• Determine the behavior of the series for $x \in \mathbb{R}$
• $\sum_{n=1}^{\infty} \frac{n^2 + 3}{n^{\alpha} + 5}x^n$ with $\alpha \in \mathbb{R}$.

Exercise 2

• Calculate the limit:
• $\lim_{x \to 0} \frac{\sin x \cos x - x + \frac{x^3}{3}}{x^5}$

Exercise 3

• Study the function and trace its qualitative properties.
• $f(x) = \log \left( \frac{x^2 - 1}{x + 3} \right)$

Exercise 4

• Calculate the integral:
• $\int \frac{1}{\sqrt{x} + \sqrt{x}} dx$

Exercise 5

• Solve the initial value problem: $\begin{cases} y' = \frac{2x}{1 + x^2}y + \sqrt{1 + x^2} \ y(0) = 1 \end{cases}$

Thevenin Equivalent Circuit

• Simplifies circuit analysis by reducing a complex circuit to a voltage source ($V_{TH}$) in series with a resistor ($R_{TH}$).

Thevenin Voltage ($V_{TH}$)

• It equals the open-circuit voltage at terminals A and B.
• Calculated by removing the load resistor $R_L$ and finding the voltage between terminals A and B.

Thevenin Resistance ($R_{TH}$)

• It equals the equivalent resistance "seen" from terminals A and B when all independent sources are turned off.
• Calculated by removing $R_L$, replacing voltage sources with short circuits and current sources with open circuits, then finding the resistance between terminals A and B.

Norton Equivalent Circuit

• Simplifies circuit analysis using a current source ($I_N$) in parallel with a resistor ($R_N$).

Norton Current ($I_N$)

• Norton current is equal to the short-circuit current at terminals A and B
• It is obtained by removing the load resistor $R_L$, short-circuiting terminals A and B, and calculating the current through the short circuit.

Norton Resistance ($R_N$)

• Equivalent to the resistance "seen" from terminals A and B when all independent sources are turned off.
• Calculated by removing $R_L$, replacing voltage sources with short circuits and current sources with open circuits, then finding the resistance between terminals A and B.

Thevenin/Norton Conversion

• $V_{TH} = I_N R_N$
• $I_N = V_{TH} / R_{TH}$
• $R_{TH} = R_N$

Linear Algebra and Vector Geometry

• Explores vectors in $\mathbb{R}^n$, vector operations, and their properties.

Vectors in $\mathbb{R}^n$

• Vectors are ordered lists of $n$ real numbers, represented as column vectors.
• $\mathbf{v} = \begin{bmatrix} v_1 \ v_2 \ \vdots \ v_n \end{bmatrix}$

Vector Operations

• Vector Addition: $\mathbf{u} + \mathbf{v} = \begin{bmatrix} u_1 + v_1 \ u_2 + v_2 \ \vdots \ u_n + v_n \end{bmatrix}$
• Scalar Multiplication: $c\mathbf{v} = \begin{bmatrix} cv_1 \ cv_2 \ \vdots \ cv_n \end{bmatrix}$

Properties Vector Operations

• Commutativity: $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$
• Associativity: $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$
• Identity Element: $\mathbf{u} + \mathbf{0} = \mathbf{u}$
• Inverse Element: $\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$
• Distributivity: $c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$ and $(c + d)\mathbf{u} = c\mathbf{u} + d\mathbf{u}$
• Associativity (Scalar): $c(d\mathbf{u}) = (cd)\mathbf{u}$
• Identity Element (Scalar): $1\mathbf{u} = \mathbf{u}$

Linear Combinations

• Vector $\mathbf{v}$ is a combination of vectors in $\mathbb{R}^n$ where $\mathbf{v} = c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \dots + c_k\mathbf{v}_k$

Vector Subspaces

• Non-empty subset of $\mathbb{R}$ that contains a vector such that the vector is closed under addition and scalar multiplication.

Vector Operations in $\mathbb{R}^2$

• The sum of $u = (x_1, y_1)$ and $v = (x_2, y_2)$ is $u+v = (x_1 + x_2, y_1 + y_2)$.
• Multiplication of $u = (x, y)$ by scalar c is $c*u = (cx, cy)$.

Properties of Vector Operations (R2 and R3)

• $u + v = v + u$ (Commutativity)
• $(u + v) + w = u + (v + w)$ (Associativity)
• There exists a vector $0 = (0, 0)$ such that $u + 0 = u$ (Élément neutre)
• There exists a vector $-u = (-x, -y)$ tel que $u+ (-u) = 0$ (Opposé)
• $a(bu) = (ab)u$ (Associativity scalaire)
• $(a + b)u = au + bu$ (Distributivity scalaire par rapport à l'addition scalaire)
• $a(u + v) = au + av$ (Distributivity scalaire par rapport à l'addition vectorielle)
• $1u = u$ (Élément neutre scalaire)

Combinaison Linéaire and Generalizing to $\mathbb{R}^n$

• Generalize vector definitions to $\mathbb{R}^n$.
• $vec w$ is $a * \vec{v_1} + c_2 * \vec{v_2}... c_k * \vec{v_k}$

Algorithmic Trading

• Uses the efficient frontier which represents investment portfolios created to maximize returns based on risk.
• Sharpe Ratio (=) (Rp - Rf) / σp where Rp = Portfolio return, Rf = Risk-free rate, σp = Portfolio standard deviation.

Order Book

• Displays the list of buy/sell orders by volume.

Order Types

• Different types of orders exist from market to stop.
• A Market order is executed immediately at market price.
• A limit order is executed at a specific price when possible.
• A stop order becomes a market order when triggered.
• A stop-limit order is a limit order when triggered.

Order attributes

• The orders persist for certain duration/Time in Force such as Day Order, Good-Til-Cancelled, Immediate-or-Cancel, Fill-or-Kill.

Order Execution Algorithms

• VWAP executes slowly based on historical volume patterns.
• TWAP executes pieces evenly over time.
• Implementation Shortfall balances immediacy and opportunity.
• POV participates in percentages to blend in with market activity.

Other Execution Algorithms

• Alternative algorithms like dark pools, stat arbs, pairs trading and smart order routers all can be used.

Linear Regression with Multiple Regression

• Omitted Variable Bias arises when a relevant variable is left out.
• Multiple Regression Model: Yi = β0 + β1X1i + β2X2i +...+ βkXki + ui

OLS Estimator in Multiple Regression

• Multicollinearity arises when regressors are correlated.
• Solutions around multicollinearity include collecting more data, omitting regressors, using variable estimation.

Nonlinear Regression

• Models include TestScorei = β0 + β1Sizei + β2Sizei^2 + ui
• Log transformation can also be applied on nonlinear regression models.

Algorithmic Game Theory

• Combines game theory and algorithm design and theory.
• Selfish Routing highlights Braess's Paradox where adding roads worsens the situation.

Cost Sharing

• VCG mechanisms implement socially positive behavior and truthful strategies.

Price of Anarchy and Stability

• Ratios between social good and social optimum.

Mechanisms

• Highlight incentive compatibility and individual rationality in algorithmic game theory.