Fourier Transform Properties

Fourier Transform Properties

• Describes how the Fourier Transform interacts with common mathematical operations
• $a \cdot f(t)+b \cdot g(t) \stackrel{\mathrm{FT}}{\longleftrightarrow} a \cdot F(\omega)+b \cdot G(\omega)$ illustrates Linearity
• $f(a t) \stackrel{\mathrm{FT}}{\longleftrightarrow} \frac{1}{|a|} F\left(\frac{\omega}{a}\right)$ describes Time Scaling
• $f\left(t-t_{0}\right) \stackrel{\mathrm{FT}}{\longleftrightarrow} e^{-j \omega t_{0}} F(\omega)$ represents Time Shifting
• $e^{j \omega_{0} t} f(t) \stackrel{\mathrm{FT}}{\longleftrightarrow} F\left(\omega-\omega_{0}\right)$ shows Frequency Shifting
• $f^{}(t) \stackrel{\mathrm{FT}}{\longleftrightarrow} F^{}(-\omega)$ describes Conjugation
• $\frac{d}{d t} f(t) \stackrel{\mathrm{FT}}{\longleftrightarrow} j \omega F(\omega)$ represents Time Differentiation
• $t f(t) \stackrel{\mathrm{FT}}{\longleftrightarrow} j \frac{d}{d \omega} F(\omega)$ indicates Frequency Differentiation
• $\int_{-\infty}^{t} f(\tau) d \tau \stackrel{\mathrm{FT}}{\longleftrightarrow} \frac{1}{j \omega} F(\omega)+\pi F(0) \delta(\omega)$ describes Integration
• $f(t) \cdot g(t) \stackrel{\mathrm{FT}}{\longleftrightarrow} \frac{1}{2 \pi} F(\omega) * G(\omega)$ illustrates Multiplication
• $f(t) * g(t) \stackrel{\mathrm{FT}}{\longleftrightarrow} F(\omega) \cdot G(\omega)$ describes Convolution
• $\int_{-\infty}^{\infty}|f(t)|^{2} d t=\frac{1}{2 \pi} \int_{-\infty}^{\infty}|F(\omega)|^{2} d \omega$ illustrates Parseval's Relation

Physics: Lorentz Transformation

• The Lorentz transformation is a coordinate transformation between inertial frames moving at constant relative velocity.
• It accounts for the principle of relativity: Laws of physics are the same for all inertial observers.
• It contrasts with the Galilean transformation, which assumes absolute space and time.

Equations

• Transformations from frame S to S' moving with velocity v along the x-axis:
  • $t' = \gamma (t - \frac{vx}{c^2})$
  • $x' = \gamma (x - vt)$
  • $y' = y$
  • $z' = z$
• $\gamma$ is the Lorentz factor: $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$, and c is the speed of light.

Inverse Transformation

• Transforms from S' to S:
  • $t = \gamma (t' + \frac{vx'}{c^2})$
  • $x = \gamma (x' + vt')$
  • $y = y'$
  • $z = z'$

Properties

• Spacetime interval is preserved: $s^2 = c^2t^2 - x^2 - y^2 - z^2 = c^2t'^2 - x'^2 - y'^2 - z'^2$
• The transformation is linear.
• Reduces to Galilean transformation when v is much smaller than c.

Análisis de Fourier

• Fourier analysis is useful for solving engineering problems involving functions varying periodically with time, like electrical signals and mechanical vibrations.

Definition

• Fourier series representation of a periodic function $f(t)$ with period $T$:
  • $f(t) = a_0 + \sum_{n=1}^{\infty} (a_n \cos(n\omega t) + b_n \sin(n\omega t))$
  • $\omega = \frac{2\pi}{T}$ is the fundamental angular frequency.
  • $a_0 = \frac{1}{T} \int_{0}^{T} f(t) dt$ is the average value of $f(t)$ over a period.
  • $a_n = \frac{2}{T} \int_{0}^{T} f(t) \cos(n\omega t) dt$
  • $b_n = \frac{2}{T} \int_{0}^{T} f(t) \sin(n\omega t) dt$

Complex Form

• Fourier series can be expressed in complex form:
  • $f(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega t}$
  • $c_n = \frac{1}{T} \int_{0}^{T} f(t) e^{-jn\omega t} dt$

Fourier Transform

• Extension of Fourier analysis to non-periodic functions:
  • $F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt$
• Inverse Fourier transform:
  • $f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j\omega t} d\omega$

Properties

• Linearity: $F{af(t) + bg(t)} = aF(\omega) + bG(\omega)$
• Scaling: $F{f(at)} = \frac{1}{|a|}F(\frac{\omega}{a})$
• Time Shifting: $F{f(t - t_0)} = e^{-j\omega t_0}F(\omega)$
• Frequency Shifting: $F{e^{j\omega_0 t}f(t)} = F(\omega - \omega_0)$
• Convolution: $F{f(t) * g(t)} = F(\omega)G(\omega)$

Applications

• Signal analysis for analyzing frequency components.
• Filter design to eliminate or attenuate certain frequencies.
• Image processing for compression and enhancement.
• Vibration analysis of mechanical structures.

Example

• For $f(t) = t$ in the interval $[0, 2\pi]$:
  • $a_0 = \pi$
  • $a_n = 0$
  • $b_n = -\frac{2}{n}$
• Resulting Fourier series:
  • $f(t) = \pi - 2\sum_{n=1}^{\infty} \frac{\sin(nt)}{n}$

Conclusion

• Essential tool for analyzing and manipulating periodic functions and signals.
• Decomposes complex functions into simpler components.

Chapitre 1: Nombres complexes

I/ Définitions

1) Vocabulaire et notations

• $\mathbb{C}$ represents the set of complex numbers.
• A complex number $z$ can be written as $z = a + ib$, where $a$ and $b$ are real numbers, and $i^2 = -1$.
• $a$ is the real part of $z$, denoted as Re($z$).
• $b$ is the imaginary part of $z$, denoted as Im($z$).
• If $b = 0$, then $z = a$ is a real number, thus $\mathbb{R} \subset \mathbb{C}$.
• If $a = 0$, then $z = ib$ is a purely imaginary number.
• Example:
  • $z_1 = 3 + 2i$, Re($z_1$) = 3, Im($z_1$) = 2
  • $z_2 = -5i$, Re($z_2$) = 0, Im($z_2$) = -5
  • $z_3 = 7$, Re($z_3$) = 7, Im($z_3$) = 0
• A complex number is zero if and only if both its real and imaginary parts are zero.
  • $z = 0 \Leftrightarrow \begin{cases} Re(z) = 0 \ Im(z) = 0 \end{cases}$

2) Représentation graphique

• A complex number $z = a + ib$ can be represented graphically by a point $M$ with coordinates $(a; b)$ in an orthonormal coordinate system $(O; \overrightarrow{u}; \overrightarrow{v})$.
• $M$ is the image of $z$, and $z$ is the affix of $M$.
• Notation: $M(z)$.
• $z$ can also be represented by a vector $\overrightarrow{OM}$ with affix $z$.

3) Conjugué d'un nombre complexe

• The conjugate of a complex number $z = a + ib$ is denoted as $\overline{z}$ and defined as $\overline{z} = a - ib$.
• Example:
  • If $z_1 = 3 + 2i$, then $\overline{z_1} = 3 - 2i$
  • If $z_2 = -5i$, then $\overline{z_2} = 5i$
  • If $z_3 = 7$, then $\overline{z_3} = 7$
• For any complex numbers $z$ and $z'$:
  • $\overline{z + z'} = \overline{z} + \overline{z'}$
  • $\overline{z \times z'} = \overline{z} \times \overline{z'}$
  • $\overline{\overline{z}} = z$
  • If $z \neq 0$, $\overline{(\frac{1}{z})} = \frac{1}{\overline{z}}$
  • If $z \neq 0$, $\overline{(\frac{z}{z'})} = \frac{\overline{z}}{\overline{z'}}$
  • $z + \overline{z} = 2Re(z)$
  • $z - \overline{z} = 2iIm(z)$
  • $z \in \mathbb{R} \Leftrightarrow z = \overline{z}$
  • $z \in i\mathbb{R} \Leftrightarrow z = -\overline{z}$

4) Module d'un nombre complexe

• The modulus of a complex number $z = a + ib$ is a non-negative real number denoted as $|z|$ and defined by $|z| = \sqrt{a^2 + b^2}$.
• If $M$ is the point with affix $z$, then $|z| = OM$.
• Examples:
  • If $z_1 = 3 + 2i$, then $|z_1| = \sqrt{3^2 + 2^2} = \sqrt{13}$
  • If $z_2 = -5i$, then $|z_2| = \sqrt{0^2 + (-5)^2} = 5$
  • If $z_3 = 7$, then $|z_3| = \sqrt{7^2 + 0^2} = 7$
• For any complex numbers $z$ and $z'$:
  • $|z| = |\overline{z}|$
  • $z \times \overline{z} = |z|^2$
  • $|z \times z'| = |z| \times |z'|$
  • If $z \neq 0$, $|\frac{1}{z}| = \frac{1}{|z|}$
  • If $z' \neq 0$, $|\frac{z}{z'}| = \frac{|z|}{|z'|}$
  • $|Re(z)| \le |z|$ and $|Im(z)| \le |z|$

5) Formule d'Euler et applications

• Euler's formula: For any real number $\theta$, $e^{i\theta} = cos(\theta) + isin(\theta)$.
• Euler's formulas:
  • $cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2}$
  • $sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i}$

6) Forme trigonométrique d'un nombre complexe

• For a non-zero complex number $z = a + ib$, the argument of $z$, denoted as arg$(z)$, is any real number $\theta$ such that:
  • $\begin{cases} a = |z|cos(\theta) \ b = |z|sin(\theta) \end{cases}$
• $\theta$ is defined up to $2\pi$.
• One writes arg$(z) = \theta [2\pi]$.
• Trigonometric form: $z = |z|(cos(\theta) + isin(\theta))$.
• Exponential form: $z = |z|e^{i\theta}$.
• Example:
  • $z_1 = 1 + i$, $|z_1| = \sqrt{2}$, $cos(\theta) = \frac{1}{\sqrt{2}}$, $sin(\theta) = \frac{1}{\sqrt{2}}$, thus $\theta = \frac{\pi}{4} [2\pi]$. $z_1 = \sqrt{2}e^{i\frac{\pi}{4}}$
  • $z_2 = -2$, $|z_2| = 2$, $cos(\theta) = -1$, $sin(\theta) = 0$, thus $\theta = \pi [2\pi]$. $z_2 = 2e^{i\pi}$
• Properties: For any non-zero complex numbers $z$ and $z'$:
  • arg$(\overline{z}) = -$arg$(z) [2\pi]$
  • arg$(zz') = $ arg$(z) + $ arg$(z') [2\pi]$
  • arg$(\frac{1}{z}) = -$arg$(z) [2\pi]$
  • arg$(\frac{z}{z'}) = $ arg$(z) - $ arg$(z') [2\pi]$
  • arg$(z^n) = n$ arg$(z) [2\pi]$, for any integer $n$

II/ Applications

1) Interprétation géométrique du module et de l'argument

• Let $A$ and $B$ be two points with affixes $z_A$ and $z_B$ respectively.
  • $|z_B - z_A| = AB$
  • arg$(z_B - z_A) = (\overrightarrow{u}, \overrightarrow{AB}) [2\pi]$

2) Transformations complexes

• A complex transformation is a mapping of the complex plane to itself.
• To each point $M$ with affix $z$, a point $M'$ with affix $z'$ is associated such that $z' = f(z)$, where $f$ is a function from $\mathbb{C}$ to $\mathbb{C}$.
• Examples:
  • Translation: $z' = z + b$, where $b$ is a complex number.
  • Homothety: $z' = az$, where $a$ is a non-zero real number.
  • Rotation: $z' = e^{i\theta}z$, where $\theta$ is a real number.

3) Equations du second degré dans $\mathbb{C}$

• Consider the quadratic equation $az^2 + bz + c = 0$, where $a, b, c \in \mathbb{R}$ and $a \neq 0$.
• Calculate the discriminant $\Delta = b^2 - 4ac$.
  • If $\Delta > 0$, the equation has two distinct real solutions: $z_1 = \frac{-b - \sqrt{\Delta}}{2a}$ and $z_2 = \frac{-b + \sqrt{\Delta}}{2a}$
  • If $\Delta = 0$, the equation has one real double solution: $z_0 = \frac{-b}{2a}$
  • If $\Delta < 0$, the equation has two complex conjugate solutions: $z_1 = \frac{-b - i\sqrt{-\Delta}}{2a}$ and $z_2 = \frac{-b + i\sqrt{-\Delta}}{2a}$

III/ Exercices

Quantum Mechanics

Core Concept

• Quantum mechanics studies matter and energy interactions at atomic and subatomic scales.
• It's needed because classical mechanics fails at very small sizes.
• It uses quantization and wave-particle duality.
• Underpins modern technologies.

Essence

• Quantization: Physical quantities are discrete.
• Wave-particle duality: Particles act like waves, and waves like particles
• Uncertainty principle: Limit to simultaneous precision of certain pairs of properties.

Schrodinger Equation

• Time-dependent form: $i\hbar \frac{\partial}{\partial t} \Psi(r,t) = \hat{H} \Psi(r,t)$

  • $i$ is the imaginary unit
  • $\hbar$ is the reduced Planck constant
  • $\Psi(r,t)$ is the wave function
  • $\hat{H}$ is the Hamiltonian operator

• Time-independent form: $\hat{H} \Psi(r) = E\Psi(r)$

  • $E$ is the energy eigenvalue

Key Principles

Superposition

• A quantum entity exists in multiple states at once until measured.
• State is a combination of possibilities with probability amplitudes.
• $\Psi = c_1\Psi_1 + c_2\Psi_2$
  • $\Psi$ is the overall state
  • $\Psi_1$ and $\Psi_2$ are individual states
  • $c_1$ and $c_2$ are probability amplitudes

Measurement

• Gaining information forces the system into a definite state.
• Probability of collapsing to a state is the amplitude squared. $P(outcome) = |\langle \Psi | \Phi \rangle |^2$
  • $|\Psi\rangle$ is the initial state
  • $|\Phi\rangle$ is the state after measurement
  • $P(outcome)$ is the probability

Entanglement

• Quantum entities are linked regardless of distance.
• Measuring a property on one instantly knows it for the other.

Applications

• Quantum computing uses superposition and entanglement to exceed classical limits.
• Quantum cryptography creates secure communications using quantum properties.
• Quantum sensors make precise measurements of fields, gravity, and time.

Key Figures

Challenges and Future Directions

• No consensus on the fundamental interpretation.
• Quantum decoherence (loss of properties) must be overcome.
• Scaling quantum technologies is a complex challenge.

Introduction to Probability

Terminology

• Experiment: a process with several possible outcomes
• Sample Space: the set of all possible outcomes
• Outcome: a singular possible result
• Event: a subset of sample space

Example

Experiment: toss a coin twice

• Sample space is {HH, HT, TH, TT}
• One Possible outcome: HT
• At least one head {HH, HT, TH}

Definition

• If all outcomes are equally likely, then
  • $P(A) = \frac{\text{number of outcomes in A}}{\text{total number of outcomes in the sample space}}$

Example

Experiment: Roll a 6-sided die

• Sample Space {1, 2, 3, 4, 5, 6}
• Event: Roll an even number {2, 4, 6}
  • $P(A) = \frac{3}{6} = \frac{1}{2}$

Properties

• Probability bounds: $0 \le P(A) \le 1$
• Probability of Sample Space: $P(S) = 1$
• Probability of mutually exclusive events $P(A \cup B) = P(A) + P(B)$

Conditional Probability

• Probability of A given B:
  • $P(A|B) = \frac{P(A \cap B)}{P(B)}$, provided $P(B) > 0$

Example

• Draw a card from 52-card deck
  • A: Draw a heart
  • B: Draw a face card (J, Q, K)
    • $P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{3/52}{12/52} = \frac{3}{12} = \frac{1}{4}$

Independence

• Events A and B are independent if: $P(A \cap B) = P(A)P(B)$
• Equivalently, $P(A|B) = P(A)$ and $P(B|A) = P(B)$

Example

Experiment: toss a coin twice.

• Event A: First toss is a head.
• Event B: Second toss is a head.
• $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{2}$
• $P(A \cap B) = P(HH) = \frac{1}{4}$
• Since $P(A \cap B) = P(A)P(B)$, events A and B are independent.

Bayes Theorem

$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$

• $P(A|B)$: posterior probability of A given B
• $P(B|A)$: likelihood of B given A
• $P(A)$: prior probability of A
• $P(B)$: prior probability of B

Example

Disease Test:

• Sensitivity: 95%
• Specificity: 90%
• Prevalence: 1% What is the probability someone tests positive and has a disease?
• A: Person has the disease. $P(A) = 0.01$
• B: Test is positive. $P(B|A) = 0.95$, $P(B|\neg A) = 0.10$
• $P(B) = P(B|A)P(A) + P(B|\neg A)P(\neg A) = (0.95)(0.01) + (0.10)(0.99) = 0.0095 + 0.099 = 0.1085$
• $P(A|B) = \frac{P(B|A)P(A)}{P(B)} = \frac{(0.95)(0.01)}{0.1085} \approx 0.0876$
• A person who tests positive has an 8.76% of having the disease.