Author: 孤筝(lvbowen040427@163.com)

Chapter 1 Wave-Particle Duality and State Description

1.1 Formation and Applications of Quantum Mechanics

1.1.1 Old Quantum Theory

Photoelectric Effect and the Photon Hypothesis

• Photon energy: $E = h\nu$
• Threshold frequency: $\nu_0 = \dfrac{W_0}{h}$; no photoelectrons are emitted when $\nu < \nu_0$
• Photoelectric equation:
  $$ E_k^{\text{max}} = \frac{1}{2}\mu v^2_m = h\nu - W_0 $$
• The photoelectric effect demonstrates the particle nature of light.

Energy-Momentum Relation of Photons and Wave-Particle Unity

• Relativistic energy-momentum relation
  

  $$ E^2=(pc)^2+(m_0c^2)^2,\quad m_0=0\ \Rightarrow\ E=c\,\lVert\vec p\rVert $$

• Photon energy
  

  $$ E=h\nu=\frac{hc}{\lambda}=\hbar\omega $$

• Photon momentum (vector form)
  Let $\mathbf n$ be the unit vector in the propagation direction, then

  $$ \vec p=\frac{E}{c}\,\mathbf n=\frac{h}{\lambda}\,\mathbf n=\hbar\vec k,\quad \vec k=\frac{2\pi}{\lambda}\,\mathbf n $$

• Unified correspondence of wave and particle pictures
  

  $$ E\ \longleftrightarrow\ \hbar\omega,\qquad \vec p\ \longleftrightarrow\ \hbar\vec k $$

Bohr Structure of the Hydrogen Atom

• Quantization of orbital angular momentum: $$ L = n\hbar,\quad n=1,2,3,\dots $$
• Energy levels: $$ E_n = -\frac{13.6\ \text{eV}}{n^2} $$
• This successfully explains the line spectrum of hydrogen.

Bohr’s Postulates

• Electrons moving in stable orbits do not radiate energy.
• Electrons absorb or emit energy when they jump between energy levels: $$ \Delta E = h\nu $$

Compton Effect

• The wavelength of a high-energy photon increases after scattering from an electron: $$ \Delta\lambda = \lambda' - \lambda = \frac{h}{m_ec}(1-\cos\theta) $$
• The experiment confirms both the particle nature of light and conservation of momentum.

Blackbody Radiation

• Energy quantization assumption: the energy of the electromagnetic field takes discrete values $E=nh\nu$.
• Planck formula: $$ u(\nu,T)=\frac{8\pi h\nu^3}{c^3}\frac{1}{e^{h\nu/kT}-1} $$
• This successfully explains the blackbody spectrum and marks the birth of quantum theory.

1.1.2 Wave-Particle Duality of Microscopic Particles

de Broglie Hypothesis

• Microscopic particles have not only particle properties but also wave properties.
• Every particle with momentum $\vec p$ corresponds to a matter wave whose wavelength and frequency are related to its momentum and energy.

de Broglie Relations

• Wavelength: $$ \lambda = \frac{h}{p} $$
• Vector form: $$ \vec p = \hbar \vec k $$
• Frequency: $$ E = h\nu = \hbar\omega $$

1.2 States and Wave Functions

1.2.1 Uncertainty Principle

• The position and momentum of a microscopic particle cannot be measured simultaneously with arbitrary precision.
• Heisenberg uncertainty relation: $$ \Delta x \cdot \Delta p_x \geq \frac{\hbar}{2} $$
• Energy-time uncertainty relation: $$ \Delta E \cdot \Delta t \geq \frac{\hbar}{2} $$
• Its origin lies in wave-particle duality and the non-commutativity of operators.

1.2.2 Wave Function

• To describe the state of a microscopic particle, we introduce the wave function $\psi(\vec r,t)$.
• Probabilistic interpretation: $|\psi(\vec r,t)|^2 dV$ gives the probability of finding the particle in the volume element $dV$.
• The wave function must satisfy the superposition principle and the Schrödinger equation.
• The total probability over all space is 1, so the probability distribution depends only on the relative magnitude of the wave function, not its absolute scale.
• Multiplying the wave function by a constant does not change the physical state it describes.
• Standard requirements for a wave function: single-valued, finite, and continuous.

1.2.3 Normalization of the Wave Function

• Normalization condition: $$ \int_{-\infty}^{\infty} \psi^* (\vec r,t) \psi (\vec r,t) dV = 1 $$
• How to normalize a wave function $$ \int_{-\infty}^{\infty} |\Psi(\vec r,t)|^2 dV = A^2 \int_{-\infty}^{\infty} |\psi(\vec r,t)|^2 dV = 1 $$ where $A$ is the normalization constant.

1.3 Schrödinger Equation

1.3.1 Wave Equation for a Free Particle

Concept
A free particle is a particle not subject to external forces. In quantum mechanics, its state is described by the wave function $\psi(\vec{r},t)$ and satisfies the Schrödinger equation.

Schrödinger equation for a free particle

where:

• $\hbar$: reduced Planck constant
• $m$: mass of the particle
• $\nabla^2$: Laplacian operator

Plane-wave solution

where:

• $\vec{k}$ is the wave vector, with $|\vec{k}| = k$
• $\omega$ is the angular frequency, satisfying $$ E = \hbar \omega = \frac{\hbar^2 k^2}{2m} $$

Momentum-wave vector relation

Plane-Wave Derivation of the Free-Particle Schrödinger Equation

1. Assume a plane-wave form

with $\vec{k}$ the wave vector, $\omega$ the angular frequency, and $A$ the amplitude constant.

2. Time derivative

Multiplying by $i\hbar$ gives

3. Spatial Laplacian

Thus,

4. Energy relation

5. Final equation

Remark

• This derivation uses only the plane-wave form and differentiation, without relying on operator definitions.
• It corresponds to the case $V=0$.

1.3.3 Stationary-State Schrödinger Equation and Stationary Wave Functions

Concept
A stationary-state wave function has separable time dependence:

Let $f(t)=e^{-i E t / \hbar}$.
Here $\phi(\vec{r})$ depends only on spatial coordinates, and $E$ is the total energy of the particle.

Derivation Starting from the time-dependent Schrödinger equation:

substitute $\psi(\vec{r},t) = \phi(\vec{r}) e^{-i E t / \hbar}$ to obtain

Time-independent form

Remark

• $\phi(\vec{r})$ is called a stationary-state wave function or eigenfunction.
• $E$ is the corresponding energy eigenvalue.

Derivation of the Schrödinger Equation from Operators

1. Start from classical energy

2. Introduce the de Broglie relations

and the plane-wave form

3. Operator representation

4. Kinetic-energy and Hamiltonian operators

5. Schrödinger equation

that is,

For a free particle ($V=0$), this reduces to

Principle of Superposition of States

Concept
If $\psi_1$ and $\psi_2$ are two possible states of the same system, then their linear combination

is also a possible state, where $c_1$ and $c_2$ are complex coefficients.

General expansion

with

• $c_n$ the expansion coefficients, or probability amplitudes;
• probabilities $|c_n|^2$ satisfying $$ \sum_n |c_n|^2 = 1 $$

Remark

• Superposition is one of the most fundamental principles of quantum mechanics.
• Different eigenstates may superpose, but a measurement yields only one eigenvalue.
• Interference in superposed states is one of the essential features that distinguishes quantum mechanics from classical mechanics.

Chapter 2 Simple Applications of the Schrödinger Equation

2.1 One-Dimensional Infinite Potential Well

2.1.1 Solving the Equation

1. Potential

2. Schrödinger equation in the well

which becomes

3. General solution

4. Boundary conditions

Hence $B=0$ and $kL = n\pi$ for $n=1,2,3,\dots$.

5. Eigenfunctions and eigenvalues

Remark

• The energy is quantized.
• The ground-state energy is nonzero, showing the zero-point energy.

2.2 Special Functions in Mathematical Physics

2.2.1 Orthogonality and Normalization

Orthogonality

Normalization

Orthonormality

2.2.2 Expansion in an Orthonormal Set

2.2.3 Fourier Series

where

2.2.4 Constructing Orthonormal Functions

The standard method is Gram-Schmidt orthogonalization:

2.2.5 Legendre Polynomials and Other Special Functions

Legendre polynomials

with orthogonality

Other common special functions

• Spherical harmonics $Y_l^m(\theta,\phi)$ appear in angular momentum problems.
• Bessel functions $J_n(x)$ appear in cylindrical symmetry problems.
• Hermite polynomials $H_n(x)$ appear in harmonic oscillator problems.

These special functions are solutions of the Schrödinger equation under different boundary conditions and symmetries.

2.3 Linear Harmonic Oscillator

2.4 Hydrogen Atom

2.4.1 Solution of the Equation (Separated into $r,\ \theta,\ \phi$ Parts)

1. Time-independent Schrödinger equation under a Coulomb potential

and

2. Separation of variables

This leads to three equations in $r$, $\theta$, and $\phi$ after separation.

3. Equation in $\phi$

4. Equation in $\theta$

with solutions proportional to the associated Legendre functions:

5. Angular part: spherical harmonics

which satisfy

6. Radial equation Let $u(r)=rR(r)$, then

7. Energy eigenvalues

with $l=0,1,\dots,n-1$.

8. Wave function

and

where $a_0=\dfrac{4\pi\varepsilon_0\hbar^2}{m e^2}$ is the Bohr radius.

2.4.2 Results and Discussion

1. Quantum numbers and their meanings

• $n$: principal quantum number
• $l$: orbital angular momentum quantum number
• $m$: magnetic quantum number

2. Degeneracy
For the Coulomb potential, the energy depends only on $n$. The degeneracy of the level with principal quantum number $n$ is $n^2$.

3. Spatial structure of the wave function

• The angular part is given by the spherical harmonics.
• The radial part $R_{nl}(r)$ has $n-l-1$ radial nodes.
• The ground state $(1,0,0)$ is spherically symmetric and has no radial node.

4. Summary
The hydrogen atom is solved by separating variables in spherical coordinates. The angular equations give spherical harmonics and angular quantum numbers, while the radial equation yields the discrete energy levels and radial eigenfunctions.

Chapter 3 Operator Representation of Dynamical Variables and Representation Theory

3.1 Relation Between Dynamical Variables and Operators

3.1.1 Mathematical Knowledge of Operators

1. Definition of an operator
   An operator is a rule acting on a function space or state space. In quantum mechanics, physical quantities are represented by operators, and the wave function is the object on which they act.

2. Linearity
   If

   $$ A(c_1\psi_1 + c_2\psi_2) = c_1 A\psi_1 + c_2 A\psi_2, $$

   then $A$ is a linear operator.

3. Commutation relations
   The commutator is defined by

   $$ [A,B] = AB - BA. $$

   If $[A,B]=0$, the two operators are said to commute.

4. Hermitian operators
   If

   $$ \langle \psi | A\varphi \rangle = \langle A\psi | \varphi \rangle, $$

   then $A$ is Hermitian. Hermitian operators have real eigenvalues and represent observables.

3.1.2 Dynamical Variables and Operators

1. Basic idea
   Every classical quantity $f(q,p)$ corresponds to a quantum operator $\hat f$.

2. Typical operator forms in the coordinate representation

   $$ \hat{x} = x, \qquad \hat{p} = -i\hbar \frac{\partial}{\partial x} $$

3. Fundamental commutation relation

   $$ [\hat{x}, \hat{p}] = i\hbar $$

4. Measurement and eigenvalue equations

   $$ \hat{A}\psi_a = a\psi_a $$

   Here $a$ is a possible measurement outcome, and $\psi_a$ is the corresponding eigenfunction.

3.2 Commutation Relations and the Uncertainty Principle

3.2.1 Commutation Relations

1. Definition

   $$ [A,B] = AB - BA $$

   If $[A,B]=0$, the two physical quantities can have simultaneous definite values.

2. Basic relation

   $$ [\hat{x}, \hat{p}_x] = i\hbar $$

3. Three-dimensional form

   $$ [\hat{x}_i, \hat{p}_j] = i\hbar \delta_{ij}, \quad [\hat{x}_i, \hat{x}_j]=0, \quad [\hat{p}_i, \hat{p}_j]=0 $$

4. Physical meaning
   Commutation relations determine whether two observables can be measured simultaneously with arbitrary precision.

3.2.2 Uncertainty Principle

1. Mathematical form

   $$ (\Delta A)^2 = \langle (A-\langle A \rangle)^2 \rangle,\qquad (\Delta B)^2 = \langle (B-\langle B \rangle)^2 \rangle $$

   which leads to

   $$ \Delta A \cdot \Delta B \geq \frac{1}{2}\left| \langle [A,B] \rangle \right| $$

2. Position-momentum uncertainty

   $$ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} $$

3. Energy-time uncertainty

   $$ \Delta E \cdot \Delta t \gtrsim \hbar $$

3.3 Representation Theory

3.3.1 Mathematical Basis

1. Concept of representation
   States and operators can be represented in different bases, such as the coordinate, momentum, and energy representations.

2. Expansion of a state

   $$ |\psi\rangle = \sum_n c_n |\phi_n\rangle,\qquad c_n = \langle \phi_n | \psi \rangle $$

3. Matrix elements

   $$ A_{mn} = \langle \phi_m | \hat{A} | \phi_n \rangle $$

4. Completeness and orthogonality

   $$ \sum_n |\phi_n\rangle \langle \phi_n| = I,\qquad \langle \phi_m | \phi_n \rangle = \delta_{mn} $$

3.3.2 Representations of States and Observables

1. Coordinate representation

   $$ \psi(x) = \langle x|\psi\rangle $$
   $$ \hat{x} \psi(x) = x \psi(x), \quad \hat{p}_x \psi(x) = -i\hbar \frac{\partial}{\partial x}\psi(x) $$

2. Momentum representation

   $$ \phi(p) = \langle p|\psi\rangle $$
   $$ \hat{p} \phi(p) = p \phi(p), \quad \hat{x} \phi(p) = i\hbar \frac{\partial}{\partial p}\phi(p) $$

3. Energy representation

   $$ |\psi\rangle = \sum_n c_n |E_n\rangle, \quad c_n = \langle E_n|\psi\rangle $$

4. Transformations between representations

   $$ \phi(p) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \psi(x) e^{-ipx/\hbar} dx $$
   $$ \psi(x) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} \phi(p) e^{ipx/\hbar} dp $$

3.4 Orbital Angular Momentum

3.4.1 Angular Momentum

1. Definition

   $$ \vec{L} = \vec{r} \times \vec{p},\qquad \hat{\vec{L}} = \hat{\vec{r}} \times \hat{\vec{p}} $$

2. Components

   $$ \hat{L}_x = y\hat{p}_z - z\hat{p}_y, \quad \hat{L}_y = z\hat{p}_x - x\hat{p}_z, \quad \hat{L}_z = x\hat{p}_y - y\hat{p}_x $$

3. Commutation relations

   $$ [\hat{L}_x, \hat{L}_y] = i\hbar \hat{L}_z, \quad [\hat{L}_y, \hat{L}_z] = i\hbar \hat{L}_x, \quad [\hat{L}_z, \hat{L}_x] = i\hbar \hat{L}_y $$
   $$ \hat{L}^2 = \hat{L}_x^2 + \hat{L}_y^2 + \hat{L}_z^2 $$

3.4.2 Conservation of Angular Momentum

1. Condition for conservation

   $$ [\hat{H}, \hat{L}_i] = 0 \quad \Rightarrow \quad \hat{L}_i \ \text{is conserved} $$

2. Spherically symmetric potential

   $$ [\hat{H}, \hat{L}^2] = 0, \quad [\hat{H}, \hat{L}_z] = 0 $$

3.4.3 Calculation of Orbital Angular Momentum

1. Eigenvalue equations

   $$ \hat{L}^2 Y_{lm}(\theta,\varphi) = l(l+1)\hbar^2 Y_{lm}(\theta,\varphi) $$
   $$ \hat{L}_z Y_{lm}(\theta,\varphi) = m\hbar Y_{lm}(\theta,\varphi) $$

2. Eigenvalues

   $$ L = \sqrt{l(l+1)} \hbar,\qquad L_z = m\hbar $$

3. Physical meaning
   The quantum numbers $l$ and $m$ determine the magnitude of the orbital angular momentum and its $z$-component, respectively.

Chapter 4 Perturbation Theory and Its Applications

4.1 Time-Independent Perturbation Theory

4.1.1 Nondegenerate Perturbation Theory

1. Basic idea

   $$ \hat{H} = \hat{H}^{(0)} + \lambda \hat{H}' $$

2. Energy corrections

   $$ E_n^{(1)} = \langle \psi_n^{(0)} | \hat{H}' | \psi_n^{(0)} \rangle $$
   $$ E_n^{(2)} = \sum_{m \neq n} \frac{|\langle \psi_m^{(0)} | \hat{H}' | \psi_n^{(0)} \rangle|^2}{E_n^{(0)} - E_m^{(0)}} $$

3. Wave-function correction

   $$ \psi_n^{(1)} = \sum_{m \neq n} \frac{\langle \psi_m^{(0)} | \hat{H}' | \psi_n^{(0)} \rangle}{E_n^{(0)} - E_m^{(0)}} \psi_m^{(0)} $$

4.1.2 Degenerate Perturbation Theory

1. Origin of the problem
   If the zeroth-order energy corresponds to multiple orthogonal eigenstates, the state is degenerate, and the nondegenerate formulas fail.

2. Method

   $$ H'_{ij} = \langle \psi_i^{(0)} | \hat{H}' | \psi_j^{(0)} \rangle $$

   Diagonalize this matrix inside the degenerate subspace.

3. Result
   The first-order energy corrections are the eigenvalues of $H'_{ij}$, and the corrected states are the corresponding linear combinations.

4.2 Time-Dependent Perturbation Theory

1. Basic framework

   $$ \hat{H}(t) = \hat{H}^{(0)} + \hat{H}'(t) $$

2. State expansion

   $$ |\psi(t)\rangle = \sum_n c_n(t) e^{-iE_n^{(0)}t/\hbar} |\psi_n^{(0)}\rangle $$

3. Transition probability amplitude

   $$ c_f^{(1)}(t) = \frac{1}{i\hbar} \int_0^t \langle \psi_f^{(0)} | \hat{H}'(t') | \psi_i^{(0)} \rangle e^{i\omega_{fi} t'} dt' $$

   where $\omega_{fi} = (E_f^{(0)} - E_i^{(0)})/\hbar$.

4. Fermi’s golden rule

   $$ W_{i \to f} = \frac{2\pi}{\hbar} \, |\langle f | \hat{H}' | i \rangle|^2 \, \rho(E_f) $$

Summary

• Time-independent perturbation theory corrects energies and wave functions for static perturbations.
• Time-dependent perturbation theory describes transitions between energy levels, such as radiation absorption and emission.

Electron Spin

Experimental Discovery of Electron Spin

1. Stern-Gerlach experiment
   Passing a beam of silver atoms through a nonuniform magnetic field produces two trajectories, revealing an intrinsic angular momentum beyond orbital angular momentum.

2. Experimental conclusions

   • The spin quantum number is $s = 1/2$.
   • The two possible spin projections are $m_s = \pm 1/2$.
   • Spin contributes an additional magnetic moment: $$ \vec{\mu}_s = -g_s \frac{e}{2m_e} \vec{S}, \quad g_s \approx 2 $$

Theory of Electron Spin

1. Quantum description

   • Spin is intrinsic angular momentum and does not depend on spatial coordinates.
   • Its operators satisfy $$ [\hat{S}_i, \hat{S}_j] = i\hbar \epsilon_{ijk} \hat{S}_k $$

2. Physical meaning

   • Spin determines the magnetic behavior of electrons.
   • Quantized spin leads to Fermi-Dirac statistics and the Pauli exclusion principle.

Spin Angular Momentum

Spin Operators

1. Spin components

   $$ \hat{S}_x, \hat{S}_y, \hat{S}_z $$

   satisfying

   $$ [\hat{S}_x, \hat{S}_y] = i\hbar \hat{S}_z, \quad \text{cyclic symmetry} $$

2. Total spin operator

   $$ \hat{S}^2 = \hat{S}_x^2 + \hat{S}_y^2 + \hat{S}_z^2 $$

   with

   $$ \hat{S}^2 |\chi_s\rangle = s(s+1)\hbar^2 |\chi_s\rangle $$

Matrix Representation of Eigenfunctions

1. Spin-$1/2$ particles

   $$ |\uparrow\rangle = \begin{pmatrix}1\\0\end{pmatrix}, \quad |\downarrow\rangle = \begin{pmatrix}0\\1\end{pmatrix} $$

2. Pauli-matrix form of the spin operators

   $$ \hat{S}_x = \frac{\hbar}{2} \sigma_x, \quad \hat{S}_y = \frac{\hbar}{2} \sigma_y, \quad \hat{S}_z = \frac{\hbar}{2} \sigma_z $$

   where

   $$ \sigma_x = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix},\quad \sigma_y = \begin{pmatrix}0 & -i\\ i & 0\end{pmatrix},\quad \sigma_z = \begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix} $$

Theory of Angular Momentum Coupling

1. Spin-orbit coupling

   $$ \hat{H}_{\text{SO}} = \xi(r)\, \vec{L} \cdot \vec{S} $$

   which produces fine-structure splitting.

2. Total angular momentum

   $$ \vec{J} = \vec{L} + \vec{S}, \quad \hat{J}^2 = (\hat{L}+\hat{S})^2 $$

   with eigenstates $|j, m_j\rangle$ satisfying

   $$ \hat{J}^2 |j, m_j\rangle = j(j+1)\hbar^2 |j, m_j\rangle, \quad \hat{J}_z |j, m_j\rangle = m_j \hbar |j, m_j\rangle $$

3. Coupling result

   • $j = l \pm s$, $m_j = -j, -j+1, ..., j$.
   • Spin-orbit coupling is an important source of the fine structure of atomic spectra.

Principle of Indistinguishability

Systems of Identical Particles

Concepts and Principles

1. Definition of identical particles
   If two particles are completely identical in physical properties such as mass, charge, and spin, and cannot be distinguished by any experiment, they are called identical particles.

2. Principle of indistinguishability
   The physical laws are invariant under exchange of identical particles. Exchanging the positions and spins of any two identical particles leaves the Hamiltonian and observables unchanged.

Hamiltonian of a System of Identical Particles

1. Form

   $$ \hat{H} = \sum_{i=1}^N \hat{T}_i + \sum_{i

2. Symmetry

   $$ [\hat{H}, \hat{P}_{ij}] = 0 $$

   where $\hat{P}_{ij}$ is the exchange operator of particles $i$ and $j$.

Wave Functions of Identical-Particle Systems

1. Symmetry requirement

   $$ \hat{P}_{ij} \Psi(\dots, \vec{r}_i, \vec{r}_j, \dots) = \pm \Psi(\dots, \vec{r}_i, \vec{r}_j, \dots) $$
   • + for bosons, whose wave functions are symmetric
   • - for fermions, whose wave functions are antisymmetric

2. Construction of many-particle wave functions

   • Bosons: symmetrized sum
   • Fermions: antisymmetrized determinant (Slater determinant) $$ \Psi(\vec{r}_1, \dots, \vec{r}_N) = \frac{1}{\sqrt{N!}} \begin{vmatrix} \psi_1(\vec{r}_1) & \cdots & \psi_1(\vec{r}_N) \\ \vdots & \ddots & \vdots \\ \psi_N(\vec{r}_1) & \cdots & \psi_N(\vec{r}_N) \end{vmatrix} $$

Pauli Exclusion Principle

1. Content of the principle
   For identical fermions with half-integer spin, no two particles may occupy the same quantum state:

   $$ \Psi(\text{same quantum state}) = 0 $$

2. Physical meaning
   It explains the arrangement of electrons in atomic orbitals and underlies atomic structure, chemical properties, and Fermi-gas behavior.

3. Examples

   • In atoms, each orbital can hold at most two electrons with opposite spins.
   • In metals, electrons form a Fermi level that determines electrical and thermal properties.

Published on 2025-09-05 at 孤筝の温暖小家, last modified on 2025-09-05

All articles on this blog are licensed under the BY-NC-SA license agreement unless otherwise stated. Please indicate the source when reprinting!

Author: 孤筝(lvbowen040427@163.com)

A couple of days ago, I migrated my blog posts from Typecho to Hugo. Just setting up the Front Matter parameters and reconfiguring image links took considerable effort.
The value of a blog lies first in its articles, followed closely by its comments. Comments are proof that the blog has made an impact in both the digital and real worlds, carrying the interactions between people. More personally, comments from all corners of the globe are precious memories and a part of what makes “me”.
Thus, it’s essential to copy the original comments to the corresponding articles on the new site.

Configuring Waline

Unlike dynamic blogs like WordPress or Typecho, static blogs can only rely on external comment systems. There are many options, each with its pros and cons. After referring to this article and checking the official websites of various comment systems, I ultimately chose Waline.
Waline’s Chinese documentation is detailed and comprehensive. After setting up the LeanCloud database and Vercel server, you can access the comment management dashboard at https://<your-server-domain>/ui/. Register as an administrator for the first time, where you can manage comments and users.

Exporting Typecho Comments

Typecho is quite old, with a smaller user base compared to more active communities like Hexo or WordPress. There’s also very little documentation available online.
The only solution I found was a plugin called Export2Valine (also mentioned in Waline’s documentation) by Yi Hong Yuan Luo, which exports Typecho comments to Valine.
However, it hasn’t been updated in three years, and testing showed it only imports the first comment. Looking at the exported jsonl file, it’s clear that all comment data was fully exported.

First, install the plugin in Typecho (make sure to rename the plugin folder to “Export2Valine”!).

Referencing this article about migrating from Typecho to Hexo, the plugin is outdated and requires some modifications.
Locate Action.php in the plugin folder and modify lines 42 onwards as follows (to track parent comments):

$arr = array(
  "objectId" => $comment["coid"],
  "QQAvatar" => "",
  "comment" => $comment["text"],
  "insertedAt" => array(
    "__type" => "Date",
    "iso" => $time
  ),
  "createdAt" => $time,
  "updatedAt" => $time,
  "ip" => $comment["ip"],
  "link" => $comment["url"],
  "mail" => $comment["mail"],
  "nick" => $comment["author"],
  "ua" => $comment["agent"],
  "url" => "/{$slug}.html"
);

if($comment["parent"]) {
  $arr["pid"] = $comment["parent"];
  $arr["rid"] = $this->getRootId($comment["coid"]);
}

No other changes are needed.
Next, go to Typecho’s admin panel → Console → Export Comments. Open the downloaded jsonl file and delete the header line:
#filetype:JSON-streaming {"type":"Class","class":"Comment"}\n\n.
Save the file, close it, and change the file extension to .json.

Fixing the JSON Format

The exported jsonl file contains escaped Chinese characters and is a single line, making it hard to read.
To convert it into a more readable, editable, and importable json format, use your editor’s Find and Replace feature to replace }\n{ with:

},
{

In Xcode, you can insert line breaks by clicking the small magnifying glass icon on the left.

Now, each line represents one comment object.

Similarly, to separate the fields within each comment object, replace "," with:

",
    "

Now, each comment object contains multiple data fields, structured like this:

{"objectId":"3",
    "QQAvatar":"",
    "comment":"\u6d4b\u8bd5\u4e00\u4e0b",
    "insertedAt":{"__type":"Date",
    "iso":"2023-06-27T09:37:07.000Z"},"createdAt":"2023-06-27T09:37:07.000Z",
    "updatedAt":"2023-06-27T09:37:07.000Z",
    "ip":"223.104.150.16",
    "link":null,"mail":"2868301418@qq.com",
    "nick":"2868301418",
    "ua":"Mozilla\/5.0 (Linux; Android 13; V2171A Build\/TP1A.220624.014; wv) AppleWebKit\/537.36 (KHTML, like Gecko) Version\/4.0 Chrome\/109.0.5414.86 MQQBrowser\/6.2 TBS\/046605 Mobile Safari\/537.36 V1_AND_SQ_8.9.63_4190_HDBM_T QQ\/8.9.63.11380 NetType\/4G WebP\/0.3.0 Ap",
    "url":"\/\u4ea4\u53cb\u6807\u51c6-\u548c\u5e73\u5171\u5904\u4e94\u9879\u539f\u5219.html"},
  {"objectId":"4",
    "QQAvatar":"",
    "comment":"\u600e\u4e48ip\u4e0d\u5bf9",
    "insertedAt":{"__type":"Date",
    "iso":"2023-06-27T09:38:15.000Z"},"createdAt":"2023-06-27T09:38:15.000Z",
    "updatedAt":"2023-06-27T09:38:15.000Z",
    "ip":"223.104.150.16",
    "link":null,"mail":"2868301418@qq.com",
    "nick":"2868301418",
    "ua":"Mozilla\/5.0 (Linux; Android 13; V2171A Build\/TP1A.220624.014; wv) AppleWebKit\/537.36 (KHTML, like Gecko) Version\/4.0 Chrome\/109.0.5414.86 MQQBrowser\/6.2 TBS\/046605 Mobile Safari\/537.36 V1_AND_SQ_8.9.63_4190_HDBM_T QQ\/8.9.63.11380 NetType\/4G WebP\/0.3.0 Ap",
    "url":"\/\u4ea4\u53cb\u6807\u51c6-\u548c\u5e73\u5171\u5904\u4e94\u9879\u539f\u5219.html"},

Common Field Descriptions

1. objectId: Unique identifier for the comment (e.g., “4” or “5”)
2. QQAvatar: QQ avatar link (currently empty)
3. comment: Comment content (contains Unicode escape sequences, e.g., \u600e\u4e48 means “how”)
4. insertedAt/createdAt/updatedAt: Timestamp (ISO 8601 format)
5. ip: Commenter’s IP address
6. link: Link provided by the commenter (may be null)
7. mail: Commenter’s email address
8. nick: Commenter’s nickname
9. ua: User agent (browser/device info)
10. url: Relative path of the commented post

Special Fields

11. pid: Parent comment ID
12. rid: Root comment ID

If "link" is null, there’s no line break between "link" and "mail". JSON is insensitive to line breaks, so this can be ignored.
Now, wrap the entire content in [ ] at the beginning and end of the file, then save it.

Modifying Comment Attributes

The file can now be imported into LeanCloud, but some adjustments are still needed.

Export2Valine sets the URL for comment associations as \/slug, e.g., "url": "\/Summary-of-the-First-Semester-of-Junior-Year.html", where \/ is an escaped /.

To link comments to the new blog’s posts, manually update the url to match the new blog’s post links.

For example, my Hugo-generated site has folders like zh-cn, zh-tw, en, and ja (due to multi-language support). Chinese posts are under /zh-cn/post/category/.
In my local blog source files, posts are organized into folders by category, e.g., /content/post/Thoughts/最近写的诗.md generates a relative URL like zh-cn/post/thoughts/最近写的诗.

If your new blog’s posts are in the root directory with unchanged names, no URL modifications are needed.
If they’re all under /post/, use Find and Replace to change:

"url":"\/

to:

"url":"\/post\/

For my case, I temporarily replaced it with:

"url":"\/zh-cn\/post\/

Similarly, comments on standalone pages like “Friends” or “Thoughts” should be updated to their new relative URLs.
For example, the Friends page:

"url":"\/links.html

should be replaced with:

"url":"\/zh-cn\/friend\/

First, apply bulk replacements for post and standalone pages where possible. Otherwise, it’ll be tedious to modify them after import.

When using Find and Replace, try to target the largest common segments to avoid accidental changes.
Remember to escape \/!!!

Importing to LeanCloud

In LeanCloud’s console → Data Storage → Import & Export, select the modified JSON file, set Class to Comment, and import.

Note: If you’ve previously tested Waline comments or attempted to import Comment, Waline may have already created the Comment class. Subsequent imports will fail silently (LeanCloud may claim success, but no new data appears).

To fix this, go to the console → Structured Data, delete the Comment class, and try importing again. The LeanCloud page may not refresh immediately—use Ctrl+F5 to force a cache refresh.

After a successful import, manually adjust the url for each comment.
For example, my posts need to be categorized under "url":"\/zh-cn\/post\/category\/. Use LeanCloud’s batch operations and filtering features to streamline this process.

Afterword

Organizing comments didn’t take too long—120 comments, mostly my own musings on the “Thoughts” page, allowed for bulk URL fixes. The handful of reader comments were scattered across just a few posts, making them easy to update via filtering. Whether that’s a good or bad thing, I’m not sure (laughs).

Whether they’re my soliloquies or others’ remarks, each comment holds unique significance. Revisiting them periodically brings new reflections.
As I said at the beginning, they’re traces of my growth, proof of my existence, and part of “me”.

And you, dear reader, are the one who gives me value.

If you have time, please leave a comment—it’ll genuinely make my day (as long as it’s kind, of course).

Published on 2025-04-19 at 孤筝の温暖小家, last modified on 2025-04-19

All articles on this blog are licensed under the BY-NC-SA license agreement unless otherwise stated. Please indicate the source when reprinting!

Author: 孤筝(lvbowen040427@163.com)

About Final Exams

For students, nothing seems more important at the end of a semester than grades. Upholding the fine tradition of last-minute cramming, I studied frantically almost every day during the final two weeks (though I occasionally slacked off—why the hell are there still experiments in week 16?). Thankfully, I didn’t fail any courses, and while my scores weren’t stellar, I’m satisfied (scoring over 70 in aerodynamics after cramming it in a day—no need to elaborate on that skill).

And so, I’ll once again make the empty promise of “next semester, I’ll study properly from the start”—a classic lie, haha.
But seriously, I really should start studying, or how else will I ever earn my Ph.D.?

About Pursuing a Ph.D.

I chose the Qian Xuesen Honors Program’s direct Ph.D. track not for the accelerated 4+3 graduation (though I’m not cut out for that anyway) nor to chase a degree that promises higher salaries or greater “social status.” I believe the person who first translated “Doctor” as “博士” (bóshì, literally “erudite scholar”) intended to emphasize the reputation of being widely learned, not the other trappings.

Though they say “cursing someone into a Ph.D. is a thunderbolt-worthy sin,” and stories of delayed graduations and disillusionment abound, I still want to try my hand at creating something entirely new—something unprecedented. Not for titles, not for academic turf, but simply because it might benefit people and humanity.

As I’ve said before:

My material demands are modest—just enough to support my family, which seems achievable given my current situation. But I’m dissatisfied with my undergraduate education; four years of study still feel like I haven’t even scratched the surface.

To cling to knowledge explored by predecessors decades or centuries ago in this rapidly evolving era? To forsake the frontiers of learning and settle for mere subsistence? I can’t do it.

I may lack talent and perhaps will never achieve much academically, but I refuse to abandon humanity’s courage to explore and innovate. At the very least, I want to stand at the forefront as a witness—to watch generations break through the shackles of matter and mind, to see civilization step out of its gentle cradle and venture into the unknown depths of space.

I can’t give up the thrill of acquiring new knowledge,
because I’ve already touched the sky.

Enough grand talk for today. Deep down, I don’t truly believe I have the power to change anything. Most likely, I’ll end up as an unremarkable academic footnote, barely graduating with a pile of mediocre papers.
But I’ll still try. I’ll still do it. Because the mountain is there.

About Family

After such serious topics, let’s lighten the mood.
In August, my mother had surgery at Wuhan Tongji Hospital, and I stayed with her for half a month (though I’m too embarrassed to call it “caretaking”). After her hysterectomy, she rested for two months and regained some energy. But then she went right back to her early-morning-to-late-night work routine. Sigh.
Otherwise, things at home are unchanged, with little to worry about.
I miss the egg-drop sweet soup my grandmother used to make in winter.

About Friends

Life at school has been uneventful, aside from an electronics competition. My friends are the same as ever, the people I like are doing well, and everyone’s living their own lives.
“At the ends of the earth, half my dear friends are scattered.”

About My Trip to Hungary

I signed up for the winter break exchange program at Óbuda University organized by the School of Space Science. I’ve been here for two weeks and am about to head back.
Learned some useless AI applications—nothing to do with my future studies, and I didn’t expect to acquire any “dragon-slaying skills” anyway.

The first few days of “white people food” were awful; I couldn’t fathom how locals survive on this. But the meals gradually improved. I tried authentic Hungarian goulash (a beef stew with potatoes, carrots, etc.) at a local spot, and it was delicious. The pork knuckle here is also incredible—huge portions and tasty (though why is it translated as “toe joint”?).

The steak, though? Forget it—served rare by default, tough to cut and chew. The ubiquitous dry bread is inedible, and the salads might as well be rocks.

Another standout was the bathrooms. Every single one I’ve seen here is spotless, with attendants, stocked toilet paper, soap, hand dryers, and paper towels (OMG). The dorm bathrooms even have bidet sprayers and bleach-soaked toilet brushes. All faucets provide 24/7 hot and cold water, and the heating is so intense that indoors often feels like a sauna.

The only downside? Almost no public restrooms, and some places charge for toilet access.

Sigh. Developed countries have money to burn on resources. It breaks my heart to think of villagers in northern China who can’t afford heating in winter or southerners who can’t run AC in summer.
Too many people in this world suffer. Comrades, we must keep striving.

I bought some authentic souvenirs to bring back, but I won’t spoil the surprise for friends who might read this.

Closing Thoughts

Flying home tomorrow for the New Year, and I’m brimming with nostalgia.
The remaining half of winter break should be spent studying Principles of Communications (maybe it will, maybe it won’t).
Wishing you a happy New Year, peace and prosperity in the years to come.

Jan 24, 2025

Published on 2025-01-28 at 孤筝の温暖小家, last modified on 2025-01-28

All articles on this blog are licensed under the BY-NC-SA license agreement unless otherwise stated. Please indicate the source when reprinting!

Author: 孤筝(lvbowen040427@163.com)

The East is Red

Midstream strokes resound through the skies, as night by night the river’s roar doth rise.

Not youth alone beholds the morning sun—ten thousand miles of land in crimson run.

Dec. 26, 2024
Xi’an

Published on 2024-12-26 at 孤筝の温暖小家, last modified on 2024-12-26

All articles on this blog are licensed under the BY-NC-SA license agreement unless otherwise stated. Please indicate the source when reprinting!

Author: 孤筝(lvbowen040427@163.com)

——This poem is dedicated to the Polaris in my life

A lovely traveler passed through a landscape unseen by others, bringing with her exotic warm breezes that swept across parched fields, rustled through treetops, and rippled the lake into shimmering waves.
The fruit trees, enchanted by this beauty, hurriedly blossomed and bore fruit; birds in the forest scoured for the few remaining flowering branches, weaving them into a crown for her.
Strolling through the falling petals, the traveler, unable to refuse such generosity, plucked a few crimson fruits; feeling shy about taking without giving, she emptied her pockets of all the flower seeds she carried, scattering them over every cracked patch of earth.
Only the field remained barren, its heart filled with shyness and guilt—what could it possibly offer this delightful maiden?
Summoning all its strength, it urged the seeds to grow swiftly, hoping she might witness the blossoms before she left.
But the traveler was never meant to stay—the scenery was beautiful, the trees and birds most attentive, yet this was no place to call home.
She departed, leaving behind wistful fruit trees, birds, and the field.
The field’s flowers never reached her hands, though perhaps the next traveler would find a garden in full bloom.
Yet none grieved,
for spring had come.

Winter gave way to spring, autumn yielded to winter.
Fruits became saplings, fledglings took their first flights; tree rings thickened, and summer rains polished wings to a gleam.
In the garden the field had tenderly nurtured, the last chrysanthemum outlived its season, bare stems swaying in the cold wind.
No longer desolate, the land had seen the traveler’s seeds sprout through the seasons, each flower blooming in its turn as promised—drinking dawn’s dew, basking in morning sun, dancing in the breeze, resting under moonlight.
In spring, the field wove a crown of gardenias and hawthorn blossoms, white as the traveler’s dress.
In summer, it fashioned a basket of crape myrtle and hibiscus, though the hibiscus wilted by dusk, much like her fleeting visit.
In autumn, it dreamed of a bed sweet with osmanthus and aster, but the wind carried the fragrance away, indifferent to its longing.
The flowers, embracing their purpose, reveled in their brief lives, preparing in the soil for journeys yet to come.
The traveler never returned, unaware that her casually scattered seeds had grown into a garden, and the field never had its chance to offer its finest tribute.
Winter gave way to spring, autumn yielded to winter.
The field longed for snow, a warm blanket for the seeds below. Should the traveler ever revisit this hidden paradise, the garden would be worthy of her.
In the drizzle of early winter, it drifted into sleep, dreaming of October’s last dandelions riding the wind, chasing the blush on her cheeks.
“Aren’t you afraid the seeds won’t survive the cold?” asked the trees and birds, as the field lay bare once more.
“It’s alright. I’ve seen spring.”
For spring had come.

Published on 2024-11-13 at 孤筝の温暖小家, last modified on 2024-11-13

All articles on this blog are licensed under the BY-NC-SA license agreement unless otherwise stated. Please indicate the source when reprinting!

Author: 孤筝(lvbowen040427@163.com)

Preface

During the National Day holiday, I traveled through Shanxi with Black Myth: Wukong, visiting three cities in southern Shanxi—Jincheng, Linfen, and Yuncheng—and composed five poems to briefly document the journey.

Southern Shanxi Journey · Part 1

The door hides broken statues, the eaves bear golden chimes.
Dabs of emerald dye the tangled woods, splashes of ink unfold jade skirts.
Fresh branches climb rotten wood, the old lead the young.
Lin Huiyin would weep—can a thousand years leave blank space?

Lone Zither
October 2, 2024
Jade Emperor Temple, Fucheng Guan Yu Temple, Qinglian Temple, Jincheng

At the Jade Emperor Temple, flash photography is banned in the Hall of Twenty-Eight Constellations, and the statues are hidden behind iron railings, their paint peeling, some even headless or limbless—a regrettable sight.
The statues at Fucheng Guan Yu Temple are mostly modern recreations, crudely made and unappetizing. The architecture, however, is worth a look.
Qinglian Temple sits high on a distant mountain, with breathtaking scenery along the way. In autumn, the mountains are lush, the waters clear, dotted with broad red and yellow leaves under a cloudless sky—a land of splendor.

Southern Shanxi Journey · Part 2

Weary, the Green Emperor has yet to take his post, morning mist rushes to break the dawn’s chill.
Just crossed seas of people to glimpse the Buddha’s face, now climb forested clouds to visit the temple.
Nine golden bodies, nine lotuses, three gates, three altars.
Before the scaled sunset gathers dusk, my light steed has leaped over layered peaks.

Lone Zither
October 3, 2024
Kaihua Temple, Iron Buddha Temple, Dinglin Temple, Jincheng

Early in the morning, I took the bus to Gaoping, first visiting Kaihua Temple before returning to Iron Buddha Temple. The latter, newly reopened, was packed, tucked deep in a small village courtyard. After queuing for over an hour, I glimpsed the Buddha’s face for just two minutes. Fortunately, I met a family of three in line—open-minded parents unbothered by anime and games. (Their twin-tailed daughter was adorable, by the way.)
Dinglin Temple’s lotus caisson ceiling had also recently reopened, and I was lucky to see it—truly magnificent.

Southern Shanxi Journey · Part 3

Prelude to Water Melody

The Buddha feasts on incense offerings, stone tablets drown in dust.
I ask the hills beneath his seat: do sutras reach the divine?
Shorn locks renounce desire, abstinence purges the halls,
smiles mask wrath. They know the laws of five aggregates,
yet fail to grasp the body of ignorance.
False cymbals, temple repairs, deceiving the credulous.
Promises of merit mimic the Tathagata’s great vehicle.
Bribes buy devotion, virtue lacks faith—
the Eight Precepts mislead the sangha.
The chime-keeper chants wealth, the false Buddha saves worldly monks.

Lone Zither
October 4, 2024
Little Western Heaven, Linfen

Eight Precepts:

1. No killing.
2. No stealing.
3. No sexual misconduct.
4. No lying.
5. No intoxication.
6. No adornments.
7. No high or broad beds.
8. No eating after noon.

Yellow Brow’s Counter:
No killing? Hatred never ends.
No stealing? Why distinguish strong from weak?
No lust? All sentience is sin.
No lies? Dreams, bubbles, shadows, void.
No wine? Sorrow ebbs and flows.
No pleasure? Beauty fades in a blink.
No sloth? Suffering binds without release.
No indulgence? All acts are joyless.

The lower temple of Little Western Heaven was unremarkable, filled with devotees praying for wealth and sons. A nun (?) sat nearby striking a chime, chanting promises of billions in daily earnings—utterly cynical.
The upper temple’s Mahavira Hall boasts astonishing hanging sculptures, grand in scale. Sadly, the crowds left little time for close inspection. I arrived too late to receive Shanxi’s official commemorative postcard—a slight regret.

Southern Shanxi Journey · Part 4

An ancient temple guards Pingyang, three quakes veil the Buddha’s light.
The towering glazed pagoda, the ethereal Arhat Hall.
Scriptures sought through merit, rain prayed for by kings.
A canon passed to the East, the golden age of Tang began.

Lone Zither
October 6, 2024
Guang Sheng Temple, Linfen

Guang Sheng Temple in Hongtong County features a Rain God Temple in its lower complex, its walls covered in murals, though poorly preserved. The dim light inside made details hard to discern. Side rooms displayed official scanned reproductions—vivid and lifelike, truly masterful.
The upper temple’s Flying Rainbow Pagoda is clad in glazed tiles, though only the first floor is open, with little to see inside.
The rear courtyard’s Tianzhongtian Hall houses three colossal Buddhas, over ten feet tall, their elegant forms a marvel.

Southern Shanxi Journey · Part 5

Encountering Wei Taoran at Stork Tower

No immortal of wine was I born,
inking brows, grinding words.
At Stork Tower, I duel Wang Zhihuan,
below Taihang, I ponder the Chairman.
You roam the world a free spirit,
I’m trapped in the ivory tower, forever young.
If heaven and earth offer no refuge,
let poetry debts buy wine.

Lone Zither
October 6, 2024
Yongle Palace, Guangren King Temple, Stork Tower, Guan Yu Temple, Yuncheng

A rushed group tour in Yuncheng left little time for details. At Stork Tower, I spotted a stall boldly labeled “Selling Original Poetry.” After touring the new tower, I revisited the collection, struck by lingering awe. Wei Taoran, inspired by a Dali poetess, returned to literature—I too once wrote many poems, mostly idle scribbles, never considering it a livelihood.
In high school, I gifted my poetry drafts to someone. After the breakup, I scarcely wrote again. First, gaming consumed me—I read little, my pen dry, my efforts unsatisfying. Second, life blurred like fog—neither piercing nor clear, I drifted in haze. Third, the ivory tower’s monotony left me numb, devoid of poetic spark.

Afterword

The three cities of southern Shanxi each have their charm.
Jincheng is bustling, with efficient buses and dedicated routes to attractions. Arriving downtown, the streets blaze with light.
Linfen is peculiar—even areas around the bus station are dark, and distant sites lack dedicated transit, making private hires costly. Its public toilets are uniquely varied—I biked past twenty-plus on shared e-scooters, no two alike.
Yuncheng thrives—shops and stalls pack main roads and alleys, with rumors of north-south markets (and delicious, honest-to-goodness buns).

Published on 2024-10-08 at 孤筝の温暖小家, last modified on 2024-10-08

All articles on this blog are licensed under the BY-NC-SA license agreement unless otherwise stated. Please indicate the source when reprinting!

Author: 孤筝(lvbowen040427@163.com)

“Mom, will there really be girls who like me in this world?”
“Of course! When I first met you, I was only in my twenties too.”

The Mid-Autumn Festival is approaching—give your family a call.

“Men have sorrow and joy, they meet or part again;
The moon is bright or dim and she may wax or wane.
There has been nothing perfect since the olden days.”
So let us wish that man may live as long as he can!
Though miles apart, we’ll share the beauty she displays.

May we all be blessed with longevity.
May we all be blessed with longevity.
May we all be blessed with longevity.

(Note: The classical Chinese poem is rendered in Xu Yuanchong’s renowned English translation, preserving both the lyrical beauty and philosophical depth of Su Shi’s original work. The repetition of “longevity” mirrors the original text’s emphasis while adapting to English poetic conventions.)

Published on 2024-09-15 at 孤筝の温暖小家, last modified on 2024-09-15

All articles on this blog are licensed under the BY-NC-SA license agreement unless otherwise stated. Please indicate the source when reprinting!

Author: 孤筝(lvbowen040427@163.com)

Preface

First Edition Preface

[[2024-09-14]] Today the makeup exam finally ended. I heard the original exam directly reused past papers. These past few days I practiced three sets of “XDU original papers” (from 2021 and two from 2023) found online. I did the 2021 paper in the morning, and in the afternoon $\frac{1}{4}$ of the questions were exact copies without any changes. I couldn’t help but laugh.

Dai Hao once said he would try his best to find the best teachers for the Qian Class. But now it seems the School of Mathematics and Statistics has no one left? Poor teaching could be excused as not focusing on education or lacking talent in teaching; but directly reusing recent past papers for exams, full of unchecked errors and omissions, made me laugh in frustration.

The exams they create have no value, and they don’t even bother to test them themselves. This is an attitude problem.

It’s fine that your university goes easy on final exams, but don’t keep fooling people with old material. You preach innovation to students, yet for yourselves, just getting by is enough. This is not the attitude for academic work, nor is it the attitude one should have for teaching.

Probability theory ends here for now. Over the past two days, I repeatedly reviewed notes, practiced problems, and corrected many errors, clarifying the knowledge structure of this course. Although the content is still relatively sparse, it should suffice as final review material. This edition will likely be the final version (probably). I’ll continue organizing Electrodynamics and Digital Signal Processing during the Mid-Autumn Festival.

Second Edition Preface

Nothing is final!!! ——Qian Xuesen

Added content on the left/right continuity of distribution functions. It seems this course is far from final…

Event Operations to Logical Operations

• $A \cup B=A+B$
• $A \cap B=A \cdot B$
• $A-B=A \bar{B}$ Event $A$ occurs and event $B$ does not occur, easily proven by Venn diagrams. $-B$ can be interpreted as $\cdot (-B)$, where $-B$ is $\bar{B}$.
• If $A \subset B$, then $A \cup B=B$, $A \cap B=A$.

After converting event operations to logical operations, most rules are shared. Using logical function operations and simplification learned in digital circuits, complex event operations can be simplified. Tips: Karnaugh maps.

Four Major Probability Formulas

Corollary

$P(A+B+C)$: Treat $A+B$ as a single event and apply the addition formula above, splitting twice to get:

Probabilities for more joint events can be derived recursively.

Complementary event: The probability that $A$ does not occur, obvious from Venn diagrams.

Non-Negativity and Normalization

Non-negativity: For any event $A$, $0 \le P(A) \le 1$. Normalization: For the total event $\Omega$, $P(\Omega)=1$.

Independence

Independence implies mutual independence.

Classical Probability Model

All elementary events have equal probability.

Eg. Coin toss, dice roll…

Classical conditional probability formula:

Bernoulli Trials (Binomial Distribution)

$n$ independent trials, each with only two outcomes: $A$ or $\bar{A}$.

$X \sim B(n,p)$

Where $p=P(A)$, $1-p=P(\bar{A})$.

Geometric Probability Model

The ratio of the length/area/volume occupied by the event to the total length/area/volume of the sample space $\Omega$. When the event’s dimension is lower than $\Omega$’s dimension, its probability is always 0. ==Warning==: A probability of 0 does not mean the event cannot occur. Eg: Randomly selecting a point inside a circle, the probability of selecting any specific point is 0, but it can still happen.

Uniform Distribution

$x \sim U(a,b)$ Approximates a linear distribution in geometric probability, with probability density:

Cumulative distribution function:

Exponential Distribution

$x \sim E(\lambda)$

Probability Density

Cumulative Distribution Function

Poisson Distribution

$X \sim \pi(\lambda)$

Normal Distribution

$x \sim N(\mu,\sigma^2)$

Probability Density

Cumulative Distribution Function

Clearly, $F(\mu)=\frac{1}{2}$, meaning $P(x \le \mu)=P(x \gt \mu)=\frac{1}{2}$.

Standard Normal Distribution

When $\mu=0,\sigma=1$, it becomes the standard normal distribution.

Corollaries

Normalization of normal distribution:

Total Probability Formula

Complete Event Group

$B_1,B_2,B_3,\cdots B_n$ form a complete event group for $\Omega$.

Total Probability Formula

Bayes’ Formula

One-Dimensional Discrete Random Variables

Probability Mass Function

Cumulative Distribution Function

One-Dimensional Continuous Random Variables

Probability Density Function

Cumulative Distribution Function

Interval Probability

$\because$ $P(x=a)=0,a \in R$ $\therefore$ The equality signs on the interval can be chosen freely.

Normalization

Two-Dimensional Discrete Random Variables

Joint Probability Mass Function

$P(X=x_i,Y=y_j)$ Create a 2D table of possible values for X and Y, filling in corresponding probabilities.

Marginal Probability Mass Function

$P(X=x_i),P(Y=y_j)$ Sum the rows/columns of the joint probability table to get $f_Y(x),f_X(y)$.

Conditional Distribution

$P(X=x_i|Y=y_j),P(Y=y_i|X=x_j)$ Divide each row/column of the joint probability table by its marginal probability. This scales the joint probabilities so each row/column sums to 1.

Independence of Two Variables

==Independence here refers to linear independence, not complete statistical independence.== Write the joint probability table as a matrix $\vec{A}$. If $\det \vec{A}=0$, X and Y are independent. Or: If the rows/columns of the joint probability table are proportional, X and Y are independent. Or: If the joint probability $\ne$ the product of marginal probabilities, i.e., $P(X=x_i,Y=y_j)\ne P(X=x_i)P(Y=y_j)$, then X and Y are not independent.

Two-Dimensional Continuous Random Variables

Joint Density Function

Normalization

Marginal Density Functions

Conditional Density

Independence

When this holds, X and Y are independent.

Distribution Function

Let $Z=X-Y$,

The distribution function $F_Z(z)=\iint_Df(x,y)dxdy$. Differentiate to get the probability density function $f_Z(z)$. ==Warning==: $F_Z(z)$ must satisfy normalization.

Expectation and Variance

Relations

When X and Y are independent, $Cov(X,Y)=0$.

Common Expectations and Variances

$(0,1)$ Distribution

$B(n,p)$ Binomial Distribution

$U(a,b)$ Uniform Distribution

$E(\lambda)$ Exponential Distribution

$P(\lambda)$ Poisson Distribution

$N(\mu,\sigma^2)$ Normal Distribution

Covariance and Correlation Coefficient

Covariance

Clearly, when $X=Y$, $Cov(X,X)=DX$.

Correlation Coefficient

Higher $|\rho|$ means stronger correlation. When $Y=X$, $X$ and $X$ are perfectly correlated, $\rho=1$. When $Y=-X$, $-X$ and $X$ are perfectly correlated, $\rho=-1$. Clearly $|\rho| \le 1$. $\rho=0$ means X and Y are uncorrelated. ==Warning==: Uncorrelated $\nRightarrow$ Independent, but Independent $\Rightarrow$ Uncorrelated.

Chebyshev’s Inequality for Probability Estimation

Central Limit Theorem

A large number of independent, identically distributed variables can be approximated by a normal distribution. If $x_1,x_2,\cdots,x_n$ are independent and identically distributed, then

Three Major Distributions

$\chi^2$ (Chi-Squared) Distribution

Upper $\alpha$ quantile $\chi^2_\alpha(n)$ Density function is in the first quadrant.

$t$ Distribution

Upper $\alpha$ quantile $t_\alpha(n)$ Density function resembles normal distribution, symmetric.

$F$ Distribution

Upper $\alpha$ quantile $F_\alpha(n_1,n_2)$ Density function is in the first quadrant.

Estimation Methods

For simple random samples that are independent and identically distributed, estimate unknown parameters.

Method of Moments

When sample size is large, approximate the sample as uniformly distributed, using sample mean to replace population mean (population moment = sample moment).

1. Calculate the expectation $EX$ (first population moment) from the given probability mass/density function.
2. Calculate the sample mean $\bar{X}$ (first sample moment) from the given sample.
3. Set $EX=\bar{X}$ and solve for $\theta_0$ as $\hat{\theta}$.

Maximum Likelihood Estimation

The estimate maximizes the probability of the observed sample. Likelihood function for the sample:

To find the maximum of $L$, take the derivative to find critical points. Since the product form is cumbersome, first take the logarithm before differentiating with respect to $\theta$.

Solve for the critical point $\theta_0$, which is the estimate $\hat{\theta}$.

Unbiasedness and Efficiency

If $E(\hat{\theta})=\theta$, then $\hat{\theta}$ is an unbiased estimator of $\theta$. If $\hat{\theta_1},\hat{\theta_2}$ are both unbiased, and $D(\hat{\theta_1}) \lt \hat{\theta_2}$, then $\hat{\theta_1}$ is more efficient than $\hat{\theta_2}$.

Interval Estimation

$X \sim N(\mu,\sigma^2)$, typically given $\bar{X}=\mu,S=\sigma$. Confidence level: $1-\alpha$, usually $\alpha=5\%$.

Confidence Interval for $\mu$

$\sigma^2$ Known

Pivotal quantity (standardized):

$\sigma^2$ Unknown

Pivotal quantity:

Confidence Interval for $\sigma^2$

Usually $\mu$ is unknown. Pivotal quantity:

Hypothesis Testing

Generally, the significance level is set at $\alpha=5\%$.

$\mu$ Test (Mean Test)

1. Hypothesis Formulation
   $H_0: \mu = \mu_0$ (null hypothesis)
   $H_1: \mu \ne \mu_0$ (alternative hypothesis)

2. Test Statistic Selection

   • When population variance $\sigma^2$ is known:
     Use $Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \sim N(0,1)$ (Z-test)
   • When population variance $\sigma^2$ is unknown:
     Use $T = \frac{\bar{X} - \mu}{S / \sqrt{n}} \sim t(n-1)$ (T-test)

3. Rejection Region Determination

   • For Z-test:
     $W = (-\infty, -z_{\alpha/2}) \cup (z_{\alpha/2}, \infty)$
   • For T-test:
     $W = (-\infty, -t_{\alpha/2}(n-1)) \cup (t_{\alpha/2}(n-1), \infty)$

4. Decision Rule
   Reject $H_0$ if the computed test statistic falls within the rejection region $W$.

$\sigma^2$ Test (Variance Test)

Sample standard deviation formula:

1. Hypothesis Formulation
   $H_0: \sigma^2 = \sigma_0^2$
   $H_1: \sigma^2 \ne \sigma_0^2$

2. Test Statistic Selection
   Use $\chi^2 = \frac{(n-1)S^2}{\sigma^2} \sim \chi^2(n-1)$ (Chi-square test)

3. Rejection Region Determination
   $W = (0, \chi^2_{1-\alpha/2}(n-1)) \cup (\chi^2_{\alpha/2}(n-1), \infty)$

4. Decision Rule
   Reject $H_0$ if the test statistic falls within the rejection region $W$.

Supplementary Notes

Properties of Distribution Functions

For different types of random variables:

• Continuous random variables: The distribution function is continuous.
• Discrete random variables: The continuity of the distribution function depends on its definition.

Left-Continuous Definition

In this case:

• $F(x) = F(x^-) = F(x-0) = P(X \lt x)$
• $F(x^+) = F(x+0) = P(X \lt x) + P(X = x)$

When $P(X = x) \ne 0$, $F(x^+) \gt F(x) = F(x^-)$, making the distribution function left-continuous but not right-continuous.

Right-Continuous Definition

In this case:

• $F(x) = F(x^+) = F(x+0) = P(X \le x)$
• $F(x^-) = F(x-0) = P(X \le x) - P(X = x)$

When $P(X = x) \ne 0$, $F(x^+) = F(x) \gt F(x^-)$, making the distribution function right-continuous but not left-continuous.

Coin Toss Example

Consider a single coin toss:

• Heads (1): Probability 0.5
• Tails (0): Probability 0.5

Random variable $X$ has the distribution:

Cumulative probabilities:

Using the left-continuous definition $F(x) = P(X \lt x)$:

Here:

• $F(0^-) = F(0) = 0$
• $F(0^+) = 0.5$
• At $x=0$, there is a discontinuity point where the function is left-continuous but not right-continuous.

Published on 2024-09-10 at 孤筝の温暖小家, last modified on 2024-09-10

All articles on this blog are licensed under the BY-NC-SA license agreement unless otherwise stated. Please indicate the source when reprinting!

Author: 孤筝(lvbowen040427@163.com)

Preface

As the saying goes, the pinnacle of customization is the default.

While the default Windows setup can efficiently handle all tasks, it’s undeniably ugly.

With a performance-overkill PC at hand, it’s entirely justified to pursue some aesthetic refinements and simplifications to satisfy my high-end taste (lol).

Below, I’ll share the customization software/solutions I’m currently using or have tried.

Current Desktop Setup

• TranslucentTB: Transparent/acrylic taskbar effects
• Sapphire: Desktop icon interaction optimization
• Wallpaper Engine: To minimize GPU and memory usage, I’ve chosen a 4K video of Noa’s memory lobby from Blue Archive as the wallpaper. The entire Wallpaper Engine process consumes about 100 MB of RAM.
• Rainmeter: Only using an audio visualizer for aesthetics because Noa is already beautiful enough 😋.
  Previously tried hardware monitoring widgets but found them useless except for inducing anxiety—so I ditched them.
• QQ customization, NetEase Cloud Music customization, cursor customization, Obsidian themes and plugins, plus Edge browser’s iTab new tab page and various practical extensions. My current workflow is perfectly comfortable.

Here are some screenshots:

Under this workflow, the browser consumes the most memory, while Wallpaper Engine is the primary GPU load. From login to full autostart, it takes under 10 seconds, all within acceptable performance limits.

Specs:

• 12400 F
• 7700 XT
• 32G DDR4
• 2K 180Hz HDR display
• Windows 11 Pro 23H2

QQ Customization

==Highly recommended!==
QQ is something most Chinese people are forced to use yet despise—ads, pop-ups, unwanted entertainment pages, and a cluttered, dysfunctional interface.
The Windows client improved somewhat with QQ 9, but the annoying elements are creeping back. Old habits die hard.
To avoid enduring this daily torture, the great open-source-minded tinkerers of the Chinese internet have launched QQ overhaul projects. Today, I’ll introduce one such masterpiece: LiteLoaderQQNT.

LiteLoader is a plugin platform for QQNT. After installation, you can download numerous plugins.
GitHub project: LiteLoaderQQNT: QQNT Plugin Loader: Lightweight · Minimalist · Open-Source · Furry

Recommended plugins:

1. PluginInstaller: LiteLoaderQQNT plugin installer for easy updates and one-click install/restart. Install this first to simplify adding other plugins.
2. LL-plugin-list-viewer: Browse, install, and update LiteLoaderQQNT plugins. Lists most plugins with direct GitHub links. Installation is buggy—some plugins require manual setup, or QQ may fail to start. Best used as a plugin/theme viewer.
3. Lite Tools: A multi-functional toolbox to avoid hunting for plugins. Key features:
   • Chat interface美化 (Telegram-style: avatars, timestamps, left-aligned messages)
   • Remove badges, VIP tags, and other clutter.
   • Right-click quick search for text/images, message-to-image conversion.
   • Highlight options and special messages.
   • Mini-program links → URL cards, remember last position, quick “+1” (repeater).
   • Message preview: Generates Telegram-like preview cards from the first link in a message.
   • Local emojis.
   • Message suffixes.
   • Cache and highlight recalled messages.
   • Custom backgrounds with brightness, transparency, and blur effects.
   • Simplify sidebar (all features toggleable).
   • Input box and message box toggles.
   • …and more.
4. QR Decode: Decodes QR codes in QQNT chat images.
5. Directly Jump: Skips QQ’s link warning page.
6. DeepL for QQNT: Integrates DeepL translation.
7. Markdown: Adds Markdown support. Only renders for other QQNT users with this plugin, so mostly useless.
8. Kill-Update: Blocks QQ update prompts (some plugins lag behind QQ updates).
9. Window On Top: Adds window pinning.

Themes aren’t covered here—a decluttered QQNT is already decent-looking.
There are also AI plugins (e.g., ChatGPT), but I’m too lazy to set up APIs (no money).

NetEase Cloud Music Customization

==Highly recommended!==
China’s music platforms love bloat: communities, short videos, VIP spam.
A music app needs just a few core features. While standalone players exist, they lack platform perks like song searches, comments, and collaborative listening. Sunk costs (playlists, artist follows, purchases) trap users in these ecosystems.

Thankfully, NetEase Cloud Music has plugins. Here’s the loader: BetterNCM

• Official site: MicroBlock | BetterNCM
• GitHub: GitHub - MicroCBer/BetterNCM
• Community guide: Feishu Doc

BetterNCM lets you download/update themes and plugins in-app—no GitHub scavenger hunts.

Recommended themes:

• Materia You: Minimalist, solid-color backgrounds. Multiple color schemes.
  

Recommended plugins:

1. RoundCornerNCM: Rounded corners (Windows 11 only).

2. MikuPlugin: Toggles UI elements (e.g., hide videos/livestreams).
   

3. Apple-Style Lyrics: Apple Music-like layout + lyric source switching.
   

4. What’s This Cover?: Adds thumbnails to song lists (may cause lag).
   

5. RuLyrics: Desktop lyrics (word-by-word, dual-line). Custom fonts/colors. Taskbar embedding (may conflict with TranslucentTB).
   

6. Explore more plugins—but check GitHub Issues for conflicts/dependencies.

Wallpaper Engine

==Essential!==
The legendary Steam小红车.

Pros:

• Vast library: Steam Workshop offers endless free wallpapers.
• Diversity: Videos, GIFs, web pages, etc. Many include extras like music visualizers.
• Easy discovery: Filter by resolution, type, age rating, etc.
• User-friendly: Steam integration means no VPNs. Most wallpapers are plug-and-play.
• Mobile sync: Android app mirrors PC wallpapers.

Warning: Complex web/4K wallpapers can hog GPU/VRAM. Tweak FPS/effects in settings.

Only con: Costs 19 RMB on Steam—a fair price.

Steam Store Page

TranslucentTB

==Highly recommended!==
Tiny tool for transparent/acrylic taskbars. Minimal resource usage.

GitHub: TranslucentTB
Chinese localization: TranslucentTB-CN

Rainmeter

==Highly recommended!==
The OG desktop widget tool. Create or import skins for:

• Hardware monitors (CPU/GPU/RAM)
• Audio visualizers
• Media players
• Anime trackers
• Image galleries
• etc.

Cons:

• Too many widgets = lag
• Some skins are resource-heavy
• Requires tinkering

Official site: Rainmeter
Chinese site: Rainmeter.cn
GitHub: Rainmeter
Chinese forum: BBS

Start11

Start menu/taskbar customizer:

• Revert to Win7/10 styles. Adjust colors, transparency, spacing, alignment.
• Enhanced search: Integrates Edge tabs into results. Option to hide web content!
• Custom start button icons, menu backgrounds, taskbar textures.

Cons:

• Paid (35 RMB lifetime, 30-day trial).
• Felt sluggish for me.
• Conflicts with TranslucentTB.

Official site: Start11
Chinese site: Start11.cn
Steam (mixed reviews): Start11 v2

Maple Customization Toolbox

File Explorer/Start menu/window美化:
Features:

• Custom fonts, light/dark modes (experimental).
• Background images for Explorer/Start Menu/Settings.
• Color schemes (title bars, headers, progress bars).
• Icon packs (desktop/Explorer).
• Window effects: Transparency, blur, acrylic, Mica.
• Title bar buttons: macOS-style, rounded tabs, scrollbars.
• Preset sharing.
• Plugins for extended functionality.

Evaluation:

• Easy to use, moderate resource usage.
• Plugins/configs add replayability.
• Incompatible with TranslucentTB.
• Explorer crashes, missing backgrounds, unthemed toolbars (may be fixed).
• Low-res backgrounds. Dark mode images can obscure text.
• Free but not fully open-source (account required).

Download: 枫の主题社

Sapphire

==Highly recommended!==
Desktop icon/layout overhaul. Revamps interaction:

• Adjust grid spacing.
• Folders-as-grids (phone-style).
• Custom icon/grid size, rounding, transparency.
• Font changes.
• Minimal mode (hide filenames).
• Animations.
• Dock栏 (Mac/phone-style, vertical/horizontal).
• Folder previews (hover to peek, click to expand in-place).
• Steam shortcut optimization.
• Double-click to hide icons (per-item toggle).
• ~100 MB RAM, negligible GPU.
• Custom wallpapers (partial Wallpaper Engine support).
• Multi-monitor.

Cons:

• Advanced folder preview crashes with many files (awaiting fixes).
• No layout saving (though crashes don’t reset it).
• Windows 11 only.
• Overrides desktop right-click (no customization yet).

GitHub: Sapphire-EnhancedDesktop

Cursor Customization

Many cursor packs are shared online. Pick your vibe. Installation is simple.

Two recommendations (subtle yet unique):

1. Genshin Nahida cursor: Bilibili video
2. Blue Archive Millennium cursor (simple/cute, currently using):
   GitHub: BlueArchive-Cursors

Published on 2024-09-07 at 孤筝の温暖小家, last modified on 2024-09-07

All articles on this blog are licensed under the BY-NC-SA license agreement unless otherwise stated. Please indicate the source when reprinting!

Author: 孤筝(lvbowen040427@163.com)

Fundamental Concepts of Digital Signal Processing

Signal Classification

1. Continuous-time signal: Analog signal, continuous in the time domain.
2. Discrete-time signal: Continuous in amplitude but discrete in time.
3. Amplitude-discrete signal: Discrete in amplitude but continuous in time.
4. Digital signal: Discrete in both amplitude and time.

Differences

The distinction between a discrete-time signal and a digital signal lies solely in the quantization error present in digital signals.

Implementation Methods of Digital Signal Processing

The primary target of digital signal processing is digital signals, achieved through numerical operations.

Software Implementation

Programs are written based on principles and algorithms and executed on general-purpose computers.

• Advantages: Flexibility
• Disadvantages: Slow computation speed, difficult to achieve real-time processing.
• Suitable for: Algorithm research and simulation.

Hardware Implementation

Hardware structures are designed according to specific requirements and algorithms, using basic components such as multipliers, adders, delay units, controllers, memory, and input/output interfaces.

• Advantages: Fast computation speed, capable of real-time processing.
• Disadvantages: Inflexibility

Hardware implementation refers to selecting an appropriate DSP chip, equipped with suitable software and language for the task, to achieve specific signal processing functions.

Dedicated Chips

Using Digital Signal Processing (DSP) chips is currently the fastest-growing and most widely applied method. DSP chips offer significant advantages over general-purpose microcontrollers:

• Internal multipliers and accumulators tailored for digital signal processing.
• Pipeline architecture, parallel structures, and multiple buses.
• Specialized instructions optimized for digital signal processing, enabling high-speed computation.

For ultra-high-speed real-time systems where DSP chips are insufficient, Field-Programmable Gate Arrays (FPGAs) or custom ASICs (Application-Specific Integrated Circuits) should be employed.

Characteristics of Digital Signal Processing

Compared to analog signal processing, digital signal processing offers:

1. Flexibility
2. High precision and stability
3. Ease of large-scale integration
4. Capability to perform functions unattainable by analog systems, such as storage, complex transformations, and operations.

Signal Dimensionality

A signal is typically a function of one or more independent variables.

• One-dimensional signal: Only one independent variable.
• Multidimensional signal: Two or more independent variables.

Discrete-Time Signals and Systems

Discrete-Time Signals

In practice, signals are generally analog. Uniform sampling (equally spaced sampling) converts them into discrete-time signals.

For an analog signal \( x_a(t) \), discrete time points \( t_n \).
With uniform sampling interval \( T \), \( t_n = nT \):

Here, \( x(n) \) is called a discrete-time signal, where \( n \) is an integer, forming a sequence:

Discrete-time signals are also referred to as sequences.

Representation Methods for Sequences

Set Notation

A set of numbers is denoted by \( \{\cdot\} \). A discrete-time signal can be represented as an ordered set of numbers.
The underlined element in the set indicates the sample value at \( n=0 \).

Formula Representation

Example:

Graphical Representation

The horizontal axis represents \( n \), and the vertical axis represents the value of \( x \), with dots atop vertical lines.

Common Standard Sequences

Unit Impulse Sequence \( \delta(n) \)

Also called the unit sample sequence, distinct from the unit impulse signal \( \delta(t) \).

Unit Step Sequence \( u(n) \)

Relationships:

Rectangular Sequence \( R_N(n) \)

\( N \) is the length of the rectangular sequence. It can be expressed using the unit step sequence:

Real Exponential Sequence

• Convergent sequence: \( |a| \lt 1 \)
• Divergent sequence: \( |a| \gt 1 \)

Sinusoidal Sequence

Here, \( \omega \) is the digital frequency (units: radians, \( rad \)), representing the rate of change (phase shift between adjacent samples).

Analog angular frequency \( \varOmega \)
If the sinusoidal sequence is derived from sampling an analog signal \( x_a(t) = \sin(\varOmega t) \):

The relationship between digital and analog frequencies is:

Given the sampling frequency \( F_s = \frac{1}{T} \):

Digital frequency is the normalized analog angular frequency relative to the sampling frequency.

Complex Exponential Sequence

Since \( n \) is an integer, both sinusoidal and complex exponential sequences are periodic with period \( 2\pi \).

Periodic Sequence

If for all \( n \), there exists a smallest positive integer \( N \) such that:

then the sequence \( x(n) \) is periodic with period \( N \).

Sequence Operations

Addition and Multiplication

Shifting, Flipping, and Scaling

Discrete-Time Systems

For a system with input \( x(n) \), output \( y(n) \), and operation \( T[\cdot] \):

Linear Systems

A system is linear if its input-output relationship satisfies the principle of superposition.

Additivity

Homogeneity (Scaling)

Time-Invariant Systems

A system is time-invariant if its operation \( T[\cdot] \) does not change over time, i.e., the system’s response is independent of when the input is applied.

Characteristics of Linear Time-Invariant (LTI) Systems

Total response = Zero-input response + Zero-state response

Unit Impulse Response

With zero initial state (no zero-input response):

For any input \( x(n) \):

Thus, the output is:

This is the convolution sum. For details, refer to Signals and Systems.

Causality of Systems

Definition: A system is causal if its output at time \( n \) depends only on the input at time \( n \) and prior inputs, not future inputs.

Necessary and Sufficient Condition:
The unit impulse response satisfies:

Stability of Systems

Definition: A system is stable if every bounded input produces a bounded output (BIBO stability).

Necessary and Sufficient Condition:
The unit impulse response is absolutely summable:

Linear Constant-Coefficient Difference Equations

Published on 2024-09-04 at 孤筝の温暖小家, last modified on 2024-09-04

All articles on this blog are licensed under the BY-NC-SA license agreement unless otherwise stated. Please indicate the source when reprinting!