#discrete fourier Transform

[comparatorclock]

Science side of tumblr, what's IDFT stand for?

[optimisticabomination]

I assumed factorial four times, but should give the same answer no matter what, right?

LGBTQIA+ people of tumblr; what's your favourite number out of these?

Option 1: 1

Option 2: 47

Option 3: 6.777778

Option 4: 78901534

Option 5: 0.0000001

Option 6: -10

Option 7: 1.5829

Option 8: -7591903938.482903

Option 9: 0

Option 10: 666

[mirroringyourstare]

Math is so much easier when they take the numbers out. I’ll do discrete, category theory, set theory, type theory, graph theory, whatever. I’ll even do quantum if you let me be abstract enough about the Fourier transform.

Please do not make me add and subtract two digit numbers, it is hard and I will cry.

[positivelyapunovexponent-good]

hey guys- want to see my absolutely pathetic attempt at taking math notes in real time in LaTeX using my phone keypad?

(this shit, signal recovery using the discrete Fourier transform, is legitimately SUPER cool. Might post cleaned up versions of some of these notes later. HEAVILY cleaned up versions)

[oburas]

Imageperiodogram[ Jon Arbuckle ] shows the squared magnitude of the discrete Fourier transform (power spectrum) of image.

[tomorrowillbeyou]

omg hiii I'm going to ryan tomorrowillbeyou world! ummm, what are your thoughts on fourier transforms and fourier space?

FOURIER TRANSOFRMSSS i love them due to their usefulness in SIGNAL PROCESSING!!!!!!!!!!!!! but i must confess to loving the DISCRETE COSINE TRANSFORM even more......

[little-phoenix-04]

Day 3/7 of productivity: 15.04.2024

Target:

Ch 6 probability questions

Discrete Fourier series (theory mainly)

+ Discrete Fourier transform (a little portion)

+ Heaps revision and codes practice

I completed all the assigned CodeChef prblms yesterday, so now I can study the theory of DSA all I want. I have a quiz coming up on 19th, a viva for lab on 18th and 2 assignments due on 22nd.

Update: I am dead tired rn. I completed more things than I planned but I feel so tired. I know I'm gonna fall asleep as soon as my head hits the pillow. Guess that means it was a good day:)

[mossfetmuncher]

DISCRETE FOURIER TRANSFORM QUEER

petition to change LGBT to DFTQ (Dykes Faggots Trannies and Queers, naturally)

[govindhtech]

What is the Fast Fourier Transform A Modern Computing Pillar

The Fast Fourier Transform

Honouring the FFT and Computing's Future: Representation to Revolution

For the first Fast Fourier Transform deployment, IBM received an IEEE Milestone award on June 11, 2025. IBM researchers created this method in 1965, transforming computers.

Since it supports JPEG and MPEG standards and reconstructs MRI and CT scan images, the FFT has a wide impact. Also needed for scientific computing (spectrum approaches for solving PDEs), music and video compression (MP3, JPEG), and telecommunications (4G/5G, WiFi). Richard Hamming called the Cooley-Tukey FFT “the most important numerical algorithm of the lifetime”.

FFT's Key Innovation

Fundamental FFT Innovation James Cooley and John Tukey introduced the FFT in 1965 as a “better way to represent information” rather than a new scientific discovery. The Fourier transform can split a time-domain signal like a wave into smaller waves with various frequencies. Before the 1960s, Fourier transform computing was too slow for real-time applications. Cooley and Tukey's technique accelerated real-time signal processing by reducing the computing cost of the Discrete Fourier Transform (DFT) from O(N^2) to O(NlogN). The revelation was that changing a computing problem's mathematical representation can change it.

Quantum Computing Lessons:

The FFT's ongoing development aids quantum algorithm development. When building new  quantum algorithms, “choosing the right representation can make the impossible possible” is crucial.

Quantum Computing: Emerging Idea Beyond improving existing methods, quantum computing changes how information is represented and abstracted. Classical computing employs bits with deterministic binary values (0s and 1s) and Boolean operations, while quantum computing uses qubits. In complex vector spaces, qubits store information as probability amplitudes (α|0⟩ + β|1⟩), where α and β are complex numbers. Quantum computing uses unitary evolution of qubit states through matrix operations instead of classical logic, generating probabilistic results.

This new computational paradigm enables Grover's method, which quadraticly speeds up unstructured search, and Shor's algorithm, which uses the Quantum Fourier Transform to exponentially speed integer factorisation. Additionally, quantum simulation can mimic quantum systems that conventional machines cannot handle.

Future Quantum-Classical Synergy. The most innovative computer future may be a mix of quantum and classical. Traditional computers are fast at control logic, data storage, and predictable computations. However, quantum systems thrive in mimicking quantum phenomena, high-dimensional linear algebra, probabilistic sampling, and landscape optimisation, where classical information representation fails.

Together, these paradigms can solve problems neither system can. VQE and QAOA are two novel hybrid classical-quantum algorithms in development. Quantum advantage, where a quantum-classical combination outperforms classical computation, is nearing, and SQD and SKQD are being developed. Supply chain optimisation, material science, finance, and drug development may use these methodologies.

Quantum technologies are expected to boost traditional computing as “coprocessors with radically different capabilities” like GPUs on CPUs. As the computing bottleneck shifts from hardware limits to algorithmic innovation, new abstractions, representations, and algorithms are needed to balance workloads among complementary architectures. The current age may be the start of a more significant algorithmic era than the FFT.

Anticipating The FFT's anniversary reminds us that innovations often come from better questions, smarter representations, or new viewpoints, not more authority. Fusion of classical and quantum domains will release new processing capability, requiring daring abstractions, inventive representations, and innovative algorithms.

[codingprolab]

A2: Sinusoids and DFT

The second programming assignment is for you to get a better understanding of some basic concepts in audio signal processing related with the Discrete Fourier Transform (DFT). You will write snippets of code to generate sinusoids, to implement the DFT and to implement the inverse DFT. There are five parts in this assignment. 1) Generate a sinusoid, 2) Generate a complex sinusoid, 3) Implement the…

[renatoferreiradasilva]

Mim Continua Ter Problems

To address the challenges outlined in the spectral approach to the Riemann Hypothesis (RH), the following research problems can be developed. These questions aim to refine the methodology, extend computational capabilities, deepen theoretical connections, and validate results rigorously:

1. Refinement of the Potential Function

Problem 1: Can physics-informed neural networks (PINNs) or other machine learning frameworks discover a potential ( V(x) ) that optimally aligns the operator’s eigenvalues with zeta zeros, while respecting physical constraints (e.g., self-adjointness, boundary conditions)?

Sub-problems:

How does the choice of neural network architecture (e.g., Fourier neural operators) affect the accuracy of the learned potential?

Can symbolic regression techniques identify an analytic form for ( V(x) ) from numerically optimized solutions?

Problem 2: Are there functional constraints (e.g., integrability, smoothness) that guarantee uniqueness of the potential ( V(x) ) for a given eigenvalue spectrum?

Investigate whether imposing symmetries (e.g., PT-symmetry) or asymptotic conditions resolves non-uniqueness.

2. Extending Spectral Computations to Higher Zeros

Problem 3: How can high-performance computing (e.g., GPU-accelerated Lanczos algorithms, distributed eigensolvers) be leveraged to compute eigenvalues of ( H ) corresponding to the ( 10^3 )-th to ( 10^6 )-th zeta zeros?

Sub-problems:

Develop adaptive discretization schemes to maintain numerical stability for large ( \text{Im}(s) ).

Optimize sparse matrix storage/operations for Schrödinger-type operators.

Problem 4: Does the spectral gap or density of ( H ) exhibit phase transitions at critical scales, and do these relate to number-theoretic properties (e.g., prime gaps)?

3. Advanced Statistical Validation

Problem 5: Can spectral form factors or ( n )-level correlation functions provide stronger evidence for the GUE hypothesis than spacing distributions alone?

Compare long-range eigenvalue correlations of ( H ) with those of random matrices and zeta zeros.

Problem 6: Do the eigenvectors of ( H ) encode arithmetic information (e.g., correlations with prime-counting functions)?

Analyze eigenvector localization/delocalization properties and their relationship to zeros.

4. Theoretical Connections

Problem 7: Can the operator ( H ) be interpreted as a quantization of a classical dynamical system (e.g., geodesic flow on a manifold), and does this link explain the spectral-zeta zero correspondence?

Explore connections to quantum chaos and the Gutzwiller trace formula.

Problem 8: Does the Riemann-von Mangoldt formula for the number of zeta zeros up to height ( T ) emerge naturally from the spectral asymptotics of ( H )?

5. Generalization to Other L-functions

Problem 9: Can the framework be adapted to study zeros of Dirichlet L-functions or automorphic L-functions by modifying ( V(x) )?

Investigate whether symmetry properties of ( H ) (e.g., modular invariance) align with functional equations of L-functions.

Problem 10: Do families of L-functions correspond to universal classes of operators (e.g., varying potential parameters), and does this align with Katz-Sarnak universality?

6. Computational and Algorithmic Improvements

Problem 11: Can hybrid quantum-classical algorithms (e.g., variational quantum eigensolvers) efficiently diagonalize ( H ) for large-scale eigenvalue problems?

Assess the feasibility of quantum advantage in spectral RH research.

Problem 12: Do multiscale basis functions (e.g., wavelets) improve the accuracy of ( V(x) ) representations compared to fixed polynomial bases?

7. Toward a Formal Proof

Problem 13: If eigenvalues of ( H ) converge to zeta zeros as discretization is refined, can this imply a rigorous spectral realization of RH?

Establish error bounds for eigenvalue approximations under discretization and optimization.

Problem 14: Can spectral deformation techniques (e.g., inverse scattering transform) reconstruct ( V(x) ) directly from the zeta zero sequence?

8. Interdisciplinary Applications

Problem 15: Does the operator ( H ) have physical interpretations (e.g., as a Hamiltonian in condensed matter systems) that could provide experimental validation?

Explore connections to quantum wires, Anderson localization, or graphene.

Summary

These problems span computational, theoretical, and statistical domains. Progress on any front would advance the spectral approach to RH by:

Strengthening the numerical evidence for a self-adjoint operator realization.

Bridging gaps between analytic number theory and quantum mechanics.

Developing tools applicable to broader problems in mathematical physics and L-function theory.

Collaboration across disciplines (e.g., machine learning, quantum computing, analytic number theory) will be critical to addressing these challenges.

[programmingandengineering]

EECS205002 Assignment 3: Sounds Good

In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. Because of their wave property, most people analyze and manipulate the sound signals using Fourier Transform (FT). In this assignment, we study a similar but simpler method, called Discrete Cosine Transform (DCT), which also transforms sound signals from the time…

[tom2tec]

FMIT ~ Free Music Instrument Tuner

FMIT is a graphical utility for tuning your musical instruments, with error and volume history and advanced features. Features: Estimation of the fundamental frequency (f0) of an audio signal, in real-time.(the f0, not the perceived pitch) Harmonics’ amplitude Waveform’s period Discrete Fourier Transform (DFT) Microtonal tuning (supports scala file format) Statistics All views are…

[codeshive]

DD2423 Lab 1: Filtering operations solved

The goal of this lab is to help you to • get an understanding of frequency analysis in image data by using the Fourier transform, expressed spatially or in the Fourier domain, • become familiar with the two-dimensional Fourier transform, by studying and testing its properties in practice, • learn the relation between the continuous and discrete Fourier transform, and design a smoothing filter for…

[hohohongsi]

i tend not to be too stressed about something that i can't deal with

i struggled today thinking about where i wanna see myself at

whether if i wanna study further at college

or

rather i should start working and actively participate in the world

it is nice to have socratic questions with everything i do

but doubting does not really help me proceed

so i should answer myself asap!

i know rushing is not a good answer

until now..

these doubts were created by

discrete time fourier transform...

i doubt that i can do well in my masters...

fighto..

[myprogrammingsolver]

Assignment 2: Fast Fourier Transform and Applications

Introduction In this assignment you will implement two versions of the Discrete Fourier Transform (DFT). One of them will be a brute force approach that follows directly from the formula. The second one will be an implementation of the Fast Fourier Transform (FFT) which follows a divide-and-conquer approach to algorithm design. In particular, the FFT we will focus on is the Cooley-Tukey FFT. At…