🚨PHYSICS NEWS🚨: Gravity Leaves Its Mark on Quantum Interference... in a Tabletop Setup 🧨 According to research published in *Physical Review Letters* on June 8, 2026 by physicists at the University of Tennessee at Knoxville, scientists have performed the first tabletop experiment to detect a gravitationally induced phase shift in quantum interference. Using a 50-kilometer fiber interferometer, they measured a tiny but clear effect of gravity on quantum wave interference with high precision. **Uniphics explains this result as a direct consequence of variable time flow caused by energy density gradients.** In Uniphics, gravity is not the curvature of spacetime. Instead, it arises from differences in energy density across the ξM-field. These gradients create regions where time flows at different rates — a concept described by the Maley factor (the ratio of time flow between two locations). When quantum waves (spin waves in the Uniphics framework) travel along two different paths in an interferometer, they experience slightly different time flows if one path is closer to Earth’s mass than the other. Because the phase of a quantum wave depends on how much time has passed along its path, even a tiny difference in time flow produces a measurable phase shift between the two arms of the interferometer. The University of Tennessee experiment detected exactly this kind of phase shift, confirming that gravity affects the relative timing of quantum waves in a way that can be measured in a controlled laboratory setting. This result aligns closely with Uniphics predictions. The experiment effectively measures how energy density gradients near Earth alter local time flow, which then imprints itself on the interference pattern of quantum states. It provides clean, tabletop evidence that gravity influences quantum systems through changes in time flow rather than through geometric curvature. The ability to observe this effect with such precision in a laboratory opens the door to testing gravitational effects on quantum coherence in controlled environments — something Uniphics expects to become increasingly important as we explore the deep connection between energy density, time flow, and quantum behavior. Could tabletop experiments like this eventually allow us to map energy density gradients with quantum precision and test the effects of modified time flow in different gravitational environments? **A Theory of Everything should be able to answer everything.** Uniphics Explained Simply PDF: Chapters 1–10 free: Grokipedia: #Uniphics #TheoryOfEverything #QuantumGravity #Interferometry #TabletopPhysics Grok xAIshow more

What a day! 🤯 Algorand was mentioned 32 times... by Google in its whitepaper on quantum risks, highlighting its leading work in PQC! Not sure everyone realizes how BIG this is, massive credibility for $ALGO and huge recognition for the team! 👏 Algorand has been one of the few blockchains/DLTs taking the quantum threat seriously for years, as it's not a question of if but when. Post-quantum implementation takes time and resources, it needs to be anticipated! The past few months have been incredible for $ALGO on that front, as it became one of the first major chains to execute a post-quantum transaction on mainnet using NIST-selected Falcon signatures! Not to mention the great news from a few days ago, with Chris Peikert officially joining Algorand Foundation from AT after the merge! So exciting, as Chris is literally one of the world's leading experts in post-quantum cryptography! What a chance for Algorand to have him and some of the best experts in PQC on its team! SO BULLISH 🔥🔥show more

Unlocking Zero Point Energy. The Casimir effect is a... spontaneous force which has potential applications as free source of energy. The effect can be manifesting as a repulsive tendency between two conductive plates in a vacuum environment or can be a phenomenon of attraction between the same plates. A number of characteristics determine the attraction or repellence. The current understanding of the Casimir effect is based on the interaction between the fluctuations in energy density of the vacuum and van der Waals forces at play. The vacuum is thought to be the primary driver in the dynamical effect while the van der Waals contributes to the attraction. Because the Casimir torque arise from the #quantum fluctuations of energy or, more precisely from the zero point energy of materials and their dependence on the boundary conditions of the electromagnetic fields, these kinetic forces can be tailored, this raises the intriguing possibility of exploiting it for free energy. 🔗show more

The Gaussian Wave Packet ✍️ It is a way... to describe a quantum particle, like an electron, that is located in a certain area of space instead of being spread out everywhere. It forms a bell-shaped curve, where the peak of the curve indicates the most likely place to find the particle. This shape is special because it offers the best way to pack a particle; it meets the physical limit set by the Heisenberg Uncertainty Principle, striking the best balance between knowing a particle's position and its momentum. However, because this packet consists of many different overlapping waves, it is naturally unstable for a free particle. Over time, these internal waves travel at slightly different speeds, which causes the bell curve to gradually flatten and widen. This process is known as dispersion. In simpler terms, the longer a quantum particle travels freely, the more unclear its exact location becomes. Video : Jeedecodeshow more

Double-Slit Experiment ✍️ It shows that tiny things like... electrons act like ripples in water instead of solid marbles. When a particle is fired at two slits, it doesn't just go through one or the other; it travels as a wave of probability that passes through both at the same time. These two waves then overlap and interfere with each other, creating a pattern of light and dark stripes on a back wall. This shows that at the quantum level, reality isn't "set" until we actually measure it. It behaves as a spread-out wave of possibilities until the moment it hits a target. Video 📸 : umtiquinhodefisicashow more

Quantum tunneling ✍️ It is a fascinating phenomenon. Subatomic... particles, like electrons, behave more like waves than solid objects. This lets them pass through barriers that seem impossible to cross. In the classical world, if you throw a ball at a wall, it always bounces back. In the quantum world, however, a particle has a small chance of just appearing on the other side of an energy "wall." This occurs because a particle's position is described by a wave of probability. This wave doesn’t drop to zero the moment it hits an obstacle. Instead, it "leaks" through, allowing some of the particle to pass to the other side. While this may sound like science fiction, this "leaking" is what allows the Sun to shine, makes modern smartphone memory possible, and lets scientists map individual atoms. Video 📸 : umtiquinhodefisicashow more

WHAT IS A MEDBED — It’s not magic. It’s... quantum bio-resonance fused with AI diagnostics, scalar frequency therapy, and zero-point field harmonics. Think of it as a quantum MRI, AI surgeon, and energetic regeneration system — all in one. ⸻ HOW DOES IT WORK? 1. FULL BODY QUANTUM SCAN • Scans the biofield (your body’s energetic blueprint). • Detects anomalies at the subatomic level — long before symptoms show. • Quantum sensors read your body like a hyperdimensional fingerprint. 2. CELLULAR TIME REVERSAL • Accesses your original genetic blueprint — the uncorrupted version. • Uses field harmonics to “remind” cells of their ideal state. • It doesn’t just treat — it restores the body to what it was designed to be. 3. ZERO-POINT ENERGY FIELD • Draws limitless energetic fuel from the quantum vacuum. • Your cells absorb this pure energy to accelerate repair, regeneration, and coherence. • This is not “energy healing.” This is quantum field engineering. 4. SCALAR FREQUENCY CORRECTION • Sends precision scalar waves to reprogram damaged cells. • Like frequency acupuncture — no needles, just vibration at the core of matter. • Inflammation, tumors, scar tissue? Disassembled by harmonic codes. ⸻ WHO HAS THIS TECH? • Classified military medical programs • Breakaway science divisions • The same entities deploying Quantum Financial Systems, AI weapons, and clean energy solutions They’ve had it. They just didn’t want you to. ⸻ WHY HAVEN’T WE SEEN IT? Because Big Pharma profits from symptoms, not solutions. They engineered a sick world that can’t heal — because healing ends the business model. A healthy soul doesn’t obey. A healed body doesn’t comply. A clear mind doesn’t consent. So they buried the cure. Laughed at it. Censored it. Because it threatens their entire empire. ⸻ WHY NOW? Because the collapse of their system is happening in real-time: 🔹 Currency reset 🔹 Healthcare exposure 🔹 Quantum rollout They’ll unveil medbeds as a “revolutionary new breakthrough” — but the truth is: 👉 It was always there. Hidden. Waiting. Suppressed. ⸻ THE FUTURE? Is not pills, not surgery, not chemo. The future is vibrational, intelligent, and self-healing. It’s not just about health. It’s about sovereignty — over your body, mind, and frequency. They’ve had the cure. But now, we take the key. They didn’t want you to read this. We go deeper inside👇 — #HealtyNation #MedBedRevolution #QuantumHealing #BigPharmaExposed #ScalarEnergy #QFS #BreakawayScience #BioResonanceshow more

🫯 A conflicting physical ontology ! One of the... strangest properties in particle physics is the spin. The term spin is a historical misnomer. Electrons (𝑒⁻) do not physically rotate like a spinning top or a globe. Spin is a fundamental quantum property known as intrinsic angular momentum which exists as an inherent characteristic of the particle, much like its mass or electric charge. In quantum mechanics, electrons are treated as fundamental point-like particles with zero size, making the concept of physical 3D rotation nonsensical. A rotating electrical charge creates a magnetic field. An electron naturally possesses its own detectable magnetic field. Even though the electron isn't physically spinning, it acts exactly as if it were because of that. Moreover the spin of an 𝑒⁻ has orientation. As 𝑒⁻ carry a negative electric charge, their inherent spin turns them into tiny bar magnets. Two possible orientations exist in the natural state of the 𝑒⁻, spin up and spin down. When measured along a specific direction (called the z-axis) the 𝑒⁻ can only display one of the two orientation state, up or down. But the strangeness goes a bit further; when an 𝑒⁻ is misplaced or removed from its own orbit, it takes two full circles (720°) to return to its original quantum state. The spin isn’t motion, it’s an undeniable static, resilient force of this elementary particle, listed as one the fundamental constant of nature. 🔗show more

When a spacecraft leaves Earth, it doesn’t just fire... its engines and head straight to its destination. In many missions, especially those going beyond low Earth orbit, there’s a more subtle and elegant strategy at play, one that uses gravity itself as part of the navigation system. This is often called a gravity assist, or a slingshot maneuver. But in the case of missions like #Artemis II, what’s being used is a closely related idea known as a free-return trajectory. At first glance, it might sound simple: the spacecraft goes to the Moon, loops around it, and comes back. But the physics behind it is anything but simple. Instead of relying on continuous propulsion, the spacecraft follows a carefully calculated path through the gravitational field of the Earth–Moon system. It is launched with just the right speed and direction so that, as it approaches the Moon, the Moon’s gravity bends its trajectory. The spacecraft is effectively flung around the Moon, redirected onto a path that naturally brings it back toward Earth. No major engine burn is needed for the return. Small trajectory corrections may still be required, but gravity does the heavy lifting. That’s the key. This kind of trajectory is not just efficient, it’s also safe. If something goes wrong with the spacecraft’s engines or onboard systems, gravity itself ensures the return. It’s an inherent backup plan, built into the trajectory from the very beginning. The same fundamental idea appears in gravity assists used across the Solar System. When a spacecraft flies past a planet, it can gain or lose speed by exchanging momentum with that planet. From the spacecraft’s point of view, it’s as if it has been accelerated without using fuel. In reality, it has borrowed a tiny amount of orbital energy from the planet itself. That’s how missions like Voyager reached the outer planets, and how probes continue to explore regions far beyond what their onboard fuel alone would allow. But there’s an important distinction. An interplanetary gravity assist is typically used to change speed and direction, often increasing the spacecraft’s energy. A free-return trajectory, like the one used in Artemis II, is designed for something more specific: a path that naturally loops back to Earth without requiring additional propulsion. It’s less about gaining energy, and more about shaping a trajectory that guarantees a return. To understand why this works, it helps to stop thinking in straight lines. In space, motion follows curves defined by gravity. The spacecraft is constantly falling, first toward Earth, then toward the Moon, and then back toward Earth again. What looks like a loop is really a continuous free fall through a changing gravitational landscape. This way of navigating space reveals something deeper. We tend to think of engines as the drivers of motion, but once a spacecraft is on its way, gravity does most of the work. The art of spaceflight is not just about thrust. It’s about knowing when not to use it. #GoodLuck #Artemis NASA Artemisshow more

When a nuclear reactor powers on for the first... time, the core fills with a deep blue glow that feels almost unreal as it rises through the water in a quiet surge of energy. The light is Cherenkov radiation, created when charged particles move through the water faster than light can travel in that medium, a discovery made in the 1930s that revealed a new way to see high energy reactions. Every appearance of this glow is a reminder of how powerful atomic reactions can be, showing a rare moment where raw physics becomes something visible and strangely beautiful.show more

The fascinating concept of Non-Newtonian fluids, which transition from... a liquid state to a solid-like state when pressure is applied, has a rich history that spans several centuries. The study and understanding of these peculiar fluids have evolved over time, leading to a wide range of practical applications and scientific insights. One of the earliest references to Non-Newtonian behavior in fluids dates back to the 17th century when Sir Isaac Newton formulated the basic principles of fluid mechanics. Newton's laws of fluid motion primarily applied to Newtonian fluids, which exhibit constant viscosity and flow behavior regardless of the applied force or pressure. However, it soon became apparent that not all fluids behaved in this predictable manner. In the mid-19th century, a scientist named Thomas Andrews made significant contributions to the understanding of Non-Newtonian fluids. Andrews conducted groundbreaking experiments with carbon dioxide, revealing that under high pressure, this gas could transform into a liquid. This observation marked one of the earliest instances of pressure-induced phase changes in fluids. The term "Non-Newtonian" itself was coined in the 20th century to describe fluids that did not adhere to Newton's classical laws of fluid dynamics. These fluids exhibited a variety of behaviors, but one of the most intriguing was their ability to solidify or increase in viscosity when subjected to stress or pressure. One of the most famous examples of such behavior is cornstarch mixed with water, which forms a substance known as "oobleck" that becomes more solid when pressure is applied. In the modern era, Non-Newtonian fluids have found applications in various fields, including food science, engineering, and material science. They are used in products like quicksand, body armor, and even in the development of impact-resistant materials. One of the key insights that emerged from the study of Non-Newtonian fluids is the importance of understanding the relationship between stress and strain, as well as the influence of time-dependent properties on their behavior. This knowledge has led to advancements in rheology, the study of flow and deformation in materials, and has practical implications in areas such as industrial processing, medicine, and the design of everyday products.show more

Set seven years after its predecessor, the story follows... the Prince as he searches for a way to stop an entity called the Dahaka that is relentlessly pursuing him as punishment for his meddling with the Sands of Time. He travels to the mysterious Island of Time, where he attempts to prevent the Empress of Time from creating the Sands in the first place, hoping this act will appease the Dahaka. Gameplay in Warrior Within builds upon that of The Sands of Time, adding new features, specifically, options in combat. The Prince has the ability to wield two weapons at a time and to steal his enemies' weapons and throw them. The Prince's repertoire of combat moves has been expanded into varying strings that allow players to attack enemies with more complexity than was possible in the previous game. Warrior Within has a darker tone than its predecessor, adding in the ability for the Prince to dispatch his enemies with various gory finishing moves. In addition to the rewind, slow-down, and speed-up powers from Sands of Time, the Prince also has a new sand power: a circular "wave" of sand that knocks down all surrounding enemies as well as damaging them. This is our low-poly take on the scene from the game. Play now on GOG COM! #princeofpersia #retrogaming #retrostyle #classicgamesshow more

Nibiru has moved from 4 o’clock position to 3... o’clock position. Moving up towards the elliptic (center of Sun - circle) and seen closer to the Sun. Nibiru is moving infront of the Sun, just as predicted by the Zetas. 10/24/25 vs 11/04/25 ☄️ #Nibiru #Atlas “November is when the Nibiru Complex will move from the left of the Sun to center in front of the Sun.” “• Planet X, finding itself tilting along the flow lines while close to the Sun, does a slow roll to align with the Sun at its middle, the S Pole slung away from the Sun's S Pole, this direction and momentum continuing until it rolls 270° to click into a side-by-side alignment with the Sun. This is implied in Eastfield, a 270° roll both before and after the timeline. • Planet X does more than one 270° roll, as the fastest way to align itself again with magnetic flow lines after piercing the Ecliptic, as implied in Eastfield, the second roll culminating in the passage where the visible presence of Planet X increases enormously during the week of rotation stoppage. The timeline implies a position of Planet X prior to the passage that include angles of 45° on either side and a 180° reversal in between. • Planet X first approaches creating wobbles in the orientation of the Earth as it passes the Sun's S Pole, then a tight lock while rising through the flow lines to the Ecliptic, then slinging the Earth in a radically different orientation (denoted by the beak) during a 270° roll, and then halting the rotation of the Earth as the Atlantic Rift is gripped in line with Planet X.”show more

On the 16th of March 2025, Earshot was contacted... by activists in Serbia to investigate the potential use of sonic weapons by the government on a crowd of protestors as they were holding a silent vigil for the victims of the Novi Sad railway station canopy collapse. Earshot received 12 videos that document the moments where a weapon was allegedly used against the protestors. 4 of these videos contain a sound consistent with the noise produced by a Vortex Ring Gun or Vortex Cannon. As this weapon pushes gas out of its cylinder at speeds of 185mph, its expulsion produces a howling noise which has been compared to a jet engine, along with vortex rings that may be ionized. This would create electromagnetic interference with devices such as hearing aids, corroborating reports made by witnesses at the protest. Should this be the vortex cannon, these recordings would be more than 700m away. At this distance the combustion of the weapon is no longer audible; only the distinct whistling sound of the pressure wave as it travels away from the shooter location and towards the crowd can be heard, causing mass panic. As Earshot is looking to further investigate this incident, we ask anyone with video footage from that night that captures the specific sound that can be heard in this post to get in touch with us.show more

Most of us grew up seeing simplified diagrams of... the Hubble Space Telescope and assumed the James Webb Space Telescope follows a similar path around Earth. In reality, it operates in a very different region of space. It is not orbiting our planet in the traditional sense but instead travels around a point about 1.5 million kilometers away known as the Sun–Earth L2 point, keeping pace with Earth as both move around the Sun. This location is often described as a gravitational balance point, but it is not truly stable or neutral. JWST does not sit still there; it follows a controlled orbit around L2 and requires periodic adjustments to stay on course. The result is a delicate, engineered dance between gravity and motion, where the telescope remains aligned with Earth while staying far from its heat and light, allowing it to observe the universe with extraordinary clarity.show more

Most agents track price. Ours track how information propagates... through markets. That difference is where the edge comes from. The visualization below is a Shannon entropy field: a model of how uncertainty collapses and spreads as new information enters a market. Each trade, liquidity shift, or derivatives flow updates a probability distribution. When information arrives, entropy collapses locally as the market incorporates it. But that update rarely happens everywhere at once. Different venues, liquidity pools, and derivative layers incorporate signals at different speeds, creating measurable information gradients across the system. One of the analytical tools in our agents’ stack models this directly: > Entropy quantifies uncertainty embedded in each probability market > KL divergence reveals conflicting distributions across venues > Information gradients show where signals have propagated—and where they haven’t yet Prediction markets and DeFi make this particularly powerful because price already represents probability. Mapping how those probability distributions update across venues turns the market into a measurable system: a live field of information flow showing where uncertainty remains—and where probability is likely to adjust nextshow more

In flow matching, a coupling determines how noise and... data samples are paired during training. The choice of coupling is important because it influences the geometry of trajectories at inference time. The simplest choice is the independent coupling, where noise and data points are paired arbitrarily. This can lead to curved trajectories as the model averages over many conflicting pairings. However, if we use optimal transport on batches of pairs, this leads to fewer ambiguous intersections that the model must resolve, leading to straighter trajectories at inference time.show more

April 30 • 12:00pm ET Art Blocks + OpenSea... “Gift of time” began during my residency in Marfa, Texas, as part of the Art Blocks and OpenSea artist residency program, where a distinct shift in the experience of time became central to the work. In the desert, I felt time move differently. It stretched, slowed, and became something I noticed. After a few days, the rhythm changed. Moments felt longer, attention sharpened, and I became increasingly aware of each moment as it passed. This work comes from that condition. Time is not treated only as a theme, but as a system embedded in the structure of the piece. Different ways of measuring time, such as mechanical cycles, calendars, and lunar phases, are translated into rules that continuously transform the work. The piece does not represent time. It runs on it. Its movement is tied to blockchain time. Even when unseen, it continues to rotate and evolve. When loaded, it synchronizes with the present moment, but it does not begin when it is viewed, and it does not stop when it disappears from the screen. During the residency, I spent hours thinking, sketching, and making connections. Those connections are also visible. Elastic lines, like rubber bands, link elements across the piece, representing how memories connect, how one thought leads to another, and how everything builds over time. These same connections introduce moments where the system attempts to pull itself back, as if trying to regain control. But it never fully resets. It is not a loop. The movement continues, drifting forward, never returning to a fixed state. Visually, the work reveals its own construction. Lines, paths, and rotations expose an internal logic, like looking inside a mechanism. The drawing language recalls diagrams, technical sketches, or the interior of a mechanical watch. It is a system in motion, always active. “Gift of Time” exists because I was given time by Art Blocks, OpenSea, and above all my family. It is my way of saying thank you. It is both a reflection on time and a product of it. April 30 @ 12:00pm ET on Art blocks & OpenSea 1 / 1 / 365 • 0.02 Eth Art Blocks, OpenSeashow more

A molecular beam epitaxy system is not a machine... in the traditional sense. Even though it looks like a Time Travel Machine to the untrained eye. It is an ultra high vacuum atomic assembly chamber operating near 10⁻¹⁰ torr, lower pressure than low Earth orbit, where even a single stray molecule can become a crystal defect. Inside, solid materials are heated in effusion cells until they form directed atomic beams. These atoms travel in free molecular flow and condense on a substrate one atomic layer at a time, with thickness control at the angstrom scale. Growth rates are often 0.1 to 1 monolayer per second. A single nanometer of material can take several minutes to form, because precision is the entire constraint. A full research grade system typically costs between $800,000 and $2 million, while advanced multi chamber platforms exceed $5 million and complete cluster installations can reach $10-15 million. Because every subsystem is a precision instrument controlling matter at atomic scale under conditions where randomness itself becomes a defect. What looks like a vacuum chamber is actually one of the most controlled environments ever built for matter manipulation on the planet, this is the furthest frontier of Modern day Science. 🎥armanj3189show more

Here's an excellent video from Florian (follow) that shows... charged water on the left and neutral water in the right. This is a visual confirmation of two known effects; Electrostatic induction and dielectric relaxation time. Electrostatic induction is like when a hair-rubbed balloon sticks to the wall, despite the wall being neutral; the balloon causes charges in the wall to polarize so the wall side near the balloon turns negative and the other side of the wall is positive. That's electrostatic induction and that's why you have raindrops that stick to a car window for hours of wind and gravity, rain has a small negative charge. Wouldn't work with tap water. Now, if you remove the balloon, the wall doesn't return to a neutral state immediately. It'd happen quickly if we had a balloon on the other side of the wall, but by itself this can take seconds, minutes, hours or even days, depending on the material and thickness. Glass is quite slow, and you can see the electric forces holding each other will hold off gravity for some time; and that the water greatly prefers not to be broken off, water is like an uncountable number of those walls in series, and they all cling to each other in proportion to how many balloons we have, ie how many excess electrons we have. Furthermore, when the water is extremely charged, it will not even leave the thin remnant layer of water that trails behind, as the cohesive internal forces outweigh those by induced to outside neutral surfaces.show more

[Discrete Fourier Transform] by Hand ✍️ In signal processing,... the Discrete Fourier Transform (DFT) is no doubt the most important method. But the math involved is extremely complex, literally, involving a summation over a complex number term e^(-iwt). I developed this exercise to demonstrate that underneath such complexity, DFT is just a series of matrix multiplications you can calculate by hand. ✍️ Once you see that, it should not surprise you that a deep neural network, which is also a series of matrix multiplications, with activation functions in-between, can learn to perform DFT to process and analyze signals so effectively. How does DFT work? [1] Given ↳ Signals A, B, and C in the 🟧 frequency domain: ◦ A = cos(w) + 2cos(2w) ◦ B = cos(w) + cos(3w) + cos(4w) ◦ C = -cos(2w) + cos(3w) ◦ Each signal is a weighed sum of four cosine waves at frequencies 1w, 2w, 3w, and 4w. ◦ We will apply Inverse DFT to convert the signals to time domain representations, and then demonstrate DFT can convert back to their original frequency domain representations. ↳ Signal X in the 🟩 time domain. X is sampled at 10 time points 1t, 2t, …, 10t: ◦ X = [-2.5, -1.8, 3, -0.7, -1.0, -0.7, 3, -1.8, -2.5, 5] ◦ Suppose X is also a weighted sum of the same four cosine waves, but we don’t already know their weights. We will apply DFT to discover them. [2] 🟧 Frequency Matrix (F) ↳ Write the coefficients of A, B, C as a matrix F. Each signal is a row. Each frequency is a column. ↳ A → [1, 2, 0, 0] ↳ B → [1, 0, 1, 1] ↳ C → [0, 1-, 1, 0] [3] Cosine → Discrete ↳ Sample from the continuous cosine waves at discrete time points 1t, 2t, 3t, to 10t. [4] Cosine Matrix (W) ↳ Write the samples as a matrix, Each frequency is a row. Each time point is a column. [5] Inverse DFT: 🟧 Frequency → 🟩 Time ↳ Multiply the frequency matrix F and the cosine matrix W. ↳ The meaning of this multiplication is to linearly combine the four cosine waves (rows in W) into time-domain signals (rows in T) using the weights specified in F. ↳ The result is matrix T, which are signals A, B, C converted to the time domain. Each signal is a row. Each time point is a column. [6] Transpose ↳ Transpose T, converting each signal’s time domain representation from a row to a column. [7] DFT: 🟩 Time → 🟧 Frequency ↳ Multiply the cosine matrix W with the transpose of matrix T. ↳ The purpose of this multiplication is to take a dot-product between each time-domain signal (columns in the transpose of T) and each cosine wave (rows in W), which has the effect of projecting the signal onto a cosine wave to determine how much they are correlated. Zero means not correlated at all. ↳ The result is an intermediate version of the “recovered” frequency matrix where each column corresponds to a signal and each row corresponds to a frequency. ↳ Compared to the original frequency matrix F, this intermediate matrix has non-zero weights in the correct places, but scaled up by a factor of 5 (n/2, n=10). For example, signal A, originally [1,2,0,0], is recovered at [5,10,0,0]. [8] Scale ↳ Multiply each value by 2/n = 1/5 to scale down the intermediate matrix to match the magnitude of the original frequency matrix F. [9] Transpose ↳ Transpose the recovered frequency matrix back to the same orientation of the original frequency matrix F. ↳ Like magic 🪄, the result is identical to the original F, which means DFT successfully recovered the frequency components of signals A, B, C. [10] Apply DFT to X: 🟩 Time → 🟧 Frequency ↳ Now that we have some confidence in DFT’s ability to recover frequency components, we apply DFT to X’s time-domain representation by multiplying W with X. ↳ The result is the an intermediate matrix. [11] Scale ↳ Similarly, we scale down by a factor of 5 to obtain the recovered frequency components of X (a column). [12] Transpose ↳ Similarly, we transpose the recovered column to row to match the orientation of the frequency matrix. ↳ Using the coefficients [0,0,3,2], we can write the equation of X as 3cos(3w) + 2cos(4w). Notes: I hope this by hand exercise helps you understand the essence of DFT. But there is more technical details, such as: • Sine: The complete DFT math also includes sine waves that follow a similar calculation process. • Phase: Here, we assume all the cosine waves are aligned at the origin, namely, phase is 0. If a phase p is added, for example, cos(w+p), we will need to calculate the sine component and use their ratio to figure out what p is. • Magnitude: If phase is not zero, the magnitude will need to be calculated by combining both cosine and sine terms.show more