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Design Engineering Tip: Don’t move objects at a constant shape. As speed increases, stretch them in the direction of travel. The faster they move, the more they deform. Combined with easing and subtle motion blur, even simple animations can feel significantly more dynamic. const x = useMotionValue(0) const v... show more
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![[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.](https://image.24vids.com/tw-1805694219031662799/ext_tw_video_thumb/1805693963464327168/pu/img/kOcpELm1j7wfqJlR.jpg)
![Two cinematic prompts for Seedance 2.0 right here 👇 [STYLE + CAMERA + ATMOSPHERE] Ultra-photorealistic cinematic 15-second action sequence from a grounded action thriller film. Nighttime at an abandoned industrial train yard. Old trains, rusted tracks, broken platforms and warehouses. Strong moonlight and industrial lighting. The visual language is gritty and realistic with hyper-realistic practical destruction. Camera is handheld-dynamic with urgent moves. [IMAGE REFERENCES] No reference images provided. Generate a fully consistent lead character: a male fugitive in his mid-30s, short hair, intense expression. He wears a dark jacket, jeans and boots. Maintain exact same appearance and clothing throughout the clip. [TIMELINE SECOND BY SECOND] 0-3s: [Dynamic handheld tracking shot from side-rear] The man runs across the train yard as a freight train derails violently after hitting a collapsed section of track. Massive train cars tip over and crash into the ground, sending metal, wood and cargo flying. The camera tracks him urgently as the destruction spreads. 3-7s: [Slow continuous orbiting camera moving around the frozen destruction] At the peak of the derailment, everything freezes in perfect realistic physics. Huge train cars, twisted metal, wooden debris and clouds of dust hang motionless in the air with accurate weight and trajectories. The camera performs a smooth, continuous orbit through the frozen chaos, moving between large suspended train cars and passing close to sharp metal fragments while the man remains visible. 7-15s: [Dynamic tracking shot as time resumes] At the 7-second mark time snaps back to forward motion with a slight realistic temporal residue. Some outer debris shows a very subtle reverse lag before continuing. The man has used the frozen moment to dive behind a concrete barrier. When motion resumes, several large pieces of metal miss him. He stays low and runs toward an open warehouse as debris finally crashes down. The camera tracks with him through the dust. [STYLE & QUALITY BOOSTERS] Photorealistic 8K, hyper-realistic rigid body destruction with correct material properties (metal, wood, train parts), perfect frozen mid-air physics, seamless transition from freeze to resume, natural dust interaction, heavy cinematic motion blur on fast debris, stable character performance, movie-level practical destruction VFX quality.](https://image.24vids.com/tw-2074131502733152605/amplify_video_thumb/2074096405480620032/img/zUYHavg8GK518E2E.jpg)
![Activate Berserk mode on Seedance 2.0 with this prompt 🤘 [STYLE + CAMERA + ATMOSPHERE] Brutal 2026 Viking action cinema with hyper-realistic grounded fight choreography and heavy physics. Single continuous dynamic tracking shot following a massive fat berserker as he rampages through a Viking village, systematically taking down villagers with brutal force and his axe until he reaches his longhouse, discovers his wife's betrayal and explodes in full berserk rage. Anamorphic lens, subtle film grain, cold northern daylight, intense motion blur, stabilized professional handheld camera. [IMAGE REFERENCES] Generate consistent character: late-30s massive fat but powerfully built Viking berserker, long wild hair and thick beard, intense wild eyes, covered in Norse tattoos and scars, heavy belly but thick muscular arms. Wears dirty fur cloak over bare scarred chest, leather pants, heavy boots and carries a large blood-stained axe. Exact same appearance, body type, clothing and weapon throughout. [TIMELINE SECOND BY SECOND] 0-4s: [Continuous Dynamic Tracking Shot] Camera tracks backward through the muddy village center as the berserker advances with growing rage. Several villagers try to stop him; he swings his heavy axe in brutal arcs, cleaving one and throwing another into a longhouse wall with raw power. Heavy impacts, bodies reacting with realistic weight. Camera tight on his massive frame. 4-8s: [Continuous Dynamic Tracking Shot with subtle lateral orbit] Pushes deeper between longhouses. Uses his massive size to bull-rush and crush one against a post, then delivers devastating axe strikes. Fabric, fur and blood move realistically. Growing intensity. Camera orbits slightly while tracking his advance. 8-12s: [Continuous Dynamic Tracking Shot - Building Rage] Approaches his longhouse area, smashing through a door with his body and axe, throws a villager into a fire pit. His movements become more unhinged. Camera tracks forward as destruction builds. 12-15s: [Continuous Dynamic Tracking Shot - Berserk Explosion] Reaches his longhouse and sees his wife with another man. Full berserk rage erupts. He charges with animalistic fury, axe swinging in devastating blows. Camera stays locked on the explosion of violence as he advances, then slowly pulls back revealing the trail of carnage through the village. Seamless continuous take, peak visceral energy. [STYLE & QUALITY BOOSTERS] Photorealistic 8K, ultra-detailed skin, fabric, fur, wood and blood textures with realistic sweat, dirt and impact, perfect coherent physics on all strikes, throws, body impacts and environmental destruction, flawless motion blur, stable character consistency, no artifacts, professional blockbuster film look, high dynamic range, intense visceral action, cold northern color palette.](https://image.24vids.com/tw-2068834853894664445/amplify_video_thumb/2068833978996387840/img/1cQxAVG9tkD-iZqU.jpg)
![Seedance 2.0 prompts are below 👇 [STYLE + CAMERA + ATMOSPHERE] Real live Tour de France motorcycle camera broadcast, Stage 14 mountain climb. Continuous single-take tracking shot filmed from a race moto running parallel then slightly behind the rider. Harsh high-altitude daylight, sweat, dust, realistic road vibration, authentic crowd energy, subtle broadcast grain and compression. [IMAGE REFERENCES] Use the provided still as the strict visual reference for TechHalla’s exact face, bald head, full dark beard, yellow RaboBank leader jersey, body proportions, bike and mountain-road environment. Do not alter his appearance. [TIMELINE SECOND BY SECOND] 0-4s: [Continuous side tracking] Camera locked parallel to TechHalla as he climbs out of the saddle with extreme effort, heavy pedaling, face clenched, sweat flying. Dense fans on both sides waving Spanish and Basque flags, shouting, running along the barriers. 4-7s: [Tracking continues, slight pull-back] A spontaneous elderly woman wearing a floral bata bursts from the left roadside and sprints at full speed, catching up until she is running right next to the still-pedaling TechHalla. 7-12s: [Tight parallel tracking] The elderly woman suddenly jumps with both hands and latches onto TechHalla’s back and jersey from behind, clinging tightly while the cyclist keeps pedaling hard up the steep gradient. TechHalla’s body reacts to the sudden extra weight — deeper lean, heavier breathing, face more strained — but he never stops pedaling. The elderly woman holds on as they move forward together. Crowd goes completely wild. 12-15s: [Camera shifts smoothly behind] TechHalla continues climbing with the elderly woman still hanging on his back, then powers harder on the pedals, pulling slightly ahead as the camera follows from the rear along the open mountain road. [STYLE & QUALITY BOOSTERS] Photorealistic 8K broadcast footage, perfect motion blur on wheels and running legs, realistic physics of the jump, body impact and weight transfer, stable character face and jersey, natural sweat and fabric stretch under the grab, coherent crowd motion, cinematic motorcycle-camera shake, no artifacts, movie-level temporal consistency. Enjoy! 😎](https://image.24vids.com/tw-2079852844522324442/amplify_video_thumb/2079852710602383360/img/wQSfcAZM1zc_L1gN.jpg)
