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What if I told you any shape can be broken down into circles? This visualization demonstrates the stunning power of the Fourier Transform. By using a series of rotating vectors—known as epicycles—we can reconstruct complex, continuous paths from simple circular motions. Each circle represents a specific frequency, phase, and...

26,160 просмотров • 2 месяцев назад •via X (Twitter)

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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.

Tom Yeh

116,622 просмотров • 2 лет назад

Meet the Xenobots, they are not machine, they are PROGRAMMABLE biological robots that can REPRODUCE. — The concept of xenobots has been a fascinating topic in the realm of developmental biology, and the latest findings have left me utterly astonished. These microscopic entities have been observed to not only self-assemble into complex forms but also exhibit a level of autonomy and adaptability that challenges our conventional understanding of multicellular organization. The researchers, led by Michael Levin, have successfully demonstrated that embryonic frog cells can spontaneously form into xenobots, which are capable of movement, self-organization, and even communication with one another. These xenobots are not merely passive aggregates of cells but rather dynamic systems that can adapt to their environment and respond to stimuli and be programmed to a degree. What's even more remarkable is that these xenobots have been found to be capable of reproduction. As Eva Jablonka, an evolutionary biologist, noted, "a xenobot may be induced to fragment and form two small ones," which raises the possibility of these entities undergoing a form of evolution. This has significant implications for our understanding of the origins of multicellularity and the evolution of complex life forms. The ability of xenobots to self-organize and adapt to their environment is thought to be mediated by a complex interplay of genetic and environmental factors. Levin suggests that the genome provides a set of rules and tendencies that allow cells to collectively "compute" their own solutions to growth and form, rather than being rigidly determined by a predetermined plan. The potential applications of this research are vast and varied, ranging from regenerative medicine to the development of novel biomaterials to a dystopian nightmare. However, what's most exciting is the prospect of gaining a deeper understanding of the fundamental principles that govern the emergence of complex life forms. As Levin notes, "we need a science of where larger-scale goals come from," and the study of xenobots may provide a key insight into this question. The discovery of xenobots and their remarkable properties has opened up a new frontier in the study of developmental biology and has significant implications for our understanding of the evolution of complex life forms.

Brian Roemmele

50,780 просмотров • 1 год назад

The Circle 🐜 Nobody told the ant about the circle. That, as far as anyone can tell, is the whole problem. A researcher named Kostowski – this was in the early 1970s, at a laboratory in Warsaw that smelled permanently of formaldehyde and institutional coffee – discovered quite by accident that if you draw a continuous line around an ant using a felt-tip pen, the ant will not cross it. It will walk right up to the line, pause with what appears to be genuine philosophical unease, and turn back. It will do this indefinitely. For hours. Sometimes for days. The ant is not stupid. Let’s be clear about that. The creature you are looking at in this photograph – this tiny, improbable machine of chitin and chemical signals, this six-legged marvel that can carry fifty times its own body weight and navigate by polarized light – has a brain roughly the size of a pinhead, and yet that brain contains approximately 250,000 neurons dedicated entirely to making sense of the world. It has survived as a species for 130 million years. It watched the dinosaurs arrive, flourish, and disappear, and then went back to work. And yet here it is. Trapped by a drawing. The reason is chemistry, not cognition. Ants navigate by pheromones – volatile chemical compounds that their legs read like a blind man reads braille. When they encounter the solvent in a felt-tip pen, something in their nervous system fires an alarm. The signal says: boundary. The signal says: edge of the known world. And the ant, loyal to its chemistry in the way that all of us are loyal to ours, obeys. This is the part that stays with you if you think about it too long. The ant’s prison has no walls. No bars. No lock. It is made entirely of information – a chemical whisper laid down by a felt-tip marker – and the ant cannot see past it, because it has no framework for doing so. The circle is not a circle to the ant. The circle is simply: where the world ends. I find myself thinking about this more than is probably healthy. We are, most of us, walking around inside our own circles. They were drawn for us gradually, by parents and teachers and early disappointments, by the limits of what we saw done and the boundaries of what we were told was possible. We bump up against them occasionally – in those moments when a job offer from another city seems too frightening, or a new idea feels somehow presumptuous – and we turn back. Not because anything is stopping us. Because the world, as far as we can tell, simply ends there. The ant in the photograph is walking the inner edge of its circle with a kind of purposeful calm that is almost admirable. It has not given up. It is still looking. It is still moving. It simply cannot conceive of a direction that leads out. Kostowski, for what it’s worth, eventually just picked the ant up and moved it. Sometimes that’s what it takes. Gandalv / Gandalv

Gandalv

28,294 просмотров • 3 месяцев назад

Okay everybody. I want you all to watch this video closely, so that I can show you what utter made up bullshit propaganda this stupid criminal kid keeps posting. It’s all made up garbage and it’s so embarrassingly fake that it’s hilarious. I’ve attached the video here again for ease of reference, plus will timestamp it with screenshots so you can see it for yourselves. This post alleges that this is a video showing IDF snipers shooting a a boy. 1. Firstly, the videos are split, different coloring, rendering, and motions. These two videos are completely unrelated and have been put one on top of the other as simple propaganda. 2. Look at both sides of the staircase. One either side are clothing stores with mannequins. Those are not even real people on the sides. It’s a small group of people coming down then up the stairs but a few mannequins being knocked over from the left of the screen to the center right. You can see the mannequins falling over like cut timber (still image pointing them out attached). One person falls down but isn’t shot as he gets up and walks away unharmed. 3. At 00:09 in the video, the man bottom right takes a few steps up and leans down… to pick up a freaking mannequin. I’ve taken a still to point him out. 4. For the next two seconds he lifts the mannequin by the head and you can clearly see it’s a mannequin. I’ve taken a still to circle the doll. 5. Slow it down and you will see one shot is fired in the unrelated video below, yet 3-4 “people” falling down from the magic bullet that just plays pinball. One real person lays down but gets up. All the others are knocked over mannequins. Hope you all enjoyed the show, and hopefully you laughed as much as I did. This entire thing is just fake bullshit. It’s the only thing these mentally ill and retarded propagandists ever post. GAZAWOOD - the PALLYWOOD saga feel free to do your thing. H/T Stealth Medical

Cheryl E 🇮🇱🇮🇱🇮🇱🎗️

19,203 просмотров • 1 год назад