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CATAMOUNTS WIN!!! Markel Townsend with the 38-yard scamper down the left side helps seal up a 45-38 victory this afternoon. T. Dickens - 28-of-33, 5 TD, 369 Yds J. Tyre - 7 Catches, 112 Yds, 1 TD M. Knight - 5 Catches, 103 Yds, 2 TDs M. Townsend -... show more
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![On This Day: The fall of the Valkyrie June 9 1966: The 2nd prototype of the North American XB-70 Valkyrie [62-02070] collides with an F-104 Starfighter over California (US). Of the three aboard both planes, only the XB-70´s pilot survives, with his co-pilot and the fighter pilot perishing. Incident occurred during a photoshoot while in close formation. Enquiry stated the F-104´s pilot was in a position where he was unable to safely distance from XB-70 (more details from the ASN Entry below) The XB-70 was a Mach 3 bomber developed in the 50s. Developments in missile technology made the concept obsolete, leading to its cancellation in 1961. At the time of the crash, the two prototypes that had been built were being used for research. Video is from an excerpt by YouTuber Mike Bell – Video´s name is “Wake Vortex and the XB-70 crash” More info on the report here “ On 8 June 1966, XB-70A No. 2 was in close formation with four other aircraft (an F-4 Phantom, an F-5, a T-38 Talon, and an F-104 Starfighter) for a photoshoot at the behest of General Electric, manufacturer of the engines of all five aircraft. The USAF summary report of the accident investigation stated that, given the position of the F-104 relative to the XB-70, the F-104 pilot would not have been able to see the XB-70's wing, except by uncomfortably looking back over his left shoulder. The report said that Walker, piloting the F-104, likely maintained his position by looking at the fuselage of the XB-70, forward of his position. The F-104 was estimated to be 70 ft (21 m) to the side of, and 10 ft (3 m) below, the fuselage of the XB-70. The report concluded that from that position, without appropriate sight cues, Walker was unable to properly perceive his motion relative to the Valkyrie, leading to his aircraft drifting into contact with the XB-70's wing”](https://image.24vids.com/tw-2063954442332873058/ext_tw_video_thumb/2063954420950310912/pu/img/8YfgZVxuW6bXPXnT.jpg)
![[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)
