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šØ Reality Isnāt Built from Particlesā¦ Itās a Locked Frequency Field What physics calls āmatterā is a misread. There are no solid particlesāonly stable, phase-locked standing waves inside a quantum-acoustic superfluid field. The real invariant isnāt mass or energy, itās Frequency Momentum: FM = Ā½ Ļ Ļ AĀ². Once... show more
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![[Discrete Fourier Transform] by Hand āļø 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. āļø 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 ā³ Signals A, B, and C in the š§ frequency domain: ā¦ A = cos(w) + 2cos(2w) ā¦ B = cos(w) + cos(3w) + cos(4w) ā¦ C = -cos(2w) + cos(3w) ā¦ Each signal is a weighed sum of four cosine waves at frequencies 1w, 2w, 3w, and 4w. ā¦ 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. ā³ Signal X in the š© time domain. X is sampled at 10 time points 1t, 2t, ā¦, 10t: ā¦ X = [-2.5, -1.8, 3, -0.7, -1.0, -0.7, 3, -1.8, -2.5, 5] ā¦ Suppose X is also a weighted sum of the same four cosine waves, but we donāt already know their weights. We will apply DFT to discover them. [2] š§ Frequency Matrix (F) ā³ Write the coefficients of A, B, C as a matrix F. Each signal is a row. Each frequency is a column. ā³ A ā [1, 2, 0, 0] ā³ B ā [1, 0, 1, 1] ā³ C ā [0, 1-, 1, 0] [3] Cosine ā Discrete ā³ Sample from the continuous cosine waves at discrete time points 1t, 2t, 3t, to 10t. [4] Cosine Matrix (W) ā³ Write the samples as a matrix, Each frequency is a row. Each time point is a column. [5] Inverse DFT: š§ Frequency ā š© Time ā³ Multiply the frequency matrix F and the cosine matrix W. ā³ 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. ā³ 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 ā³ Transpose T, converting each signalās time domain representation from a row to a column. [7] DFT: š© Time ā š§ Frequency ā³ Multiply the cosine matrix W with the transpose of matrix T. ā³ 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. ā³ The result is an intermediate version of the ārecoveredā frequency matrix where each column corresponds to a signal and each row corresponds to a frequency. ā³ 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 ā³ 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 ā³ Transpose the recovered frequency matrix back to the same orientation of the original frequency matrix F. ā³ Like magic šŖ, 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: š© Time ā š§ Frequency ā³ Now that we have some confidence in DFTās ability to recover frequency components, we apply DFT to Xās time-domain representation by multiplying W with X. ā³ The result is the an intermediate matrix. [11] Scale ā³ Similarly, we scale down by a factor of 5 to obtain the recovered frequency components of X (a column). [12] Transpose ā³ Similarly, we transpose the recovered column to row to match the orientation of the frequency matrix. ā³ 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: ā¢ Sine: The complete DFT math also includes sine waves that follow a similar calculation process. ā¢ 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. ā¢ 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)
