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In Excel and Word 95 and prior editions a weak protection algorithm is used that converts a password to a 16-bit verifier and a 16-byte XOR obfuscation array [1] key. [4] Hacking software is now readily available to find a 16-byte key and decrypt the password-protected document. [5] Office 97, 2000, XP and 2003 use RC4 with 40 bits. [4]
A decimation-in-time radix-2 FFT breaks a length-N DFT into two length-N/2 DFTs followed by a combining stage consisting of many butterfly operations. More specifically, a radix-2 decimation-in-time FFT algorithm on n = 2 p inputs with respect to a primitive n -th root of unity ω n k = e − 2 π i k n {\displaystyle \omega _{n}^{k}=e^{-{\frac ...
The Cooley–Tukey algorithm, named after J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size = in terms of N 1 smaller DFTs of sizes N 2, recursively, to reduce the computation time to O(N log N) for highly composite N (smooth numbers).
An FFT analyzer computes a time-sequence of periodograms. FFT refers to a particular mathematical algorithm used in the process. This is commonly used in conjunction with a receiver and analog-to-digital converter. As above, the receiver reduces the center-frequency of a portion of the input signal spectrum, but the portion is not swept.
Despite its strengths, the Fast Fourier Transform (FFT) has limitations, particularly when analyzing signals with non-stationary frequency content—where the frequency characteristics change over time. The FFT provides a global frequency representation, meaning it analyzes frequency information across the entire signal duration.
A special case occurs when, by design, the length of the blocks is an integer multiple of the interval between FFTs. Then the FFT filter bank can be described in terms of one or more polyphase filter structures where the phases are recombined by an FFT instead of a simple summation.
Lectures on Image Processing: A collection of 18 lectures in pdf format from Vanderbilt University. Lecture 6 is on the 1- and 2-D Fourier Transform. Lectures 7–15 make use of it., by Alan Peters; Moriarty, Philip; Bowley, Roger (2009). "Σ Summation (and Fourier Analysis)". Sixty Symbols. Brady Haran for the University of Nottingham.
Both transforms are invertible. The inverse DTFT reconstructs the original sampled data sequence, while the inverse DFT produces a periodic summation of the original sequence. The Fast Fourier Transform (FFT) is an algorithm for computing one cycle of the DFT, and its inverse produces one cycle of the inverse DFT.