| rdfs:comment
| - The sins of a person's life are passed to priests discretely, during confession. However, as God wishes to evaluate sins over an entire lifetime, and therefore continuously over an interval, discrete sampling of Sins is not sufficient to determine the fate of an individual. Rather, God must wrack His brain over the Cos's. It is awkward but not uncommon for a new arrival at the Pearly Gates to argue that using that cigar on that intern did not cause that dot-com bust. (That is, "sometimes a cigar is just a cigar.")
- A mathematical method that analyzes a sound sample and determines what frequencies and harmonics are present in the sound. The algorithm works by processing a group of samples representing a short period of time of the waveform being transformed; therefore, the output of the transform represents the average frequency content over the period of time being sampled. A tradeoff that the implementer must make is deciding on the length of the time period being sampled; shorter periods of time produce a more instantaneous view of the waveform, but the resulting analysis has less resolution than a transform performed on a longer sample. Computing devices that use the Fourier transform (in particular the fast Fourier transform algorithm) have almost totally taken over the spectrum analyzer market,
- An intuitive way to think about the Fourier transform is by thinking as a non-periodic function as a periodic function of infinite wavelength. As such, it will have a Fourier series, generally with an infinite number of terms. Since the wavelength is infinite, the frequency of the sine and cosine waves do not have to be integer multiples of the original frequency, and so the frequencies that compose the signal can be continuous, rather than discrete. As such, the Fourier transform gives the amplitude and phase shift of the wave at each frequency. Because of this, the more concentrated the signal, the more spread out the Fourier transform, and vice versa.
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| abstract
| - The sins of a person's life are passed to priests discretely, during confession. However, as God wishes to evaluate sins over an entire lifetime, and therefore continuously over an interval, discrete sampling of Sins is not sufficient to determine the fate of an individual. Rather, God must wrack His brain over the Cos's. It is awkward but not uncommon for a new arrival at the Pearly Gates to argue that using that cigar on that intern did not cause that dot-com bust. (That is, "sometimes a cigar is just a cigar.") St. Peter always strives to be non-partisan in such claims and instead to decide by algorithm who wades into the River Styx. When priests refer to the Inverse Fourier Transform being done at the Pearly Gates to evaluate a person's life, they mean St. Peter's Inversion Formula, which recompiles the data collected across an individual's entire lifetime. The following formula is effective, but fairly expensive for mortals to compute, requiring the services of several cardinals on several Sunday Masses. (God, of course, has the Earth at his disposal.) In the prior equation, (the so-called fudge factor) is determined at the baptism and stored until required for processing later on. In recent years, however, the Vatican has suggested the adoption of Discrete Fourier Transforms, which require fewer church services to compute. The length of funerals is thus cut in half, greatly reducing the length of a service. There is a remarkable theorem, due to Bishop Karl Ludwig Weierstrass, that makes possible the calculation of Fourier Transforms using Gregorian Chants. However, its practical application has been never accomplished, because of mysterious reasons.
- A mathematical method that analyzes a sound sample and determines what frequencies and harmonics are present in the sound. The algorithm works by processing a group of samples representing a short period of time of the waveform being transformed; therefore, the output of the transform represents the average frequency content over the period of time being sampled. A tradeoff that the implementer must make is deciding on the length of the time period being sampled; shorter periods of time produce a more instantaneous view of the waveform, but the resulting analysis has less resolution than a transform performed on a longer sample. Computing devices that use the Fourier transform (in particular the fast Fourier transform algorithm) have almost totally taken over the spectrum analyzer market, since they are more precise and work faster than older devices that relied on sweeping a filter across the spectrum. The Walsh function is another method of spectral analysis. It is seldom used in music, but is seen in other types of signal processing, such as speech recognition and radar target detection.
- An intuitive way to think about the Fourier transform is by thinking as a non-periodic function as a periodic function of infinite wavelength. As such, it will have a Fourier series, generally with an infinite number of terms. Since the wavelength is infinite, the frequency of the sine and cosine waves do not have to be integer multiples of the original frequency, and so the frequencies that compose the signal can be continuous, rather than discrete. As such, the Fourier transform gives the amplitude and phase shift of the wave at each frequency. Because of this, the more concentrated the signal, the more spread out the Fourier transform, and vice versa. The Fourier transform of is equal to This is sometimes represented using angular frequency, rather than frequency, as The inverse Fourier transform, which takes a function of frequency and transforms it into a function of time, is equal to
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