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I have come across a paper where it is stated that if the infinity assumption in the FT is removed, the uncertainty doesn't hold.
Is this a sensible argument?
Thank you.
Is this a sensible argument?
Thank you.
The uncertainty principle is a fundamental principle in quantum mechanics that states that it is impossible to simultaneously know the exact position and momentum of a particle. This means that the more precisely we know the position of a particle, the less we know about its momentum, and vice versa.
The Fourier Transform is a mathematical tool used to decompose a function into its constituent frequencies. The uncertainty principle applies to the Fourier Transform because it shows that there is a limit to how precisely we can know the frequency and time domain of a signal. This is due to the fact that the Fourier Transform is based on the concept of superposition, which means that a signal is represented as a sum of infinitely many sinusoidal functions.
Removing infinity in the Fourier Transform is important because it allows us to accurately represent signals in both the frequency and time domain. If we were to keep infinity in the Fourier Transform, it would result in an infinite uncertainty and make it impossible to accurately determine the frequency and time domain of a signal.
Infinity is removed in the Fourier Transform by using a windowing function, such as the Gaussian function, to limit the range of frequencies that are considered in the transform. This effectively removes the infinite number of frequencies and allows for a more accurate representation of the signal in both the frequency and time domain.
The main implication of removing infinity in the Fourier Transform is that it allows for a more accurate representation of signals in both the frequency and time domain. This is important in many applications, such as signal processing and data analysis, where precise information about a signal is needed.