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If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. The DFT is therefore said to be a frequency domain representation of the original input sequence. It has the same sample-values as the original input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. In mathematics, the discrete Fourier transform ( DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. Its similarities to the original transform, S(f), and its relative computational ease are often the motivation for computing a DFT sequence. #HEISENBERG UNCERTAINTY PRINCIPLE FOURIER TRANSFORM SERIES#The respective formulas are (a) the Fourier series integral and (b) the DFT summation. The spectral sequences at (a) upper right and (b) lower right are respectively computed from (a) one cycle of the periodic summation of s(t) and (b) one cycle of the periodic summation of the s(nT) sequence. Fig 2: Depiction of a Fourier transform (upper left) and its periodic summation (DTFT) in the lower left corner. The FFT algorithm computes one cycle of the DFT and its inverse is one cycle of the DFT inverse. The inverse DFT (top) is a periodic summation of the original samples. Right column: The DFT (bottom) computes discrete samples of the continuous DTFT. Its Fourier transform (bottom) is a periodic summation ( DTFT) of the original transform. Center-right column: Original function is discretized (multiplied by a Dirac comb) (top). The inverse transform is a sum of sinusoids called Fourier series. Fourier transform (bottom) is zero except at discrete points. Center-left column: Periodic summation of the original function (top). Left column: A continuous function (top) and its Fourier transform (bottom). Quantum level.Fig 1: Relationship between the (continuous) Fourier transform and the discrete Fourier transform. Limitations when interpreting the impact of the HUP on the nature of the That we need to discern between measurement limitations and inherent ByĬomparison with other fields where Fourier Transform theory is used, we propose ![]() Manifestation of the Nyquist-Shannon Sampling Theorem at the quantum level. ![]() We show that Brillouin zones in Solid State physics are a It is a better principle to describe the measurement limitations of Quantum The Nyquist-Shannon Sampling Theorem of Fourier Transform theory, and show that ![]() We introduce the Quantum Mechanical equivalent of We note that the HUP is not a quantum mechanical It also establishes that the HUP is purely a relationshipīetween the effective widths of Fourier transform pairs of variables (i.e.Ĭonjugate variables). Uncertainty Theorem of Fourier Transform theory demonstrates that the HUPĪrises from the dependency of momentum on wave number that exists at the #HEISENBERG UNCERTAINTY PRINCIPLE FOURIER TRANSFORM PDF#Millette Download PDF Abstract: The derivation of the Heisenberg Uncertainty Principle (HUP) from the Download a PDF of the paper titled The Heisenberg Uncertainty Principle and the Nyquist-Shannon Sampling Theorem, by Pierre A. ![]()
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