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Fourier Transform Signal Theory Zdzisław Papir Periodicity of Fourier series Limiting behaviour of Fourier series Limiting form of Fourier series Fourier transform pairs Existence of Fourier transform

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Periodicity of Fourier series Signal Theory Zdzisław Papir sawtooth signal time t period T = 1

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Limiting behaviour of Fourier series Signal Theory Zdzisław Papir x(t)x(t) time t -T/2 xT(t)xT(t) Periodic extension of a signal window x T (t) through Fourier series +T/2

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Limiting behaviour of Fourier series Signal Theory Zdzisław Papir

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Limiting behaviour of Fourier series

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Signal Theory Zdzisław Papir Limiting behaviour of Fourier series

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Signal Theory Zdzisław Papir Limiting behaviour of Fourier series amplitude of the 1st spectrum line of an exponential puls Fourier series window

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Signal Theory Zdzisław Papir Limiting behaviour of Fourier series Fourier series window T amplitude spectrum – exponential pulse

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Signal Theory Zdzisław Papir Limiting behaviour of Fourier series Fourier series window 3T amplitude spectrum – exponential pulse

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Signal Theory Zdzisław Papir Limiting behaviour of Fourier series amplitude spectrum – exponential pulse Fourier series window 10T

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Signal Theory Zdzisław Papir Limiting behaviour of Fourier series amplitude spectrum – exponential pulse Fourier series window 100T

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Signal Theory Zdzisław Papir Squeezing Fourier series coefficients in FREQUENCY: Limiting behaviour of Fourier series Squeezing Fourier series coefficients in AMPLITUDE:

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Signal Theory Zdzisław Papir Riemann integral ab x f(x)f(x)

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Limiting form of Fourier series Signal Theory Zdzisław Papir Fourier series coefficients: FORWARD FOURIER TRANSFORM:

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Limiting form of Fourier series Signal Theory Zdzisław Papir Fourier series: INVERSE FOURIER TRANSFORM:

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Signal Theory Zdzisław Papir Fourier Integral Theorem

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Signal Theory Zdzisław Papir Fourier transform Fourier integral theorem Forward Fourier transform

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Signal Theory Zdzisław Papir Inverse Fourier transform

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Fourier transform pairs Signal Theory Zdzisław Papir TRANSFORM INVERSEFORWARD TRANSFORM PAIRS

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Fourier transform pairs Signal Theory Zdzisław Papir FORWARD FOURIER TRANSFORM:

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Fourier transform pairs Signal Theory Zdzisław Papir FOURIER TRANSFORM: time t

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Fourier transform pairs Signal Theory Zdzisław Papir FOURIER TRANSFORM: T/2-T/2 1 frequency f

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Signal Theory Zdzisław Papir FOURIER TRANSFORM: T/2-T/2 Fourier transform pairs frequency f

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Signal Theory Zdzisław Papir Existence of Fourier transform Dirichlet conditions are necessary for Fourier transform existence. Signal x(t) must have only a finite number of maxima and minima, as well as a finite number of discontinuities over the entire range [–, + ]. Signal x(t) is also allowed to be unbounded provided that it is absolutely integrable:

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Signal Theory Zdzisław Papir Summary Fourier series is a spectral decomposition of periodic signal or produces a periodic extension of signal window. Fourier transform is a tool for spectral decomposition of nonperiodic signals. Fourier transform is a limiting case of Fourier series with signal window being extended up to infinity. Dirichlet conditions are necessary for Fourier transform existence. In engineering applications it is commonly assumed that signals of limited energy are Fourier transformable.

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