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Seismic Reflection Data Processing and Interpretation A Workshop in Cairo 28 Oct. – 9 Nov. 2006 Cairo University, Egypt Dr. Sherif Mohamed Hanafy Lecturer Title: 1D Fourier Transform
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1D Fourier Transform Theory and Practice Real Data Examples Using SU and MatLab
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Simple Harmonic Motion A simple harmonic motion is fully described in time domain by its amplitude, frequency, and phase difference. A simple harmonic motion is fully described in time domain by its amplitude, frequency, and phase difference. A simple harmonic motion is fully described in space domain by its amplitude, wavelength, and phase shift. A simple harmonic motion is fully described in space domain by its amplitude, wavelength, and phase shift.
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A wave in time domain
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A wave in space domain
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The connection between time and space domains is Velocity
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Simple Harmonic Motion Where A is the amplitude w is the angular frequency t is the time
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Consider the following simple harmonic motions
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Adding some of these simple harmonic motions together will give us a more complex harmonic motions
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If we have the sinusoidal wave in time domain, could we know the frequencies making it? Yes, using Fourier transform
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What dose “transform” means? Transform is taking a group of data as input to give another group of data as output. The output results can not be calculated unless all the input is available and used at once
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Jean Baptiste Joseph Fourier (1768- 1830), a French mathematician and physicist said that; “Any signal in the time domain is the summation of a specific number of simple sinusoidal waves”
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Fourier Transform is given by :- Inverse Fourier Transform is given by :-
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Discrete Fourier Transform Discrete Inverse Fourier Transform
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Note
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DFT
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Fast Fourier Transform If number of data is = 2 n, where n is a positive integer number. Then we can use fast Fourier transform FFT is incredibly more efficient, often reducing the computation time by hundreds
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Practical solution of FFT 1. Transform the 1 signal of N points into N signals of 1 point
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With the help of binary numbers, things are much easier
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2. Find the frequency spectra of the 1 point time domain signals Nothing could be easier; the frequency spectrum of a 1 point signal is equal to itself
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3. Combine the N frequency spectra in the exact reverse order that the time domain decomposition took place Unfortunately, the bit reversal shortcut is not applicable, and we must go back one stage at a time. In the first stage, 16 frequency spectra (1 point each) are synthesized into 8 frequency spectra (2 points each). In the second stage, the 8 frequency spectra (2 points each) are synthesized into 4 frequency spectra (4 points each), and so on. The last stage results in the output of the FFT, a 16 point frequency spectrum
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Synthetic Examples
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Real Examples
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End of this lecture Thank You for you attention All examples on this lecture is based on my work
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