1. Fourier Analysis and its
Applications
D. McLean Snyder III
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2. A method for solving some differential equations
What Is Fourier Series?
A method for solving some differential equations
An approximation for a complex function with an infinite sine and cosine series
A foundation of Fourier Transformation which is used for various analyses such as sounds and images
From: “Elementary Differential Equations and Boundary Value Problems(Ninth Edition)”, William E. Bryce and Richard C. Prima, John Wiley and Sons, Inc. 2009
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3. The General Formula for a Fourier Series
From:”Fourier Series”, University of Hawaii,
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4. The full rectifier can be approximated with Fourier series.
Full rectifier as the series
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From:”Fourier Series”, University of Hawaii,

5. The Computational Result
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6. One Dimensional Fourier Transformation
An example function:
The test function has four different frequencies and these generate several periods as a wave function.
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7. The time series of the function
Data set “sine.d” before run through Ftrans
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8. 1 3 2 4
This is the Fourier transformed graph. Four peaks are found in the plot.
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9. Time series
Fourier Transform
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10. Fourier Transform using Sine Functions
Fourier Transforms using
Cosine Functions
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11. Graph with six sine functions
Graph with six cosine functions
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12. 2D Fourier Transformation (Image Processing)
One of the most popular uses of the Fourier Transform is in image processing.
Fourier Transforms represents each image as an infinite series of sines and cosines.
Images consisting of only cosines are the simplest
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13. Cosine Image and its Transform
The higher frequency colors on each image generate the patters of dots in their Fourier Transform.
From: “Introduction to Fourier Transforms in Image Processing”,The University of Minnesota ,
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14. For all REAL (not imaginary or complex) images, Fourier Transforms are symmetrical about the origin.
From: “Introduction to Fourier Transforms in Image Processing”,The University of Minnesota ,
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15. What happens when you rotate the image?
The Fourier Transform creates a much more complex image.
What causes the “+” shaped vertical and horizontal components?
From: “Introduction to Fourier Transforms in Image Processing”,The University of Minnesota ,
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16. Fourier Transforms are INFINITE series of sines and cosines
Fourier Transforms are INFINITE series of sines and cosines. The edges of the arrays affect each other.
From: “Introduction to Fourier Transforms in Image Processing”,The University of Minnesota ,
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17. Image with the edges covered by a gray frame
Putting a frame around the image creates a more accurate Fourier Transform
Image with the edges covered by a gray frame
Transform of original image
Transform of gray framed image
Actual transform of original image framed image
From: “Introduction to Fourier Transforms in Image Processing”,The University of Minnesota ,
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18. Effect of noise on a Image
From: “Introduction to Fourier Transforms in Image Processing”,The University of Minnesota ,
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19. From: “Introduction to Fourier Transforms in Image Processing”,The University of Minnesota ,
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20. Fourier Transforms of more general images have very little structure
The more symmetrical baboon has a more symmetrical Fourier Transform
From: “Introduction to Fourier Transforms in Image Processing”,The University of Minnesota ,
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21. Data set for a two dimensional map
0, 0, 0, 0, 0, 0, , 0, 0, 0,
0, 0, 0, 0, 0, 0, , , 0,
0, 0, 0, 100, 100, 100, 100, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, , 0, 0,
0, 0, 0, 0, 0, 0, 0, , 0, 0
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22. Two Dimensional Fourier Transform of the data
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23. Data set for two dimensional map with ‘noise' around the edges
50, 50, 50, 50, 50, 50, 50, 50, 50, 50,
50, 0, 0, 0, 0, 0, 0, 0, 0, 50,
500, 0, 0, 0, 0, 0, 0, 0, 0, 50,
50, 0, 0, 100, 100, 100, 100, 0, 0, 50,
50, 50, 50, 50, , 50, 50, 50,50, 50
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24. Two Dimensional Fourier Transform with noise
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25. Data set of a Two Dimensional map with random numbers
49, 29, 13, 69, , 62, 03, 97, 0, 44,
18, 4,46,66, 41, 39, 44, 57, 27, 59,
26, 30, 98, 74, , 89, 84, 1, 98, 46,
0, 40,35, , 100, 100, 100, 76, 4, 48,
98, 15, 46, 100, 100, 100, 100, 34, 55, 86,
73, 29, 40, 100, 100, 100, 100, 35, 34, 9,
7, 61, 99, 100, 100, 100, 100, 40, 67, 61,
25, 77, 53, 84, , 63, 18, 13, 69, 31,
81, 52, 20, 91, , 63, 6, 8, 23, 73,
21, 59, 76, 68, , 44, 20, 48, 53, 19
Values used came from the middle two terms of phone numbers from a random page in the telephone directory
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26. Two Dimensional Fourier Transform with Random Noise
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27. Original Fourier Transform versus Transform with Random Noise
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28. Summary
Fourier series and transformation are used for various scientific and engineering applications, such as heat conduction, wave propagation, potential theory, analyzing mechanical or electrical systems acted on by periodic external forces, and shock wave analysis
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