Fourier analysis-Fourier Transformation

1. Fourier Transformation

The continuous Fourier transformation can be seen as the continuous form of the "Fourier series in the plural form:

 

From this theorem, we can see that the function can be perfectly reconstructed by its Fourier transformation through the integral operation.

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Another example:

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Ii. Properties of Fourier Transformation

Introduce the following marks:

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Partial Properties of Fourier transformation and its inverse transformation are given below:

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Iii. Fourier transformation of convolution

From the above theorem, we can see that another method to obtain the convolution operation result is to perform Fourier transformation on F and G, and then perform inverse transformation on its Fourier transformation to obtain f * g. In many cases, it is difficult for F and G to directly perform convolution. Because of the proposal of Fast Fourier Transform (FFT), the computation of Fourier transform becomes simple, so ....

In the field of filter, Fourier transform is widely used. We will introduce it in Discrete Fourier Transform (DFT.

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Iv. sampling theorem

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Because many signals are stored in discrete form, the application of Fourier transform is described in the Discrete Fourier Transform (DFT) section.

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References:

1. Basis of wavelet and Fourier Analysis

<This book is well written and focuses on Intuitive understanding, but lacks a physical layer explanation>

2. Digital Image Processing

3. Practical Digital Signal Processing

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