Gray-scale image--Fourier series of discrete periodic signal in frequency domain filter Fourier transform

Study Dip 19th Day

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0. Opening nonsensenonsense began to think that Oppenheim will introduce all Fourier family transformation, but the lack of digital image processing to use the DFT, but the sense of understanding the Fourier series, Fourier transform, discrete-time Fourier transform, DFT is basically no obstacle, After writing this, we can return to Gonzalez's book, especially the filter template behind the design of image processing, should be well researched. Remember before to interview a company, technical side asked me, the spatial convolution of the time complexity is how much, I "said not simplified is ' w*h*n*n '", is the simplest image size multiplied by the window size, and then he asked why not convert to the frequency domain, I said Fourier transform (DFT) is very slow, he said "no Ah, FFT also quickly ah, only need n log (n) ", I was a little confused, because Fourier I did not quite understand, and then I said ' This I do not understand ', it is obvious that they did not want me ... Want to know the specific reason of the classmate I later blog will explain in detail why, think of their own dishes. 1. Periodicity of discrete-time complex exponential sequencesfor the periodicity of discrete complex exponential sequences, it must be explained that sampling, sampling is the key to discretization, from the continuous signal to the discrete signal, we do the operation is sampling, for example, the signal f (x) sampling, sampling interval of 1, then get the sequence {f (0), F (1), F (2), F (3) 4) ... f (n)}, the actual example: we sampled sin (pi*x) and sin (3*pi*x), the sine function is an object of the complex exponential function, and the sampling interval is 0.5, then we get the following result:

you can see that the result is the same, as you can see from the formula:1.1
Sampling for equal intervals (0.5) of sine functions with periodic intervals of pi:

for the constant interval sampling of the complex exponent with frequency interval of 2*PI, the result is consistent, and this cause will also lead to the important difference between the discrete period Fourier series and the continuous periodic Fourier series. 2. Fourier series synthesisthe Fourier series of a discrete periodic sequence is the same as the purpose of a continuous period Fourier series-in order to decompose the signal, the original signal is synthesized linearly by a set of basic signals of the harmonic relation, the base signal contains the frequency information, and the signal information can be analyzed from another angle. The discrete-period Fourier series is characterized by its limited number of series, and there is no convergence problem, the specific reasons will be described in detail later. A periodic signal sequence with a period of N is expressed as:2.1
The minimum positive integer N that is formed is the period of the above sequence, w=2*pi/n, is the frequency of the fundamental, such as the following complex exponential sequence:2.2
The Upper cycle is n, and all frequencies of the harmonic set are multiples of 2*pi/n. Based on the periodicity of the discrete-time complex exponential sequence we mentioned above, we can get:2.3
when the K change is an integer multiple of n, an identical sequence is obtained, mathematically supported by the fact that the equation 1.1 and equation 2.2 are linked. So, a linear combination of a finite term (n-term) of a relationship of a basic wave:2.4
k only changes in n successive integer intervals, not necessarily 0 to N, or 1 to n+1, or M to n+m, so rewrite into the following formula:2.5
Formula 2.5 is the discrete-time Fourier series, and the coefficient AK is the coefficient of the Fourier series. Fourier series Analytical Formula

after the synthesis is determined, the next step is to determine the coefficients of the various series, here and the continuous periodic signal Fourier series The same method, the following expression can be seen, the original signal x total n is known, each item has an n terms of the series item combination, each series item corresponds to an unknown coefficient, the series term can be calculated, that is, , there are n equations, n unknowns, and these equations are linear independent, so the coefficient is a unique solution set, which is also the discrete periodic signal Fourier series no convergence problem is willing to calculate the limit, but the real solution of the equation, and the discrete periodic signal sequence is definitely can be and, so the Fourier series must exist.

But in order to correspond with the continuous form, we do not adopt the method of solving the equation, but use the method of responding to the continuous periodic function, because of the following properties:

3.2

The derivation is as follows:

F (n ) is the original sequence, S (n) is the set of series, an for the corresponding coefficients, according to the formula can be obtained S0 value everywhere is 1 (although the value is written n and N, but through the complex exponent of the original formula can be obtained all items are 1, because the exponent of E is 0, or the integer multiples of 2*pi), S*k (n) is the SK (n) conjugate, so they fit is S0 , so combined with the above formula, we get the final conclusion of the derivation below.

Combining the above derivation process into two steps, the key features used are also 3.2:

Get analysis Results

So we get the discrete-period signal Fourier series:

PropertiesThe properties of the Fourier series of discrete periodic signals are similar to those in continuous cases and are summarized in the following table in the original book:
to add a Paswal theorem:
SummaryThe difference between Fourier series and continuous conditions of discrete-time periodic signals is that there is no convergence problem in discrete cases, there is no Gibbs phenomenon, the series term is finite n, and the coefficient of Fourier series of discrete time period sequence is periodic, and the period is N.

Gray-scale image--Fourier series of discrete periodic signal in frequency domain filter Fourier transform