Gray image--Fourier series of Fourier transform in frequency domain filter

Source: Internet
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Study Dip 18th Day

Reprint please indicate the source of this article: Http://blog.csdn.net/tonyshengtan, Welcome to reprint, found that the blog is reproduced in some forums, the image can not be normal display, unable to express my views, to this expression is not satisfied ....

0. Opening nonsensenonsense start, the story is this, when I went to college, learned the signal and the system, at that time has learned the high mathematics, also knew the Fourier transform formula, but, the formula is how to come, what uses, not clear, learns the signal and the system, knows the Fourier uses in where, But it's not clear why it can be used in these places, the book's Memory is: Fourier transform or its family transformation, the main purpose is to convert the signal to the frequency domain, and then blah ... In fact, things are really like the description of the book, but, can be completely changed to accept the way to tell everyone, but the person who wrote the book refused, so bitter read the person. There is an article known as "the choking tutorial of Fourier analysis (full version)"we can search by themselves, positive article no mathematical deduction, completely from the perceptual introduction of the Fourier transform, there are many diagrams, there are moving pictures, you can understand, but the perceptual after the rational thinking from the mathematical perspective of the Fourier transform family, for image processing, Fourier is a tool that must be used and mastered, The image processing is also from the signal processing, so I choose to see Oppenheim's "Signal and system", from which to re-understand the Fourier transform family. In This article, we start with the simplest Fourier series. The approximate structure of knowledge such as:
1. Fourier seriesFourier series should be proposed by a lot of people, but everyone just use a similar structure, but there is no systematic proof, in fact, Fourier also did not prove that he just the Fourier series formally used in his research, Later a group of people wave upon wave to prove that there is a Fourier series of the accuracy and existence conditions, so the Fourier series and subsequent Fourier transform began to promote human progress. 1.0 Signal Decompositionthe person who studies the signal has this kind of experience, there is a signal in front of you, a mess of things, you do not know what is a thing, take a simple cycle signal to say:
such a lump of signal f (t), the general people do not see what information, the cycle can be seen, but the other basic see nothing, but if the periodic signal decomposition, then the signal is by sin (t), cos (t), sin (2t), cos (2t), sin (3t), cos ( 3t), sin (4t), cos (4t) combined (called base), with these, we can easily predict the future time of the signal accurate values, these trigonometric functions, the nature of the complete signal composition. The decomposition of large, complex things into small, simple things, easy to understand and grasp, this is a kind of signal problems of the research method, so that the signal decomposition has generated such a branch. 1.1 Target Signala signal can be broken down into a combination of countless other signals, but which one is what we need, which requires us to artificially add some control to the nature of the target

    • Property 1: These basic signals can form a fairly broad class of useful signals

    • Property 2: Linear time-invariant system response to each basic signal should be very simple, so that the system to the response of any input signal has a convenient expression

1.2 orthogonalityIn fact, in addition to the above two point nature, orthogonality is also an important property, for a vector, which has a base vector, if the base vector is orthogonal, then its correlation is 0, that is, a vector consisting of a base vector, each base vector is unrelated, in a simple two-dimensional Cartesian coordinate system, the vector (A, B) , wherein two groups of bases are (1,0) and (0,1) its orthogonal, because its vertical, or the inner product of 0 can prove its orthogonal, the value of a, B has no effect on each other. function orthogonal meaning: If two functions ψ1 (R) and ψ2 (R) satisfy the condition: ∫ψ1 (R) *ψ2 (R) dτ=0, the two functions are said to be orthogonal to each other. The official explanation:"orthogonality" is the term borrowed from geometry. If the two lines intersect at right angles, they are orthogonal. In vector terms, these two lines are independent of each other. Moving along a line, the line is projected to the same position on the other line, and the term is used to denote some kind of non-phase dependency or decoupling. If one of the two or more species changes, it will not affect other things. These things are orthogonal. The signal is decomposed into a combination of signals, and any of the two of these signals are orthogonal, so the information stored in each of these groups is not affected. 1.3 Complex exponential signalcontinuing with the objective function we are looking for in 1.1, the predecessors found that the complex exponential signal has the required properties of 1.1, and found that for a linear time-invariant system to enter a complex exponential signal, the output is still a complex exponential signal, but the amplitude changes. (handwriting formula because there are too many formulas)for continuous signals:for discrete signals:for a continuous linear time-invariant system, the system unit Impact response is H (t), the input x (t) is a continuous signal, the output is defined by its convolution (does not understand the convolution of a blog)
To move the e^st to the front:
assuming that the integral converges in the above equation, the response of the system to the input is:
where H (s) is a complex constant.
for a system, if the output equals input multiplied by constant, then this input is called the characteristic function of the system, this constant is called eigenvalue, it can be seen that the complex exponential function is the characteristic function of the system. For discrete-time linear invariant systems, the unit impulse response is h[n] and the input sequence is:where z is a complex number, the system output can be determined by the convolution:Similarly, assuming the above summation converges, the output is:
among them:as a complex constant. consistent with the successive results, the complex exponent is the characteristic function of the discrete time invariant linear system. for a linear time-invariant system, a more general input can be decomposed into a combination of feature functions. make x (n) A linear combination of three complex exponential signals:
depending on the nature of the characteristic function, the response of each component of the system is:
according to the superposition properties of the linear system:
The above results indicate that if a signal is decomposed into a combination of its characteristic functions:
then its response can be very easy to express:
for discrete sequences, they are decomposed into characteristic complex exponential linear combinations:
The result is exactly the same as in a continuous situation

1.4 Harmonicsfor a periodic signal:The fundamental period is the smallest non-0 positive T that satisfies this equation, and w0=2*pi/t, called the fundamental frequency. Basic periodic function--complex exponential function signal:, the complex exponential signal set of the harmonic relationship (harmonically related) associated with this signal is: k is an integer. Each of these signals has a fundamental wave frequency of w0 integer multiples, T is the period of each signal, for k>=2 after k<=-2 its fundamental period is the approximate number of T, so use a complex exponential linear combination of harmonic relations to form the original signal:
When k=0, a constant, k=1,-1, the fundamental frequency equals w0, called the fundamental component, or the primary wave component, k=2,-2, the fundamental wavelength is half of t, and the frequency is 2*w0. Moreover, the harmonics of each other signal are orthogonal to each other.
1.5 Fourier series to the students who have not fainted here. Congratulations to you. The above equation is the Fourier series, which is it:

for the Fourier series, there are two other representations, but it is not as common as in the above formula.     Derivation process:
    • 1: Because X (t) is the real signal x* (t) =x ( t) so in 1 of the equation.
    • 2: Use-K instead of K, the result of the formula is unchanged, the form of 2.
    • 3: Compare formulas 1 and 2 to find the result of Equation 3.
    • 4: Sum rewrite delimit sum range
    • 5: Replace a-k with the conjugate of AK
    • 6: The inner brackets are conjugated to each other, so get the figure Chinese sub 6


when AK is represented as polar coordinates and Cartesian coordinates, respectively:Polar Coordinates:Cartesian Coordinates:
1.5.1 coefficient derivation processsince the Fourier series form has been determined, the rest of the task is to determine the coefficient of each item is the value of AK, we will use the aforementioned orthogonal properties, that is, in one cycle, the harmonic signal multiplied after the integration of 0, because the harmonic set in the Fourier series are orthogonal to each other, So simply multiply the series by the AK term and then the integral within the period, mathematically deduced as follows: for the coefficients of the nth thought, the sequence of sums and integrals is multiplied by multiplying the signal components of the nth term on both sides of the series.

the above results are Euler transform, get the following results, get the real part and the imaginary parts of the integral, easy to see k=n when the integration result is not 0, the other items due to each other orthogonal, the integral result must be 0, so the time for k!=n is 0;
so we get the final result of the above an, the coefficient {AK} is often called the X (t) Fourier series coefficients, or X (t) spectral coefficients:


Convergence of 1.5.2The problem of the convergence of the series is not described in detail here, because the detailed proof of convergence, the closed derivation process is more complex, here only the conditions for the existence of Fourier series (Dirichlie conditions):
    1. In any period, X (t) must be absolutely integrable.
    2. In any finite interval, X (t) has a finite number of fluctuations, that is, the maximum and minimum values of X (t) are limited in any single period.
    3. Within any finite interval of x (t), there are only a finite number of discontinuous points, and at these discontinuous points, the function is finite.
a,b,c, respectively, correspond to the inverse example of the map:
1.5.3 PropertyNature Upload a picture of it, Oppenheim's book is taken, it is clear
2. SummaryThe Fourier series is the simplest member of the Fourier family, and here we analyze the history of the Fourier series from a mathematical perspective, and we introduce the continuous non-periodic Fourier transform, the discrete Fourier transform, and the discrete-time Fourier transform

Gray image--Fourier series of Fourier transform in frequency domain filter

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