The nature and physical significance of convolution (comprehensive understanding of convolution)

The nature and physical significance of convolution (comprehensive understanding of convolution)

The essence and physical meaning of convolution

Hint: the understanding of convolution is divided into three parts 1) signal angle 2) mathematician understanding (layman) 3) relationship with polynomial

1. sources

Convolution is actually the birth of the impact function. The "Impact function" is a symbol proposed by Dirac to solve some of the physical phenomena of instantaneous action. The ancients said: "To say a bunch of principles than to give a good example," the momentum of the physical phenomenon is very good to explain the "impact function." In t time, the force of F on an object, if the action time t is very small, the force F is very large, but let the product of the FT is constant, that is, the impulse is unchanged. So in the coordinate system with t-axis, F to do ordinate, just like an area of the same rectangle, the bottom is squeezed narrow, height is squeezed high, in mathematics it can be squeezed to infinity, but even if it is infinitely thin, infinitely high, but it still keep the area unchanged (it has not been squeezed no!) ), in order to confirm its existence, it can be integral, the integral is to seek the area! So the "convolution" of this mathematical monster was born.

Convolution is the "signal and system" in the discussion of the system to the input signal response and proposed.


2. significance

Signal processing is to map a signal space to another signal space, usually the time domain to the frequency domain, (as well as the z domain, s domain), the signal energy is the function of the norm (the concept of signal and function equivalence), we all know that there is a paserval theorem that the norm is unchanged before and after the map, in mathematics called Paufan shot, In fact, the transformation of signal processing is basically Paufan, as long as the Paserval theorem is Paula Fan mapping (is the energy invariant mapping).

How the convolution occurs in signal processing. Suppose B is a system whose T-moment input is X (t), the output is Y (t), and the response function of the system is H (t), the relationship between the output and the input should, in theory, be

Y (t) =h (t) x (t),

However, the actual situation is that the output of the system is not only related to the response of the system at the T moment, but also to its response before the T time, but the system has an attenuation process, so the input to the output of the T1 (<t) moment can usually be expressed as X (t) h (T-T1), the process may be discrete, It can also be sequential, so the output of the T-moment should be the superposition of the response of the system response function at various moments before the T-moment, which is convolution, which is represented by a mathematical formula

Y (s) =∫x (t) h (s-t) DT,

The discrete case is the progression.

3. calculation

Convolution is an integral operation that can be used to describe the input and output relationships of a linear time-invariant system: the output can be obtained by means of an input and a function (impulse response function) Characterizing the system's characteristics. (The following is a $ symbol for integrals from negative infinity to positive infinity)

1) One dimensional convolution:

Y (t) =g (k) *x (k) = $g (k) x (T-K)

The function x (k) is reversed with respect to the origin, then the distance T is moved to the right, and then the two functions multiply and the integral, the output at the T is obtained. The output curve is obtained by repeating the above procedure for each T value.

2) Two dimensional convolution:

H (x, y) =f (u,v) *g (u,v) =$ $f (u,v) g (X-U,Y-V)

The G (U,V) is rotated 180 degrees around its origin, then the origin is shifted, and the U-axis is shifted on the X, and the V-axis is shifted y. The two functions then multiply the integrals to get the output at one point.

4. Humorous jokes--on the physical meaning of convolution

There is a seven-product magistrate, like to use flogged to punish those Ishika, and there is a convention: if not committed a large crime, only play a board, released home, to show am father.

There is a rogue, want to get ahead but no hope, thought: Since the good name, out of notoriety also become AH. How do you get a bad reputation? Speculation Bai! How to Hype? Look for celebrities! He naturally thought of his chief executive, magistrate.

Rogue then in broad daylight, standing in front of the county government to sprinkle a bubble urine, the consequences are imagined, naturally be invited into the lobby by a board, and then head home, lie down for a day, hey! There's nothing on your body! The next day followed suit, ignoring the kindness of the chief executive and the decency of the office, the third day, the fourth day ... Every day to county palace to lead a board back, still beaming, adhere to one months long! The name of the rogue has been the same as the stench of the mouth of the palace, spread all over Happo!

Magistrate adults quieted nose, staring at the case of gavel, twist the brow thinking a question: These 30 Big board how not good to make pinch? ...... Think originally, the Master C.P. Group when, mathematics but got full marks, today at least to solve this problem:

--Man (System!) ) by the board (Pulse!) Later, what will be the performance (output!) ）？

--Crap, pain!

I'm asking: what will be the performance?

--look at the pain to what extent. Like this rascal's physique, every day to get a board what things will not have, even hum is impossible, you also see his smug face (output 0); If you punch him 10 boards at a time, he may frown, bite, and do not hum (output 1); hit 20 boards, he will be hurt facial distortion, To hum like a pig (output 3); Hit 30 boards, he may howl like a donkey, a runny nose to beg you to spare his life (output 5); hit 40 boards, he will defecate, barely hum aloud (output 1); hit 50 boards, he can't even hum it (output 0)--dead!

Magistrate roll out the coordinate paper, with the number of flogged as the x-axis, to the degree of hem (output) for the y-axis, draw a curve:

--Oh, alas! The curve is like a mountain, and it doesn't make sense. Why did the rogue even get a 30-day Big board but not shout around life?

--hehe, you hit the time interval (δτ=24 hours) too long, so that the rogue suffered the degree of pain a day, no superposition, is always a constant; If you shorten the flogged time interval (recommended δτ=0.5 seconds), then his pain level can be quickly superimposed Wait until the knave gets to 30 big plates (T=30), the pain degree reaches the limit which he can shout, will receive the best punishment effect, the more dozen will not show your kindness.

-Or do you not understand why the pain is superimposed when the time interval is small?

-This is related to the response of the human (linear time-invariant system) to the board (pulse, input, excitation). What is a response? After a person is hit by a board, the feeling of pain will slowly disappear (decay) within a day (presumably, for different people), without the possibility of sudden disappearance. In this way, as long as the time interval of the board is very small, each board caused by the pain is too late to complete decay, will have a different contribution to the ultimate degree of pain:

The pain level caused by the T-Big Board =σ (pain caused by the first big Board * attenuation factor)

[Attenuation coefficient is (t-τ) function, carefully taste]

Mathematically expressed as: Y (t) =∫t (TAU) H (t-τ)

The pain of the people, it is too cruel to convolution. Do other things conform to this rule besides people?

--Oh, Magistrate adults after all kindness. In fact, in addition to people, many things also follow this way. Think about it, why does the wire bend once without folding, quickly bend many times but will easily fold off?

--Well, the moment still confused, let the officer slowly want to--but one thing is clear--somebody, will pee that rogue grabbed, hard dozen 40!

It can also be understood that: T (TAU) is the first board, H (t-τ) is the first tau board caused by pain to the pain of the moment, all the boards add up is ∫t (tau) H (t-τ)

The application of 4 convolution in specific disciplines

Image processing: With a template and an image of the convolution, for the image of a point, so that the template's origin and the point coincident, and then the template on the point and the corresponding point on the image multiplied, and then the points of the product added, the point is obtained the convolution value. This is done for each point on the image. Because most templates are symmetric, the template does not rotate. Convolution is an integral operation that is used to calculate the area of two overlapping areas of a curve. Can be considered as weighted sum, can be used to eliminate noise, feature enhancement.

The pixel value of a point is replaced by a weighted average of the pixel value of the dots around it.

Convolution is a linear operation, and the common mask operation in image processing is convolution, which is widely used in image filtering.

Convolution is used in data processing to smooth, convolution has smoothing effect and widening effect.

Circuit Science: Convolution method is based on the characteristics of linear constant circuit (homogeneous, superposition, time invariance, integration, etc.), with the aid of the Unit impulse response H (t) of the circuit to solve the system response tool, the system's excitation can be expressed as a convolution of the impulse function and the function of excitation, and convolution is the integral concept in higher mathematics. The magnitude of the impact function in the concept is determined by the area of each rectangular element.

One of the most important cases of convolution is the convolution theorem in signal and linear systems or digital signal processing. By using this theorem, the convolution operation in time domain or space domain can be equivalent to the multiplication operation of frequency domain, and the fast algorithm such as FFT is used to realize effective calculation and save the operation cost.

Signal Processing:

1) The convolution is essentially filtering the signal;

2) convolution is to use the impact function to express the excitation function, and then according to the impact response to solve the system's 0 state response.

Convolution is summation (integral). For linear time-invariant systems, the input can be decomposed into a number of different intensities of the impulse and the form (for the time domain is integral), then the output is the effect of these impulse respectively to the system-generated response and (or integral). So the physical meaning of convolution is to express the relationship between input, system impulse response, and output in time domain.

Signal angle: Convolution represents the linear system to the input signal response, its output is equal to the system impact function and signal input convolution, only in accordance with the superposition principle of the system, there is the concept of system impact function, so convolution becomes the system of input in the mathematical calculation of the inevitable form, the impact function is actually the green function solution of the problem. The point excitation source is the solution of a linear problem, and the green function is the system impact response. So in the linear system, the impact response of the system has a definite connection with the convolution.

Mathematics: The convolution is a multiplication that defines two functions, or a method that reflects the operation between two sequences or functions. For discrete sequences, it is the multiplication of two polynomial. The physical meaning is the linear superposition of impulse response, the so-called impulse response can be regarded as a function, and the other function is orthogonal to the impulse signal.

In reality: convolution represents the move of a signal to another frequency, such as modulation, which is a frequency volume.

Physical: Convolution can represent a system's modulation or contamination of a physical quantity or input.

In reality: convolution represents the move of a signal to another frequency, such as modulation, which is a frequency volume.

Image metaphor: Convolution I think it's like a file, it's mainly smoothing out some non-smooth functions or operators.

The task of signal processing is to find a set corresponding to the signal set, and then analyze the signal in another set, Fourier transformation is a kind of, it establishes the time domain each signal function and the frequency domain each spectrum function one by one correspondence relation, this is the element correspondence. So the correspondence between the operations, in the time domain addition corresponds to the frequency domain of addition, this is the embodiment of FT linearity, then the time domain multiplication corresponds to what, the last obtained expression we call it convolution, is the corresponding frequency domain convolution.

In short, convolution is an overlapping relationship, meaning that the resulting results reflect the overlapping portions of the two convolution functions. So, a function with a known frequency band convolution another wide function, that is, the latter is filtered, the latter overlapping bands to pass the filter well.

5. convolution and polynomial

One of the important operations in signal processing is convolution. When a beginner convolution is often in a continuous situation, two functions f (x), g (x) convolution, is ∫f (U) g (x-u) du. Of course, it is not difficult to prove some of the properties of convolution, such as exchange, Union, and so on, but for convolution operations, the beginner is unclear.

In fact, from a discrete situation to see convolution, perhaps more clearly, for two series F[n],g[n], it is generally possible to define its convolution as s[x]=∑f[k]g[x-k].

A typical example of convolution, in fact, is the multiplication of the polynomial multiplied by the middle school.

For example (x*x+3*x+2) (2*x+5) The general calculation Order is as follows:

(x*x+3*x+2) (2*x+5)

= (x*x+3*x+2) *2*x+ (x*x+3*x+2)
= 2*x*x*x+3*2*x*x+2*2*x+ 5*x*x+3*5*x+10
Then merge the coefficients of the similar terms,

2x*x*x

3*2+1*5x*x

2*2+3*5x

2*5
2*x*x*x+11*x*x+19*x+10

In fact, it is known from linear algebra that the polynomial forms a vector space whose base is optionally {1,x,x*x,x*x*x,...} Thus, any polynomial can correspond to a coordinate vector in an infinite-dimensional space, such as (x*x+3*x+2) corresponds to (1 3 2), (2*x+5) corresponds to (2,5). In linear space, the convolution operation between two vectors is not defined, but only the addition and multiplication of two kinds of operations, and in fact, the multiplicative of polynomial cannot be explained in linear space, so the theory of visible linear space is limited. However, if the coordinate vector is processed according to our definition of the vector convolution, (1 3 2) * (2 5) There are (1 3 2) * (2 5) = (2 11 19 10).

Back to the expression of the polynomial, (x*x+3*x+2) (2*x+5) =2*x*x*x+11*x*x+19*x+10, the result is exactly the same as what we get in the traditional way. In other words, the polynomial multiplies, which is equivalent to the convolution of the coefficient vectors. In fact, the truth is very simple, The convolution operation is actually a coefficient of x*x*x, x*x,x,1, that is to say, he has made the addition and summation mixed together. (The traditional approach is to do multiplication first, and then add when merging similar terms) take the x*x coefficient for example, get x*x, or use X*x by 5, or 3x by 2x, that is

2 3 1
_ 2 5
6+5=11

In fact, this is the inner product of the vector. So, the convolution operation can be regarded as a series of inner product operations. Since it is a series of inner product operations, we can try to represent the above process with a matrix.
[2 3 1 0 0 0]
[0 2 3 1 0 0]==a
[0 0 2 3 1 0]
[0 0 0 2 3 1]
[0 0 2 5 0 0] ' = = X
b= ax=[2 11 19 10] '

With a line view of AX, each line of B is an inner product. Each line of a is a moving position of the sequence [23 1]. Clearly, in this particular context, we know that convolution satisfies the law of exchange, binding, because, well-known, polynomial multiplication satisfies the commutative law, the binding law. In the general case, it is actually established.

Here, we find that the polynomial, in addition to the formation of a specific linear space, there is a special relationship between the base and the base, it is this connection, given the polynomial space with a special nature.

When learning vectors, generally will give this example, a has three apples, 5 oranges, B has 5 apples, three oranges, then there are a few apples, oranges. The teacher repeatedly warned that oranges are oranges, apples are apples, can not be mixed together. So there are (3,5) + (5,3) = (8,8). Yes, oranges and apples are no problem, but it's not easy to say if you think about oranges or oranges and apples.

Again, if you just define a complex number pair (A, a, b), it is simply too simple to look at C2 at a linear space level. In fact, just add one (a, b) * (c,d) = (AC-BD,AD+BC). It is well known that the content of complex functions is very rich and colorful. In addition, recall a basic theorem in signal processing, the product of the frequency domain, which corresponds to the convolution of the time domain or the spatial signal. Exactly the same as the situation here. What kind of implicit relationships exist behind this, and you need to continue with the details.

From this point of view, the high convolution operation is nothing more than an abstraction of an elementary operation. The mathematics in the middle school, in fact, contains many advanced content (such as commutative algebra). Warm so know the new, the word is not wrong. In fact, this truth is not complicated, how many years of human reproduction, but in the past N decades, people only know that men and women seduced sperm, but can reproduce offspring. Sperm, the discovery of eggs, the study of reproductive mechanisms, that is, the last few years of things.

Confucius said that the Tao in the daily human relations, it seems that we should look at the eyes of the surrounding, and even ourselves, to know it, and know its why.

Reference:

Http://hi.baidu.com/a__g/blog/item/10873722cab331ac4723e8f7.html

Http://blog.chinaunix.net/u2/76475/showart_1682636.html

The nature and physical significance of convolution (comprehensive understanding of convolution)