On the bloody example, essence and physical meaning of convolution

As a hard-pressed engineering student, "Signal and system" + "digital signal processing" is not the past, a variety of headache concepts and mathematical formula: Fourier variation, Laplace change, Z-transform, convolution, cyclic convolution, autocorrelation, cross-correlation, discrete fourier changes, discrete fourier time changes ...

Some time ago in the discovery of an interesting example, vividly explain the physical meaning of convolution, and the more accurate explanation, below, the text came:

For example, your boss ordered you to work, you went downstairs to play billiards, and later by the boss found that he was very angry, fan you a slap (note, this is the input signal, pulse), so your face will gradually (cheap) puffed up a bag, your face is a system, and the drum up the bag is your face to spank the response, OK, so it's connected to the meaning of the signaling system.

Here are some assumptions to ensure the rigor of the argument: assuming your face is linear and unchanging, that is, whenever the boss slaps you in the same position on your face (which seems to require your face to be smooth enough, if you say you have a lot of acne, even the entire skin everywhere is not guide, it is too much difficulty, I have nothing to say. haha), your face will always muster up a package of the same height at the same time interval, and assume the size of the bulging package as the system output. Well, then, the following can go into the core content-convolution!

If you go to the underground every day to play billiards, then the boss will slap you every day, but when the boss slapped you, you 5 minutes on the swelling, so long, you even adapt to this life ... If one day, the boss is unbearable, Start your process with a 0.5-second interval of uninterrupted fan this is the problem, the first fan you drum up the package has not swelling, the second slap on the face of the package may be up to twice times high, the boss constantly fan you, pulse constantly in your face, the effect is constantly superimposed, so that these effects can be summed, the result is the height of the bag on your face over time A function of the change (to be understood); If the boss is a bit more aggressive, the frequency is getting higher, so that you can not tell the time interval, then the sum becomes integral.

Can you understand, at some fixed moment in the process, how much the pack on your face is blowing up and what is it about? And before each hit you are related! But the contribution is not the same, the earlier The slap, the smaller the contribution, so that is to say, the output of a moment is a few times before the input multiplied by the respective attenuation coefficient after the superposition to form a certain point of output, and then put the output points of different moments together to form a function, this is convolution, The function after convolution is a function of the size of the package on your face that changes over time.

Originally your bag can be swelling in a few minutes, but if the continuous dozen, a few hours can not eliminate the swelling, this is not a smooth process? Reflected to the Cambridge University formula, F (a) is the first slap, G (x-a) is the first slap in the X-moment of the role of the degree, multiply it and then stack it OK, people say that is not the reason? I think this example is already very image, you have a more specific and profound understanding of the convolution? Anyway, the small part is to be frightened to understand.

Well, to finish this example, let's analyze the substance and physical meaning of it.

Three parts to understand:
1. The angle of the signal
2. Mathematician's understanding (layman)
3. relationships with the polynomial

Convolution this is a "signal and system" in the discussion of the system to the input signal response and proposed. Because it is the analog signal discussed, so often with cumbersome arithmetic down, very simple problem of the essence is often overwhelmed by a large pile of formulas, then convolution what is the physical meaning?
Convolution is represented as Y (n) = x (n) *h (n)
Using a discrete sequence to understand the convolution would be a little more vivid, and we would represent the sequence of Y (n) as Y (0), Y (1), Y (2) and so on; This is the signal that the system responds to.
Similarly, the sequence of the corresponding moment of X (n) is x (0), X (1), X (2) ... and so on;
In fact, if we have not learned the signal and the system, in common sense, the system's response is not only related to the current moment of the system input, but also with the input of a number of times before, because we can understand that this is the input signal before the moment through a process (this process can be decremented, weakened, or other) on the current moment of the system output, then obviously, we have to take into account the current moment of signal input response and the response of the signal input at a number of previous moments "residual" effect of a superposition effect.
Assuming that the response for the 0-moment system is Y (0), and if the response is unchanged at 1, then the response at 1 is changed to Y (0) +y (1), which is called the summation of the sequence and (not the same as the sequence). But often this is not the case in the system, because the 0-moment response is not likely to remain unchanged in 1, then, how to express this change, by the H (t) this response function and x (0) is expressed by multiplying, expressed as X (m) xh (N-M), the specific expression is not multi-tube, as long as remember that there is By introducing this function, we can express how much the Y (0) weakens at 1, and then the value of y (0) is the true value of the 1 moment, then the real system response can be obtained by summing up and calculating.
The extension point, the system response of a moment is often not necessarily determined by the current moment and the previous moment of the two responses, it may also be the former moment before, before the previous moment, before the previous time, and so on, then how to constrain this range, is through the h (N) function in the expression after the change of H (n-m) To constrain the range of M in. To be blunt, the system response at the current moment is related to the "residual impact" of how many previous moments of response.
When these factors are considered, it can be described as a system response, and these factors are described by an expression (convolution) as a clever and fascinating aspect of mathematics.

++++++++++++++++++++++ I'm a smart split line +++++++++++++++++++++++++++++++

Convolution is a human-defined operation, which is an algorithm for calculating the convenience of a rule. The Integral transformation (Fourier transform and Laplace transform) of the ordinary product of two functions establishes a relationship with the convolution of the two function integral transformations, so that we can find the transformation of the product of the two functions as long as we will find the transformation of two functions and use convolution.
Convolution is used in data processing to smooth, convolution has smoothing effect and widening effect.
Talking about volume integral of course, first of all, the impact function----This inverted tadpole (don't think too much, simple point), convolution is actually born for it. 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, we can make 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 do ordinate, just like an area of the same rectangle, the bottom is squeezed narrow, height is crowded high, in mathematics it can be squeezed to infinity, but even if it is infinitely thin, infinitely high, but it still maintain an area of 1 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. Say it is a mathematical monster because the pursuit of the perfect mathematician always in the mind to bend, a thin to the infinite small guy, can occupy a place in the integral, must be the Kango to clear the mathematical world. But physicists and engineers really like it because it solves a lot of practical problems that mathematicians couldn't solve at the time. Finally the pursuit of perfection of the mathematician eventually figured out that mathematics is derived from the actual, and ultimately serve the reality is true. As a result, they tailor a set of operating rules for it. So, Mom! You and I both feel dizzy roll integrals have been produced.

At present, one of the most important applications of Fourier transform is that the convolution equation can be transformed into a product form of two functions to solve. Volume integral is an important member of the integral equation family.

++++++++++++++++++++++ hahaha, you know a little bit +++++++++++++++++++++++++++++++

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 that represents the integral from negative infinity to positive infinity.
One dimensional convolution: Y (t) =g (k) *x (k) = $g (k) x (T-K)
The function x (k) is reversed with respect to the origin, then the distance T is moved to the right, and then the two functions multiply and the integral, the output at the T is obtained. The output curve is obtained by repeating the above procedure for each T value.
Two dimensional convolution: h (x, y) =f (u,v) *g (u,v) =$ $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.
The convolution in image processing is slightly different from the definition above. Convolution with a template and an image, for a point on the image, the origin of the template and the point coincident, and then the point on the template and the corresponding point on the image multiplied, and then the point 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.
The pixel value of a point is replaced by a weighted average of the pixel value of the dots around it. (In fact, it's the mean value method.)

++++++++++++++++++++++ said finished, home New Year ++++++++++++++++++++++++++++++

The physical meaning of convolution, the interpretation of the true humor!
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!) ）？
--Nonsense, 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, Hum hum like a pig (output 3); Hit 30 boards, he may howl like a pig, a runny nose begging 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!
You can also understand this:
T (Tau) is the first board, H (t-τ) is the pain of the first tau board caused by the pain to the t moment, all the boards add up is ∫t (tau) H (t-τ)
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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. It is recommended that you take a look at the content of the definite integral. (Haha, but also to fill the high number, the small part is not very bad ah!) ）

It is particularly important to note that the magnitude of the impact function in the concept is determined by the area of each rectangular element.
In general, convolution is the use of impulse function to express the excitation function, and then according to the impact response to solve the system 0 State response.
Convolution is essentially filtering the signal.
Convolution should be summed, which is integral, for linear time-invariant systems, 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.
Convolution is a good way to solve the 0 state response of LTI system to arbitrary excitation in time domain, which can avoid solving complex differential equations directly.
Mathematically, convolution is a multiplication that defines two 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. This is the frequency volume
From the perspective of mathematics, convolution is a method that reflects the operation between two sequences or functions.
Physically, convolution can represent a system's modulation or contamination of a physical quantity or input;
From the signal point of view, convolution represents the linear system to the input signal response, the 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. Therefore, in the linear system, the system impact response and convolution have an inevitable connection. But the convolution itself is just a mathematical calculation method.
++++++++++++++++++++++ I'm the evil dividing line ++++++++++++++++++++++++++++++

Correlation is divided into self-related and cross-correlation, self-correlation represents the signal itself and delay after a period of similar degree, cross-correlation represents the similarity of two signals. Convolution is an operation, and the related operations can be obtained by convolution.
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What is the relevant practical significance? What is the difference between it and convolution?
Correlation is to find the similarity of two signals, correlation can be obtained by convolution, as if G (t) *g (-T) is g (t) and its own correlation. Because one of the signals is to be flipped when convolution, g (t) *g (-T) is the equivalent of asking for correlation
I analyze it from a mathematical point of view.
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).
What is said above means that 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 correspondence between the elements.
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.
For the time domain, multiplication can be achieved using multipliers, but multiplication in the frequency domain can be done through the convolution operation of the time domain.
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.
Convolution I feel like a file, it is mainly to smooth some non-smooth functions or operators.
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Convolution is a linear operation, and the common mask operation in image processing is convolution, which is widely used in image filtering.
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.
The convolution itself is an operation, but when a signal passes through a linear system, the output signal is the convolution of the input signal and the system shock response.
Convolution is a linear operation that occurs on the basis of a signal and a linear system.
Autocorrelation refers to the dependence of the instantaneous value of the signal at 1 moments with the instantaneous value of another 1 time, which is the time domain description of 1 random signals.
the understanding of convolution--layman (mathematician)
I wrote so "wonderful" math science no one to see, but let not engage in mathematics to write the math accounted for the upper hand, cup with Ah, is really a cup. Is my math level too high or is your level of math appreciation too low? Or did I write too professionally? This is a little unprofessional to come back.
Tang teacher with the infusion process to explain convolution is indeed a bit of meaning, easier to accept, the old evil method is more concise and understandable, but the old evil method can explain how to define the convolution, but can not explain why to define the convolution.
If I remember correctly, convolution came first from the signaling system theory, which was later developed by mathematicians, and its power was far beyond the inventor's original intention.
Let's look at how 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 is supposed to be y (t) =h (t) x (t), however, the actual situation is that the output of the system is not only related It is also related to its response before the T-moment, but the system has a decay process, so the input to the T1 (<t) moment can usually be expressed as X (t) h (T-T1), a process that may be discrete or continuous, 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 the mathematical formula Y (s) =∫x (t) h (s-t) DT, which is the progression in discrete cases.
I smattering on the signal processing, but don't pull my pigtail. We know that the integral transform can turn convolution into the usual product operation, and the physical meaning of the integral transformation is that the function of the time domain can be transformed into a function on the frequency domain by this transformation, which is reversible. The convolution is converted to Y (u) =x (U) by integral transformation. The integral transform of y,x,h is y,x,h respectively. In signal processing people are concerned with Y (U), but X (U) and H (U) are often not easy to find, and X (t) and H (t) are relatively easy to get (really?). ), in order to find the correspondence between Y (U) and Y (t) to get Y (u), people invented convolution.
Signal processing experts, am I right? As for the role of convolution in mathematics, it is said that the words are long, and then the table.
convolution and polynomial
One of the important operations in signal processing is convolution. When it comes to a beginner's convolution, it's 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 this,
(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,
2 x*x*x
3*2+1*5 x*x
2*2+3*5 x
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,
For example, (x*x+3*x+2) corresponds to
(1 3 2),
(2*x+5) corresponds to
(2,5).
The convolution operation between two vectors is not defined in the linear space, but only the addition, multiply two operations, and in fact, the multiplication of the polynomial can not be described in the linear space. How limited is the theory of visible linear space.
But if we deal with the coordinate vector according to our definition of the vector convolution, (don't look at it, here's an explanation)
(1 3 2) * (2 5)
Then there are
2 3 1
_ _ 2 5
--------
2
2 3 1
_ 2 5
-----
6+5=11
2 3 1
2 5
-----
4+15 =19
_ 2 3 1
2 5
-------
10
Or say,
(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
It seems magical, and 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, pondering, the reason 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 coefficient of x*x as an 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 (that is, the quantity of the product). 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 row of a is a moving position of the sequence [2 3 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, as long as one (a, b) * (c,d) = (AC-BD,AD+BC) is changed, the content of the complex function is so rich and well known.
In addition, recall a basic theorem in signal processing, the product of frequency domain, equivalent to the convolution of time domain or spatial signal. Exactly the same as the situation here. What kind of implicit contact exists behind this, and you need to continue with the details.
From this point of view, the high convolution operation is actually just an abstraction of a primary operation. The mathematics in the middle school, in fact, also contains many advanced content (such as Exchange algebra). It is not absurd to know the new words.
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.

On the bloody example, essence and physical meaning of convolution