The physical meaning of convolution

Original link: http://www.cnblogs.com/ylhome/archive/2010/01/07/1641121.html

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?

The 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 time system input, but also with the input of a number of previous moments, Because we can understand that this is the input signal of the previous moment after a process (this process can be decremented, weakened, or otherwise) on the current system output, then obviously, we have to consider the output of the current moment of the signal input response and the previous time signal input response "residue" Effect of an overlay effect.

Assuming a 0-moment system response is Y (0), and if the response is unchanged at 1, then the 1-moment response becomes Y (0) +y (1), which is called the cumulative sum 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 (m-n), the specific expression is not multi-tube, as long as remember that there is a 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 is obtained by adding 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 (m-n) To constrain the range of M in. That is to be blunt, is the current moment of the system response and how many times before the response of the "residual impact" related.

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.

For non-mathematics students, as long as they know how to use convolution can be, the study of what is convolution is not very meaningful, it is a micro-multiplication accumulation of the limit form. The convolution itself is just a mathematical operation. As with the "butterfly operation", how to prove that this is the work of the people in the mathematics department.

In the signal and system, the 0 State response Y (T) of f (t) can be solved by the convolution integral of f (t) and its unit impulse response H (t), i.e. Y (t) =f (t) *h (t). Both the signal and the system should know that the convolution of the time domain equals the product of the frequency domain, that is, Y (s) =f (s) xh (s). (S=JW, the function after Laplace transform is actually the frequency domain expression of the signal). One thing you have to understand is that in communication systems, what we care about and what we want to study is the frequency domain of the signal, not the time domain, because the frequency of the signal is the amount of information that is carried. So, we need the expression Y (s), but in fact, we can not easily get the two expressions F (s) and H (s), but it is very easy to get f (t) and H (t), so in order to find the correspondence between Y (s) and Y (t), it is necessary to use the convolution operation.

Complex frequency domain. S=JW, in which J is a plural unit, the complex frequency domain is used. The popular explanation method is, because the system has inductance x=jwl, the capacitance x= "1/JWC", the physical meaning is, the system H (s) has the different attenuation to the different frequency component, namely this attenuation is occurs in the frequency domain, therefore in order to distinguish with the time domain, introduces the complex number operation. But the form of computation in the complex frequency domain still satisfies the ohm theorem, KCL, KVL, superposition method. The negative frequency. The reason for the negative frequency, which is only the result of mathematical operations, only in the mathematical operation, there is no negative frequency in practice.

The convolution process is equivalent to splitting the signal into an infinite number of shock signals, and then the impact response is superimposed.

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 (the pain caused by the first big Board * attenuation coefficient) [attenuation coefficient is (t-τ) function, carefully taste]. The mathematical expression is: Y (t) =∫t (TAU) H (t-τ) dτ.

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!

The physical meaning of convolution