What is "convolution"?

It is also quite beneficial to see a few blog posts on the science web about convolution. Indeed, convolution is an extremely important operation, and its definition is actually natural. But because we are too rigidly in the form of textbooks, so that students feel that this is a thing falling from the sky. I felt it the first time I touched the convolution. However, in the late period of gradual contact, formed some of their own humble opinion, and in the classroom often mentioned. Only in this, think the exchange of cheap.

First, what is the definition of convolution? In fact, the name of convolution is a bit of a sense of alienation. However, if we call it the "weighted average product", it will be much easier to accept. Indeed, the discrete form of convolution is the weighted average that is used by everyone, whereas a continuous form can be considered as a weighted average of a continuous function. If we observe or calculate a set of data. However, due to the noise pollution is not smooth, we want to be treated manually. The simplest method, then, is the weighted average. For example, we want to revise the data X_j, which can be weighted to mean
w/2*x_{j-1}+ (1-w) X_J+W/2 *x_{j+1}.
Here, W is the selected weight, if selectable 0.1, and so on.
This is actually a bit of a correction of the intermediate data with both sides of the data. The above formula, which is actually two sequences in the making of a discrete convolution, where a sequence is
...... 0,0,w/2,1-w,w/2,0,0 ...,
Another sequence is
...., X_1,x_2,x_3,......
To generalize these simple ideas is the general convolution. To change the sequence to a function is the definition of our usual convolution. At this point, you might consider a weighted average of one function for another. However, one plays the weight role and the other plays the average role.

So why is convolution important? The product is everywhere, and convolution is everywhere. The reason is that convolution is the product of frequency domain!
If you know something about the Fourier transformation, you know that a function can be viewed in two ways: the time domain and the frequency domain. Fourier transforms are like the demon Mirror of a journey to the monkey, and any function in front of it shows another side. Therefore, many times if we do not see a function clearly, it is in the demon mirror to see, do a Fourier transformation, will be enlightened. The nature of the function, after Fourier transformation, will also have its corresponding properties. For example, the smoothness of a function is Fourier transformed, which tends to be 0 at infinity. Then, the product of the function after Fourier transformation, is convolution! So, convolution is actually the other side of the product, but this side needs to be demon mirror to see, so let's feel a little strange. Convolution, the Fourier transformation and product are closely linked. So:
Where there is a convolution, there will be a Fourier transformation, where there is a Fourier transformation, there will be convolution!
It is not difficult to understand the importance of convolution if you think about the application scope of Fourier transformation.

Speaking for a long while, our students, and even the mathematics of college graduates, why the convolution feel is not testing and magic it? As I mentioned in my previous blog post, the mathematics education in our university seems to be more dismissive of the Fourier transformation. In fact, we have inherited the tradition of high school mathematics education and like to have a great fuss in the skillful place. In higher mathematics, two local students spend a lot of time: one is the median theorem, the other is indefinite integral. Because these two places are most prone to the skillful topic. However, the Fourier transformation of such a place with new ideas is insufficient. As I said earlier: where there is a convolution, there will be a Fourier transformation. It is not difficult to understand that the fundamental reason for the students to be unfamiliar with the convolution is the contempt of Fourier transformation in teaching.

Related topics: big Talk convolution
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What is "convolution"?