(turn)---again convolution

One of the important operations in signal processing is convolution. When a beginner convolution, it 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, it may be clearer to look at convolution from a discrete situation,

For two sequence f[n],g[n], it is generally possible to define the 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).

　　

In a linear space, there is no convolution operation between two vectors, but only addition, multiply by two operations, and in fact, the multiplication of polynomial can not be described in linear space. How limited is the theory of visible linear space.

But if we deal with the coordinate vectors as defined by our upper face vector convolution,

(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. 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, 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). 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.

(turn)---again convolution