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轉自：https://blog.csdn.net/dkcgx/article/details/46652021

轉自：https://blog.csdn.net/Reborn_Lee/article/details/83279843

conv（向量卷積運算）
所謂兩個向量卷積，說白了就是多項式乘法。 比如:p=[1 2 3],q=[1 1]是兩個向量，p和q的卷積如下： 把p的元素作為一個多項式的系數，多項式按升冪（或降冪）排列，比如就按升冪吧，寫出對應的多項式：1+2x+3x^2;同樣的，把q的元素也作為多項式的系數按升冪排列，寫出對應的多項式：1+x。
卷積就是“兩個多項式相乘取系數”。 （1+2x+3x^2）×（1+x）=1+3x+5x^2+3x^3 所以p和q卷積的結果就是[1 3 5 3]。

記住，當確定是用升冪或是降冪排列後，下面也都要按這個方式排列，否則結果是不對的。 你也可以用matlab試試 p=[1 2 3] q=[1 1] conv(p,q) 看看和計算的結果是否相同。
conv2（二維矩陣卷積運算）
a=[1 1 1;1 1 1;1 1 1]; b=[1 1 1;1 1 1;1 1 1]; >> conv2(a,b)
ans =
1 2 3 2 1

2 4 6 4 2

3 6 9 6 3

2 4 6 4 2

1 2 3 2 1
>> conv2(a,b,‘valid‘)
ans =
9
>> conv2(a,b,‘same‘)
ans =

4 6 4

6 9 6

4 6 4
>> conv2(a,b,‘full‘)
ans =

1 2 3 2 1

2 4 6 4 2

3 6 9 6 3

2 4 6 4 2

1 2 3 2 1

convn（n維矩陣卷積運算）

>> a=ones(5,5,5)
a(:,:,1) =

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1
a(:,:,2) =
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
a(:,:,3) =
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
a(:,:,4) =
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
a(:,:,5) =
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
>> b=ones(5,5,5);
>> convn(a,b,‘valid‘)
ans =
125
>> convn(a,b,‘same‘)
ans(:,:,1) =
27 36 45 36 27

36 48 60 48 36

45 60 75 60 45

36 48 60 48 36

27 36 45 36 27
ans(:,:,2) =
36 48 60 48 36 48 64 80 64 48 60 80 100 80 60 48 64 80 64 48 36 48 60 48 36
ans(:,:,3) =
45 60 75 60 45 60 80 100 80 60 75 100 125 100 75 60 80 100 80 60 45 60 75 60 45
ans(:,:,4) =
36 48 60 48 36 48 64 80 64 48 60 80 100 80 60 48 64 80 64 48 36 48 60 48 36
ans(:,:,5) =
27 36 45 36 27 36 48 60 48 36 45 60 75 60 45 36 48 60 48 36 27 36 45 36 27
>> convn(a,b)
ans(:,:,1) =
1 2 3 4 5 4 3 2 1

2 4 6 8 10 8 6 4 2

3 6 9 12 15 12 9 6 3

4 8 12 16 20 16 12 8 4

5 10 15 20 25 20 15 10 5

4 8 12 16 20 16 12 8 4

3 6 9 12 15 12 9 6 3

2 4 6 8 10 8 6 4 2

1 2 3 4 5 4 3 2 1
ans(:,:,2) =
2 4 6 8 10 8 6 4 2 4 8 12 16 20 16 12 8 4 6 12 18 24 30 24 18 12 6 8 16 24 32 40 32 24 16 8 10 20 30 40 50 40 30 20 10 8 16 24 32 40 32 24 16 8 6 12 18 24 30 24 18 12 6 4 8 12 16 20 16 12 8 4 2 4 6 8 10 8 6 4 2
ans(:,:,3) =
3 6 9 12 15 12 9 6 3 6 12 18 24 30 24 18 12 6 9 18 27 36 45 36 27 18 9 12 24 36 48 60 48 36 24 12 15 30 45 60 75 60 45 30 15 12 24 36 48 60 48 36 24 12 9 18 27 36 45 36 27 18 9 6 12 18 24 30 24 18 12 6 3 6 9 12 15 12 9 6 3
ans(:,:,4) =
4 8 12 16 20 16 12 8 4 8 16 24 32 40 32 24 16 8 12 24 36 48 60 48 36 24 12 16 32 48 64 80 64 48 32 16 20 40 60 80 100 80 60 40 20 16 32 48 64 80 64 48 32 16 12 24 36 48 60 48 36 24 12 8 16 24 32 40 32 24 16 8 4 8 12 16 20 16 12 8 4
ans(:,:,5) =
5 10 15 20 25 20 15 10 5 10 20 30 40 50 40 30 20 10 15 30 45 60 75 60 45 30 15 20 40 60 80 100 80 60 40 20 25 50 75 100 125 100 75 50 25 20 40 60 80 100 80 60 40 20 15 30 45 60 75 60 45 30 15 10 20 30 40 50 40 30 20 10 5 10 15 20 25 20 15 10 5
ans(:,:,6) =
4 8 12 16 20 16 12 8 4 8 16 24 32 40 32 24 16 8 12 24 36 48 60 48 36 24 12 16 32 48 64 80 64 48 32 16 20 40 60 80 100 80 60 40 20 16 32 48 64 80 64 48 32 16 12 24 36 48 60 48 36 24 12 8 16 24 32 40 32 24 16 8 4 8 12 16 20 16 12 8 4
ans(:,:,7) =
3 6 9 12 15 12 9 6 3 6 12 18 24 30 24 18 12 6 9 18 27 36 45 36 27 18 9 12 24 36 48 60 48 36 24 12 15 30 45 60 75 60 45 30 15 12 24 36 48 60 48 36 24 12 9 18 27 36 45 36 27 18 9 6 12 18 24 30 24 18 12 6 3 6 9 12 15 12 9 6 3
ans(:,:,8) =
2 4 6 8 10 8 6 4 2 4 8 12 16 20 16 12 8 4 6 12 18 24 30 24 18 12 6 8 16 24 32 40 32 24 16 8 10 20 30 40 50 40 30 20 10 8 16 24 32 40 32 24 16 8 6 12 18 24 30 24 18 12 6 4 8 12 16 20 16 12 8 4 2 4 6 8 10 8 6 4 2
ans(:,:,9) =
1 2 3 4 5 4 3 2 1 2 4 6 8 10 8 6 4 2 3 6 9 12 15 12 9 6 3 4 8 12 16 20 16 12 8 4 5 10 15 20 25 20 15 10 5 4 8 12 16 20 16 12 8 4 3 6 9 12 15 12 9 6 3 2 4 6 8 10 8 6 4 2 1 2 3 4 5 4 3 2 1

conv

Convolution and polynomial multiplication

Syntax

w = conv(u,v)

w = conv(u,v,shape)

Description

w = conv（u，v）返回向量u和v的卷積。如果u和v是多項式系數的向量，則對它們進行卷積相當於將兩個多項式相乘。

w = conv(u,v,shape) returns a subsection of the convolution, as specified by shape. For example, conv(u,v,‘same‘) returns only the central part of the convolution, the same size as u, and conv(u,v,‘valid‘) returns only the part of the convolution computed without the zero-padded edges.

w = conv（u，v，shape）返回卷積的子部分，由形狀指定。 例如，conv（u，v，‘same‘）僅返回卷積的中心部分，與u的大小相同，而conv（u，v，‘valid‘）僅返回計算後的卷積部分而沒有零填充邊。

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Polynomial Multiplication via Convolution

Create vectors u and v containing the coefficients of the polynomials x^2 + 1 and 2x + 7.

u = [1 0 1];
v = [2 7];

Use convolution to multiply the polynomials.

w = conv(u,v)

w = 1×4

     2     7     2     7

w contains the polynomial coefficients for 2x^3 + 7x^2 + 2x + 7.

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Vector Convolution

Create two vectors and convolve them.

u = [1 1 1];
v = [1 1 0 0 0 1 1];
w = conv(u,v)

w = 1×9

     1     2     2     1     0     1     2     2     1

The length of w is length(u)+length(v)-1, which in this example is 9.

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Central Part of Convolution

Create two vectors. Find the central part of the convolution of u and v that is the same size as u.

u = [-1 2 3 -2 0 1 2];
v = [2 4 -1 1];
w = conv(u,v,‘same‘)

w = 1×7

    15     5    -9     7     6     7    -1

w has a length of 7. The full convolution would be of length length(u)+length(v)-1, which in this example would be 10.

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matlab中卷積convolution與filter用法