转自：https://blog.csdn.net/dkcgx/article/details/46652021
转自：https://blog.csdn.net/Reborn_Lee/article/details/83279843
conv（向量卷积运算）
所谓两个向量卷积，说白了就是多项式乘法。 比如:p=[1 2 3],q=[1 1]是两个向量，p和q的卷积如下： 把p的元素作为一个多项式的系数，多项式按升幂（或降幂）排列，比如就按升幂吧，写出对应的多项式：1+2x+3x^2;同样的，把q的元素也作为多项式的系数按升幂排列，写出对应的多项式：1+x。
卷积就是“两个多项式相乘取系数”。 （1+2x+3x^2）×（1+x）=1+3x+5x^2+3x^3 所以p和q卷积的结果就是[1 3 5 3]。
记住，当确定是用升幂或是降幂排列后，下面也都要按这个方式排列，否则结果是不对的。 你也可以用matlab试试 p=[1 2 3] q=[1 1] conv(p,q) 看看和计算的结果是否相同。
conv2（二维矩阵卷积运算）
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维矩阵卷积运算）
>> 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的卷积。如果u和v是多项式系数的向量，则对它们进行卷积相当于将两个多项式相乘。
w = conv( returns a subsection of the convolution, as specified by u,v,shape)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）返回卷积的子部分，由形状指定。 例如，conv（u，v，\'same\'）仅返回卷积的中心部分，与u的大小相同，而conv（u，v，\'valid\'）仅返回计算后的卷积部分而没有零填充边。
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.
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.
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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