Convolution and polynomial multiplication
https://www.mathworks.com/help/matlab/ref/conv.html?s_tid=gn_loc_drop
conv
Convolution and polynomial multiplication
Description
w = conv(
returns the convolution of vectors u,v
)u
and v
. If u
and v
are vectors of polynomial coefficients, convolving them is equivalent to multiplying the two polynomials.
Examples
Polynomial Multiplication via Convolution
Create vectors u
and v
containing the coefficients of the polynomials and .
u = [1 0 1];
v = [2 7];
Use convolution to multiply the polynomials.
w = conv(u,v)
w = 2 7 2 7
w
contains the polynomial coefficients for .
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 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 = 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.
Input Arguments
u,v
— Input vectors
vectors
Input vectors, specified as either row or column vectors. The
output vector is the same orientation as the first input argument, u
.
The vectors u
and v
can be different
lengths or data types.
Data Types: double
| single
Complex Number Support: Yes
shape
— Subsection of convolution
'full'
(default) | 'same'
| 'valid'
Subsection of the convolution, specified as 'full'
, 'same'
,
or 'valid'
.
'full' |
Full convolution (default). |
'same' |
Central part of the convolution of the same size as |
'valid' |
Only those parts of the convolution that are computed |
Convolution
@向量的卷积 重叠面积
The convolution of two vectors, u
and v
, represents the area of overlap under the points as v
slides across u
. Algebraically, convolution is the same operation as multiplying polynomials whose coefficients are the elements of u
and v
.
Let m = length(u)
and n = length(v)
. Then w
is the vector of length m+n-1
whose k
th element is
The sum is over all the values of j
that lead to legal subscripts for u(j)
and v(k-j+1)
, specifically j
=
max(1,k+1-n):1:min(k,m)
. When m
=
n
, this gives
w(1) = u(1)*v(1)
w(2) = u(1)*v(2)+u(2)*v(1)
w(3) = u(1)*v(3)+u(2)*v(2)+u(3)*v(1)
...
w(n) = u(1)*v(n)+u(2)*v(n-1)+ ... +u(n)*v(1)
...
w(2*n-1) = u(n)*v(n)
https://www.zhihu.com/question/22298352?rf=21686447
“
:
卷积就是带权的积分
:
从概率论的角度来理解吧,举例为X Y 两组连续型随机变量,那么令Z=X+Y ,当X Y两组变量独立时,就能推导出卷积公式了,fz=fx*fy的意义就是在于两组变量叠加出来的概率密度,也就是算两信号X Y混叠起来的时候的响应
:
他的女儿是做环保的,有一次她接到一个项目,评估一个地区工厂化学药剂的污染(工厂会排放化学物质,化学物质又会挥发散去),然后建模狮告诉她药剂的残余量是个卷积。她不懂就去问她爸爸,prof就给她解释了。假设t时刻工厂化学药剂的排放量是f(t) mg,被排放的药物在排放后Δt时刻的残留比率是g(Δt) mg/mg;那么在u时刻,对于t时刻排放出来的药物,它们对应的Δt=u-t,于是u时刻化学药剂的总残余量就是∫f(t)g(u-t)dt,这就是卷积了。
”
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