在数值分析 和泛函分析 领域中，离散小波变换 （Discrete Wavelet Transform，DWT）是小波被离散采样的小波变换 。与其他小波变换一样，它与傅里叶变换相比的一个关键优势是时间分辨率：它既能捕获频率信息，又能捕获位置（时间上的位置）信息。
第一个离散小波变换由匈牙利 数学家哈尔 发明，离散小波转换顾名思义就是离散的输入以及离散的输出，但是这里并没有一个简单而明确的公式来表示输入及输出的关系，只能以阶层式架构来表示。
首先我们定义一些需要用到的信号及滤波器。
x[n]：离散的输入信号，长度为N。
g
[
n
]
{\displaystyle g[n]}
h
[
n
]
{\displaystyle h[n]}
↓
{\displaystyle \downarrow }
举例说明: 清楚规定以上符号之后，便可以利用阶层架构来介绍如何将一个离散信号作离散小波转换：
架构中的第1层(1st stage)
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L
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g
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{\displaystyle x_{1,L}[n]=\sum _{k=0}^{K-1}x[2n-k]g[k]}
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{\displaystyle x_{1,H}[n]=\sum _{k=0}^{K-1}x[2n-k]h[k]}
架构中的第2层(2nd stage)
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{\displaystyle x_{2,L}[n]=\sum _{k=0}^{K-1}x_{1,L}[2n-k]g[k]}
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x
1
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h
[
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{\displaystyle x_{2,H}[n]=\sum _{k=0}^{K-1}x_{1,L}[2n-k]h[k]}
可继续延伸
⋮
{\displaystyle \vdots }
⋮
{\displaystyle \vdots }
架构中的第
α
{\displaystyle \alpha }
α
−
t
h
{\displaystyle \alpha -th}
x
α
,
L
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g
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{\displaystyle x_{\alpha ,L}[n]=\sum _{k=0}^{K-1}x_{\alpha -1,L}[2n-k]g[k]}
x
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H
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{\displaystyle x_{\alpha ,H}[n]=\sum _{k=0}^{K-1}x_{\alpha -1,L}[2n-k]h[k]}
注意：若输入信号
x
[
n
]
{\displaystyle x[n]}
α
{\displaystyle \alpha }
x
α
,
L
[
n
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{\displaystyle x_{\alpha ,L}[n]}
x
α
,
H
[
n
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{\displaystyle x_{\alpha ,H}[n]}
N
2
α
{\displaystyle {\frac {N}{2^{\alpha }}}}
此时的输入信号变成
x
[
m
,
n
]
{\displaystyle x[m,n]}
首先对n方向作高通、低通以及降频的处理
v
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x
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g
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{\displaystyle v_{1,L}[m,n]=\sum _{k=0}^{K-1}x[m,2n-k]g[k]}
v
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x
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[
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{\displaystyle v_{1,H}[m,n]=\sum _{k=0}^{K-1}x[m,2n-k]h[k]}
接著对
v
1
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L
[
m
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{\displaystyle v_{1,L}[m,n]}
v
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{\displaystyle v_{1,H}[m,n]}
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L
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g
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{\displaystyle x_{1,LL}[m,n]=\sum _{k=0}^{K-1}v_{1,L}[2m-k,n]g[k]}
x
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h
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{\displaystyle x_{1,HL}[m,n]=\sum _{k=0}^{K-1}v_{1,L}[2m-k,n]h[k]}
x
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v
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g
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{\displaystyle x_{1,LH}[m,n]=\sum _{k=0}^{K-1}v_{1,H}[2m-k,n]g[k]}
x
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H
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{\displaystyle x_{1,HH}[m,n]=\sum _{k=0}^{K-1}v_{1,H}[2m-k,n]h[k]}
经过(1)(2)两个步骤才算完成2-D DWT的一个stage。
在讨论复杂度之前，先做一些定义，当x[n]*y[n]时，x[n]之长度为N，y[n]之长度为L：
⇒
I
D
F
T
N
+
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[
D
F
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(
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[
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)
]
{\displaystyle \ \Rightarrow IDFT_{N+L-1}\left[DFT_{N+L-1}(x[n])DFT_{N+L-1}(y[n])\right]}
其中，
I
D
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+
L
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1
{\displaystyle \ IDFT_{N+L-1}}
D
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+
L
−
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{\displaystyle \ DFT_{N+L-1}}
(1)一维离散小波转换之复杂度(没有分段卷积(sectioned convolution))：
3
2
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N
+
L
−
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log
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≈
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log
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{\displaystyle {\frac {3}{2}}(N+L-1)\log _{2}{(N+L-1)}\approx {\frac {3}{2}}N\log _{2}N}
(2)当 N >>> L 时，使用 “分段卷积(sectioned convolution)”的技巧：
将x[n]切成很多段，每段长度为
N
1
{\displaystyle \mathbb {N} _{1}}
S
=
N
N
1
{\displaystyle S={\frac {N}{N_{1}}}}
N
>
N
1
>>
L
{\displaystyle N>N_{1}>>L}
则
x
[
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∗
g
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s
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{\displaystyle \ x[n]*g[n]=x_{1}[n]*g[n]+x_{2}[n]*g[n]+...+x_{s}[n]*g[n]}
x
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{\displaystyle \ x[n]*h[n]=x_{1}[n]*h[n]+x_{2}[n]*h[n]+...+x_{s}[n]*h[n]}
复杂度为：
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S
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{\displaystyle {\begin{aligned}{\frac {3}{2}}S(N_{1}+L-1)\log _{2}{(N_{1}+L-1)}&\approx {\frac {3}{2}}SN_{1}\log _{2}(N_{1}+L-1)\\&\approx {\frac {3}{2}}N\log _{2}(N_{1}+L-1)\\&\approx {\frac {3}{2}}N\log _{2}N_{1}\\\end{aligned}}}
在这里要注意的是，当N>>L时，一维离散小波转换之复杂度是呈线性的(随N)，
O
(
N
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{\displaystyle {\mathit {O(N)}}}
(3)多层(Multiple stages )的情况下：
1.若
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{\displaystyle \ x_{a,H}[n]}
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{\displaystyle {\begin{aligned}Complexity&\approx \left(N+{\frac {N}{2}}+{\frac {N}{4}}+{\frac {N}{8}}+...+2\right){\frac {3}{2}}\log _{2}N_{1}\\&=(2N-2){\frac {3}{2}}\log _{2}N_{1}\\&\approx 3N\log _{2}N_{1}\\\end{aligned}}}
2.若
x
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[
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{\displaystyle \ x_{a,H}[n]}
C
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{\displaystyle {\begin{aligned}Complexity&\approx \left(N+2{\frac {N}{2}}+4{\frac {N}{4}}+8{\frac {N}{8}}+...+{\frac {N}{2}}2\right){\frac {3}{2}}\log _{2}N_{1}\\&=(N\log _{2}N){\frac {3}{2}}\log _{2}N_{1}\\\end{aligned}}}
(4)二维离散小波转换之复杂度(没有分段卷积(sectioned convolution))：
⇒
M
3
2
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+
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−
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)
log
2
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+
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+
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2
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log
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{\displaystyle \ \Rightarrow M{\frac {3}{2}}(N+L-1)\log _{2}{(N+L-1)}+(N+L-1){\frac {3}{2}}(M+L-1)\log _{2}{(M+L-1)}}
上式中，第一部分需要M个一维离散小波转换并且每个一维离散小波转换的输入有N个点；第二部分需要N+L-1个一维离散小波转换并且每个一维离散小波转换的输入有M个点。
C
o
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p
l
e
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i
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y
≈
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+
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=
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)
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log
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{\displaystyle {\begin{aligned}Complexity&\approx {\frac {3}{2}}MN\log _{2}N+{\frac {3}{2}}MN\log _{2}M\\&={\frac {3}{2}}MN(\log _{2}N+\log _{2}M)\\&={\frac {3}{2}}MN\log _{2}MN\\\end{aligned}}}
(5)二维离散小波转换之复杂度，使用 “分段卷积(sectioned convolution)”的技巧：
假设原始尺寸为
M
×
N
{\displaystyle M\times N}
M
1
×
N
1
{\displaystyle M_{1}\times N_{1}}
C
o
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{\displaystyle {\begin{aligned}Complexity&\approx {\frac {MN}{M_{1}N_{1}}}{\frac {3}{2}}M_{1}N_{1}\log _{2}M_{1}N_{1}\\&={\frac {3}{2}}MN\log _{2}M_{1}N_{1}\\\end{aligned}}}
所以若是使用分段折积，则二维离散小波转换之复杂度是呈线性的(随MN)，
O
(
M
N
)
{\displaystyle {\mathit {O(MN)}}}
(6)多层(Multiple stages )与二维的情况下：
首先x[m,n]的尺寸为
M
×
N
{\displaystyle M\times N}
1.若
x
a
,
H
1
[
n
]
,
x
a
,
H
2
[
n
]
,
x
a
,
H
3
[
n
]
{\displaystyle \ x_{a,H_{1}}[n],x_{a,H_{2}}[n],x_{a,H_{3}}[n]}
x
a
,
L
[
n
]
{\displaystyle \ x_{a,L}[n]}
t
o
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=
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M
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+
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+
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+
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.
)
3
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≈
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=
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log
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{\displaystyle {\begin{aligned}totalcomplexity&=\left(MN+{\frac {MN}{4}}+{\frac {MN}{16}}+...\right){\frac {3}{2}}\log _{2}M_{1}N_{1}\\&\approx {\frac {4}{3}}MN{\frac {3}{2}}\log _{2}M_{1}N_{1}\\&=2MN\log _{2}M_{1}N_{1}\\\end{aligned}}}
2.若
x
a
,
H
1
[
n
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,
x
a
,
H
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[
n
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,
x
a
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[
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{\displaystyle \ x_{a,H_{1}}[n],x_{a,H_{2}}[n],x_{a,H_{3}}[n]}
t
o
t
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l
c
o
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p
l
e
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=
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M
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+
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+
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3
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=
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]
3
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{\displaystyle {\begin{aligned}totalcomplexity&=\left(MN+4{\frac {M}{2}}{\frac {N}{2}}+16{\frac {M}{4}}{\frac {N}{4}}+...\right){\frac {3}{2}}\log _{2}M_{1}N_{1}\\&=\left[MN\log _{2}(min(M,N))\right]{\frac {3}{2}}\log _{2}M_{1}N_{1}\\\end{aligned}}}
压缩、去除杂讯：使用低通滤波器，将小波转换的高频滤掉，即保留
x
1
,
L
L
[
m
,
n
]
{\displaystyle x_{1,LL}[m,n]}
边缘侦测：使用高通滤波器，将小波的低频滤掉，即保留
x
1
,
H
L
[
m
,
n
]
{\displaystyle x_{1,HL}[m,n]}
x
1
,
L
H
[
m
,
n
]
{\displaystyle x_{1,LH}[m,n]}
R语言小波分析wavelet （页面存档备份 ，存于互联网档案馆 ）作为 JPEG2000 的内部架构
模式辨认：由于可以利用低频的部分得到原图的缩略版，加上模式通常为整体的特性，借由在缩略图上进行工作，小波转换可以有效减少寻找模式与比对模式的运算时间
滤波器设计：小波转换保留部分时间资讯，可以据此资讯加上讯号的强度资讯，保留特定时点的资讯而同时去除杂讯 
![{\displaystyle g[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5e1d771a2385e9aeb71838a40425bb07c89525)
![{\displaystyle h[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89981bbbb05ffd469eeadb828c18359965985e46)
![{\displaystyle x_{1,L}[n]=\sum _{k=0}^{K-1}x[2n-k]g[k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e89171201a089a787ca8eecf3cfcc0a997ca2687)
![{\displaystyle x_{1,H}[n]=\sum _{k=0}^{K-1}x[2n-k]h[k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f25f3aaaa0a147dc3d27be4ffa8ec69966f512b)
![{\displaystyle x_{2,L}[n]=\sum _{k=0}^{K-1}x_{1,L}[2n-k]g[k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/778ad7dd66c5fb41a730549fa7a229eb16187cbf)
![{\displaystyle x_{2,H}[n]=\sum _{k=0}^{K-1}x_{1,L}[2n-k]h[k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a655d7ed52b22f9cdddc4212e23b84da4fd260f1)
![{\displaystyle x_{\alpha ,L}[n]=\sum _{k=0}^{K-1}x_{\alpha -1,L}[2n-k]g[k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b3237f86646ab171917e1d6fbab78eb53ad7496)
![{\displaystyle x_{\alpha ,H}[n]=\sum _{k=0}^{K-1}x_{\alpha -1,L}[2n-k]h[k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36145d18b7c0b1f46f34d3321d019964fb06b9c6)
![{\displaystyle x[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/864cbbefbdcb55af4d9390911de1bf70167c4a3d)
![{\displaystyle x_{\alpha ,L}[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bee354c65204f381107da41675caf54fdd0f8af0)
![{\displaystyle x_{\alpha ,H}[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a985602660a89b0b360223e0d07cd3e8c522b0c8)
![{\displaystyle x[m,n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9745486e9c980ded7be564ca034810c0502193a5)
![{\displaystyle v_{1,L}[m,n]=\sum _{k=0}^{K-1}x[m,2n-k]g[k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb581d19be3dc84b6f66476669af51bb565f115)
![{\displaystyle v_{1,H}[m,n]=\sum _{k=0}^{K-1}x[m,2n-k]h[k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adf51b66814e213300e947da548d3bdc2d77c4ea)
![{\displaystyle v_{1,L}[m,n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2c2f9e57456cfe1f1bb72f65e453ddcd3e328b6)
![{\displaystyle v_{1,H}[m,n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6aaed46292fa3e843bdcc62f065f819508c5965)
![{\displaystyle x_{1,LL}[m,n]=\sum _{k=0}^{K-1}v_{1,L}[2m-k,n]g[k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5699b1611ae488b196dfeb4b12a5ffbe17a7b756)
![{\displaystyle x_{1,HL}[m,n]=\sum _{k=0}^{K-1}v_{1,L}[2m-k,n]h[k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1bc0b122f88e2ec5cd699913f5291c491ac9a9f)
![{\displaystyle x_{1,LH}[m,n]=\sum _{k=0}^{K-1}v_{1,H}[2m-k,n]g[k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6657c6da1e42df215ace930f89aae0da4fa6b5f6)
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![{\displaystyle \ x_{a,L}[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11c7a580e0467ccd1a109d8d7862ff79c29cd707)
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![{\displaystyle X_{1,L}(z)={\frac {1}{2}}[X(z^{\frac {1}{2}})G(z^{\frac {1}{2}})+X(-z^{\frac {1}{2}})G(-z^{\frac {1}{2}})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a5ca6b242399e9142d1785f12bed631cf884fa)
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![{\displaystyle X_{0}(z)={\tfrac {1}{2}}[X(z)G(z)+X(-z)G(-z)]G_{1}(z)+{\tfrac {1}{2}}[X(z)H(z)+X(-z)H(-z)]H_{1}(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a41efe9f8250241456370898ec832227ac1daa37)
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![{\displaystyle x_{1,LL}[m,n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d8608023ad34350b051b07146e3fcab335cc9cd)
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