Problem Statement 2D adaptive filters

general block diagram 2d adaptive filter.

in digital signal processing, linear shift invariant system can represented convolution of signal filter s impulse response, given expression:

y
(

n

1


,

n

2


)
=

∑


m

1





∑


m

2




w
(

m

1


,

m

2


)
x
(

n

1


−

m

1


,

n

2


−

m

2


)


{\displaystyle y(n_{1},n_{2})=\sum _{m_{1}}\sum _{m_{2}}w(m_{1},m_{2})x(n_{1}-m_{1},n_{2}-m_{2})}

if system model desired response



d
(

n

1


,

n

2


)


{\displaystyle d(n_{1},n_{2})}

, adaptive system can obtained continuously adjusting weight values according cost function



f
[
e
(

n

1


,

n

2


)
]


{\displaystyle f[e(n_{1},n_{2})]}

evaluates error between 2 responses.

e
(

n

1


,

n

2


)
=
d
(

n

1


,

n

2


)
−
y
(

n

1


,

n

2


)


{\displaystyle e(n_{1},n_{2})=d(n_{1},n_{2})-y(n_{1},n_{2})}








w

j
+
1


(

n

1


,

n

2


)
=

w

j


(

n

1


,

n

2


)
+
f
[
e
(

n

1


,

n

2


)
]


{\displaystyle w_{j+1}(n_{1},n_{2})=w_{j}(n_{1},n_{2})+f[e(n_{1},n_{2})]}