LMS for complex data and coefficients (such as quadrature
communication systems) takes the form
y
k
=
W
k
H
X
k
y
k
W
k
H
X
k
e
k
=
d
k
−
y
k
e
k
d
k
y
k
W
k
+
1
=
W
k
+2μ
e
k
*
X
k
W
k
+
1
W
k
2
μ
e
k
*
X
k
It is derived in exactly the same way as LMS, using the
following complex vector differentiation formulas
dd
W
P
H
W=0
W
P
H
W
0
dd
W
W
H
P=2P
W
W
H
P
2
P
dd
W
W
H
RW=2RW
W
W
H
R
W
2
R
W
or by differentiating with respect to the real and imaginary
parts separately and recombining the results.
"A good introduction in adaptive filters, a major DSP application."