Digital Signal Processing Reference
In-Depth Information
From now on, we will consider that the factor 2 will be incorporated to
the step size, so that the iterative process to attain the Wiener solution is
given by
w
(
n
+
1
)
=
w
(
n
)
−
μ [
Rw
(
n
)
−
p
]
(3.49)
In order to evaluate the effectiveness of such iterative procedure, we must
analyze the equilibrium points of the steepest-descent algorithm as well as
its properties of convergence. In view of this objective, we can note that (3.49)
corresponds to a linear dynamical system whose state variables are the filter
coefficients. The equilibrium points of a discrete-time dynamical system are
the points that are invariant to the iterative process. In other words, these
points are the solutions of the following equation:
w
(
n
+
1
)
=
w
(
n
)
→
μ [
Rw
(
n
)
−
p
]
=
0.
(3.50)
which can be simplified to yield the equilibrium point
R
−
1
p
.
w
e
=
(3.51)
Thus, the system has a single equilibrium point, which is not surprising,
since the system is linear and this point is exactly the Wiener solution. How-
ever, the convergence is not guaranteed since we have not yet verified under
what conditions this equilibrium point is stable. In order to proceed with
this verification using concepts of dynamical system theory, let us first write
(3.49) in the following form:
w
(
n
+
1
)
=
[
I
−
μ
R
]
w
(
n
)
+
μ
p
(3.52)
The stability of such a system depends on the eigenstructure of the following
matrix [139]:
B
=
I
−
μ
R
.
(3.53)
If all the eigenvalues of
B
lie inside the unit circle, the Wiener solution will
be a stable equilibrium point. On the other hand, if a single eigenvalue is
outside the unit circle, the entire scheme will be compromised. Clearly, the
stability of the steepest-descent algorithm is strongly dependent on the step
size μ, which serves as a sort of control parameter. The eigenvalues of
B
are
λ
B
=
1
−
μλ
R
(3.54)
where λ
B
and λ
R
stand for a generic pair of eigenvalues. If all eigenvalues of
B
are to be inside the unit circle, it is necessary that the following condition
hold:
