Digital Signal Processing Reference
In-Depth Information
(a)
(b)
(c)
(R)
=
1, = 0.
1
(R)
=
1, = 1.
5
(R) =
1
, =
2
.0
5
3
3
2.5
2.5
3
2
2
2
1.5
1.5
1
1
1
0
0.5
0.5
1.5
0
0
−1
0
1
−1
0
1
−2
−1
0
1
2
w
1
(n)
w
1
(n)
w
1
(n)
0
0
5
4
(R) = 1
(R) = 1
−50
−50
3
−100
−100
2
−150
−150
1
−200
−200
0
0
50
100
150
200
0
10
20
30
0
2
4
6
8
10
Iteration number
Iteration number
Iteration number
Fig. 3.1
Behavior of the steepest descent algorithm with
1. In the top plots showing the
contour curves of the error surface and the trajectory of the algorithm, the principal axes (i.e., the
eigenvectors of
R
x
) are shown centered at
w
opt
, with the dashed line representing the eigenvector
associated to
χ(
R
x
)
=
λ
max
. The inial guess is
w
(
−
1
)
=
0
.
a
μ
=
0
.
1.
b
μ
=
1
.
5.
c
μ
=
2
.
05
where the conditioning number will be a parameter that will be varied. This means
that
λ
max
=
μ
(
,
)
to ensure
stability. To simplify the description, we define each convergence mode for the SD
algorithm as
1, so the step size
needs to be chosen in the interval
0
2
mode
max
=
1
−
μλ
max
=
1
−
μ,
mode
min
=
1
−
μλ
min
=
1
−
μ/χ(
R
x
)
(3.26)
We will show the evolution of the estimate
w
in relation to the contour plots of
the error surface and we study the performance dynamics of the algorithm using the
mismatch
, which is defined as
(
n
)
2
10 log
10
w
(
n
)
−
w
opt
.
(3.27)
w
opt
2
I
L
.InFig.
3.1
we study the SD algorithm and use different step sizes to represent three different
regimes: 0
We start with
χ(
R
x
)
=
1, which in this case means that
R
x
=
I
L
,the
contour curves are circles, so the direction normal to these curves (i.e., the direction
of the gradient) points towards the minimum and the algorithm evolves following
a straight line trajectory (i.e., at equal speed along both principal axes). On the
left panel we see the case
< μ <
1
/λ
max
,1
/λ
max
< μ <
2
/λ
max
and
μ >
2
/λ
max
.As
R
x
=
9. Since the
modes are positive, the convergence is overdamped so there is no change of sign
μ
=
0
.
1, so mode
max
=
mode
min
=
0
.
