Environmental Engineering Reference
In-Depth Information
Ψ
(t)
Ψ
(t)
a
1
1
a+b
0
1
0
t
b
t
-1
-
1
(a)
(b)
Figure 4.26
Basic Haar wavelet Ψ
(
t
)
with unit scale and zero displacement (a) and with scale
a
and displacement
b
(b)
forms is that while in (4.147) the function
y
is multiplied by a function having
finite support -
i.e.
, it vanishes outside a finite interval - in the Fourier sine trans-
form of
y
(
t
)
(
t
)
is multiplied by a sine wave which is different from zero in intervals
[
for all
t
. In addition, while the Fourier transform is a function of
only one variable - frequency - the CWT is a function of two variables: scale and
displacement.
If
y
t
,
t
+
T
](
T
>
0
)
is assumed to be a wide sense stationary random process (at least within
a given time window), the expected value of
W
y
(
t
)
(
a
,
b
)
is zero, and its variance
Var
{
W
y
(
a
,
b
)}
does not depend on the displacement
b
[22],
i.e.
,
W
y
E
{
W
y
(
a
,
b
)} =
0
,
Var
{
W
y
(
a
,
b
)} =
E
{
(
a
,
b
)} =
V
y
(
a
).
(4.148)
4.4.2.5 Templates
=
(
,
)
When the grindability index Γ
Γ
i
is known, let
V
1
a
Γ
i
be the variance of the
CWT for the feed ore flow
F
f
(
t
,
Γ
i
)
,
V
2
(
a
,
Γ
i
)
for the power draw
P
d
(
t
,
Γ
i
)
,and
V
3
(
a
,
Γ
i
)
for the return ore flow
F
r
(
t
,
Γ
i
)
.
3.
These CWT variances for known grindability indexes serve as templates for esti-
mating the current grindability of the ore in the SAG mill. They may be determined
off-line using historical plant data [22]. Since the expected value of the measured
variable
y
Figure 4.27 shows
V
1
(
a
,
Γ
i
)
for hard
(
i
=
1
)
,medium
i
=
2 and soft ores
i
=
is not taken into account by (4.147) because of (4.148), the expected
value has been added to the CWT variance, so
V
j
(
t
)
(
a
,
Γ
i
)
shall be considered to in-
clude this addition [22].
Also shown in Figure 4.27 is the CWT of the sample variance template
S
1
(
a
,
Γ
)
determined on-line using the current measurement of feed ore flow
F
f
-in-
cluding its mean - in a given time window when the current grindability index is
unknown and must be estimated. Then the current grindability index is the one for
which the sample template
S
1
(
t
,
Γ
i
)
(
a
,
Γ
)
is closest to,
e.g.
, as determined by the mini-
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