Biomedical Engineering Reference
In-Depth Information
Using Westheimer's parameter values with any arbitrary saccade magnitude given by
g
K
D
y
¼
and Eq. (13.10) gives
y
ð
T
mv
Þ¼
55
:
02
D
y
ð
13
:
13
Þ
Equation (13.13) indicates that peak velocity is directly proportional to saccade magnitude.
As illustrated in the main sequence diagram shown in Figure 13.6, experimental peak veloc-
ity data have an exponential form and do not represent a linear function as predicted by the
Westheimer model. Both Eqs. (13.7) and (13.13) are consistent with linear systems theory.
That is, for a step input to a linear system, the duration (and time to peak overshoot) stays
constant regardless of the size of the input, and the peak velocity increases with the size of
the input.
EXAMPLE PROBLEM 13.3
Show that Eq. (13.12) follows from (13.11).
Solution
Beginning with Eq. (13.11), we have
g
zo
n
e
zo
n
t
zo
n
cos o
d
t
þ
p
1
ð
ð
c
Þ þ
o
d
sin o
d
t
þ
ð
c
Þ
Þ
z
2
K
¼
þ
e
zo
n
t
zo
n
o
d
sin o
d
t
þ
ð
c
Þ þ
o
2
d
cos o
d
t
þ
ð
c
Þ
0
The terms multiplying the sinusoids are removed, since they do not equal zero. Therefore,
cos o
d
t
þ c
d
z
2
o
2
o
2
ð
Þ
2zo
n
o
d
sin o
d
t
þ c
ð
Þ ¼
0
n
which reduces to
z
2
o
2
n
o
2
d
sin o
d
t
þ
ð
c
Þ
¼
Þ
¼
tan o
d
t
þ c
ð
Þ
ð
13
:
14
Þ
2zo
n
o
d
cos o
d
t
þ c
ð
into Eq. (13.14), we have
z
2
Substituting o
d
¼ o
n
1
z
2
o
2
n
z
2
2z
2
2z
o
2
n
1
1
¼
p
¼
tan o
d
t
þ c
ð
Þ
ð
13
:
15
Þ
z
2
2zo
n
o
n
1
z
2
1
z
1
z
1
With tan c ¼
p
, we factor out
p
in Eq. (13.15), substitute tan c, giving
z
2
z
2
tan c ¼
2z
2
2z
1
z
1
1
2z
2
1
2z
2
p
p
¼
1
¼
1
tan o
d
t
þ c
ð
Þ
ð
13
:
16
Þ
z
2
z
2
1
Now
tan o
d
ðÞþ
tanc
tan o
d
t
þ c
ð
Þ ¼
ð
13
:
17
Þ
1
tan o
d
ðÞ
tan c
