Biomedical Engineering Reference
In-Depth Information
1 radian/s is investigated. The pole at zero
contributes a value of 0 toward the logarithmic gain. The pole at 2 contributes a value of -1 dB
To determine the constant
K,
the magnitude at o
¼
q
1
¼
2
2
20 log
j
2
þ
1
2
(
1
20 log
þ
¼
1 dB) toward the logarithmic gain. The reason that
the contribution of the pole at 2 was computed was that the point 1 radian/s was within the range
of
a decade of the break frequency. At o
¼
1 radian/s, the nonzero terms are
20 log
j
o
20 log
j
G
ðÞ
j ¼
20 log
K
2
þ
1
From the gain-frequency graph, 20 log
j
G
ðÞ
j ¼
17 dB. Therefore,
17
¼
20 log
K
1
or
K
¼
8. The model now consists of
8
G
ð
j
oÞ¼
j
o
2
þ
jw
1
The last term to investigate is whether there is a time delay in the system. At the break
frequency o
¼
2.0 radians/s, the phase angle from the current model should be
2
tan
1
o
135
fðÞ¼
90
¼
90
45
¼
This value is approximately equal to the data, and thus there does not appear to be a time
delay in the system.
Example Problem 13.9 illustrated a process of thinking in determining the structure and
parameters of a model. Carrying out an analysis in this fashion on complex systems is
extremely difficult, if not impossible. Software packages that automatically carry out estima-
tion of poles, zeros, a time delay, and a gain of a transfer function from data are available,
such as the System Identification toolbox in MATLAB. There are also other programs that
provide more flexibility than MATLAB in analyzing complex systems, such as the FORTRAN
program written by Seidel [43].
There are a variety of other inputs that one can use to stimulate the system to elicit a
response. These include such transient signals as a pulse, step, and ramp, and noise signals
such as white noise and pseudo random binary sequences. The reason these other tech-
niques might be of interest is that not all systems are excited via sinusoidal input. One such
system is the fast eye movement system. Here we typically use a step input to analyze the
system (see [17]).
In analyzing the output data obtained from step input excitation to determine the trans-
fer function, we use a frequency response method using the Fourier Transform and the fast
Fourier Transform. The frequency response of the input is known. The frequency response
of the output is calculated via a numerical algorithm called the fast Fourier Transform
(FFT). To calculate the FFT of the output, the data must first be digitized using an A/D
converter and stored in disk memory. Care must be taken to antialias (low-pass) filter
