Biomedical Engineering Reference
In-Depth Information
1.2
1
0.8
0.6
0.4
0.2
0
−
0.1
−
0.05
0
Time (sec)
0.05
0.1
FIGURE 11.16
MATLAB results showing the step response of a linear cell membrane. Dotted lines repre-
sent the step input. Continuous gray line shows the theoretical step response. Black line shows the simulated
step response. Note that the theoretical solution and simulated results are superimposed.
The results for the cell membrane step response are shown in Figure 11.16. Note that the
simulated (black) and theoretical (gray) outputs are precisely matched.
11.6.3 Frequency Domain Representation of Linear Systems
We have already considered the special case of linear systems in the frequency domain
for periodic inputs. Recall that the system output of a linear system to a periodic stimulus
is fully described by the system transfer function. The output of a linear system is also
expressed in the time domain by the convolution integral (Figure 11.15b). The impulse
response is the mathematical model that describes the linear system in the time domain.
These two descriptions for the input-output relationship of a linear system are mutually
related. Notably, for the aperiodic signal case, the transfer function is the Fourier transform
of the impulse response
Time Domain Frequency Domain
h
ð
t
Þ ,
H
ð
ð
11
:
34
Þ
o
Þ
where
(o) is the system transfer function. Since the impulse response is a complete model
of a linear system and since the Fourier transform is invertible (we can always go back and
forth between
H
(o)), the transfer function contains all the necessary information to
fully describe the system. The advantage of the transfer function comes in its simplicity of
use. Rather than performing a convolution integral, which can be quite intricate in many
h
(
t
) and
H
