Biomedical Engineering Reference
In-Depth Information
applications, the output of a linear system in the frequency domain is expressed as a prod-
uct of the system input and its transfer function:
Y ð
o
Þ¼ X ð
o
Þ H ð
o
Þ
. This result is reminis-
cent of
the result
for the Fourier series (Eqs.
(11.30) and (11.31)). Specifically,
the
convolution property
of the Fourier transform states that a convolution in time corresponds
to a multiplication in the frequency domain
Time Domain Frequency Domain
y ð t Þ¼ x ð t Þ * h ð t Þ, Y ð
ð
11
:
35
Þ
o
Þ¼ X ð
o
Þ H ð
o
Þ
and thus the output of a linear system,
Y
(o), is expressed in the frequency domain by the
product of
(o)(Figure 11.15c). Note that this result is essentially the convolution
theorem (Eqs. (11.14) and (11.15); for proof, see Example Problem 11.19) applied to the out-
put of a linear system (Eq. (11.32)). In many instances, this is significantly easier to compute
than a direct convolution in the time domain.
X
(o) and
H
EXAMPLE PROBLEM 11.19
Prove the convolution property of the Fourier transform.
Solution
ð
ðð
y ð t Þ e j o t dt ¼
x ðtÞ h ð t d t e j o t dt
Y ðoÞ¼ FT y ð t fg¼
Make a change of variables,
u ¼ t t,
du ¼ dt
ðð
ð
ð
x ðtÞ h ð u Þ d t e j u þtÞ du ¼
x ðtÞ e j ot
h ð u Þ e j o u du ¼ X ðoÞ H ðoÞ
¼
d t
11.6.4 Analog Filters
Filters are a special class of linear systems that are widely used to manipulate the proper-
ties of a biological signal. Conceptually, a filter allows the user to selectively remove an
undesired signal component while preserving or enhancing some component of interest.
Although most of us are unaware of this, various types of filters are commonplace in every-
day settings. Sunblock, for instance, is a type of filter that removes unwanted ultraviolet
light from the sun in order to minimize the likelihood of sunburn and potentially reduce
the risk of skin cancer. Filters are also found in many audio applications. Treble and bass
controls in an audio system are a special class of filter that the user selectively adjusts to
boost or suppress the amount of high-frequency (treble) and low-frequency (bass) sound
to a desired level and quality.
Filters play an important role in the analysis of biological signals, in part because
signal measurements in clinical settings are often confounded by undesirable noise.
Such noise distorts the signal waveform of interest, making it difficult to obtain a reliable
diagnosis. Ideally, if one could completely remove unwanted noise, one could signifi-
cantly improve the quality of a signal and thus minimize the likelihood of an incorrect
diagnosis.
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