Biomedical Engineering Reference
In-Depth Information
FIGURE 5-61
Comparison
between the step
responses of a
number of low-pass
filters.
In a digital implementation of the conventional method, the rectification process
is achieved by taking the absolute value of the signal before it is processed through a
low-pass algorithm such as the Butterworth filter discussed earlier in this chapter. The
cutoff frequency of the low-pass filter determines the modulation depth of the envelope,
with lower cutoff frequencies resulting in smaller modulation depths.
The Hilbert transform is a mathematical function that can represent a time waveform as
the product of a slowly varying envelope and a carrier containing fine structure information.
In mathematical terms, the filtered waveform,
x
i
(
t
)
, can be represented as
x
i
(
t
)
=
a
i
(
t
)
cos
f
i
(
t
)
(5.55)
where
a
i
(
represents the fine structure waveform.
Note that
f
i
(
t
)
is the instantaneous phase of the signal, and its derivative produces the
instantaneous frequency (the carrier frequency).
Figure 5-62 shows the outputs of a number of different envelope detectors used in a
cochlear implant. It can be seen that the Hilbert transform method reproduces the envelope
more accurately than the conventional envelope detector even with a filter bandwidth of
400 Hz.
t
)
represents the envelope, and cos
f
i
(
t
)
5.5.8 Spectral Estimation
The traditional method of frequency analysis is based on the Fourier transform, which is
generally evaluated using the discrete Fourier transform (DFT) or the fast Fourier transform
(FFT) if the number of points corresponds to
k
2
N
, where
N
is an integer.
=
