85
Wavelet Transform
600
500
400
300
200
100
0 0
1 0
2 0
3 0
(b)
Frequency
STFT
40
50
FIGURE 5.7 (continued) (b) The magnitude of STFT for a = 8.
weight to the central part of the captured events and therefore reduce the negative effect of the overlap between the neighboring events. Such windows are
becoming more popular in analysis of the biomedical signals in which events
often overlap and one may need to attempt to separate the events from each other
using triangular windows.
Despite the favorable properties of the STFT, this transform is not the best solution to address the time and frequency localization of the signal events. The first simple factor that identifies the shortcomings of the STFT is the choice of the window
length. A too short window may not capture the entire duration of an event, while
a too long window may capture two or more events in the same shift. For instance,
consider the signal in Example 5.3 and a window with duration 0.1. Such a time
window would never capture any of the events mentioned earlier even in its entirety.
At the same time, a time window with duration 10 could never capture only one of
the events without including at least a part of another event. In practice, we would
like to capture all events with any duration without supervising the transformation
(i.e., without manually adjusting the length of the window). This disadvantage of the
STFT calls for another transform in which all window sizes are tested.
Another disadvantage of the STFT deals with the nature of the basis functions
used in the FT, i.e., complex exponentials. The term e −j2 πft describes sinusoidal variations in real and complex spaces. Such sinusoidal functions exist in all times and are
not limited in time span or duration. However, by definition, events are variations
that are typically limited in time, i.e., they start at a certain point in time and end
at another. The fact that the sinusoidal basis functions that are time unlimited are
used to analyze the time-limited variations (i.e., events) explains why the STFT may
not be the best solution for even detection. At this point in our search for an ideal
transformation for even detection, it makes perfect sense to use time-limited basis
functions to decompose and analyze the time-limited events.
These two disadvantages of the STFT lead us to the definition of the WT that is
extremely useful for biomedical applications.
Wavelet Transform
600
500
400
300
200
100
0 0
1 0
2 0
3 0
(b)
Frequency
STFT
40
50
FIGURE 5.7 (continued) (b) The magnitude of STFT for a = 8.
weight to the central part of the captured events and therefore reduce the negative effect of the overlap between the neighboring events. Such windows are
becoming more popular in analysis of the biomedical signals in which events
often overlap and one may need to attempt to separate the events from each other
using triangular windows.
Despite the favorable properties of the STFT, this transform is not the best solution to address the time and frequency localization of the signal events. The first simple factor that identifies the shortcomings of the STFT is the choice of the window
length. A too short window may not capture the entire duration of an event, while
a too long window may capture two or more events in the same shift. For instance,
consider the signal in Example 5.3 and a window with duration 0.1. Such a time
window would never capture any of the events mentioned earlier even in its entirety.
At the same time, a time window with duration 10 could never capture only one of
the events without including at least a part of another event. In practice, we would
like to capture all events with any duration without supervising the transformation
(i.e., without manually adjusting the length of the window). This disadvantage of the
STFT calls for another transform in which all window sizes are tested.
Another disadvantage of the STFT deals with the nature of the basis functions
used in the FT, i.e., complex exponentials. The term e −j2 πft describes sinusoidal variations in real and complex spaces. Such sinusoidal functions exist in all times and are
not limited in time span or duration. However, by definition, events are variations
that are typically limited in time, i.e., they start at a certain point in time and end
at another. The fact that the sinusoidal basis functions that are time unlimited are
used to analyze the time-limited variations (i.e., events) explains why the STFT may
not be the best solution for even detection. At this point in our search for an ideal
transformation for even detection, it makes perfect sense to use time-limited basis
functions to decompose and analyze the time-limited events.
These two disadvantages of the STFT lead us to the definition of the WT that is
extremely useful for biomedical applications.