Databases Reference
In-Depth Information
If the window function
g
is a Gaussian, the STFT is called the
Gabor transform
.
The problem with the STFT is the fixed window size. Consider Figure
15.1
. In order to
obtain the low-pass component at the beginning of the function, the window size should be
at least
t
0
so that the window will contain at least one cycle of the low-frequency component.
However, a window size of
t
0
or greater means that we will not be able to accurately localize
the high-frequency spurt. A large window in the time domain corresponds to a narrow filter
in the frequency domain, which is what we want for the low-frequency components—and
what we do not want for the high-frequency components. This dilemma is formalized in the
uncertainty principle, which states that for a given wind
ow
g
(
t
)
(
t
)
, the product of the time spread
is lower bounded by
√
1
t
2
ω
σ
and the frequency spread
σ
/
2, where
t
2
2
dt
|
)
|
g
(
t
2
t
σ
=
|
(2)
2
dt
g
(
t
)
|
ω
2
2
d
|
G
(ω)
|
ω
2
σ
ω
=
|
(3)
2
d
G
(ω)
|
ω
t
Thus, if we wish to have finer resoluti
o
n in time, that is, reduce
σ
, we end up with an increase
2
ω
in
or a lower resolution in the frequency domain. How do we get around this problem?
Let's take a look at the discrete STFT in terms of basis expansion, and, for the moment,
let's look at just one interval:
σ
∞
g
∗
(
e
−
jm
ω
0
t
dt
F
(
m
,
0
)
=
f
(
t
)
t
)
(4)
−∞
e
j
2
ω
o
t
, and so on. The first three basis functions
are shown in Figure
15.2
. We can see that we have a window with constant size, and within
this window, we have sinusoids with an increasing number of cycles. Let's conjure up a
different set of functions in which the number of cycles is constant, but the size of the window
keeps changing, as shown in Figure
15.3
. Notice that although the number of cycles of the
sinusoid in each window is the same, as the size of the window gets smaller, these cycles occur
in a smaller time interval; that is, the frequency of the sinusoid increases as the size of the
window gets smaller. The lower frequency functions cover a longer time interval, while the
higher frequency functions cover a shorter time interval, thus avoiding the problem that we
had with the STFT. If we can write our function in terms of these functions and their translates,
we have a representation that gives us time and frequency localization and can provide high
frequency resolution at low frequencies (longer time window) and high time resolution at high
frequencies (shorter time window). This, crudely speaking, is the basic idea behind wavelets.
e
j
ω
o
t
The basis functions are
g
(
t
)
,
g
(
t
)
,
g
(
t
)
F I GU R E 15 . 2
The first three STFT basis functions for the first time interval.
