Signal Processing - September 2017 - 186

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can be implemented with the SDFT (at
the cost of increased complexity), while
the SWIFT is limited to the exponential
window. The SDFT is also more directly
comparable to the FFT.

Rectangular
Hanning
Exponential
α

-10

(dB)

-20

The a SWIFT

-30

The spectral leakage of the SWIFT
can be further mitigated by removing
the exponential window's discontinuity at m = 0. The discontinuity can
be removed by modifying the window function to be the difference of
two exponentials:

-40
-50
-60
-π

-1π
2

0
(ω)

1π
2

π

w a [m] = '

FIGURE 2. The normalized Fourier transform of four windows [rectangular (blue, N = 20), Hanning
(black, N = 20), exponential (green, x = 14.4) ] and of a (red, x slow = 14.4, x fast = 2.89).

between time and frequency resolution.
Conversely, the SWIFT's output, as a
type of DTFT, is continuous in the frequency domain, providing the SWIFT
with great flexibility in tuning the frequencies of interest.

Time-frequency tradeoff
When operating multiple SWIFTs in
parallel, each time constant can be tuned
to the frequency bin of interest without
increasing computational complexity,
e.g., x can be set as a multiple of the
period such that x = c/f, where c is a
unitless constant and f is the center frequency. Conversely, to achieve a similar
effect with parallel SDFTs, one must add
additional comb filters for each SDFT
bin, further increasing computational
complexity and memory requirements.
This allows the SWIFT algorithm to be
implemented with a multiresolution property, similar to a wavelet transform,
which provides better time resolution at
higher frequencies and better frequency
resolution at lower frequencies.

Stability
The SWIFT is guaranteed stable,
whereas the SDFT is only marginally
stable. The SWIFT is guaranteed stable because its pole resides within the
z-domain's unit circle. In contrast, the
SDFT's pole resides on the z-domain's
unit circle, which can lead to instabilities if numerical rounding causes the
pole to move outside the unit circle.
186

To guarantee stability, the SDFT must
add a damping factor, but this causes the
SDFT's output no longer to be exactly
equivalent to the N-point DFT. Other
SDFTs have been developed that are
both accurate and guaranteed stable, but
at the cost of increased computational
complexity [5].

Spectral leakage
The SWIFT's exponential window
reduces spectral leakage compared to the
SDFT's rectangular window, as shown
in Figure 2. It is difficult to compare the
leakage of finite-length windows to infinite-length windows directly; therefore,
instead of requiring that each window
have the same length, we required that
each window have the same halfmass, i.e.,
the length of the window in which half
the area is contained. For instance, a rectangular window of length N = 20 and an
exponential window with x = 14.43 both
have a halfmass of ten. The exponential
window has a narrower main lobe and
smoother falloff compared to the rectangular window. We can further reduce the
SWIFT's spectral leakage with another
window, which we will introduce in the
aSWIFT algorithm.
Despite these advantages, there may
be situations in which the traditional
SDFT is called for. For instance, the
sharpness of the SWIFT/ aSWIFT's
window may be too narrow for some applications that require tracking a broad
oscillation. Additionally, any window
IEEE SIGNAL PROCESSING MAGAZINE

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September 2017

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e m/x slow - e m/x fast m # 0
,
0
m20
(7)

where x slow2 x fast 2 0. We will refer
to this as the a function, which goes
smoothly to zero at m = 0 .
Figure 2 compares the spectral
leakage of the a window with the exponential, rectangular, and Hanning
windows. We have chosen to compare
the a SWIFT to the Hanning SDFT,
which is among the simplest windowed
SDFTs and was presented by Jacobsen
and Lyons in 2003 [2]. As compared to
the exponential window, the a window
has a similarly narrow main lobe but
significantly faster fall off at surrounding frequencies. On the other hand, the
Hanning window has a significantly
wider main lobe, but its side lobes fall
off faster than the a function's.

Derivation
The aSWIFT cannot be derived using
the same method as the SWIFT because
w a [0] = 0, and so the aSWIFT cannot
be written as a difference equation in
the form of X n (~) a = aX n - 1 (~) a + x [n].
However, the aSWIFT can be solved as
the difference between two SWIFTs with
different time constants through the linearity property of the Fourier transform:
X n (~) a = X n (~) slow - X n (~) fast, (8)
where X n (~) a is the a SWIFT a nd
X n (~) slow and X n (~) fast are individual
SWIFTs with x 's equal to the slow and
fast time constants, respectively. We call
this form of the aSWIFT the parallel
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