Signal Processing - September 2017 - 187
1
1
0
0
t = 25
5
-1
0
t = 26
10
15
Sample
(a)
20
25
0
20
40
60
Imag
80
100
Sample
(b)
×
e -1/τslowe j ω
X n -1(ω)slow
z -1
+
1
x [n]
Imag
-1
30
Real
+
ω
0
-
+
e -1/τfaste j ω
-1
-1
0
Real
(c)
1
×
X n (ω)α
×
x [n]
z -1
x n -1(ω)fast
z -1
×
β = e -1/τslowe j ω β -γ
γ = e -1/τfaste j ω
(d)
β +γ
+
- βγ
×
z -1
X n (ω)α
z -2
(e)
FIGURE 3. (a) The signal windowing for the a SWIFT algorithm: the data samples and window used for the first computation (blue) and second computation (green). (b) The impulse response and (c) the pole/zero map for a single-bin a SWIFT with x slow = 50 samples, x fast = 10 samples, and ~ = r/10
radians/sample. (d) The parallel a SWIFT filter structure and (e) the direct a SWIFT filter structure.
form. The aSWIFT can be seen operating on an example signal in Figure 3(a).
Transfer function and direct form
We can solve for the z-domain transfer
function of (8) by substituting in (6) one for
each of the slow and fast SWIFTs, to yield
H aSWIFT (z) =
(b - c) z -1
,
1 - (b + c) z -1 + bcz -2
where
b=e
-1/x slow
e j~
c=e
-1/x fast
j~
e .
Numerical simulation
The a SWIFT can also be implemented
as an IIR filter in either the parallel or
direct form, as shown in Figure 3(d)
and (e). Both filters produce identical
impulse responses and pole/zero maps.
However, the parallel form is more efficient than the direct form, requiring
three fewer memory locations and two
fewer real multiplies to compute the
next X n (~) a.
To demonstrate the differences between
the three types of SFTs, Figure 4 depicts
each transform operating on a chirp signal. Each transform's center frequency is
50 Hz, which the chirp crosses 5 s into
the simulation (denoted by the dashed
black line). To facilitate comparison, each
window is set to have the same halfmass.
Both the SDFT's rectangular window
(N = 100) and the SWIFT's exponential
window (x = 72.1) have a halfmass of
50 samples. As compared to the SDFT,
both the SWIFT and aSWIFT have narrower peaks and lower spectral leakage.
In addition, both the SWIFT and SDFT
have noise in their outputs, which is
reduced in the aSWIFT.
Each transform peaks at slightly different times, as well. The SDFT, with a
rectangular window, peaks 0.05 s (or 50
samples) after the chirp passes 50 Hz.
Computational efficiency
(9)
From this form, we can easily analyze the poles/zeros of the system. We
can then derive the discrete difference
form of the aSWIFT from the inverse
z-transform of (9):
X n (~) a = (b + c) X n - 1 (~) a
- bcX n - 2 (~) a
+ (b - c) x [n - 1],
IIR filter implementation
(10)
which we call the direct form. The
a SWIFT's impulse response and pole/
zero map are shown in Figure 3(b) and
(c), with x slow = 50 samples, x fast = 10
samples, and ~ = r/10 rad/sample.
Like windowed SDFTs, the aSWIFT
compromises computational efficiency to reduce spectral leakage. However, the aSWIFT is far more efficient
than comparable windowed SDFTs.
Table 2 compares the computational
costs and memory requirements of
the a SWIFT and the Hanning-windowed SDFT.
Table 2. A single-bin comparison of the computational cost and memory requirements of
computing the next Xn[k]/Xn(~) using the Hanning-windowed SDFT and aSWIFT.
Method
Real multiplies
Real adds
Memory (floats)
Hanning SDFT
18
14
N +15
a SWIFT
Eight
Eight
Eight
IEEE SIGNAL PROCESSING MAGAZINE
|
September 2017
|
187
Table of Contents for the Digital Edition of Signal Processing - September 2017
Signal Processing - September 2017 - Cover1
Signal Processing - September 2017 - Cover2
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Signal Processing - September 2017 - Cover3
Signal Processing - September 2017 - Cover4
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