Improvement of frequency resolution using a sub-bin structure in the discrete Fourier transform

Manabu Sato; Yuuki Kimura; Junpei Masuta; Tetsuo Kosaka; Izumi Nishidate

doi:10.1364/AO.426045

Applied Optics
Vol. 60,
Issue 21,
pp. 6290-6301
(2021)
•https://doi.org/10.1364/AO.426045

Improvement of frequency resolution using a sub-bin structure in the discrete Fourier transform

Manabu Sato, Yuuki Kimura, Junpei Masuta, Tetsuo Kosaka, and Izumi Nishidate

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Manabu Sato, Yuuki Kimura, Junpei Masuta, Tetsuo Kosaka, and Izumi Nishidate, "Improvement of frequency resolution using a sub-bin structure in the discrete Fourier transform," Appl. Opt. 60, 6290-6301 (2021)

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- Instrumentation and Measurements
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About this Article
History
- Original Manuscript: March 30, 2021
- Revised Manuscript: June 17, 2021
- Manuscript Accepted: June 21, 2021
- Published: July 20, 2021

Abstract

To improve frequency resolution, assignments of sub-bins within the bin interval of the conventional discrete Fourier transform are proposed by decreasing the number of sampled data. The simulated and experimental results of the basic sub-bin spectrum characteristics are presented. Using a signal oscillator, digital oscilloscope, and PC, the sub-bin line spectrum between two conventional bins is measured, and nine line spectra of sub-bins corresponding to the simulated results are measured. The linearity of the sub-bin spectrum intensity for amplitudes of 0.001–10 V of the input signal is also measured. The Doppler signals from the optical interferometer using a He–Ne laser are measured using a mixer to tune the frequency. A sub-bin spectrum of 104.89 kHz between two bin frequencies of 104.49 and 105.47 kHz is measured using 256 data points at a sampling frequency of 250 kHz and an acquisition time of 1.024 ms. The precision of displacement for the acquisition time is improved by one order to 32 nm for a full displacement of 2.342 µm. The sub-bins, as a composition of bin structure, are applicable to various fields.

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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SB no.	$R^{'}$	$N^{'}$	$i_{0}$	$R^{'}$	$f$ (Hz)
	0	256	$k_{0} = 107$		0.41797
1	0.1	196	81	0.10204	0.41837
2	0.2	160	66	0.20000	0.41875
3	0.3	167	69	0.30539	0.41916
4	0.4	174	72	0.40230	0.41954
5	0.5	131	54	0.48092	0.41985
6	0.6	138	57	0.59420	0.42029
7	0.7	145	60	0.69655	0.42069
8	0.8	133	55	0.78947	0.42105
9	0.9	121	50	0.90083	0.42149
	1.0	256	$k_{0} = 108$		0.421875

$A = 0.1$	Sub-Bin No. 4			Sub-Bin No. 5
$C_{SL} Φ_{i 0 + 1}$		$2.379 E + 02$			$3.049 E + 09$
$Φ$	$Φ_{i 0}$	$Φ_{i 0 + 1}$	$Φ_{i 0 + 2}$	$Φ_{i 0}$	$Φ_{i 0 + 1}$	$Φ_{i 0 + 2}$
	4.380 E-01	$1.622 E + 02$	4.910 E-01	3.133 E-15	$1.637 E + 02$	2.631 E-15
Ratio	$1.398 E + 14$	9.906 E-01	$1.866 E + 14$	1	1	1
Theory				0	$1.637 E + 02$	0
$A = 1.0$	$Sub - Bin No . 4$			Sub-Bin No. 5
$C_{SL} Φ_{i 0 + 1}$		$2.379 E + 03$			$3.049 E + 10$
$Φ$	$Φ_{i 0}$	$Φ_{i 0 + 1}$	$Φ_{i 0 + 2}$	$Φ_{i 0}$	$Φ_{i 0 + 1}$	$Φ_{i 0 + 2}$
	$4.380 E + 01$	$1.622 E + 04$	$4.910 E + 01$	3.133 E-13	$1.637 E + 04$	2.631 E-13
Ratio	$1.398 E + 14$	9.907 E-01	$1.866 E + 14$	1	1	1
Theory				0	$1.637 E + 04$	0

M	1	2	3	4	5
Group 1	nos. 1, 9	nos. 1–2, 8–9	nos. 1–3, 7–9	nos. 1–4, 6–9	nos. 1–5, 5–9
Group 2	nos. 1’, 9’	nos. 1’–2’, 8’–9’	nos. 1’–3’, 7’–9’	nos. 1’–4’, 6’–9’	nos. 1’–5’, 5’–9’

$A = 0.1$	Sub-Bin No. 4			Sub-Bin No. 5
$C_{SL} Φ_{i 0 + 1}$		$2.302 E + 02$			$6.674 E + 03$
$Φ$	$Φ_{i 0}$	$Φ_{i 0 + 1}$	$Φ_{i 0 + 2}$	$Φ_{i 0}$	$Φ_{i 0 + 1}$	$Φ_{i 0 + 2}$
	4.069E-01	$1.697 E + 02$	6.80E-01	6.12E-04	$1.71 E + 02$	6.96E-04
Ratio Theory	$6.645 E + 02$	9.939E-01	$9.768 E + 02$	1	1	1
$A = 1.0$	Sub-Bin No. 4			Sub-Bin No. 5
$C_{SL} Φ_{i 0 + 1}$		$2.379 E + 03$			$8.161 E + 04$
$Φ$	$Φ_{i 0}$	$Φ_{i 0 + 1}$	$Φ_{i 0 + 2}$	$Φ_{i 0}$	$Φ_{i 0 + 1}$	$Φ_{i 0 + 2}$
	$4.919 E + 01$	$1.696 E + 04$	$5.24 E + 01$	3.61E-03	$1.71 E + 04$	8.45E-02
Ratio	$1.363 E + 04$	9.901E-01	$6.205 E + 02$	1	1	1

SB no.	$R^{'}$	$N^{'}$	$i_{0}$	$R^{'}$	$f$ (Hz)
	0	256	$k_{0} = 107$		0.41797
1	0.1	196	81	0.10204	0.41837
2	0.2	160	66	0.20000	0.41875
3	0.3	167	69	0.30539	0.41916
4	0.4	174	72	0.40230	0.41954
5	0.5	131	54	0.48092	0.41985
6	0.6	138	57	0.59420	0.42029
7	0.7	145	60	0.69655	0.42069
8	0.8	133	55	0.78947	0.42105
9	0.9	121	50	0.90083	0.42149
	1.0	256	$k_{0} = 108$		0.421875

Abstract

Data Availability

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Figures (18)

Tables (4)

Equations (26)

Applied Optics