Implementation of a likelihood ratio test for laser Doppler velocimeter burst detection

Patrick M. Howard; Robert V. Edwards

doi:10.1364/AO.36.007629

Applied Optics
Vol. 36,
Issue 30,
pp. 7629-7638
(1997)
•https://doi.org/10.1364/AO.36.007629

Implementation of a likelihood ratio test for laser Doppler velocimeter burst detection

Patrick M. Howard and Robert V. Edwards

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Patrick M. Howard and Robert V. Edwards, "Implementation of a likelihood ratio test for laser Doppler velocimeter burst detection," Appl. Opt. 36, 7629-7638 (1997)

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Abstract

A likelihood ratio test for laser Doppler velocimeter burst detection is derived and implemented with the appropriate photon detection statistics. This detector utilizes the first point of the autocorrelation function of the laser Doppler velocimeter photodetector signal and relies on a simple identity for this point. The detector looks at the ratio of two probability functions of the data (burst and no burst) to make its decision. Because the detector does not depend on the absolute strength of the signal, detection down to very low (-10-dB) signal-to-noise levels can be achieved. Because the autocorrelation function is insensitive to the phase of the signal, the detector will operate reliably with signals containing multiple overlapping bursts.

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ln η	N_P	n_b	n_d	n_f	P_D	P_F	E
-1	64	321	288	544	0.8972	0.4382	0.4590
0	64	321	288	544	0.8972	0.4382	0.4590
1	64	321	248	110	0.7726	0.0886	0.6840
2	64	321	225	14	0.7009	0.0113	0.6897
3	64	321	217	6	0.6760	0.0048	0.6712
4	64	321	213	4	0.6636	0.0032	0.6603
5	64	321	213	4	0.6636	0.0032	0.6603
6	64	321	72	2	0.2243	0.0016	0.2227
7	64	321	19	2	0.0592	0.0016	0.0576
8	64	321	12	2	0.0374	0.0016	0.0358
-1	128	321	147	279	0.4579	0.2549	0.2030
0	128	321	147	278	0.4579	0.2540	0.2039
1	128	321	129	56	0.4019	0.0503	0.3515
2	128	321	117	13	0.3645	0.0116	0.3529
3	128	321	111	9	0.3458	0.0080	0.3378
4	128	321	110	7	0.3427	0.0062	0.3365
5	128	321	109	5	0.3396	0.0044	0.3351
6	128	321	107	1	0.3333	0.0009	0.3325
7	128	321	105	1	0.3271	0.0009	0.3262
8	128	321	97	1	0.3022	0.0009	0.3013
-1	256	321	72	133	0.2243	0.1297	0.0946
0	256	321	72	133	0.2243	0.1297	0.0946
1	256	321	63	18	0.1963	0.0171	0.1792
2	256	321	63	12	0.1963	0.0114	0.1849
3	256	321	59	5	0.1838	0.0047	0.1791
4	256	321	58	2	0.1807	0.0019	0.1788
5	256	321	58	2	0.1807	0.0019	0.1788
6	256	321	52	1	0.1620	0.0009	0.1611
7	256	321	52	1	0.1620	0.0009	0.1611
8	256	321	45	1	0.1402	0.0009	0.1393
-1	512	321	34	70	0.1059	0.0698	0.0362
0	512	321	34	53	0.1059	0.0528	0.0531
1	512	321	32	14	0.0997	0.0138	0.0859
2	512	321	32	4	0.0997	0.0039	0.0958
3	512	321	32	4	0.0997	0.0039	0.0958
4	512	321	31	3	0.0966	0.0029	0.0936
5	512	321	31	2	0.0966	0.0020	0.0946
6	512	321	31	2	0.0966	0.0020	0.0946
7	512	321	31	2	0.0966	0.0020	0.0946
8	512	321	31	2	0.0966	0.0020	0.0946
-1	1024	321	16	36	0.0498	0.0359	0.0139
0	1024	321	16	36	0.0498	0.0359	0.0139
1	1024	321	16	31	0.0498	0.0310	0.0189
2	1024	321	16	22	0.0498	0.0220	0.0279
3	1024	321	15	14	0.0467	0.0138	0.0330
4	1024	321	15	8	0.0467	0.0079	0.0389
5	1024	321	15	13	0.0467	0.0030	0.0438
6	1024	321	15	3	0.0467	0.0030	0.0438
7	1024	321	15	3	0.0467	0.0030	0.0438
8	1024	321	15	3	0.0467	0.0030	0.0438

ln η	N_P	n_b	n_d	n_f	P_D	P_F	E
-1	64	328	215	498	0.6555	0.4034	0.2521
0	64	328	215	498	0.6555	0.4034	0.2521
1	64	328	206	452	0.6280	0.3661	0.2619
2	64	328	100	133	0.3049	0.1077	0.1971
3	64	328	43	45	0.1311	0.0365	0.0946
4	64	328	13	15	0.0396	0.0122	0.0275
5	64	328	9	11	0.0274	0.0089	0.0185
6	64	328	3	6	0.0091	0.0049	0.0043
7	64	328	2	6	0.0061	0.0049	0.0012
8	64	328	0	4	0.0000	0.0032	-0.0032
-1	128	328	109	248	0.3323	0.2203	0.1120
0	128	328	109	248	0.3323	0.2203	0.1120
1	128	328	106	242	0.3232	0.2144	0.1087
2	128	328	44	56	0.1341	0.0470	0.0871
3	128	328	17	12	0.0518	0.0099	0.0420
4	128	328	6	6	0.0183	0.0049	0.0134
5	128	328	6	6	0.0183	0.0049	0.0134
6	128	328	2	5	0.0061	0.0041	0.0020
7	128	328	2	5	0.0061	0.0041	0.0020
8	128	328	2	5	0.0061	0.0041	0.0020
-1	256	328	51	118	0.1555	0.1091	0.0464
0	256	328	51	116	0.1555	0.1073	0.0482
1	256	328	31	44	0.0945	0.0385	0.0560
2	256	328	18	13	0.0549	0.0110	0.0439
3	256	328	7	8	0.0213	0.0066	0.0147
4	256	328	4	5	0.0122	0.0041	0.0081
5	256	328	2	4	0.0061	0.0033	0.0028
6	256	328	1	4	0.0030	0.0032	-0.0002
7	256	328	1	4	0.0030	0.0032	-0.0002
8	256	328	0	4	0.0000	0.0032	-0.0032
-1	512	328	28	66	0.0854	0.0636	0.0218
0	512	328	27	63	0.0823	0.0603	0.0221
1	512	328	23	53	0.0701	0.0494	0.0208
2	512	328	2	6	0.0061	0.0049	0.0012
3	512	328	2	4	0.0061	0.0033	0.0028
4	512	328	1	4	0.0030	0.0033	-0.0002
5	512	328	0	3	0.0000	0.0024	-0.0024
6	512	328	0	3	0.0000	0.0024	-0.0024
7	512	328	0	3	0.0000	0.0024	-0.0024
8	512	328	0	3	0.0000	0.0024	-0.0024
-1	1024	328	15	35	0.0457	0.0347	0.0111
0	1024	328	15	35	0.0457	0.0347	0.0111
1	1024	328	15	35	0.0457	0.0347	0.0111
2	1024	328	10	11	0.0305	0.0101	0.0203
3	1024	328	5	7	0.0152	0.0060	0.0092
4	1024	328	1	5	0.0030	0.0041	-0.0011
5	1024	328	0	4	0.0000	0.0032	-0.0032
6	1024	328	0	4	0.0000	0.0032	-0.0032
7	1024	328	0	4	0.0000	0.0032	-0.0032
8	1024	328	0	2	0.0000	0.0016	-0.0016

ln η	N_P	n_b	n_d	n_f	P_D	P_F	E
-1	64	321	288	544	0.8972	0.4382	0.4590
0	64	321	288	544	0.8972	0.4382	0.4590
1	64	321	248	110	0.7726	0.0886	0.6840
2	64	321	225	14	0.7009	0.0113	0.6897
3	64	321	217	6	0.6760	0.0048	0.6712
4	64	321	213	4	0.6636	0.0032	0.6603
5	64	321	213	4	0.6636	0.0032	0.6603
6	64	321	72	2	0.2243	0.0016	0.2227
7	64	321	19	2	0.0592	0.0016	0.0576
8	64	321	12	2	0.0374	0.0016	0.0358
-1	128	321	147	279	0.4579	0.2549	0.2030
0	128	321	147	278	0.4579	0.2540	0.2039
1	128	321	129	56	0.4019	0.0503	0.3515
2	128	321	117	13	0.3645	0.0116	0.3529
3	128	321	111	9	0.3458	0.0080	0.3378
4	128	321	110	7	0.3427	0.0062	0.3365
5	128	321	109	5	0.3396	0.0044	0.3351
6	128	321	107	1	0.3333	0.0009	0.3325
7	128	321	105	1	0.3271	0.0009	0.3262
8	128	321	97	1	0.3022	0.0009	0.3013
-1	256	321	72	133	0.2243	0.1297	0.0946
0	256	321	72	133	0.2243	0.1297	0.0946
1	256	321	63	18	0.1963	0.0171	0.1792
2	256	321	63	12	0.1963	0.0114	0.1849
3	256	321	59	5	0.1838	0.0047	0.1791
4	256	321	58	2	0.1807	0.0019	0.1788
5	256	321	58	2	0.1807	0.0019	0.1788
6	256	321	52	1	0.1620	0.0009	0.1611
7	256	321	52	1	0.1620	0.0009	0.1611
8	256	321	45	1	0.1402	0.0009	0.1393
-1	512	321	34	70	0.1059	0.0698	0.0362
0	512	321	34	53	0.1059	0.0528	0.0531
1	512	321	32	14	0.0997	0.0138	0.0859
2	512	321	32	4	0.0997	0.0039	0.0958
3	512	321	32	4	0.0997	0.0039	0.0958
4	512	321	31	3	0.0966	0.0029	0.0936
5	512	321	31	2	0.0966	0.0020	0.0946
6	512	321	31	2	0.0966	0.0020	0.0946
7	512	321	31	2	0.0966	0.0020	0.0946
8	512	321	31	2	0.0966	0.0020	0.0946
-1	1024	321	16	36	0.0498	0.0359	0.0139
0	1024	321	16	36	0.0498	0.0359	0.0139
1	1024	321	16	31	0.0498	0.0310	0.0189
2	1024	321	16	22	0.0498	0.0220	0.0279
3	1024	321	15	14	0.0467	0.0138	0.0330
4	1024	321	15	8	0.0467	0.0079	0.0389
5	1024	321	15	13	0.0467	0.0030	0.0438
6	1024	321	15	3	0.0467	0.0030	0.0438
7	1024	321	15	3	0.0467	0.0030	0.0438
8	1024	321	15	3	0.0467	0.0030	0.0438

ln η	N_P	n_b	n_d	n_f	P_D	P_F	E
-1	64	328	215	498	0.6555	0.4034	0.2521
0	64	328	215	498	0.6555	0.4034	0.2521
1	64	328	206	452	0.6280	0.3661	0.2619
2	64	328	100	133	0.3049	0.1077	0.1971
3	64	328	43	45	0.1311	0.0365	0.0946
4	64	328	13	15	0.0396	0.0122	0.0275
5	64	328	9	11	0.0274	0.0089	0.0185
6	64	328	3	6	0.0091	0.0049	0.0043
7	64	328	2	6	0.0061	0.0049	0.0012
8	64	328	0	4	0.0000	0.0032	-0.0032
-1	128	328	109	248	0.3323	0.2203	0.1120
0	128	328	109	248	0.3323	0.2203	0.1120
1	128	328	106	242	0.3232	0.2144	0.1087
2	128	328	44	56	0.1341	0.0470	0.0871
3	128	328	17	12	0.0518	0.0099	0.0420
4	128	328	6	6	0.0183	0.0049	0.0134
5	128	328	6	6	0.0183	0.0049	0.0134
6	128	328	2	5	0.0061	0.0041	0.0020
7	128	328	2	5	0.0061	0.0041	0.0020
8	128	328	2	5	0.0061	0.0041	0.0020
-1	256	328	51	118	0.1555	0.1091	0.0464
0	256	328	51	116	0.1555	0.1073	0.0482
1	256	328	31	44	0.0945	0.0385	0.0560
2	256	328	18	13	0.0549	0.0110	0.0439
3	256	328	7	8	0.0213	0.0066	0.0147
4	256	328	4	5	0.0122	0.0041	0.0081
5	256	328	2	4	0.0061	0.0033	0.0028
6	256	328	1	4	0.0030	0.0032	-0.0002
7	256	328	1	4	0.0030	0.0032	-0.0002
8	256	328	0	4	0.0000	0.0032	-0.0032
-1	512	328	28	66	0.0854	0.0636	0.0218
0	512	328	27	63	0.0823	0.0603	0.0221
1	512	328	23	53	0.0701	0.0494	0.0208
2	512	328	2	6	0.0061	0.0049	0.0012
3	512	328	2	4	0.0061	0.0033	0.0028
4	512	328	1	4	0.0030	0.0033	-0.0002
5	512	328	0	3	0.0000	0.0024	-0.0024
6	512	328	0	3	0.0000	0.0024	-0.0024
7	512	328	0	3	0.0000	0.0024	-0.0024
8	512	328	0	3	0.0000	0.0024	-0.0024
-1	1024	328	15	35	0.0457	0.0347	0.0111
0	1024	328	15	35	0.0457	0.0347	0.0111
1	1024	328	15	35	0.0457	0.0347	0.0111
2	1024	328	10	11	0.0305	0.0101	0.0203
3	1024	328	5	7	0.0152	0.0060	0.0092
4	1024	328	1	5	0.0030	0.0041	-0.0011
5	1024	328	0	4	0.0000	0.0032	-0.0032
6	1024	328	0	4	0.0000	0.0032	-0.0032
7	1024	328	0	4	0.0000	0.0032	-0.0032
8	1024	328	0	2	0.0000	0.0016	-0.0016

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Tables (2)

Equations (41)

Applied Optics