Optical frequency hopping for channel presence detection in optical networks

By using heterodyne sampling and frequency hopping techniques through an optical channel monitor (OCM), the problem of slow optical channel attenuation detection speed in existing technologies is solved, and a fast response to optical channel attenuation is achieved.

CN121333403APending Publication Date: 2026-01-13II VI DELAWARE INC
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Patent Information

Application Number
CN202411524818.3
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Priority Date
2024-07-12
Filing Date
2024-10-30
Publication Date
2026-01-13

AI Technical Summary

Technical Problem

Existing technologies for detecting optical channel attenuation in optical networks cannot respond quickly to sudden drops in optical channel power within a shorter timeframe than the traditional segmented linear optical signal scanning time.

Method used

An optical channel monitor (OCM) is used to detect the frequency jump response of the optical signal by using heterodyne sampling and frequency hopping technology, combined with the controller, to generate an error signal to quickly identify optical channel attenuation.

Benefits of technology

This enables rapid detection of optical channel attenuation in a shorter time, improving the response speed and stability of optical networks.

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Abstract

The present disclosure relates at least to optical frequency hopping for channel presence detection in an optical network. An optical channel monitor (OCM) for optical channel attenuation detection in a fiber optic communication system and a method of using the OCM are disclosed, comprising: (a) heterodyne sampling of an optical signal from an optical fiber with light from a light source, the frequency of the light from the light source varying at a constant slope over time; (b) suspending the sampling in step (a); (c) after step (b), heterodyne sampling of the optical signal with light from the light source by frequency hopping; (d) concurrently with step (c), sampling, by the controller, the response of the optical signal to each frequency jump in step (c); and (e) in response to determining, by the controller, that at least one sample in step (d) exceeds a predetermined tolerance or range of the sample, the controller generates an error signal.
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Description

Technical Field

[0001] This disclosure relates to systems and methods for detecting the presence of channels in optical networks. Background Technology

[0002] Referring to the example in Figure 1, an example of a prior art test scan for detecting channel loss of a sampled optical signal using the internal laser of an optical channel monitor (OCM) (including an OCM controller and an internal OCM laser) may include a piecewise linear optical (or laser) signal 2, as shown in Figure 1. In the example, the prior art test scan may include a sampled optical signal mixed (or heterodyned) with the piecewise linear optical (or laser) signal 2. Hereinafter, the "sampled optical signal" may be simply referred to as the "optical signal".

[0003] In a non-limiting example, the piecewise linear optical signal 2 may have a slope of 125 MHz / µs, and the OCM controller may be programmed, operated, and / or configured to cause the internal laser of the OCM to switch to different laser modes after scanning a predetermined frequency range (such as, for example, 100 GHz as shown in Figure 1).

[0004] In the example, the segmented linear optical signal 2 may include, for example, segments 4-1 to 4-5, wherein each segment may cover a frequency range of, for example, 100 GHz, which is scanned within a period of, for example, 1 ms, with a spacing 6 between adjacent segments 4. In this example, the segmented linear optical signal 2 may include a spacing 6-1 between segments 4-1 and 4-2, a spacing 6-2 between segments 4-2 and 4-3, a spacing 6-3 between segments 4-3 and 4-4, and a spacing 6-4 between segments 4-4 and 4-5.

[0005] In the example, the piecewise linear optical signal 2 may also include frequency overlap 8 (e.g., 10 GHz) between adjacent segments 4, such as overlap 8-1 between segments 4-1 and 4-2, overlap 8-2 between segments 4-2 and 4-3, overlap 8-3 between segments 4-3 and 4-4, and overlap 8-4 between segments 4-4 and 4-5. In the example, the OCM controller may be programmed, operated, and / or configured to have a splicing algorithm that can be used to cause the internal laser of the OCM to output a piecewise linear optical signal 2 with a spacing 6 and overlap 8 between adjacent segments 4.

[0006] During the scanning of the piecewise linear optical signal 2, while the piecewise linear optical signal 2 is being applied to the optical signal, the OCM controller can continuously, periodically, or aperiodically sample the mixed (or heterodyne) response of the optical signal and the piecewise linear optical signal 2, and compare each sample with a predetermined tolerance or range. In the example, the sampled response could be the power at each sampling frequency.

[0007] If, based on comparison, the OCM controller determines that the power at one or more sampling frequencies exceeds a predetermined tolerance or range for the one or more samples (e.g., the power of the sample is ≤ 80% of the expected sampling power), the OCM controller may output an appropriate error signal for the event, which can be used to investigate and / or correct the conditions that led to the error signal.

[0008] In the example, the piecewise linear optical signal 2 may include 50-80 segments 4, where each segment 4 may include multiple channels, and the total scan time of the piecewise linear optical signal 2 may be approximately 500 ms. After scanning the optical signal with the piecewise linear optical signal 2 from a start frequency 10 to an end frequency 12, the OCM controller can cause the OCM laser to repeat another scan of the optical signal with the piecewise linear optical signal 2 from the start frequency 10 to the end frequency 12. In the example, the OCM controller can scan the optical signal continuously, periodically, or non-periodically using the piecewise linear optical signal 2.

[0009] Optical networks are disrupted when the power of a set of channels (e.g., 16-24) suddenly drops. It is desirable to detect these sudden power drops within a timeframe faster than the total scan time (approximately 500 ms) of the piecewise linear optical signal 2 shown in Figure 1. Summary of the Invention

[0010] This document discloses a method for detecting optical channel attenuation in an optical communication system, the method comprising: (a) heterodyne sampling an optical signal with light from a light source, the frequency of which changes with time at a constant slope; (b) pausing sampling in step (a); (c) after step (b), heterodyne sampling the optical signal with light from the light source by frequency hopping; (d) simultaneously with step (c), sampling the response of the optical signal to each frequency hop in step (c) by a controller; and (e) in response to the controller determining that at least one sampling frequency hop in step (d) exceeds a predetermined tolerance or range for the sampling frequency hop, the controller generates an error signal. In the example, each step of heterodyne sampling may include mixing the optical signal with light from a light source, the light from the light source (1) changing its frequency with time at a constant slope in step (a) and / or (2) changing its frequency by frequency hopping in step (c).

[0011] This document also discloses an optical channel monitor (OCM) programmed, operated, and / or configured to perform a method for detecting optical channel attenuation in an optical communication system, the method comprising: (a) heterodyne sampling (or mixing) an optical signal (e.g., an optical signal sampled from an optical fiber) with light from a light source; (b) pausing the sampling in step (a); (c) after step (b), heterodyne sampling the optical signal with light from the light source by frequency hopping; (d) simultaneously with step (c), sampling the response of the optical signal to each frequency hop in step (c) by a controller; and (e) in response to the controller determining that at least one sampling frequency hop in step (d) exceeds a predetermined tolerance or range for the sampling frequency hop, the controller generates an error signal. In the example, each step of heterodyne sampling may include mixing the optical signal with light from a light source, the light from the light source (1) changing its frequency over time at a constant slope in step (a) and / or (2) changing its frequency by frequency hopping in step (c). Attached Figure Description

[0012] Figure 1 is a non-limiting embodiment or example of a prior art piecewise linear optical signal, which can be mixed or heterodyned with the optical signal of an optical network to detect power drops in the optical signal;

[0013] Figure 2 This is a non-limiting embodiment or example schematic diagram of an optical network, which includes optical fiber and an exemplary optical channel monitor, such as an optical heterodyne channel monitor (OCM) coupled to the optical fiber, which can be used to sample optical signals from the optical fiber in accordance with the principles of this disclosure;

[0014] Figure 3A This is a non-limiting embodiment or example diagram of an optical test signal based on the principles of this disclosure, showing the inter-segment optical signal scan between adjacent segments that can be obtained by OCM and from... Figure 2 Frequency hopping scanning of mixed or heterodyned optical signals sampled in the optical fiber is shown.

[0015] Figure 3B yes Figure 3A An enlarged view of a portion of an exemplary optical test signal shows details of a frequency-hopping scan, which can be performed by OCM mixing or heterodyne between adjacent segments of a piecewise linear optical signal scan; and

[0016] Figure 4 This is a method that conforms to the principles of this disclosure. Detailed Implementation

[0017] Various non-limiting embodiments will now be described with reference to the accompanying drawings, wherein similar reference numerals correspond to similar or functionally equivalent elements or features.

[0018] As used herein, spatial or directional terms such as “left,” “right,” “inner,” “outer,” “upper,” “lower,” etc., relate to the disclosure shown in the accompanying drawings. However, it should be understood that this disclosure may assume various alternative orientations, and therefore these terms should not be considered limiting. Furthermore, all figures used herein, in the specification and claims, to indicate dimensions, physical properties, processing parameters, amounts of components, reaction conditions, etc., should be understood to be modified in all cases by the terms “approximately” or “about.” Therefore, unless stated to the contrary, the numerical values ​​specified in the following specification and claims may vary depending on the desired characteristics sought to be obtained through this disclosure.

[0019] At least, and not in an attempt to limit the application of the equivalence principle to the scope of the claims, each numerical value should be interpreted, at least according to the number of significant figures reported and by applying ordinary rounding techniques. Furthermore, all ranges disclosed herein should be understood to include the starting and ending range values ​​and any and all subranges contained therein. For example, the specified range “1 to 10” should be considered to include any and all subranges between the minimum value of 1 and the maximum value of 10 (including the endpoints); that is, all subranges that begin with a minimum value of 1 or more and end with a maximum value of 10 or less, such as 1 to 3.3, 4.7 to 7.5, 5.5 to 10, etc. “A” or “an” refers to one or more.

[0020] As used herein, the terms “coupled,” “coupled,” and similar terms refer to two or more elements that are connected, linked, fastened, connected, linked, or otherwise associated with each other (e.g., mechanical, electrical, fluid, optical, electromagnetic). In various examples, these elements may be directly or indirectly associated together. As an example, element A may be directly associated with element B. As another example, element A may be indirectly associated with element B, for example, through another element C. It should be understood that not all associations between the various disclosed elements need to be represented. Therefore, couplings other than those depicted in the figures may also exist.

[0021] As used herein, when used with a list of items, the phrase "at least one of" means that different combinations of one or more of the listed items may be used, and it may be necessary to use only one of each item in the list. For example, "at least one of items A, B, and C" may include, but is not limited to, item A or items A and B. The example may also include items A, B, and C, or items B and C. In other examples, "at least one of" may include, for example, but not limited to, two of items A, one of items B, and ten of items C; four of items B and seven of items C; and other suitable combinations.

[0022] In this disclosure, each item, component, circuit, and / or system described as "programmed, operated, and / or configured" may be formed from the following:

[0023] (1) One or more discrete passive optical and / or electrical components, such as, but not limited to, laser sources, optical waveguides, resistors, capacitors, inductors, transistors, operational amplifiers, and some combination thereof as determined by the application; or

[0024] (2) One or more controllers, processors, memories, storage components, input components, output components, and communication interfaces, all connected via a bus in some combination as determined by the application; or

[0025] (3) A certain combination of (1) and (2) determined by the application.

[0026] refer to Figures 2-3B The optical network or system 20 according to the principles of this disclosure may include optical fiber 22, such as an erbium-doped fiber amplifier (EDFA) having an input 24 coupled to receive an optical (e.g., laser) signal from an upstream optical signal source 26 and an output 28 coupled to provide an optical signal propagating in the optical fiber 22 to a downstream optical signal receiver 29.

[0027] According to the principles of this disclosure, the optical system 20 may include an optical channel monitor (OCM) 30, which includes an OCM controller 32 and an OCM laser or light source 34. The OCM 30 may also include additional undescribed elements, which are omitted for brevity, that enable or facilitate the OCM 30 to perform the various functions or operations described in this disclosure.

[0028] The OCM30 can be coupled at 36 (e.g., via an optical splitter) to a sampled optical signal output from the optical signal source 26 for propagation in the optical fiber 22. However, Figure 2 The description of the OCM 30 coupled to the optical fiber 22 should not be construed as limiting, as it is envisioned that the OCM 30 can be coupled to any suitable and / or desired location of the disclosed optical system 20, said location being between the optical signal source 26 and the optical fiber 22 (e.g., ...). Figure 2 (as shown) or between optical signal receiver 29 and include these locations.

[0029] Under the control of the OCM controller 32, the optical test scan 40 generated by the OCM laser 34 can be mixed (or heterodyned) with the optical signal sampled from the fiber optic cable 22. The optical test scan 40 may include, for example: Figure 3AThe segmented linear optical signal portion 42 shown is similar to the segmented linear optical signal 2 shown in FIG1 and described in the background section of this disclosure, and may further include a frequency hopping scan portion 44 in each interval 6 between adjacent segments 4 of the segmented linear optical signal portion 42. In this disclosure, similar elements or features of the segmented linear optical signal 2 and the segmented linear optical signal portion 42 in FIG1 will be described using similar reference numerals.

[0030] In the example, each frequency hopping section 44 may include a number or more channels 46 that are mixed (or heterodyneeded) with the optical signal sampled from the optical fiber 22 by the OCM controller 32. Figure 3A and 3B In the example shown, each frequency hopping section 44 may include channels 46-1 to 46-10 separated by 0.5THz hops between the frequencies of 191.0THz and 191.45THz. However, this example should not be construed as limiting, as the number of channels 46 and / or the spacing or hop (e.g., 0.5THz) between each pair of adjacent channels 46 may vary. Figure 3A and 3B The options shown may be the same or different, and can be selected as those deemed suitable and / or desired for a specific application. In other words, Figure 3A and 3B The 0.5THz interval or jump between adjacent channels 46 shown should not be interpreted as restrictive, as it is conceivable that the interval or jump between each pair of adjacent channels 46 can be the same or different, such as 0.05THz, 0.4THz, 0.6THz, 1THz, etc.

[0031] In addition, Figure 1, Figure 3A and Figure 3B The frequency ranges relative to time (in milliseconds) for each graph shown are for illustrative purposes only and should not be construed as limiting, as the frequency range scanned by the piecewise linear optical signal portion 42, which includes a frequency-hopping scan portion 44 (comprising the number or quantity of channels 46 and / or the intervals or hops between each pair of adjacent channels 46) in each interval 6 between adjacent segments 4 of the piecewise linear optical signal portion 42, can be selected as one or more bands deemed suitable and / or desired for the channel under test. For example, assuming C-band optical signal testing of an optical fiber, whose wavelength is typically in the range of approximately 1530 nm to 1565 nm, the piecewise linear optical signal portion 42 could range from a starting frequency 10 of 191 THz to an ending frequency 12 of 196 THz. However, this example is for illustrative purposes only and should not be construed as limiting.

[0032] Furthermore, if the fiber 22 is subjected to multi-band testing, such as two or more of the C-band, L-band, and / or S-band, then the segmented linear optical signal portion 42 may be selected as appropriate and / or desired for each band, the segmented linear optical signal portion 42 including a frequency range between a start frequency 10 and an end frequency 12, the amount of channels 46, and / or the frequency spacing between each channel 46 in each interval 6 between adjacent segments 4 of the segmented linear optical signal portion 42.

[0033] refer to Figure 4 And continue to refer to Figures 2-3B A method for detecting optical channel attenuation in an optical fiber communication system, based on the principles of this disclosure, includes step S1, heterodyne sampling (or mixing) of an optical signal (e.g., sampled from optical fiber 22) using light from a light source (e.g., laser 34 of OCM 30), the frequency of which changes with time at a constant slope. In step S2, sampling in step S1 is paused. After step S2, in step S3, heterodyne sampling of the optical signal is performed using light from the light source by frequency hopping.

[0034] Simultaneously with step S3, in step S4, the controller (e.g., OCM controller 32) samples the response of the optical signal to each frequency jump in step S3. In step S5, in response to the controller determining that at least one sample in step S4 exceeds a predetermined tolerance or range of the sample, the controller generates an error signal. The error signal may be a human-detectable optical and / or audible alarm or signal, and / or an electronic signal detectable by another electronic system that can generate a subsequent human-detectable optical and / or audible alarm or signal.

[0035] The method may further include step S6, wherein the scan of step S1 is resumed after step S4 (when each sample is within the predetermined tolerance or range of the sample) or step S5 (when at least one sample exceeds the predetermined tolerance or range of the sample). Finally, in step S7, steps S2 to S6 are repeated at least once.

[0036] In the example, in step S6, the scan can be resumed at a frequency that overlaps with the scan frequency at which the scan was paused in step S2. In the example, each instance of the scan cycle in step S1 and / or step S6 can be the same or different. In the example, the difference between adjacent frequency hopping in step (c) can be the same or different.

[0037] Other non-limiting examples or aspects of this disclosure are set forth in the following illustrative and exemplary numbered clauses:

[0038] Article 1: A method for detecting optical channel attenuation in an optical fiber communication system, the method comprising: (a) heterodyne sampling an optical signal with light from a light source, the frequency of which changes with time at a constant slope; (b) pausing sampling in step (a); (c) after step (b), heterodyne sampling the optical signal with light from the light source by frequency hopping; (d) simultaneously with step (c), sampling the response of the optical signal to each frequency hop in step (c) by a controller; and (e) in response to the controller determining that at least one sample in step (d) exceeds a predetermined tolerance or range of said sample, the controller generates an error signal. In the example, each step of heterodyne sampling may include mixing an optical signal (which may be sampled from an optical fiber) with light from a light source, the light from the light source (1) changing its frequency with time at a constant slope in step (a) and / or (2) changing its frequency by frequency hopping in step (c).

[0039] Article 2: The method according to Article 1, wherein step (a) includes sampling continuously, periodically or aperiodically, and comparing each sample in step (a) with a predetermined tolerance or range of the sample.

[0040] Article 3: The method described in Article 1 or Article 2 further includes (f) resuming sampling in step (a) after step (d) or step (e), i.e., resuming or continuing sampling in step (a) where sampling was paused in step (b). In an example of step (f), sampling in step (a) may be resumed after step (d) when each sample is within a predetermined tolerance or range of the sample. In another example of step (f), sampling in step (a) may be resumed after step (e) when at least one sample exceeds a predetermined tolerance or range of the sample.

[0041] Article 4: The method described in any of Articles 1-3 further includes (g) repeating steps (b)-(f) at least once.

[0042] Article 5: The method according to any one of Articles 1-4, wherein, in step (f), sampling is resumed at a frequency that overlaps with the frequency at which sampling was paused in step (b).

[0043] Article 6: The method according to any one of Articles 1-5, wherein each instance of step (b) occurs after the sampling period in step (a) or after the sampling period following step (f).

[0044] Article 7: The method described in any of Articles 1-6, wherein each instance of the sampling period in step (a) or step (f) is identical.

[0045] Article 8: The method described in any of Articles 1-7, wherein each instance of the sampling period in step (a) or step (f) is different.

[0046] Article 9: The method described in any of Articles 1-8, wherein the difference between adjacent frequency jumps in step (c) is the same.

[0047] Article 10: The method described in any of Articles 1-9, wherein the difference between adjacent frequency jumps in step (c) is different.

[0048] Article 11: An optical channel monitor (OCM) programmed, operated, and / or configured to perform the following methods includes: (a) heterodyne sampling an optical signal with light from a light source, the frequency of which changes over time at a constant slope; (b) pausing sampling in step (a); (c) after step (b), heterodyne sampling the optical signal with light from the light source by frequency hopping; (d) simultaneously with step (c), sampling the response of the optical signal in step (c) to each frequency hop by a controller; and (e) in response to the controller determining that at least one sample in step (d) exceeds a predetermined tolerance or range of the sample, the controller generates an error signal. In the example, each step of heterodyne sampling may include mixing an optical signal (which may be sampled from an optical fiber) with light from a light source, the light from the light source (1) changing its frequency over time at a constant slope in step (a) and / or (2) changing its frequency by frequency hopping in step (c).

[0049] Article 12: The OCM as described in Article 11, wherein step (a) includes sampling continuously, periodically or aperiodically, and comparing each sample in step (a) with a predetermined tolerance or range of the sample.

[0050] Article 13: According to Article 11 or 12 of the OCM, the method further includes (f) resuming the sampling in step (a) after step (d) or step (e), i.e., resuming or continuing the sampling in step (a) where sampling was paused in step (b). In an example of step (f), the sampling in step (a) may be resumed after step (d) when each sample is within a predetermined tolerance or range of the sample. In another example of step (f), the sampling in step (a) may be resumed after step (e) when at least one sample exceeds a predetermined tolerance or range of the sample.

[0051] Article 14: The OCM according to any one of Articles 11-13, wherein the method further comprises (g) repeating steps (b)-(f) at least once.

[0052] Article 15: OCM according to any one of Articles 11-14, wherein, in step (f), sampling is resumed at a frequency that overlaps with the frequency at which sampling was paused in step (b).

[0053] Article 16: OCM according to any one of Articles 11-15, wherein each instance in step (b) occurs after the sampling period in step (a) or after the sampling period following step (f).

[0054] Article 17: OCM according to any of Articles 11-16, wherein each instance of the sampling period in step (a) or step (f) is identical.

[0055] Article 18: OCM according to any of Articles 11-17, wherein each instance of the sampling period in step (a) or step (f) is different.

[0056] Article 19: OCM according to any of Articles 11-18, wherein the difference between adjacent frequency jumps in step (c) is the same.

[0057] Article 20: OCM according to any of Articles 11-19, wherein the difference between adjacent frequency jumps in step (c) is different.

[0058] Although this disclosure has been described in detail for illustrative purposes, based on embodiments currently considered to be most practical and preferred, it should be understood that the foregoing details are for this purpose only, and that this disclosure is not limited to the disclosed embodiments. Rather, this disclosure is intended to cover modifications and equivalent arrangements within the spirit and scope of the appended claims. For example, it should be understood that this disclosure contemplates that, to the extent possible, one or more features of any embodiment may be combined with one or more features of any other embodiment.

Claims

1. A method for detecting optical channel attenuation in an optical fiber communication system, the method comprising: (a) Heterodyne sampling of an optical signal using light from a light source, wherein the frequency of the light from the light source changes with time at a constant slope; (b) Pause sampling in step (a); (c) After step (b), the optical signal is heterodyne sampled using light from the light source by frequency hopping; (d) Simultaneously with step (c), the controller samples the response of the optical signal to each frequency jump in step (c); and (e) In response to the controller determining that at least one sample in step (d) exceeds the predetermined tolerance or range of the sample, the controller generates an error signal.

2. The method of claim 1, wherein step (a) comprises sampling continuously, periodically or aperiodically, and comparing each sample in step (a) with a predetermined tolerance or range of the sample.

3. The method according to claim 1, further comprising: (f) After step (d) or step (e), resume the sampling in step (a).

4. The method of claim 3, further comprising: (g) Repeat steps (b)-(f) at least once.

5. The method according to claim 3, wherein, In step (f), sampling is resumed at a frequency that overlaps with the frequency at which sampling was paused in step (b).

6. The method of claim 3, wherein each instance of step (b) occurs after a sampling period in step (a) or after a sampling period following step (f).

7. The method of claim 6, wherein each instance of the sampling period in step (a) or step (f) is identical.

8. The method of claim 6, wherein each instance of the sampling period in step (a) or step (f) is different.

9. The method of claim 1, wherein the difference between adjacent frequency jumps in step (c) is the same.

10. The method of claim 1, wherein the difference between adjacent frequency jumps in step (c) is different.

11. An optical channel monitor (OCM) programmed, operated, and / or configured to perform a method comprising the following steps: (a) Heterodyne sampling of an optical signal using light from a light source, wherein the frequency of the light from the light source changes with time at a constant slope; (b) Pause sampling in step (a); (c) After step (b), the optical signal is heterodyne sampled using light from the light source by frequency hopping; (d) Simultaneously with step (c), the controller samples the response of the optical signal to each frequency jump in step (c); and (e) In response to the controller determining that at least one sample in step (d) exceeds the predetermined tolerance or range of the sample, the controller generates an error signal.

12. The OCM of claim 11, wherein step (a) comprises sampling continuously, periodically or aperiodically, and comparing each sample in step (a) with a predetermined tolerance or range of the sample.

13. The OCM of claim 11, wherein the method further comprises: (f) After step (d) or step (e), resume the sampling in step (a).

14. The OCM of claim 11, wherein the method further comprises: (g) Repeat steps (b)-(f) at least once.

15. The OCM of claim 13, wherein in step (f), sampling is resumed at a frequency that overlaps with the frequency at which sampling was paused in step (b).

16. The OCM of claim 13, wherein each instance of step (b) occurs after a sampling period in step (a) or after a sampling period following step (f).

17. The OCM of claim 16, wherein each instance of the sampling period in step (a) or step (f) is identical.

18. The OCM of claim 16, wherein each instance of the sampling period in step (a) or step (f) is different.

19. The OCM of claim 11, wherein the difference between adjacent frequency jumps in step (c) is the same.

20. The OCM of claim 11, wherein the difference between adjacent frequency jumps in step (c) is different.