In many underwater acoustics problems such as search and rescue, it is desirable to establish the source of the sonar pings. In this paper a methodology is proposed to watermark sonar pulses prior to transmission by embedding them with a chain of low power chirplets represented by a graph. Watermark detection is formulated using stretch processing where chirplet energies are focused on a single beat frequency only if the graph is traversed along the same path used at the source. The algorithm is tested by simulation in shallow water at varying ranges.
1. Introduction
The ocean is full of sounds. From whale calls to shipping noise to research equipment left on the ocean floor, the underwater acoustic channel is unusually crowded. In commercial aviation, signals from underwater locator beacons are looked for in search and rescue efforts. Recent events have highlighted that frequency, or even ping rate and duration for that matter, are not enough to help narrow the search.1 Identifying the source of a sonar ping can be achieved by watermarking. The watermark is in effect a digital fingerprint that can be used to tie the received waveform to the source identity. In this work we propose a new approach to sonar watermarking using the concept of chirplet graphs2 to design the watermark followed by stretch processing3 to detect it. Prior work on sonar watermarking has appeared in Ref. 4.
2. Watermark design using chirplet graphs
Chirplets are a family of multiscale, windowed chirps defined over intervals Ik = [(k − 1)2−i, k2−i] where k = 1, 2,…, 2i. For example, assuming unit time interval, at scale i = 3 the chirplets are defined over 8 equal length segments, I = [{0, 1/8}, {1/8, 1/4},…, {7/8, 1}]. A chirplet “codebook” specifies a family of chirplets defined over a discrete set of allowable slopes and offsets (αk, βk) M. Let T be the length of the main pulse. This length is divided into N segments each of length Tc = T/N. The kth chirplet is defined over the kth subinterval (k −1)Tc ≤ t ≤ kTc,
The chirplet graph, originally proposed for the detection of nonstationary signals of unknown structure in noise,2 is described by the set (V, E) where V are the vertices and E are the edges. The vertices are pairs of allowable (αk, βk) in each interval. The edges define the permitted interconnections between the vertices as not every chirplet is allowed in an interval. Collectively, (V, E) defines the codebook. The time-domain representation of the chirplet graph is shown in Fig. 1. This signal defines the watermark which will later additively modify the main pulse. The specific (V, E) of the graph is the key which the detector needs for decoding.
A chirplet graph is a sequence of short, low energy chirps that are randomly selected from an (α, β) map, i.e., the codebook. Individually, no chirplet is detectable by itself. However, when chirplet energies are combined in a stretch processor, the entire pulse can be detected as a single graph.
A chirplet graph is a sequence of short, low energy chirps that are randomly selected from an (α, β) map, i.e., the codebook. Individually, no chirplet is detectable by itself. However, when chirplet energies are combined in a stretch processor, the entire pulse can be detected as a single graph.
The watermarked signal sw(t) is the addition of the main linear frequency modulated (LFM) pulse and the chirplet graph in Fig. 1,
where A and Bk control the power of the LFM pulse and the watermark, respectively. The two power levels are quantified by the signal-to-watermark ratio (SWR) and watermark-to-noise ratio (WNR). They measure the power of the watermark relative to the main pulse and the ambient noise, respectively.
In terms of complexity, the chirplet graph approach and that in Ref. 4 are not substantially different. The receivers in both approaches are likelihood ratio detectors and the measurement vectors are the output of correlation detectors. The manner by which the keys are generated are different. In Ref. 4, the key can be reproduced at the detector using a seed. In the chirplet graph approach, a binary mask is used to select the embedded chirplets. This mask is functionally similar to the seed and depending on the length of the chirplet graph varies in size. However, the key size is modest. For example, a b bit mask can encode a graph of length 2b. Key size of 10 bits or higher covers graph sizes of several thousands. For similar channels, the approach in Ref. 4 requires channel compensation whereas the stretch processor has proven more robust, specifically to Doppler.
The received watermarked pulse in response to a time-varying acoustic channel is given by
where t is the geotime, τl(t) is the time-varying delay, al(t) is the time varying amplitude, and L is the number of the path. There are a multitude of techniques that can mitigate the effect of the channel by producing a time-varying estimate of the channel.5 Our experiments actually show that over the ranges often encountered in shallow water, the proposed algorithm is robust even without channel mitigation. The delay spread relative to the pulse lengths is small and does not adversely affect stretch processing. The Doppler spread does and its effect has been examined. The algorithm can be tuned to protect the detector against the Doppler effect by controlling the location of the beat frequency at the stretch processor output.
3. Chirplet detection using stretch processing
The detection of the watermark is formulated as a composite hypothesis test. The parameter space Θ is divided into two complementary sets Θ0 and Θ1 where Θ1 = {(αi, βi), i = 1,…, N} defines the alternative hypothesis H1 indicating the presence of the watermark and H0 is the null hypothesis corresponding to an unmarked signal:
where θ is the embedded chirplet graph. The detection of the individual chirplets alone, even if possible, is not sufficient to identify the watermark. Detecting the chirplets in the proper order is key. Therefore, a measurement vector v = {v1, v2,…, vN} is computed over the entire chirplet graph and a test statistic ρ is computed from it which is then used in the following likelihood ratio test:
The null hypothesis is rejected if Λ(ρ) < c. If the receiver already knows the parameter space Θ1 of the embedded chirplets, Eq. (6) reduces to simple hypothesis testing. Graph traversal only along the path guided by Θ1 would result in rejecting H0. This test can also be formulated as a sequential probability ratio test and could shorten watermark detection.
Stretch processing is used to focus the total watermark energy distributed over the graph at a selected beat frequency. The integration of energy is achieved by traversing the graph in the specific order dictated by the embedding key captured in Θ1. The output of the stretch processor in the kth subinterval is then given by
where fd is the Doppler shift. The first term in Eq. (7) is the tone at the beat frequency shifted by possible Doppler. This tone is the same for all intervals. The second term is the “crosstalk” between the dechirper and the main LFM pulse. Crosstalk is simply another LFM pulse with offset (fb + β − βk + fd) and slope (α − αk). The dechirped chirplets all map to a single tone outside the crosstalk whereby their energies are coherently combined. The Doppler shift is handled by providing a guard band between the beat frequency and the crosstalk spectrum greater than or equal to the expected Doppler shift. The test statistic vk for the kth subinterval is the energy at the output of the stretch processor
With n(t) as a complex Gaussian, |rk(t)|2 has a noncentral χ2 distribution under H1.
The middle term in Eq. (9) is the crosstalk energy but is filtered out by positioning the beat frequency outside the crosstalk spectrum.
4. Simulation results
The acoustic channel described in Table 1 is simulated in software. The main LFM pulse is swept from 9.5 to 10.5 kHz for 1 s and then watermarked by 10 chirplets at random offsets and slopes. The beat frequency is at 5 kHz and is positioned 500 Hz away from the crosstalk spectrum. The bandpass of the filter to detect the beat frequency is selected to cover the expected Doppler shift but does not extend into the crosstalk spectrum. Figure 2 shows that combining the energies at the expected beat frequency across the chirplet graph reveals the presence of the watermark. The response at the beat frequency with Doppler is shown in Fig. 3. There is a 200 Hz shift in the beat frequency but the shift is well within the 500 Hz gap to the crosstalk spectrum. The receiver operating characteristic (ROC) curves are shown in Fig. 4 for ranges 5250, 6000, and 6750 m and corresponding to WNR from −14 to −24 dB, respectively. Closer ranges result in perfect detection. At 5250 m, there is near perfect detection of the watermark at WNR = −14 dB. At 6000 m where WNR = −19 dB, above 95% detection rate is achieved at 5% false alarm rate. However, at 6750 m, the performance deteriorates considerably. For detection rates greater than 80%, the false alarm rate is close to 30%, signifying that the acceptable performance envelope for the given watermark power is reached. This is not surprising as WNR at 6750 m is down to −24 dB.
Acoustic channel and sonar signal parameters.
Acoustic channel at a glance . | |
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Range | 4500, 5250, 6000, 6750 m |
Ocean depth | 200 m |
Source/receiver depth | 10 m |
Source level | 210 dB re 1 μPa |
Source signal | 1 s LFM, 9500–10 500 Hz |
Motion | 0, 5 m/sec |
Ambient noise | Sea State 3 (55 db re 1 μPa) |
Volume attenuation | Thorpe |
Max eigenrays | 5 |
Pings | 500 |
Acoustic channel at a glance . | |
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Range | 4500, 5250, 6000, 6750 m |
Ocean depth | 200 m |
Source/receiver depth | 10 m |
Source level | 210 dB re 1 μPa |
Source signal | 1 s LFM, 9500–10 500 Hz |
Motion | 0, 5 m/sec |
Ambient noise | Sea State 3 (55 db re 1 μPa) |
Volume attenuation | Thorpe |
Max eigenrays | 5 |
Pings | 500 |
(Color online) Coherently combined stretch processor outputs. The spike at 5 kHz is the beat frequency indicating the presence of the watermark. The larger peak is the crosstalk spectrum.
(Color online) Coherently combined stretch processor outputs. The spike at 5 kHz is the beat frequency indicating the presence of the watermark. The larger peak is the crosstalk spectrum.
(Color online) Same as Fig. 2 with source motion. The beat frequency is less dominant and shifted by the receiver moving away at 5 m/s. The shift of 200 Hz is still within the passband of the detection filter.
(Color online) Same as Fig. 2 with source motion. The beat frequency is less dominant and shifted by the receiver moving away at 5 m/s. The shift of 200 Hz is still within the passband of the detection filter.
(Color online) Watermark detection ROC curves at three ranges. The SWR is set to 27 dB at the source.
(Color online) Watermark detection ROC curves at three ranges. The SWR is set to 27 dB at the source.
5. Conclusions
A combination of chirplet graphs on the design side and stretch processing on the detector side is used to watermark sonar waveforms. The watermark energy distributed over the graph is recoverable only by authorized detectors while attempts at detecting a single chirplet fail. The technique is shown to work in simulated shallow water.