Left Right Ambiguity Resolution

Left/right target discrimination is important for targeting. Passive acoustic line arrays composed of simple monopole elements are naturally L/R ambiguous. Various techniques are possible with today’s technologies. The most recent is the development of miniaturized passive acoustic elements that support formation of acoustic dipoles as well as monopoles and, therefore, through judicious combination, cardioid responses. Cardioid processing forms the deepest nulls when the elements (real or virtual) used to create the dipole response are a quarter wavelength in separation. These elements have been in development for a long time.  Systems using these elements are more costly in both recurring hardware and in processing than those using L/R ambiguous hardware and processing.

Older technology elements similar in principle to those described above were structured in vertical arrays as a part of sonobuoys. These arrays became quite cheap since they were expendable. However, in an effort to acquire L/R ambiguity resolution ability at low cost for passive towed arrays, at least two earlier techniques were tried. The first uses so-called array wander to discriminate between left and right targets from a line array. The target response measured from the output of left and right beamformers gives a discernible indication (differing signal levels) that breaks the ambiguity. The technique hinges on accurately measuring the array shape as input for the beamformers.

The second approach harkens back to sonobuoy processing. Here, two (or more) line arrays are towed at some separation. Both line arrays are beamformed together. Left and right cardioid responses  are created by forming dipole and monopole virtual elements between physical elements of the two arrays. These cardioids  are beamformed into left and right array responses. Targets have enhanced response from the corresponding beamformer and suppressed response from the opposite beamformer. Although knowing the array shapes helps beamformer fidelity it is not the primary mechanism for left/right  discrimination.

Similar techniques apply for hull mounted line arrays at low-frequency where sound diffracts around the hull. Since these arrays do not wander, L/R target discrimination similar to the first approach relies on the differing left and right target track responses to platform motion. This technique is similar only in so far as the array motion is used to break the ambiguity. The second approach requires deterministic element equalization to form dipoles and monopoles. Obviously, the deepest nulls form when the dipole elements are quarter wavelength separated in either the hull mount or towed cases. This may occur at a frequency other than that corresponding to the array design separation(s).

Shnidman’s Equation

It is notoriously difficult to evaluate receiver operating characteristic curves exactly. This is due to the complexity of these expressions in terms of incomplete Gamma, incomplete Toronto, confluent hypergeometric functions or other special functions. Albersheim fit a set of parametric curves to the exact expression of an ROC curve for a receive chain with a filter, linear detector and independent sample integration. His fit applied to within 0.2 dB of the exact curves for a wide range of detection and false alarm probabilities, non-coherent independent sample averaging and signal to noise ratios. However, his fit modeled only non-fluctuating statistical signals.

Shnidman provides an empirical fit applicable to non-fluctuating and Swerling 1 – 4 fluctuating signals and square law detection. Square law detection is usually implemented given today’s floating point signal processing.  The trade-off is that the equations are accurate to only 0.5 dB over a smaller range (1 – 100 versus 1 – 8096) of independent samples. The expressions are valid for a wider range of detection probabilities (0.99 versus 0.9 at the upper end of the range) and a wider range of false alarm probabilities (10^-9 versus 10^-7 at the lower end of the range). Richards provides a complete summary with algorithms for both Albersheim’s and Shnidman’s curves. Either is easy to program into an Excel spreadsheet using that program’s Visual Basic for Applications or into Matlab.

One can use this receiver operating characteristic curve to relate correct and false classification probabilities through detection and false alert probabilities (using auxiliary expressions) to detector output signal to noise ratio and, via a range equation (or propagation curve), to a detectable range.