Arrays and Beamforming — Focusing on a Direction

Multiple sensors measure differences in arrival time

Sound arriving from far away reaches the hydrophones in an array at slightly different times. Those arrival-time differences depend on the direction of the sound. Comparing the signals therefore lets us estimate the direction of arrival, which is difficult to obtain from a single hydrophone.

Arrays can be fixed, moored, or towed, but the basic idea is always the same: place multiple sensors at known positions.

Correct the delays, then add the signals

The most basic form of beamforming is delay-and-sum. First, assume the sound comes from a chosen look direction and correct each element's signal for the corresponding difference in arrival time. After that correction, signals from the look direction line up and reinforce when added, while signals from other directions do not line up as well.

This align-and-add operation focuses the receiver's sensitivity on a direction and can improve signal-to-noise ratio (SNR) in that direction.

Use an idealized approximation to understand directivity index (DI)

For an introductory estimate, the directivity index of an N-element array is sometimes approximated as DI ≈ 10 log10(N). This is an idealized approximation: about 10 dB for 10 elements and about 20 dB for 100 elements. It gives a useful first sense of how increasing element count can improve SNR.

The approximation assumes (1) the noise at different elements is uncorrelated and arrives equally from all directions, (2) every element has the same weight, and (3) the target signal is correctly aligned in phase across all elements. In practice, directional surface noise, sensitivity differences between elements, weighting, and phase disturbances from multipath usually make the improvement smaller than this estimate.

Excessive element spacing creates false directions

If element spacing is too large relative to wavelength, waves arriving from different directions can produce the same phase pattern across the array. This is spatial aliasing. It creates strong responses away from the true arrival direction; these false lobes are called grating lobes.

The spatial counterpart of the Nyquist criterion for time sampling (fs ≥ 2 fmax) requires the element spacing d to satisfy d ≤ λ/2. Keeping a simple uniform array's spacing at approximately λ/2 or less helps prevent spatial aliasing and suppress grating lobes.

Connect this chapter to the Chapter 6 simulator

The Chapter 6 simulator has sliders for water temperature, salinity, depth, frequency, and range, as well as the number of elements N. Beyond directivity index (DI), these values connect to earlier chapters as follows.

• Water temperature, salinity, and depth: through the Mackenzie equation in Chapter 2, they change sound speed c and therefore affect wavelength λ = c/f and round-trip time 2R/c.
• Frequency: through the absorption coefficient in Chapter 4, it affects TL. Absorption tends to rise with frequency, making SNR more likely to decrease.
• Range: through spreading and absorption in Chapter 4, it affects TL. Active sonar includes the outbound and return losses as 2TL.

By moving these sliders, you can examine the Chapter 2–4 relationships together under one set of conditions.

Takeaways from this chapter

• An array uses arrival-time differences measured by multiple hydrophones at known positions.
• Delay-and-sum corrects the time offsets for a look direction and then adds the signals.
• DI ≈ 10 log10(N) is an idealized approximation for estimating SNR improvement.