Multibeam Sonar Theory of Operation

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Introduction to Multibeam Sonar: Projector and Hydrophone Systems Multibeam Sonar Theory of Operation where: or equivalently: S(t) = A(t) cos (φ(t)) (3.10) φ(t) = 2πft (3.11) The phase φ(t) is simply a measure of where a signal is in its oscillations. An instantaneous measure of a signal will yield S(t), as described above, but what you are really interested in is A(t), the amplitude. If measurements of S(t) and φ(t) are made simultaneously, A(t) can be extracted using Equation 3.12. The SEA BEAM 2100 measures both signal strength S(t) and phase φ(t). Both pieces of information are converted from continuous analog signals to a series of discrete digital measurements through the analog-to-digital converters. These measurements are made at the sampling rate, which is between 1 and 3 milliseconds. All of the signal and phase data for all hydrophones from one of these samples is called a time slice. Page 3-24 Copyright © 2000 L-3 Communications SeaBeam Instruments (3.12) To form a steered beam, the signals from the hydrophones S1(t) through SN(t) must be combined with the correct adjustments for the path differences between different elements. Recall from Equation 3.7 that the path difference of a signal from steered beam angle θ between two elements of an array spaced d apart is given by: d sin θ (3.13) If the N hydrophones are numbered i = 0 through i = N 1, the path differences required to properly combine the signals of each are given by: id sin θ (3.14) For computational purposes, it is easier to speak in terms of the phase difference between hydrophone elements rather than the path difference. The phase difference is a measure of the fractional number of wave oscillations that occur in the path difference. The phase difference in one complete oscillation or cycle is 2π. The fractional difference in phase caused by the path difference id sin θ is given by: 2π id sin θ (3.15) λ The in-phase signal received from an angle θ by each hydrophone is the measured signal adjusted by the appropriate phase delay. Using Equations 3.9 and 3.15, you can find the signal of hydrophone element i with the appropriate phase delay required to form a steered beam at angle θ, called Bi(θ), in terms of the instantaneous measurements Si and φi within a time slice: (3.16) (3.17) No portion of this document may be reproduced without the expressed written permission of L-3 Communications SeaBeam Instruments
Introduction to Multibeam Sonar: Multibeam Sonar Theory of Operation Projector and Hydrophone Systems The combined steered beam at angle θ for a particular time slice is found by summing the contributions of all hydrophones, with appropriate shading coefficients si: (3.18) (3.19) The calculations required to form M steered beams from the data of N hydrophones within one time slice can be represented using matrix algebra. The phase delay required for hydrophone i at a beam steered to angle θm as Dmi is defined as: Then the operation required to find a steered beam becomes: The logic required to form M steered beams can then be expressed as a matrix multiplication: (3.20) (3.21) To create M steered beams out of data from N hydrophones requires on the order of M × N operations. These computations must be completed for each time slice in real time— before data appears from the next time slice. The time between time slices is only a few milliseconds— far too short for the matrix multiplication to be performed. Fortunately, some computation short cuts can be used. Equation 3.19 is similar to a familiar equation— that of a Fast Fourier Transform (FFT 1 ): where k is an integer. This is a useful similarity if we make the substitutions: (3.22) 1 The subject of FFTs is vast and deep. Interested readers will find more information in Discrete-Time Signal Processing by A. V. Oppenheim and R. W. Schafer: Prentice Hall, 1989. Copyright © 2000 L-3 Communications SeaBeam Instruments Page 3-25 No portion of this document may be reproduced without the expressed written permission of L-3 Communications SeaBeam Instruments
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