ABSTRACT

N Number of sensors in a line array receiver, where {xn(ti), n = 1, 2, …, N}, or number of rings in a cylindrical array

NLMS Normalized least mean square

D fi θ,( )

dn fi θ,( ) j2π i 1-( )fs

M ------------------τn θ( )exp=

D f θs φs, , )( )

ε x s ε+=

k

n Index for space samples of line array sensor time series {xn(ti) ,n = 1,2, …,N }

P(f, θs) Beam power pattern in the frequency domain for a line array steered at azimuth angle θs and expressed by P(f, θs) = B(f, θs) × B∗(f, θs)

π 3.14159

r Index for the rth ring of a cylindrical or spherical array of sensors

R Radius of a receiving circular array

R(fi) Spatial correlation matrix with elements Rnm(f, dnm) for received sensor time series

ρnm(f, δnm) Cross-correlation coefficients given from ρnm(f, dnm) = Rnm(f, dnm)/ (f) Signal vector whose nth element is expressed by sn(ti) = sn[ti + τn(θ)]

S Spatial correlation matrix for the plane wave signal sn(ti)

S(fi, θ) Spatial correlation matrix for the plane wave signal in the frequency domain; it has as its nth row and mth column defined by, Snm(fi, θ) = As(fi)dn(fi, θ) (fi, θ)

STCM Steered covariance matrix

STMV Steered minimum variance

SVD Singular value decomposition method

Power spectral density of noise, εn(ti)

Row vector of received N sensor time series {xn(ti), n = 1, 2, …, N}

Xn(f) Fourier transform of xn(ti)

Xn(fi, θs) Pre-steered sensor time series in frequency domain xn(ti, τn(θs)) Pre-steered sensor time series in the time domain X2(f) Mean acoustic intensity of sensor time sequences at frequency bin f

τn(θ, φ) Time delay between (n – 1)st and nth sensor of a multi-dimensional array for incoming plane waves with direction of propagation of azimuth angle θ and elevation angle φ

TL Propagation loss for the range separating the source (reflected signals) and the array

τn(θ) Time delay between the first and the nth sensor of the line array for an incoming plane wave with direction of propagation θ

W(θs) Diagonal matrix with the off diagonal terms being zero and the diagonal terms being the weights of a spatial window to reduce the sidelobe structure of a circular array beamformer

wr, m The (r, m)th term of a 3-D spatial window of a multi-dimensional plane wave beamformer

ω Frequency in radians/second

Result of the signal blocking matrix Cof the GSC adaptive line array beamformer being applied to pre-steered sensor time series

Line array adaptive beamforming weights or solution to the constrained minimization problem that allows signals from the look direction θ to pass with a specified gain

The aim of this chapter is to bring together some of the recent theoretical developments on beamformers and to provide suggestions of how modern technology can be applied to the development of current and next-generation ultrasound systems and integrated active and passive sonars. It will focus on the development of an advanced beamforming structure that allows the implementation of adaptive and synthetic aperture signal processing techniques in ultrasound systems and integrated active-passive sonars deploying multi-dimensional arrays of sensors.