MATERIA PRIMA (LECHE)

A.1 LIST OF ABBREVIATIONS

AAR Adaptive Angular Response method - see chapter 5, sub-section 5.2.3 AIC Akaike’s Information Criterion - see chapter 5, sub-section 5.2.9

BASS-ALE Broad-bAnd Signal subspace Spatial spectrAL Estimation methods - see chapter 5, sub-section 5.2.7

CRLB Cramér-Rao Lower Bound - see chapter 5, sub-section 5.2.1

CSS Coherent Signal Subspace methods - see chapter 5, sub-section 5.2.7

CW Continuous Wave (monotonie)

DF direction finding

DPT Discrete Fourier Transform

DML Deterministic Maximum Likelihood method - see chapter 5, sub­ section 5.2.6

DOA Direction Of Arrival (of signals relative to the array) ERP Effective radiated power - see footnote 2 in chapter 2

ESPRIT Estimation of Signal Parameters via Rotational Invariance Techniques - see chapter 5, sub-section 5.2.5

FIR Finite Impulse Response (refers to a non-recursive digital filter)

FM Frequency Modulation - see in connection with the response to a linear FM pulse in chapter 2, section 2.3 and in appendix B .l

HF High Frequency ( 3 - 3 0 MHz) IDFT Inverse Discrete Fourier Transform

n R Infinite Impulse Response (refers to a recursive digital filter) IMP Iterative Multi-Parameter method - see chapter 5, sub-section 5.2.6 LS Least Squares — see in connection with LS-ESPRIT in chapter 5, sub­

section 5.2.5

MDL Minimum Description length - see chapter 5, sub-section 5.2.9

ME Maximum Entropy (Burg’s method) - see chapter 5, sub-section 5.2.4 ML Maximum Likelihood method - see chapter 5, sub-section 5.2.6

MN Minimum Norm method (also known as the TK method) - see chapter 5, sub-section 5.2.4

187 List of a b b r ev ia tio n s an d s y m b o ls

MLM Maximum Likelihood Method (former name for Capon’s method) - see chapter 5, sub-section 5.2.4

MUSIC M ultiple Signal Classification method - see chapter 5, sub-section 5.2.4 MVDR Minimum Variance Distortionless Response (Capon’s method) - see

chapter 5, sub-section 5.2.4

MVIR Minimum Variance Inverse Response - see chapter 5, sub-section 5.2.4 MVPR M inim um Variance Protected Response - see chapter 5, sub­

section 5.2.4

PDF Probability Density Function

RMS Root Mean Square - see in connection with rms sidelobes in chapter 4, sections 4.2 to 4.4

SML Stochastic Maximum Likelihood method - see chapter 5, sub­ section 5.2.6

STCM STeered Covariance Matrix technique - see chapter 5, sub-section 5.2.7 SNR Signal to Noise Ratio

SPM Sectoral Phase Mode - see chapter 3, section 3.5

TAM Toeplitz Approximation Method - see chapter 5, sub-section 5.2.5 TLS Total Least Squares - see in connection with TLS-ESPRIT in chapter 5,

sub-section 5.2.5

TSCM Tempo-Spatial Covariance Matrix - see chapter 5, sub-section 5.2.7 UHF Ultra High Frequency (300 - 3000 MHz)

VHF Very High Frequency (30 - 300 MHz)

A .2 List o f s y m b o ls ________________________________________________________ iillilllillllll 1 8 8 lllllll

A.2 LIST OF SYMBOLS

inverse of a matrix

complex conjugate of a scalar, a vector or a matrix transpose of a vector or a matrix

transpose conjugate of a vector or a matrix inverse of the transpose conjugate matrix

I • I determinant of a matrix. Euclidean norm o f a vector, absolute value of a scalar

II • lip Frobenius norm of a matrix: IIAIIp = tr(A^A)

[ • ] ceiling of (smallest integer larger than) an expression [^]mod2;r modulo-2;r value of a real number x

0 null vector

0 null matrix

Om complex weighting applied to the m ’th array element - see {222} and (235) in chapter 2

Omkit) steering impulse response of the m ’th array sensor in the direction of the it’th far-field source - see chapter 5, sub-section 5.2.2

A (ç)} amplitude pattern of a sectoral phase mode - see chapter 3, section 3.5 A signal amplitude - see appendix C.l

A(û}) array steering matrix, equals to A (ç>= ^, m) - see chapter 5, sub­ section 5.2.2

Amk mk^Xh element of the steering matrix A

A mode-space steering matrix - see chapter 5, sub-section 5.3.2 À combined ESPRIT steering matrix - see chapter 5, sub-section 5.2.5 Ax, Ay ESPRIT sub-array steering matrix - see chapter 5, sub-section 5.2.5

A steering vector - see chapter 5, sub-section 5.2.4

A(ç>, (û) functional form of steering matrix - see footnote 1 in chapter 5, sub­ section 5.2.2

argmaxf(^) maximising vector y in the expression ^{q>)

<p

argminfF(y) minimising vector y in the expression ^{<p)

<p

{ bq} vectors defined in appendix C.3

B constraint matrix in an MVIR algorithm - see chapter 5, sub­ section 5.2.4,

TAM sub-matrix comprising the top rows o f A - see (52.80) in chapter 5,

spatial-smoothing sub-matrix comprising the top M" rows of A - see (5.4J) in chapter 5

1 8 9 Iillilllillllll _________________________________________ List o f a b b r ev ia tio n s a n d s y m b o ls

b vector of least Euclidean norm whose first element equals to one, that belongs to the noise subspace - see chapter 5, sub-section 5.2.4

c speed of propagation

Cd, Cs asymptotic DML and SML DOA error covariance matrices - see

chapter 5, sub-section 5.2.6

CfiqicOy 6) ^ ’th order coefficient for the ;x’th phase-mode - see chapter 2, section 2.4

Cq the vector , • • •, Coq, • • •, Cy^qf - see (C3.6) in appendix C.3 C^n unaliased component of - see (C.4.1) in appendix C.4 cov (u) covariance matrix of a random vector u

d inter-element spacing in a uniformly-spaced linear array - see (3.22) in chapter 3

(inverse) DFT coefficients of { _ see (3.621) in chapter 3 D block matrix defined by (E.43) in appendix E.4

2>^(z) transfer function o f FIR block in UR mode-alignment filter - see (3.6.18) in chapter 3

det[ • ] determinant of a matrix

diag(z) diagonal matrix whose main diagonal comprises the elements of z !E general cost function in a minimisation process,

DOA estimation error defined by (C.l.I) in appendix C.1

E delay-only steering vector - see expression (5222) in chapter 5, sub­ section 5.2.3

M xM diagonal mode-phasing matrix defined by (43.13) in chapter 4 E mode-phasing matrix as defined by (53.2) in chapter 5, sub-section 5.3.2 En a column of E - see definition (532) sub-section 5.3.2

S i ’ ) expectation operator exponential function

fu (u ) probability density function of a random vector u

/u(mI^) conditional probability density function of a random vector £/, given the second random vector 0 - see chapter 5, sub-section 5.2.1

Fi6,(p,cû) far-field (steady-state) array radiation pattern - see (222), (2.3.5) in chapter 2

Fm((p) (nominal) mode-space beam pointing in direction In m lM - see (3.23) in chapter 3

Fmi(p) nominal mode-space beam Fmiç) at a modified array radius - see chapter 4, section 4.4

jFmiÇ) perturbed mode-space beam pointing in direction In m lM - see chapter 4, sections 4.2,4.3,4.4

A .2 List o f s y m b o ls ________________________________________________________ nililiilllllll 1 9 0 lllllll

lÆI^, \AFm^ rms error pattern - see chapter 4, sections 4 .2,4.3,4.4 F L x l array far-field pattern vector - see (45.4) in chapter 4

F y ‘^((p) a sectoral phase mode pointing in direction Inm lM , having an effective mode number of v - see chapter 3, section 3.5

G LxM element pattern matrix - see {455} in chapter 4, block matrix defined by (EA7) in appendix E.4

G ,G ”,G ”G frequency and elevation averaged matrices, defined by (4.73) - (4.75) in chapter 4

g(0,(p) (firequency-independent) element (voltage) pattern shared by all the array elements

gm(0, (p, CO) element (voltage) pattern for the m’th array element ge(0) g^(ç>) separable element pattern - see appendix C.3

Ç array gain factor defined by (42.8) in chapter 4

H hermitian square root of R* - see chapter 5, sub-section 5.52, and also (E.13) in appendix E .l

hi(0) the f’th angular Fourier coefficient of an element pattern - see (22.4) in chapter 2

h the vector [h.i h.i ho h i - •• hjY - see (C35) in appendix C.3 { Ik (inverse) DFT of { - see Fig. 3 .6 5 in chapter 3

TÇin complex weights in an FIR mode-alignment filter - see (3.2.4) in chapter 3

j^(z) transfer function of HR mode-alignment filter - see (3.620) in chapter 3 H^{s) transfer function of an analogue deconvolution filter for phase mode

coefficients - see chapter 3, section 3.6

i index referring to the f’th angular Fourier coefficient of an element pattern - see (22.4),

index in the range 0 < i < M - l referring to the f’th eigenvalue (in decreasing order) or corresponding eigenvector of the covariance matrix I the order of the highest non-vanishing term in the angular Fourier-series

representation (22.4) for the element-pattems

I identity matrix

Iq first column of I

I v ( x ) modified Bessel function of the first kind of order v and argument x

Int ( x) integer part of the real number x

j square root of -1

J Fisher information matrix - see chapter 5, sub-section 5.2.1

J d\ Js^ CRLB matrices for the deterministic and stochastic signal model - see chapter 5, sub-sections 5.2.1 and 5.2.6