# Notation

## Notation

ix Notation Alphabetic an adjA A == (aij) AT A-I [AlB] IAI A(t) A(w) bn b chirp( t) comb(t) Real coefficients of a Fourier cosine series Adjoin...

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ix

Notation Alphabetic an adjA A == (aij)

AT

A-I

[AlB]

IAI A(t)

A(w) bn b chirp( t) comb(t)

Real coefficients of a Fourier cosine series Adjoint of matrix A Matrix with element at i t h row and jth column Transpose of A Inverse of A Augmented matrix formed from matrices A and B Determinant of A Amplitude modulation (amplitude envelope) Amplitude spectrum Real coefficients of a Fourier sine series Data vector of linear system Ax = b Unit chirp function [with complex form exp( -io:t 2 ) ] Comb function [= ~ (t - nT)]

s

n

cond(A) en

C c D detA e

f(t)

f(z) If I

Ilf(t) II Ilfill Ilfil12 Ilxll p Ilxll oo IIAII F(w) Fr(w) Fi(w) Fi

g(t) g(t I to,w) H(t) I

Im[f]

Condition number of matrix A (=IIAII x IIA- I II) Complex coefficients of a complex Fourier series for example Capacitance or the contour followed by a path of integration in the z-plane Data vector associated with linear system x = Mx + c Fractal dimension or diagonal matrix Determinant of A (also denoted by I A I) Error vector Arbitrary real function - typically object function or system input Function of a complex variable modulus of complex variable or function f Norm (e.g. a Euclidean norm) of a function f(t) Norm of an array or vector fi Euclidean norm of array or vector fi p-norm of vector x 'Infinity' or uniform norm of a vector x Norm of matrix A Complex spectrum of function f(t) Real component of spectrum Imaginary component of spectrum Discrete complex spectrum of discrete function I, Arbitrary function Green's function Tophat function Unit or identity matrix Imaginary part of complex variable or function f

x

k

L L1 M

n(t) ni

n.

p(t) Pi P(w) Pi P(X) P(a I b) P(w) Pi Pr(x) q q(t) R Ri Re[j]

s(t) Si

sgn(t) sinc(t) t

U U1 U(t) u(x, t) Vi

WAL(n, t) X Xi T

x

Xi Xo Z

z* E

\I

Wavenumber (= 27f/ A) Lower triangular matrix Lower triangular matrix with 1's along the leading diagonal Iteration matrix associated with linear system X = M X + c Noise function Discrete noise function Noise spectrum Instrument function or Impulse Response Function Discrete Impulse Response Function Transfer Function (Fourier transform of Pi) Discrete Transfer Function (DFT of Pi Probability density function also denoted by Pr[x(t)] Conditional probability of obtaining a given b Power spectrum (=1 F(w) 12 ) where F(w) is the Fourier transform of f(t) Discrete power spectrum Probability occurrence of X Fourier dimension Quadrature signal Resistance Denotes the i t h row of a matrix Real part of complex variable or function f Real or complex (analytic) signal Discrete real or complex signal Sign function Sine function (= sin (t) / t) Time Upper triangular matrix Upper triangular matrix with l's along the leading diagonal Unit step function Solution to a partial differential equation (e.g. wavefield) Eigenvectors of linear system AVi = AiVi Walsh function Solution vector of linear system Ax = b Eigenvectors of linear system AXi = AiXi Transpose of vector x i t h element of vector x Initial value Complex number of the form a + ib Complex conjugate a - ib of a complex number a + ib In (e.g. x E [a, b) is equivalent to a x < b) Forall (e.g. f(t) = 0, \It E (a, b])

s:

xi Greek a

f(q) 6(t) 6ij

6t

()(t) A Ai

7jJ(t) ¢n p(M) w

n

Chirping parameter Gamma function Dirac delta function Kronecker delta Small increment (denoted also by t1t) Instantaneous phase Wavelength Eigenvalues of linear system AXi = Aixi Instantaneous frequency Delta sequence function Spectral radius of (iteration) matrix M Angular frequency or the relaxation parameter Bandwidth of a spectrum

Operators

6 iJ

t» PI PI~ 1

fI jq

L L-1 S W W- I ®

8 ~ +--+

Cosine transform operator Linear differential operator Fractional differential operator One dimensional Fourier transform One dimensional inverse Fourier transform Hilbert transform operator Fractional integral operator (e.g. Riemann-Liouville fractional integral) One sided Laplace transform operator Inverse Laplace transform operator (Bromwich integral) Sine transform operator Wavelet transform Inverse wavelet transform Convolution operation (continuous or discrete and causal or otherwise, depending on the context specified) Correlation operation (continuous or discrete and causal or otherwise, depending on the context specified) Transformation into Fourier space Transformation into some transform space (as defined)