Part 1 Signal analysis: Complex analysis; The delta function; The Fourier series; The Fourier transform; Other integral transforms. Part 2 Computational linear algebra: Matrices and matrix algebra; Direct methods of solution; Iterative methods of solution; Eigen values and Eigen vectors. Part 3 Programming and software engineering: Principles of software engineering; Modular programming in C. Part 4 DSP: Methods, algorithms and building a library: Digital filters and the FFT; Frequency domain filtering with noise; Statistics, entropy and extrapolation; Digital filtering in the time domain; Random fractal signals; Solutions; Graphics utility.
Notation
Alphabetic
an Real coefficients of a Fourier cosine series
adjA Adjoint of matrix A
A = (aij) Matrix with element at ith row and jth column
AT Transpose of A
A-1 Inverse of A
[A|B] Augmented matrix formed from matrices A and B
|A| Determinant of A
A(t) Amplitude modulation (amplitude envelope)
A(?) Amplitude spectrum
bn Real coefficients of a Fourier sine series
b Data vector of linear system Ax = b
chirp (t) Unit chirp function [with complex form exp(-iat2)]
comb(t) Comb function =Sndt-nT]
cond(A) Condition number of matrix A (=||A|| × ||A-1||)
cn Complex coefficients of a complex Fourier series for example
C Capacitance or the contour followed by a path of integration in the z-plane
c Data vector associated with linear system x = Mx + c
D Fractal dimension or diagonal matrix
detA Determinant of A (also denoted by | A |)
e Error vector
f(t) Arbitrary real function - typically object function or system input
f(z) Function of a complex variable
| f | modulus of complex variable or function f
||f(t)|| Norm (e.g. a Euclidean norm) of a function f(t)
||fi|| Norm of an array or vector fi
||fi||2 Euclidean norm of array or vector fi
||x||p p-norm of vector x
||x||8 'Infinity' or uniform norm of a vector x
||A|| Norm of matrix A
F(?) Complex spectrum of function f(t)
Fr(?) Real component of spectrum
Fi(?) Imaginary component of spectrum
Fi Discrete complex spectrum of discrete function fi
g(t) Arbitrary function
g(t|t0,?) Green's function
H(t) Tophat function
I Unit or identity matrix
Im[f] Imaginary part of complex variable or function f
k Wavenumber (= 2p/?)
L Lower triangular matrix
L1 Lower triangular matrix with l's along the leading diagonal
M Iteration matrix associated with linear system x = Mx + c
n(t) Noise function
ni Discrete noise function
Ni Noise spectrum
p(t) Instrument function or Impulse Response Function
pi Discrete Impulse Response Function
P(?) Transfer Function (Fourier transform of pi)
Pi Discrete Transfer Function (DFT of pi
P(x) Probability density function also denoted by Pr[x(t)]
P(a|b) Conditional probability of obtaining a given b
P(?) Power spectrum (=| F(?) |2) where F(?) is the Fourier transform of f(t)
Pi Discrete power spectrum
Pr(x) Probability occurrence of x
q Fourier dimension
q(t) Quadrature signal
R Resistance
Ri Denotes the ith row of a matrix
Re[f] Real part of complex variable or function f
s(t) Real or complex (analytic) signal
si Discrete real or complex signal
sgn(t) Sign function
sinc(t) Sinc function (= sin(t) / t)
t Time
U Upper triangular matrix
U1 Upper triangular matrix with l's along the leading diagonal
U(t) Unit step function
u(x,t) Solution to a partial differential equation (e.g. wavefield)
vi Eigenvectors of linear system Avi = ?ivi
WAL(n, t) Walsh function
x Solution vector of linear system Ax = b
xi Eigenvectors of linear system Axi = ?ixi
xT Transpose of vector x
xi ith element of vector x
x0 Initial value
z Complex number of the form a + ib
z* Complex conjugate a - ib of a complex number a + ib
? In (e.g. x ? [a, b) is equivalent to a = x < b)
? Forall (e.g. f (t)= 0, ?t ? (a,b])
Greek
a Chirping parameter
G(q) Gamma function
d(t) Dirac delta function
dij Kronecker delta
dt Small increment (denoted also by ?t)
?(t) Instantaneous phase
? Wavelength
?i Eigenvalues of linear system Axi = ?ixi
?(t) Instantaneous frequency
?n Delta sequence function
?(M) Spectral radius of (iteration) matrix...
