INF5410 Array signal processing.
Intro, ch. 1 and 2
Andreas Austeng
January 2005
Andreas Austeng
INF5410 Array signal processing. Intro, ch. 1 and 2
Introduction
Ch. 1: Introduction
Ch. 2: Signals in Space and Time
Introduction
Terminologi
Cooking spagetti???
One microphone.
Directive?
Two microphones.
Less noise?
Directive array? How to combine?
An array of microphones / colanders.
One microphone+colander is directive!
How ... ???
Andreas Austeng
INF5410 Array signal processing. Intro, ch. 1 and 2
Introduction
Ch. 1: Introduction
Ch. 2: Signals in Space and Time
Introduction
Terminologi
The different chapters in J & D.
Ch. 1: Introduction.
Ch. 2: Signals in Space and Time.
Physics: Waves and wave equation.
c, λ, f , w , ~k, . . .
Ideal and “real” conditions.
Ch. 3: Apertures and Arrays.
Ch. 4: Beamforming.
Classical, time and frequency domain algorithms.
Ch. 7: Adaptive Array Processing.
Andreas Austeng
INF5410 Array signal processing. Intro, ch. 1 and 2
Introduction
Ch. 1: Introduction
Ch. 2: Signals in Space and Time
Introduction
Terminologi
Norsk terminologi
Bølgeligningen
Planbølger, sfæriske bølger
Propagerende bølger, bølgetall
α
~ - sinking/sakking
Dispersjon
Attenuasjon eller demping
Refraksjon
Ikke-linearitet
Diffraksjon; nærfelt, fjernfelt
Gruppeantenne ( = array)
Kilde: Bl.a. J. M. Hovem: “Marin akustikk”, NTNU, 1999.
Andreas Austeng
INF5410 Array signal processing. Intro, ch. 1 and 2
Introduction
Ch. 1: Introduction
Ch. 2: Signals in Space and Time
Chapter 1
Array signal processing
Goal of signal processing: To extract as much information as
possible from our environment.
ASP: Branch of signal processing; focusing on signals
conveyed by propagating waves.
Array: a group of sensors located at distinct spatial locations.
Fig 1.2:
Goal:
To enhance the signal-to-noise ratio beyond that of a single
sensor’s output.
To characterize the field by determining the number of sources
of propagation.
To track the enery sources as they move in space.
Andreas Austeng
INF5410 Array signal processing. Intro, ch. 1 and 2
Introduction
Ch. 1: Introduction
Ch. 2: Signals in Space and Time
Propagating waves
Wave equation
This is the equation in array signal processing:
∇2~s =
∂2s
∂2s
∂2s
1 ∂2s
+
+
=
∂x 2 ∂y 2 ∂z 2
c 2 ∂t 2
s is a general scalar field (electromagnetics: electric or
magnetic field, acoustics: sound pressure ...).
c is the speed of propagation.
Monochromatic solution in Cartesian coordinates, plane wave:
s(~x , t) = A exp{j(ωt − ~k · ~x )}
Andreas Austeng
INF5410 Array signal processing. Intro, ch. 1 and 2
Introduction
Ch. 1: Introduction
Ch. 2: Signals in Space and Time
Propagating waves
Wave equation and arbitrary solutions
The wave equation is linear
Solution may be a sum of complex exponentials
Almost any signal may be expressed as a sum of complex
exponentials using Fourier theory
Therefore any signal, no matter its shape, may be a solution
to the wave equation - and the shape will be preserved as it
propagates
Propagating waves are therefore ideal carriers of information
Modified by the boundary conditions - to determine which
components that are excited
Propagation is determined by the deviations of the medium
from ideal
Andreas Austeng
INF5410 Array signal processing. Intro, ch. 1 and 2
Introduction
Ch. 1: Introduction
Ch. 2: Signals in Space and Time
Propagating waves
Plane waves
Propagating plane wave: s(t − α
~ · ~x )
Propagating sinusoidal plane wave: sin(ωt − ~k · ~x )
Slowness vector: α
~ = ~k/ω, |~
α| = 1/c
Dispersion relation: ω = c · k
Wavenumber vector: ~k = ω~
α, |~k| = 2π/λ
Frequency and wavelength: c = λ · ω/2π = λ/f
Andreas Austeng
INF5410 Array signal processing. Intro, ch. 1 and 2
Introduction
Ch. 1: Introduction
Ch. 2: Signals in Space and Time
Propagating waves
Wave equation in spherical coordinates
wavefront
φ: azimuth
θ: elevation
z
k
sφ,θ
φ
y
θ
xm
x
Transducer array
Andreas Austeng
INF5410 Array signal processing. Intro, ch. 1 and 2
Introduction
Ch. 1: Introduction
Ch. 2: Signals in Space and Time
Propagating waves
Doppler effect
Nonrelativistic:
‘
w =w
1−α
~ · ~vsensor
1−α
~ · ~vsource
.
Relativistic (v c)
w ‘ = w (1 − α
~ · (~vsensor − ~vsource )) .
Andreas Austeng
INF5410 Array signal processing. Intro, ch. 1 and 2
Introduction
Ch. 1: Introduction
Ch. 2: Signals in Space and Time
Propagating waves
Array Processing Implications
Whenever the wave equation apply, the following is valid:
Propagating signals are functions of a single variable, s(·),
with space and time linked by the relation t − α
~ · ~x .
The speed of propagation depends on physical parameters of
the medium.
Signals propagates in a specific direction ζ~0 represented
equivalently by either α
~ or ~k.
Spherical waves desctibe the radiation pattern of most
sources.
The Superposition Prinsiple applies, allowing several
propagating waves to occur simultaneously without
interaction.
Andreas Austeng
INF5410 Array signal processing. Intro, ch. 1 and 2
Introduction
Ch. 1: Introduction
Ch. 2: Signals in Space and Time
Propagating waves
When does wave equation apply?
The medium must be homogeneous (== constant
propagation speed). If not; refraction, c = c(~x ).
Regardless of amplitude and frequency, the wave must
interact with the medium in the same way. If not;
nonlinearities, c = c(s(t)), and dispersion, c = c(w ) (==
frequency dependent propagation).
The medium must be lossless. If not;
attenuation, c = c< + c= .
Andreas Austeng
INF5410 Array signal processing. Intro, ch. 1 and 2
Introduction
Ch. 1: Introduction
Ch. 2: Signals in Space and Time
Propagating waves
Norsk terminologi
Bølgeligningen
Planbølger, sfæriske bølger
Propagerende bølger, bølgetall
α
~ - sinking/sakking
Dispersjon
Attenuasjon eller demping
Refraksjon
Ikke-linearitet
Diffraksjon; nærfelt, fjernfelt
Gruppeantenne ( = array)
Kilde: Bl.a. J. M. Hovem: “Marin akustikk”, NTNU, 1999.
Andreas Austeng
INF5410 Array signal processing. Intro, ch. 1 and 2