analytic signals and radar processing wayne m. lawton department of mathematics national university...
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ANALYTIC SIGNALS AND RADAR PROCESSING
Wayne M. Lawton
Department of Mathematics
National University of Singapore
Block S14, 10 Kent Ridge Road
Singapore 119260
REVIEW BASIC CONCEPTS
OBJECTIVES
Analytic signal representation
Accuracy of analytic signal representation
FORMULATE ISSUES
Wigner-Ville distribution
Warp transformations and radar echos
Matched filtering, radar images, and ambiguity
Signal design and ambiguity localization
Fourier and Hilbert transform operators
FOURIER TRANSFORM OPERATOR
Hilbert space
Fourier transform operator
(R)L2
dth(t)f(t) h)(f,
and extended by continuity to (R)L2
of complex-valued functions
with scalar product
(R)L(R)L: 22 F
(R)L(R)Lf 21 ,dtef(t)f)(y)( tyi-2 F
Unitary property*-1h)(f,h)f,( FFFF
HILBERT TRANSFORM OPERATOR
Hilbert transform operator (R)L(R)L: 22 H
,dtf)(s)(|t|
lim
S
0 )ts(f(t)
H f smooth, compact support
and extended by continuity to (R)L2
Unitary, and f)(y)()ysgn(f)(y)( FFH
ANALYTIC SIGNAL REPRESENTATION
Construct
Iwhere
(R)L(R)L:iI 22 HAis the identity operator
Then 2HL 2R A , where
2RL
functions,2H is Hardy subspace of functions f that satisfy
0,0f)(y)( yF
is subspace of real-valued
f admits an analytic extension
to the upper half of the complex plane
Furthermore, ff Ae and h)f,(h)(f, 21 FF
WIGNER-VILLE DISTRIBUTION
)(RL(R)L:W 222R
dviuv-2ev/2)f(tv/2)f(tu)W(f)(t,
W(h))(W(f),|h)(f,| 2 Moyal
Describes time/frequency distribution of signal energy
)-,-W(f)())-f(W(e i2
group of orientation preserving diffeomorphisms of D~ circle Cayley}{R
Unitary representation
WARP TRANSFORMATIONS
i)(zi)(zz
(R))L(UD:U 2
))(('(t))f(g(U(g)f)(t) -1-1 tgg
Let an antenna (in an inertial frame) transmit a signal f that propagates at the speed of light c in the direction of a point scatterer whose distance (from the antenna) function d satisfies
Then special relativity implies that the echo signal reflected by the point scatterer is proportional to U(g)f where
RADAR ECHOS
(t),-t(t)g-1
c|(t)d'|Rt
sup
c(t)/c)-2d(t(t)
The radar echo signal is the sum of echos from point scatterers
MATCHED FILTERING, RADAR IMAGES, AND AMBIGUITY
(x)dfU(x)hx
D
U(g)f)(h,)(~ gThus the radar image, computed from matched filtering, is
)((x)g)d(x f
x
-1f g
D
Since U is unitary, the radar image equals the a convolution
ACCURACY OF ANALYTIC SIGNAL REPRESENTATION
Accuracy depends on transmitted signal f and warp U(x)
U(x)]fi[H, )fU(x)-U(x)(U(x)]f,[ AAA
Error equals the commutator
This vanishes for all h if and only if x is a linear fractional transformation
dct
batt
Error can be bounded using a wavelet expansion of f
SIGNAL DESIGN AND AMBIGUITY LOCALIZATION
Ideally, the radar image equals the scattering measure
~
Severe ambiguity constraints makes this impossible
This requires f
However, a priori knowledge about
adequate ambiguity localization.
enables
REFERENCESSanjay K. Mehta, Signal Design Issues for the Wigner Distribution Function and New Twin Processor for the Measurent of Target and/or Channel Structures, PhD Dissertation, University of Rochester, 1991
Elias Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, New Jersey, 1993