Characteristic Functions in Radar and Sonar
Southeastern Symposium on System Theory
March 18,2002
Stephen R. Addison
Department of Physics and Astronomy
University of Central Arkansas
Conway, AR 72035
saddison@mail.uca.edu
Presented By
John E. Gray
Code B-32
Naval Surface Warfare Center Dahlgren
Division
Dahlgren, VA 22448
grayje1@nswc.navy.mil
1
Outline
• FOUR QUESTIONS
• WHAT ARE CHARCTERISTIC FUNCTIONS?
• SINGLE VARIABLE APPLICATIONS OF CHARCTERISTIC
FUNCTION.
• MULTI-DIMENSIONAL CHARCTERISTIC FUNCTIONS
• RAYLEIGH PROBLEM
2
FOUR QUESTIONS
å
å
å
Question 1: Given we know the density for x is PÝxÞ, what is the distribution for u = fÝ xÞ?
å
å
Question2: Givenwe knowthe densityfor x is PxÝxÞ and the densityfor y is PyÝyÞ, what is the distributionfor
å å å
u = x + y?
å
å
Question 3: Given we know the density for x is P x ÝxÞ and the density for y is P y ÝyÞ, what is the distribution for
å åå
u = x y?
å
å
Question 4: Given we know the density for x is P x ÝxÞ and the density for y is P y ÝyÞ, what is the distribution for
å åx
u = å?
y
3
DEFINTIONS OF CHARCTERISTIC
FUNCTION
MP ÝSÞ = e jSx = X
DEFINTION:
1.
2.
3.
4.
PROPERTIES
K
?K
e jSx PÝxÞ dx
MÝ0Þ = 1.
MD Ý?SÞ = MÝSÞ.
|MÝSÞ| ² 1.
|MÝSÞ| ² MÝ0Þ.
ALTERNATIVE DEFINITION:
K
K
MP ÝSÞ = X ?K e PÝxÞ dx = X ?K >
jSx
n
K ÝjSxÞ
PÝxÞdx
n=0 n!
=>
n
K ÝjSÞ
n=0 n!
Öx n ×
4
DEFINTIONS OF CHARCTERISTIC
FUNCTION
TWO METHODS FOR
CALCULATION
OF MOMENTS
K
Öx × = X ?K x n PÝxÞ dx
n
n
Öx × =
n
1 / MP ÝSÞ
/S n
jn
P S =0
differentiation is easier than integration
Fourier transform pair relationship between the PDF and CF:
PÝxÞ =
1
2^
K
X?K e ?jSx MP ÝSÞdx
5
SINGLE VARIABLE APPLICATIONS OF
CHARCTERISTIC FUNCTION
one would often like to know the density of a new variable,
å
say û, that is a function of the old variable: å
u= fÝ xÞ
Probability Density Function:
Characteristic Function:
Apply CF to PDF:
PDF for Transformed variable:
1
2^
PÝuÞ =
Mu ÝSÞ = e
PÝuÞ =
1
2^
K
X?K e ?jSu Mu ÝSÞ dS
jSfÝxÞ
K
K
= X ?K e jSfÝxÞ PÝxÞ dx
K
X?K X ?K e jSfÝxÞ e ?jSx PÝxÞ dS dx
K
PÝuÞ = X ?K NÝu ? fÝxÞÞ PÝxÞ dx
This solves the problem for Exercise 1.
6
SINGLE VARIABLE APPLICATIONS OF
CHARCTERISTIC FUNCTION
Interpretation of Delta Function of Function
NÝgÝxÞÞ = >i NÝx ? x iÞ g v Ýx1 Þ
|
i
|
NÝfÝxÞ ? uÞ = >i NÝx ? x i Þ fv Ýx1 Þ
|
PÝuÞ = >i
i
|
PÝxi Þ
|fv Ýx iÞ |
7
SINGLE VARIABLE APPLICATIONS OF
CHARCTERISTIC FUNCTION
Translation
å å
å
(Translation): Let u = ax + b, what is PÝuÞ given we know that the PDF of x is PxÝxÞ? Solving
for zero of the equation gives
x = u ?a b
and note dx = 1a , then the distribution is
1 PxÝ u ? b Þ
u?b =
#
PÝxÞ|
PÝuÞ = 1
x= a
a
a
a
8
SINGLE VARIABLE APPLICATIONS OF
CHARCTERISTIC FUNCTION
Square Law Detector
å å
å
(Square Law Detector): Let u = x 2 , what is PÝuÞ given we know that the PDF of x is P x ÝxÞ?
Answer: Solve for zeros
x1 = u
and
x2 = ? u
note 2xdx = du, then * becomes
PÝuÞ =
å
If x is NÝ0, aÞ, then the PDF is
1 P x ÝuÞ| x ,x = 1 PÝ u Þ + PÝ? u Þ
1 2
2 u
2 u
P u ÝuÞ =
#
? u
1
e 2a2 BÝuÞ.
a Ý2^uÞ
where B is the unit step function
BÝxÞ = 1 Ýx ³ 0Þ
= 0 ÝotherwiseÞ
#
9
SINGLE VARIABLE APPLICATIONS OF
CHARCTERISTIC FUNCTION
The coordinate transform ation y = R sinÝj! Þ a P DF f j Ýj Þ is onto but not one-to-one ov er
interv al ß?K, Kà. Thus ref: * has an infinite num ber of zeros. It is m ore conniv ent to determ ine the C F
directly, so the transform ation of the P DF is
M S Ýg Þ =
K
X ?K e jgR sin Ýj Þ f j Ýj Þdj .
The ex ponential can be w ritten as
K
>
J n Ýg RÞe jn j = e jgR sin Ýj Þ ,
n =?K
so the C F is giv en by
K
M S Ýg Þ =
>
n =?K
J n Ýg RÞ X
K
?K
e
jn j
K
fÝj Þdj =
>
J n Ýg RÞF ÝnÞ;
n =?K
w hich can be rew ritten as
K
M j Ýg Þ = J 0 Ýg RÞ +
> J n Ýg RÞ ßF ÝnÞ + Ý? 1Þ n F Ý?nÞ à
n =1
w here F ÝnÞ is the F ourier transform of the P DF f j Ýj Þ ev aluated at g = n. Depending on the problem , the
C F is sufficient, but som etim es it is still useful to k now the P DF . N ow if w e apply the identity (A bram ow itz)
n
K
Þ
B Ý1 ? |g |Þ,
X ?K e ?jgt J n ÝtÞ dt = 2Ý? jÞ T n Ýg
Ý1 ? g 2 Þ
w here T n Ýx Þ is the n-th order C hebyshev polynom ials, the P DF of the coordinate transform ation is
f y ÝyÞ =
w here a n = Ý? jÞ n F ÝnÞ.
a 0 + > Kn =1 Ýa n + a ?n ÞT n Ý
2
2
^ ÝR ? y Þ
y
R
Þ
B Ý1 ?
y
Þ,
R
#
10
MULTI-VARIABLE APPLICATIONS OF
CHARCTERISTIC FUNCTION
Repeating the same procedure for the single dimensional case
to the multi-dimensional case yields:
K
K
PÝu, vÞ = X ?K X ?K X X Nßu ? fÝx, yÞàNßv ? gÝx, yÞàPÝx, yÞ dxdy
This due to Cohen, is sufficient to allow us to solve the problem
of determining many of the two dimensional combinations of
random variables such as those in Exercises 2-4.
11
MULTI-VARIABLE APPLICATIONS OF
CHARCTERISTIC FUNCTION
Let û = fÝx!,yÞ, the CF is
K
K
MuÝSÞ = X
ejSfÝx,yÞ PÝx,yÞ dxdy,
X
?K ?K
so the density becomes
K
1
e?jSuMuÝSÞ dS
PÝuÞ =
X
2^ ?K
K K K
1
ejSßu?fÝx,yÞà PÝx,yÞ dydxdS
=
X
X
X
2^ ?K ?K ?K
integrating out S gives delta functions
K
PÝuÞ = X
K
X
?K ?K
Nßu ? fÝx,yÞàPÝx,yÞ dx.
# (C)
12
MULTI-VARIABLE APPLICATIONS OF
CHARCTERISTIC FUNCTION
EXAMPLE:
Translation
å å å
Let z = x + y, with distributions P x ÝxÞ and P y ÝyÞ (assuming they are uncorelated), then the
distribution of û which is denoted by P u ÝuÞ is ref: C
P z ÝzÞ =
=
=
K
K
X ?K X?K Nßz ? x ? yàP xy Ýx, yÞ dxdy
K
X ?K Pxy Ýx, z ? xÞ dx
K
X ?K Px ÝxÞP y Ýz ? xÞ dx (uncorrelated)
#
13
MULTI-VARIABLE APPLICATIONS OF
CHARCTERISTIC FUNCTION
EXAMPLE:
Multiplication
å åå
Let z = x y, with distribution PxyÝx,yÞ or distributions PxÝxÞ and PyÝyÞ (assuming they are
uncorelated), then the distribution of û which is denoted by PzÝzÞ is ref: C
K
PzÝzÞ = X
If we note that >i NÝx ? x i Þ
K
X
?K ?K
Nßz ? xyàPxyÝx,yÞ dxdy.
1
, so
|f Ýx iÞ|
v
NÝy ? zx Þ
Nßz ? xyà =
|x |
so
K
f zÝzÞ = X
?K
1 P z ,x dx.
|x| xy x
#
14
MULTI-VARIABLE APPLICATIONS OF
CHARCTERISTIC FUNCTION
EXAMPLE:
Division or Monopulse Ratio
å åx
Let z = å , with distribution PxyÝx,yÞ or distributions PxÝxÞ and PyÝyÞ (assuming they are
y
uncorelated), then the distribution of z! which is denoted by PzÝzÞ is
K K
PzÝzÞ = X X Nßz ? xy àPxyÝx,yÞ dydx.
?K ?K
If we note that
Nßz ? xy à = NÝy ? zxÞ|x|
so
K
f zÝzÞ = X |x|PxyÝzx,xÞdx.
?K
#
15
MULTI-VARIABLE APPLICATIONS OF
CHARCTERISTIC FUNCTION
RAYLEIGH PROBLEM
Sum of two dimension random amplitudes and phases:
X=>
N
!
x
i=1 i
=>
N
i=1
â i cosÝS! i Þ
Arose in Scattering off Rough Surfaces,
Applications in Radar, Sonar, Acoustics, Communications, Physics, etc.
16
MULTI-VARIABLE APPLICATIONS OF
CHARCTERISTIC FUNCTION
INSTANCE OF TRACKING PROBLEM
Transformations from spherical to C artesian coordinates
x! = Ý r! Þ cos S! sinÝ j! Þ ,
y = Ý r! Þ sin S! sinÝ j! Þ ,
z! = Ý r! Þ cos Ý j! Þ .
Rewritten as:
x! =
y=
!r
2
!r
2
#(
sin S! + j! + sin S! ? j!
cosÝS! + j! Þ ? cosÝS! + j! Þ
17
MULTI-VARIABLE APPLICATIONS OF
CHARCTERISTIC FUNCTION
1
Then the distribution for z is: f z ÝzÞ = X ?1
>Kn=1 Ýan +a ?n ÞT n ÝjÞ
P r Ý jz Þ a 0 +
^|j|
Ý1?j 2Þ
dj.
two sum case z = z 1 + z 2
fzÝzÞ
K
=X
?K
1
X?1
(uncorrelated)
=
1
X?1
K
X?K Pz Ýz1ÞPz Ýz1 ? zÞ dz1
1
#
2
K
Pr1Ý j11 Þ a0 + >n=1
Ýan +a?n ÞTnÝj1Þ
z
^|j1 |
Ý1 ? j21Þ
K
1?z
Pr2Ý z2+z
j2 Þ a0 + >n=1Ýan + a?n ÞTnÝj2Þ
^|j2 |
Ý1? j22Þ
dj1 ×
dj2dz1
Case n is obtained by repeated application of Case 2
18
Conclusions
• Demonstrated how to use CF to bypass difficulties in computing
combinations of random variables using a method based on Cohen
• Using Fourier analysis and application of definition of Dirac delta
function, combined PDF is obtained with considerable simplification
• Motivation engineering applications that occur in radar/sonar
applications that involve combinations of probability density functions
• Have demonstrated simple method to obtain PDF for different
combinations of random variables
• Rayleigh problem for i=2 solved which implies general solution for
case i=n
19