Prof. David R. Jackson ECE Dept. Spring 2014 Notes 36 ECE 6341 1.
Prof. David R. Jackson ECE Dept. Spring 2014 Notes 30 ECE 6341 1.
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Prof. David R. Jackson ECE Dept. Spring 2014 Notes 30 ECE 6341 1
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Transcript of Prof. David R. Jackson ECE Dept. Spring 2014 Notes 30 ECE 6341 1.
1
Prof. David R. JacksonECE Dept.
Spring 2014
Notes 30
ECE 6341
2
2-D Stationary Phase Method
2D stationary phase point:
,, j g x y
S
I f x y e dx dy
0 0
0 0
, 0
, 0
x
y
g x y
g x y
Assume 0 0, ,x y x y
0 0 0 0
2 2
0 0
0 0
, ,
1 1
2 2
x y
xx yy
xy
g x y g x y g x x g y y
g x x g y y
g x x y y
3
2-D Stationary Phase (cont.)
Denote
Then
0 0
0 0
0 0
1,
21
,2
,
xx
yy
xy
g x y
g x y
g x y
2 20 0 0 00 0,
0 0~ ,
j x x y y x x y yj g x yI f x y e e dx dy
4
2-D Stationary Phase (cont.)
Let 0
0
x x x
y y y
1, 0
1, 0
1, 0
1, 0
xx
xxx
yy
yyy
g
g
g
g
x
y
2 2
0 0,0 0~ , x y
j x y xyj g x yI f x y e e dx dy
whereand
We then have
5
2-D Stationary Phase (cont.)
Let
s x t y
2 2
0 0,0 0
1~ ,
x y
stj s t
j g x yI f x y e e ds dt
We then have
2 2
0 0,0 0~ , x y
j x y xyj g x yI f x y e e dx dy
6
2-D Stationary Phase (cont.)
2 2
0 0,0 0
1~ ,
x y
stj s t
j g x yI f x y e e ds dt
22 2
2 2 2
22 2
2
42
42
xx y x x y
xx y x y
tst ts t s t
t ts t
Complete the square:
2I
7
2-D Stationary Phase (cont.)
Now use
2 2
2
2
4 2
2
x xy x y
tt j sj t
I e e ds dt
The integral I is then
2xt
s s
ds ds
2 22
24
2
y x yx
tj t
j sI e e ds dt
so
8
2-D Stationary Phase (cont.)
2 22
24
2
y x yx
tj t
j sI e dt e ds
The integral I is then in the form of the product of two integrals:
9
2-D Stationary Phase (cont.)
Integral in s:
2
2
4
1
x
x
x
j s
s
j u
j
I e ds
e du
e
2 t sI I IThis has the form
u s
Use
2 /4j x je dx e
Recall that
10
2-D Stationary Phase (cont.)
Define:
Integral in t:
2
2 14y x y
j t
tI e dt
2
14
x y
11
2-D Stationary Phase (cont.)
Hence
Then we have 2
yj t
tI e dt
0 0, 4 4
0 0
1 1~ , x y
j jj g x yI f x y e e e
4
1 yj
tI e
u t
Use
12
2-D Stationary Phase (cont.)
We then have
0 0,0 0
4
2
~ ,
1 1
14
x y
j g x y
j
x y
I f x y e
e
13
2-D Stationary Phase (cont.)
or
0 0, 40 0
2
~ ,
1
4
x yjj g x y
x y
I f x y e e
14
2-D Stationary Phase (cont.)
, 0 0 orboth
Important special case:
In this case:
1
2 2
1 1 04 4x y
and
so
2
04
15
2-D Stationary Phase (cont.)
0 0, /20 0 2
1~ ,
4
j g x y jI f x y e e
, 0
, 0
and
and
where
We then have:
16
2-D Stationary Phase (cont.)
0 0,0 0 2
1~ ,
4
j g x yI f x y e j
, 0
, 0
and
and
where
Hence, the final result is
