Digital Signal Processing Reference
In-Depth Information
heorem 7. 2
[17] “Let A k ( t ) be defined as follows:
∑
(
N
−
12
)/
(
)
+ ( )
a
+
2
a
cos
n
θφ if
( ),
t
N
isodd
k
k
k
def
n
=
1
At
()
=
(7. 2 0)
N
/
2
k
∑
1
2 θφ 
+ ( )
2a k
cos
n
−
( )
t
,
if
N
is even.
k
n
=
1
hen the PDF p H k ( t ) (·) and CDF F H k ( t ) (·) of H k ( t ) are given by
2
2
2
x
x
+
At I
()
2
x
()
σ 2
AAt
k
p
k () ()
x
=
exp
−
k
,
(7. 21)
Ht
0
2
2
σ
σ
k
k
k
2
At
() ,
2
x
Fx
k () ()
=−
1
Q
k
k
,
(7. 2 2)
Ht
1
2
2
σ
σ
k
where I 0 (·) is the modified Bessel function of order zero and Q m (·,·) is the Marcum
Q-function.”
Proof: See appendix A in [ 17 ] .
It is very interesting to see that (7.21) is exactly the same form as a Ricean distribution,
except that A k ( t ) is a summation of multiple cosine functions. When N = 1, (7.21) reduces
to the well-known Ricean PDF. As in the conventional Ricean case, we define Ω k ( t ) to be
the total power of (7.21) as follows:
def
=
2
2
2 σ
Ω k
()
t
=
E Ht At
k
()
()
+
.
(7. 2 3)
Ht
()
k
k
k
Also, the K -factor of (7.21) can be defined as follows:
2
At
() .
K k
()
t
=
k
(7. 2 4)
2
σ
k
It follows that (7.21) can be rewritten as
+ ( )
(( )
( )
2
21
x
K
()
()
t
− − +
1
K
()
t
x
KK
()
t
()
()
t
t
1

k
k
k
k
p
k () ()
x
=
exp
K
( )
t
I
2
x
. (7. 2 5)
Ht
k
0
Ω
t
Ω
()
t
Ω
k
k
k
 
Previous Page
Next Page
Adaptive Signal Processing in Wireless Communications
Search WWH ::




Custom Search
Home