July 2015
Intermediate to advanced
600 pages
17h 41m
English
Content preview from Trends in Digital Signal Processing
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June 16, 2015 13:47 PSP Book - 9in x 6in 03-Yong-Ching-Lim-c03
90 How to Perform Very Wideband Digital Filtering in Modern Software Defined Radios
˜
H
NY Q
(
ω − ω
m
)
=
"
1 −
#
#
#
ω−ω
m
2π
/
M
#
#
#
, ω ∈ [ω
m
−
2π
M
, ω
m
+
2π
M
]
0 , Otherwise
.
Due to the underlying triangular pulse shape, the approximated
spectrum by the complex elements K
m
, K
m+1
for ω ∈ [ω
m
ω
m+1
],
can be written as
{
(
ω
m+1
− ω
)
· K
m
+ ω · K
m+1
}
. Using Eq. (3.18), and
only considering region ω ∈ [ω
m
ω
m+1
], we can define the error
transfer function on the interval ω ∈ [ω
m
ω
m+1
]as
T
E,m
(
ω
)
def
=
(
ω
m+1
− ω
)
· K
m
+ ω · K
m+1
− S
(
ω
)
. (3.26)
Set K
m
= S
(
ω
m
)
and K
m+1
= S
(
ω
m+1
)
, and expand S
(
ω
)
using
Taylor series around ω = ω
m
,wefind
S
(
ω
)
= S