When designing digital filters for signal processing applications one is often interested in creating objects h∈RNh∈RN in order to alter some of the properties of a given vector x∈RMx∈RM (where 0<M,N<∞0<M,N<∞). Often the properties of xx that we are interested in changing lie in the frequency domain, with X=F(x)X=F(x) being the frequency domain representation of xx given by
x
↔
F
X
=
A
X
e
j
ω
φ
X
x
↔
F
X
=
A
X
e
j
ω
φ
X
(1)where AXAX and φXφX are the amplitude and phase components of xx, and F(·):RN↦R∞F(·):RN↦R∞ is the Fourier transform operator defined by
F
{
h
}
=
H
(
ω
)
≜
∑
n
=
0
N
-
1
h
n
e
-
j
ω
n
∀
ω
∈
[
-
π
,
π
]
F
{
h
}
=
H
(
ω
)
≜
∑
n
=
0
N
-
1
h
n
e
-
j
ω
n
∀
ω
∈
[
-
π
,
π
]
(2)So the idea in filter design is to create filters hh such that the Fourier transform HH of hh posesses desirable amplitude and phase characteristics.
The filtering operator is the convolution operator (**) defined by
(
x
*
h
)
(
n
)
=
∑
m
x
(
m
)
h
(
n
-
m
)
(
x
*
h
)
(
n
)
=
∑
m
x
(
m
)
h
(
n
-
m
)
(3)An important property of the convolution operator is the Convolution Theorem[1] which states that
x
*
h
↔
F
X
·
H
=
(
A
X
·
A
H
)
e
j
ω
(
φ
X
+
φ
H
)
x
*
h
↔
F
X
·
H
=
(
A
X
·
A
H
)
e
j
ω
(
φ
X
+
φ
H
)
(4)where AX,φXAX,φX and AH,φHAH,φH represent the amplitude and phase components of XX and HH respectively.
It can be seen that by filteringxx with hh one can apply a scaling operator to the amplitude of xx and a biasing operator to its phase.
A common use of digital filters is to remove a certain band of frequencies from the frequency spectra of xx. Consider the lowpass filter from Figure 1; note that only the desired amplitude response is shown (not the phase response). Other types of filters include band-pass, high-pass or band-reject filters, depending on the range of frequencies that they alter.
Once a filter design concept has been selected (such as that from Figure 1), the design problem becomes finding the optimal vector h∈Rnh∈Rn that most closely approximates our desired frequency response concept (we will denote such optimal vector by h☆h☆). This approximation problem will heavily depend on the measure by which we evaluate all vectors h∈RNh∈RN to choose h☆h☆.
In this document we consider the discrete lplp norms defined by
∥
a
∥
p
=
∑
k
|
a
k
|
p
p
∀
a
∈
R
N
∥
a
∥
p
=
∑
k
|
a
k
|
p
p
∀
a
∈
R
N
(5)as measures of optimality, and consider a number of filter design problems based upon this criterion. The work explores the Iterative Reweighted Least Squares (IRLS) approach as a design tool, and provides a number of algorithms based on this method. Finally, this work considers critical theoretical aspects and evaluates the numerical properties of the proposed algorithms in comparison to existing general purpose methods commonly used. It is the belief of the author (as well as the author's advisor) that the IRLS approach offers a more tailored route to the lplp filter design problems considered, and that it contributes an example of a made-for-purpose algorithm best suited to the characteristics of lplp filter design.
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Ziemer, Rodger E. and Tranter, William H. and Fannin, D. Ronald. (1998). Signals and Systems: Continuous and Discrete. (Fourth). Prentice Hall.
