Digital Signal Processing Reference
In-Depth Information
and certain entries, which is added to the problem space such that the particles tend to shy away
from such a region. The width of the barren layers is calculated based on a worst case scenario that
may happen in the particles movements in the search space. However, the entries of barren layers
are different for different problems and depend on the topology of the search space and the fitness
function used in the problem.
This chapter discusses in detail the design, realization and discrete PSO of FRM IIR digital filters. FRM
IIR digital filters are designed by FIR masking digital subfilters together with IIR interpolation digital
subfilters. The FIR filter design is straightforward and can be performed by using hitherto techniques.
The IIR digital subfilter design topology consists of a parallel combination of a pair of allpass networks
such that its magnitude-frequency response matches that of an odd order elliptic minimum Q-factor
(EMQF) transfer function. This design is realized using the bilinear-LDI approach, with multiplier
coefficient values represented as finite-precision CSD numbers.
The above FRM digital filters are optimized over the discrete multiplier coefficient space, resulting in
FRM digital filters which are capable of direct implementation in digital hardware platform without any
need for further optimization. A new PSO algorithm is developed to tackle three different problems. In
this PSO algorithm, a set of indexed LUTs of permissible CSD multiplier coefficient values is generated
to ensure that in the course of optimization, the multiplier coefficient update operations constituent in the
underlying PSO algorithm lead to values that are guaranteed to conform to the desired CSD wordlength,
etc. In addition, a general set of constraints is derived in terms of multiplier coefficients to guarantee
that the IIR bilinear-LDI interpolation digital subfilters automatically remain BIBO stable throughout
the course of PSO algorithm. Moreover, by introducing barren layers, the particles are ensured to
automatically remain inside the boundaries of LUTs in course of optimization.
2. The conventional PSO algorithm
Let us consider an optimization problem consisting of
N
design variables, and let us refer to each
solution as a particle. Let us further consider a swarm of
K
particles in the
N
-dimensional search space.
The position of the
k
-th particle in the search space can be assigned a
N
-dimensional position vector
X
k
=
{
x
k
1
,
x
k
2
, . . . ,
x
kN
}
.
In this way, the element
x
kj
(for
j
=
1, 2, . . . ,
N
) represents the
j
-th
coordinate of the particle
X
k
.
The PSO optimization fitness function maps each particle
X
k
in the search space to a fitness
value. In addition, the particle
X
k
is assigned a
N
-dimensional velocity vector
V
k
=
{
v
k
1
,
v
k
2
, . . . ,
v
kN
}
. The PSO optimization search is directed towards promising regions by
taking into account the velocity vector
V
k
together with the best previous position of the
k
-th
particle
X
best
k
=
{
x
best
k
1
,
x
best
k
2
, . . . ,
x
best
k N
}
, and the best global position of the swarm
G
best
=
{
g
best
1
,
g
best
2
, . . . ,
g
best
N
}
(i.e. the location of the particle with the best fitness value).
The conventional PSO is initialized by spreading the particles
X
k
through the search space in a random
fashion. Then, the particles make movements through the search space towards regions characterized
by high fitness values with corresponding velocities
V
k
. The movement of each particle is governed
by the best previous location of the same particle
X
best
k
, and by the global best location
G
best
. The
velocity of particle movement is determined from the previous best location of the particle, the global
best location, and the previous velocity.
