Linear Programming for the Design of IIR Filters - IEEE Xplore

Linear Programming for the Design of IIR Filters
Mauricio F. Quelhas, Antonio Petraglia, Mariane R. Petraglia
Federal University of Rio de Janeiro, EPOLI, COPPE, Rio de Janeiro, Brazil
{quelhas, antonio, mariane}@pads.ufrj.br
Abstract—This paper introduces a novel procedure for the
design of IIR discrete-time filters. The solution is obtained in
the minimax sense through an optimization procedure termed
Sequential Linear Programming (SLP). The method is based
on pole-zero mapping, and the stability constraints are easily
incorporated to the optimization procedure. Whereas the
methodology is formulated for the design of filters that satisfy
magnitude specifications, the procedure can be readily modified
to suit other criteria, such as phase/group delay deviation,
maximally flatness, time response, coefficient spread, or a
combination of these. Efficiency of the technique is evaluated
through illustrative examples and comparisons with other approaches.
I. INTRODUCTION
Elliptic filters are a common choice when complexity is
the main concern for physical realizations. This is mainly
owed to the Chebyshev characteristics in both passband and
stopband responses. On the other hand, these filters intrinsically
present high phase/group delay deviations that might
cause loss of information carried by the signals to be processed.
However, in some circumstances, as in communication
applications, approximately linear phase may be an
acceptable choice.
For those cases, several structures for IIR filters have
been proposed, such as, lattice wave digital filters [1]-[3]
and filters having unequal numerator and denominator orders.
Designs with small number of poles, in particular, are
suitable for both digital [4]-[8] and analog discrete-time
filters [9]-[11]. In applications where reduced phase response
distortion is required, group delay equalization [12]-
[14] and design techniques that handle simultaneously magnitude
and phase specifications [15]-[20] have been applied.
The last case is advantageous mainly because the designer
is able to determine exactly the desired delay deviation,
or phase slope. In particular, filter design techniques
that simultaneously satisfy magnitude and group delay
specifications produce filters with lower complexity than
allpass equalized filters. Some of these methodologies
search for the optimum solution in the least-squares sense
[16], whereas other ones focus on minimax solution [15]. As
for the optimization routine different techniques are employed,
including Second-Order Cone Programming
(SOCP) [19], FIR model reduction [17] and eigenfilters
[18].
All the aforementioned methods search for the best set
of numerator and denominator coefficients that satisfy the
given specifications. Assuring filter stability has been a recurrent
issue, but the methods reported so far can only provide
sufficient conditions for stability [13], [15], [16]. As a
consequent drawback, viable regions for stability are discarded
from the possible solutions. Besides, good initial
estimates are required for optimization robustness and convergence
rate.
This paper presents a novel methodology for the design
of discrete-time IIR filters based on pole-zero mapping.
Consequently, constraints for the radii of the poles are easily
merged to the optimization routine to ensure filter stability.
For the optimization routine Sequential Linear Programming
[21] is efficiently applied. In this work, a LP problem is
solved in each iteration, whose solution is used as the initial
estimate for the next iteration. Linear cost function and linear
constraints for the LP problem are derived by a technique
based on Taylor series expansion [8], [16].
Despite the fact that only magnitude response is considered
in the formulation, different specifications may be introduced
to the LP problem such as phase/group delay deviation,
maximally flatness, time response, transfer function
coefficient spread, or a combination of these requirements.
Moreover, the method may be applied to the design of alternative
structures, such as frequency response masking filters.
Some of these will be considered in future work.
The formulation of the Linear Programming problem is
developed in Section II. A detailed algorithm is provided at
the end of this section. In Section III, illustrative examples
are shown to verify the method efficacy for various specifications,
including the simultaneous design of magnitude and
group delay responses. Concluding remarks are provided in
Section IV.
II. OPTIMIZATION PROCEDURE
The proposed procedure consists in finding the optimum
allocation for the N poles, M zeros and the scaling factor, G,
of the IIR discrete-time filter transfer function:
H ( z)
= G ⋅ z
N −M
M 2
∏
m=
1
N 2
∏
n=
1
( z − q
m
jφ
e m )( z − q
jθ
( z − r e n )( z − r e
n
− jφm
me
− jθn
n
)
)
(1)

such as to satisfy the magnitude response specifications:
passband edge frequency (ω p ) and ripple (δ p ), stopband edge
frequency (ω s ) and attenuation (δ s ).
To achieve high stopband attenuation, the zeros can be
allocated on the unit circle by simply setting q m
= 1, m = 1,
2, …,M/2. The exception is for filters that have a number of
zeros much higher than that of poles, in which case the introduction
of zeros in the passband is necessary to avoid the
extra-ripple response [6]-[8]. For this reason, the following
equations will be written in terms of q m
.
For conciseness of presentation, the filters are assumed
to have even numbers of poles and zeros, as implied by (1).
It should be noted, however, that this formulation introduces
no loss of generality, since the equations can be straightforwardly
rewritten to include odd numerator and denominator
orders. As a result, the optimization procedure has N + M/2
+ 1 unknown parameters, comprising one scaling factor, N/2
radii of poles, N/2 angles of poles, and M/2 angles of zeros.
Since the magnitude response evaluated at a frequency ω k
of
a IIR filter
h
k
jω
= H ( e k )
(2)
is a nonlinear function of those parameters, a Taylor series
expansion is used [7], [8], such as to obtain the linear approximation
h h ∇ h
(3)
T
k
≈
k,0 +
k,0 ⋅Δp
on a grid of K points along the bands of interest. Equation
(3) also defines the optimization recursion scheme, in which
h k,0
is the magnitude response before an update has been
performed, the vector Δp is defined as p – p 0
, where p = [G,
r l
, θ l
, q l
, φ l
] T and p 0
= [G 0
, r l,0
, θ l,0
, q l,0
, φ l,0
] T , and the gradient
vector is
⎡∂hk ∂hk ∂hk ∂hk ∂h
⎤
k
∇ hk
,0
= ⎢ ⎥ (4)
⎣∂G ∂rn ∂θn ∂qm ∂ϕm
⎦
T
hk
= hk,0
for n = 1, 2, …, N/2 and m = 1, 2, …, M/2.
As mentioned previously, the objective of this work is to
find the optimum IIR filter in the minimax sense. Accordingly,
a sequential linear programming (LP) algorithm is
applied to find the optimum set of parameters of the IIR
filter that minimizes the cost function
= max w d − h
δ k k k
(5)
where w k
is a weighting factor and d k
is the desired magnitude
response at ω k
. Equivalently,
δ
≤ dk
w
k
− h
k
δ
≤
w
− (6)
which are the constraints of the following dual-form linear
programming problem : find the filter parameters and response
deviation, such as to maximize
ρ = −δ
(7a)
k
subject to
T δ
−∇
h ⋅Δp
− ≤−e
∇
k,0 k,0
w1
h
δ
⋅Δp
− ≤e
T
k,0 k,0
w1
(7b)
for k =1, 2, …, K. In the above formulation, the a priori error
e k,0
is the difference between the desired and the magnitude
response before the update d k
– h k,0
takes place.
The linear set of constraints (7b) results from a Taylor
series approximation, as in (3), which holds for small deviations
on the initial parameters. To this end, two restrictions
are incorporated to the LP problem (7), by including two
constraints
( 1− l)
⋅ p + l)
⋅ p
(8)
0 ≤ p ≤ ( 1
to ensure that the parameters are not updated by more than a
given fraction l of its initial values, p 0
. In this work, l = 0.2
was adopted, such as to meet an adequate tradeoff between
convergence rate – as l increases, so does the convergence
rate, but at the cost of loss of robustness in terms of success
of reaching the optimum parameter set.
Additionally, it is necessary that the radii of the poles are
lower than 1 to assure filter stability. The phases of poles
and zeros should also lie in the interval [0,π]. The radii of
the zeros are usually equal to 1, in order to satisfy high stopband
attenuation specifications. In some cases, to avoid extra-ripple
response in the passband [5],[7], zeros are allocated
inside the unit circle yielding minimum phase. The
aforementioned constraints can be written as
0≤
rn
≤1−ε
0 ≤θn
≤π
0≤
qm
≤1
0 ≤ϕ
≤π
m
for n = 1, 2, …, N/2, m = 1, 2, …, M/2 and ε is a small number,
e.g., 10 -4 , in order to keep r n
lower than 1.
The LP problem described so far may be applied to any
linear programming optimization tool such as SeDuMi. The
function linprog of Matlab® was employed in this work. It
should also be noted that, because of the Taylor approximation
the optimal solution is not expected to be reached after
one evaluation of the optimization procedure. This means
that after obtaining a solution, it must be used as the initial
value of another LP problem, until the difference between
the solutions of two successive LP problems lie bellow a
given threshold – 10 -5
in this work. The algorithm below
summarizes the procedure for obtaining the optimal parameters
of the IIR filter:
1. Give filter specifications.
2. Initial allocation of poles, zeros and scaling factor.
3. Model the LP program with current parameters
{G 0
, r n,0
, θ n,0
, q m,0
, φ m,0
}, including constraints.
4. Find new parameters {G, r n
, θ n
, q m
, φ m
} by solving
LP problem.
0
(9)

5. Is ||{G, r n
, θ n
, q m
, φ m
} – {G 0
, r n,0
, θ n,0
, q m,0
, φ m,0
}|| lower
than the threshold?
• If not, make {G 0
, r n,0
, θ n,0
, q m,0
, φ m,0
} =
{G, r n
, θ n
, q m
, φ m
} and go back to step 3.
• If yes, go to step 6.
6. Determine filter transfer function H(z).
7. End optimization procedure.
The proposed approach can be applied to the design of IIR
filters to meet a given set of magnitude specifications, including
multiband and multiripple responses. Illustrative
examples are shown next to verify the efficacy of the proposed
technique.
III. SIMULATION RESULTS
Example 1. The first illustrative example considers a bandpass
filter with the following specifications [8], [22]: passband
in the range 0.28π ≤ ω ≤ 0.54π with 0.1 dB maximum
ripple, and stopbands at 0 ≤ ω ≤ 0.2π and 0.62π ≤ ω ≤ π,
with attenuation larger than 60 dB. The filter designed in
[22] met the specifications with M = N = 12. In [8] two
bandpass filters were designed, one having M = 16 and N =
8, and the other M = 10 and N = 12. With the aid of the optimization
procedure proposed here, two bandpass filters
were designed with the same orders as in [8]. The main difference
between the results reported in [8] and the ones obtained
by the proposed methodology is that the latter provides
the result in the true minimax sense, whereas the
method in [8] searches for the allocation of poles and zeros
that provide equiripple response.
0
-20
For the designed filters the passband weights were equal
to 1, whereas the stopband weights were 6 and 8, respectively,
for the first (M = 16 and N = 8) and second M = 10
and N = 12) configurations. Figure 1 displays in solid and
dashed lines the magnitude frequency responses of both
design solutions produced by the proposed algorithm. The
former was designed with smaller number of poles to reduce
phase distortion and sensitivity in the passband, whereas the
latter was designed with focus on the reduction of complexity,
as it requires smaller number of multiplications per output
sample.
It is important to mention that the filters were designed
with the radii of zeros equal to 1. As a result the stopbands
are not equiripple, but the number of optimization parameters
is significantly reduced in 16 and 10 parameters, respectively,
for each design.
Example 2. In this second example, the proposed method is
used for the design of a lowpass filter such as to minimize
the group delay deviation while still satisfying the following
magnitude specifications [15], [16]: passband ripple of 0.2
dB, passband cutoff frequency at 0.4π, stopband edge frequency
at 0.56π, and stopband attenuation larger than 30 dB.
Figure 2 displays the magnitude response of the filter designed
with 4 poles and 8 zeros, in solid black line, as well
as the responses for the designs reported in [15], [16] having
4 poles and 15 zeros, in solid and dashed gray lines, respectively.
The orders of the filters designed with the proposed
approach were chosen such as to obtain a group delay deviation
as low as those achieved in [15] and [16]. Figure 3 presents
the group delay responses of the 3 filters, whereas Fig.
4 displays the same responses subtracted from their respective
maximum values, to emphasize the group delay deviations
produced by each filter design.
Magnitude
0.1
0.05
0
-0.05
Magnitude
-40
-60
-0.1
0 0.1 0.2 0.3 0.4
Normalized frequency
-80
-100
0 0.2 0.4 0.6 0.8 1
Normalized frequency
Figure 1. Magnitude responses for Example 1.
Figure 2. Magnitude responses for Example 2.
It can observed from the plots that the proposed method
was able to design an IIR filter with lower complexity, group
delay mean value and group delay deviation than those of
[15] and [16], while attending the magnitude specifications.
IV. CONCLUSIONS
This paper advanced an alternative approach for the IIR filter
design that searches for the best allocation of poles and
zeros, instead of the conventional optimization based on the
search of filter coefficients. The method employs LP for the
optimization procedure. Although the formulation only considers
magnitude specifications, the method may be readily
modified to combine both magnitude and group delay criteria.
Constraints for the radii of the poles are easily incorporated
to the LP problem, assuring filter stability. A Taylor

series expansion was applied to provide the LP linear cost
function and constraints. Simulation examples were shown
and the obtained results were compared to other approaches
reported in the literature to illustrate the efficacy of the proposed
method.
Group delay
Group delay deviation
13
12
11
10
9
8
0 0.1 0.2 0.3 0.4
Normalized frequency
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
Figure 3. Group delay responses for Example 2.
-0.7
0 0.1 0.2 0.3 0.4
Normalized frequency
Figure 4. Group delay deviation for Example 2.
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