Designing Special IIR-Filters by Approximating the Magnitude ...

Designing Special IIR-Filters by Approximating the Magnitude Frequency
Characteristic of an Analogue Prototype System
Peter Möhringer, Tristan Schneider,
Faculty of Electrical Engineering
University of Applied Sciences
Fachhochschule Würzburg-Schweinfurt
D-97421 Schweinfurt, Ignaz-Schön-Str. 11
Germany
pmoehrin@fh-sw.de
Abstract - The design of special IIR-filters using
optimization procedures is a well known practice. In
the following, a combination of approximation and
transformation is proposed for the design of digital
IIR-filters with special demands for the magnitude
frequency characteristic. The design is realized by
four steps. The desired squared magnitude
characteristic via frequency is defined for the digital
filter, first. Then the prescription is transformed to
the analogue domain by inverse bilinear transform,
this is followed by an optimization procedure of the
analogue frequency response. Finally the optimized
result is transformed back to the digital domain
yielding the desired IIR-filter.
I. INTRODUCTION
Starting with the definition of the desired
magnitude frequency characteristic digital IIR
filters are designed using the bilinear transform to
allow an optimization in the analogue domain. The
appropriately selected function yields stable
systems, automatically and the nonlinear distortion
of the frequency axis is equalized by using the
bilinear transform back and forth. First, from digital
to analogue domain and second, after the
optimization, back to the digital domain again. A
program for MATLAB was developed and the
capability of the method was tested by optimizing a
digital to analogue converter equalizing filter.
[1,2,3]
II. THE DESIGN METHOD
First we define the magnitude characteristic of the
desired filter in the digital domain. This design
goal HW(jω) is squared and then transformed to the
analogue domain by means of the bilinear
transformation following equ. (1). In the analogue
domain we choose an appropriate complex
function, e. g. Hist(s 2 ) as proposed in [2]. The
chosen function is a separable product of a stable
system H(s) and an instable system H(-s). See equ.
(2). Optimizing Hist(s 2 ) using a least squares
method to approximate the required frequency
characteristic and hereafter separating the stable
system H(s) from the product are the next steps.
Fig. 1 shows the poles and the zeroes of an
example system combination H(s)H(-s). The stable
system is selected by taking the poles in the left
open half plane and splitting the double zeroes on
the imaginary axis for H(s) and H(-s).
z −1
1 + s
s = ; z =
(1)
z + 1 1 − s
(2)
(3)
Fig. 1: Poles and zeroes of the analogue system
Hist(s 2 )
omega-analog
6
5
4
3
2
1
bilinear transform axis distortion
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
omega-digital
Fig. 2. Distortion of the frequency axis

D →
A
s =
z − 1
z + 1
equ .( 4)
ω A = tan( ω D T / 2)
A → D
1 + s
z =
1 − s
jω
DT
e
1 +
=
1 −
jω
A
jω
A
equ .( 5)
ω D
2
= arctan( ω A )
T
Equations (4) and (5)
In the following step,
the analogue filter is
transformed back to
the digital domain,
again using the
bilinear transform and,
thus equalizing the
distortion of the
frequency axis. See
Fig. 2 and equ. (4) and
equ. (5) The effect of
distortion of the
frequency axis and its
equalization are
explained in Fig. 3.
The poles and zeroes
of a resulting digital
system HDist are shown
in Fig. 4.
Fig. 3. Distortion and equalization of the
frequency axis.
top: desired digital characteristic, normalized
middle: transformed to the analogue domain
bottom: back in the digital domain, fa=44.1kHz
Although the bilinear transform is commonly not
applicable for the design of equalizing filters, the
way of using the transform back and forth allows a
wide variety of designs, avoiding the distortion of
the frequency axis.
Fig. 4 poles and zeroes of a resulting digital
system.
III. DESIGN OF A POST DAC FILTER
To show the capability of the proposed method, a
digital to analogue converter equalizing filter (DAC
filter) was designed
H DA ( jω)
T
sin ω
=
2
T
ω
2
(6)
and measured. The magnitude frequency
characteristic of the DAC HDA(jω), equ. (6), shall
be equalized in the passband and the step function
shall be smoothed, thus suppressing the aliasing
spectra. The desired frequency response HW(jω) is
defined in equ. (7) and the squared response is
shown in Fig. 5.
H
0
W
T
sin ω
( jω)
=
2
T
ω
2
−1
ωωa/2
IV. MEASURED RESULTS
Finally the example IIR-filter was implemented on
a signal processor test board and frequency
response as well as pulse response were measured
and compared to the desired solution. Fig. 6 shows
the measuring circuit. The DAC of the test board
was used for the distortion and the tested filter
frequency response is used for the equalizing. The
plot in Fig. 7 shows the product result of DAC
magnitude distortion and the equalizer. As can be
seen, the desired characteristic is closely met.

Fig. 5. The squared frequency response, the
optimized result and the difference
Fig. 6. Measuring circuit for time and frequency
response
V. CONCLUSION
As was shown by example, the combination of a
well known classic analogue filter design method
and the bilinear transform used back and forth
yields IIR digital filters closely meeting the desired
prescriptions. The method was programmed in
Fig. 7. Measured frequency characteristic,
product of DAC and filter frequency response
MATLAB and results were tested by implementing
the resulting filters on a signal processor board and
measuring the overall performance.
VI. ACKNOWLEDGMENT
Special thanks for review and support in layout to
H. Endres, P. Dückert and Stefanie Möhringer
VII. REFERENCES
[1] Schüßler H.W.: „Digitale Systeme zur
Signalverarbeitung“, Springer Verlag 1973
[2] Möhringer P.: „Beitrag zum Entwurf von
Filtern zur A/D- und D/A-Wandlung”,
Dissertation Uni Erlangen/Nürnberg 1982
[3] Schneider, T.: „Entwurf rekursiver Digitalfilter
mit besonderer Anforderung an den
Frequenzgang“, Diplomarbeit, University of
Applied Sciences FH-WS 2007
VIII. THE AUTHORS
Prof. Dr.-Ing. Peter Möhringer, University of
Applied Sciences, Fachhochschule Würzburg-
Schweinfurt, is head of the communication systems
branch of the faculty of electrical engineering.
Dipl.-Ing. (FH) Tristan Schneider developed the
program, provided the examples and did the
measurements in the frame of his diploma thesis.
IX. APPENDIX
Poles and zeroes of the tested equalizing filter.
Analogue prototype and the digital IIR filter result.
Analogue prototype of degree 5
Zeroes: Poles:
0 + 5.6707j -6.1801
0 + 5.971j -0.82268 - 2.2j
0 - 5.6707j -0.82268 + 2.2j
0 - 5.971j -5.5708 + 0.3j
-5.5708 - 0.3j
Resulting digital IIR filter of degree 5
Zeroes: Poles:
-1 -0.51101
-0.77874 + 0.62735j -0.11843 - 0.6871j
-0.77874 - 0.62735j -0.11843 + 0.6871j
-0.79825 + 0.60233j -0.47248 + 0.020903j
-0.79825 - 0.60233j -0.47248 - 0.020903j