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Double Conversion Method for Synthesis of Inverse Filters
Shahrokh Saeedi1, Student Member, IEEE, Juseop Lee2, Member, IEEE and Hjalti H. Sigmarsson1, Member, IEEE
1: School of Electrical and Computer Engineering, The University of Oklahoma, Norman, OK 73019, United States
2: College of Informatics, Korea University, Seoul 136-701, Korea
Abstract — A novel method for the inverse filter synthesis of
the same type from an original filter is presented. The proposed
procedure utilizes a double conversion, frequency inversion and
swapping the transfer and reflection functions, which are
commonly used for lowpass to highpass filter transformation.
Due to the use of double conversions, the resulting filter always
has the same type as the original filter. Using this procedure, the
available general Chebyshev synthesis method can be used to
synthesize inverse Chebyshev and elliptic filters. As an
illustration, the coupling matrix of a third-order inverse
Chebyshev bandpass filter is synthesized using the presented
method. Then the original filter and its inverse filter are
fabricated using microstrip technology. The measured and
simulated filter results are shown to be in good agreement.
Index Terms — Elliptic, filter synthesis, filter transformation,
generalized Chebyshev, inverse Chebyshev.
I. INTRODUCTION
These days, filters can be found virtually in any RF and
microwave communication system (cell phones), navigation
system (GPS), object detection system (radar), TV broadcasting, and test and measurement systems. Until the early
1970s, extraction of lumped electrical elements from the
characteristic polynomials that mathematically describe the
filter response, were mostly used for any microwave filter
design [1]. This conventional design technique was adequate
for the required applications and available technologies of that
time. However, in the early 1970s since the revolution caused
by the usage of satellites in telecommunication systems for the
first time, more stringent requirements have been put on the
specifications of channel filters in terms of both in-band and
out-of-band parameters. These requirements spurred the
development of technologies for implementation of microwave components at higher frequencies as well as a new
method for filter design called the coupling matrix method [2].
Since then, microwave bandpass and bandstop filters are
mainly designed using this method. Due to the capabilities of
this method, extensive research has been devoted to filter
synthesis. Therefore, varieties of direct and indirect coupling
matrix synthesis methods for both bandpass and bandstop
filters with maximally-flat, generalized Chebyshev, and
elliptic responses have been reported in the literature [3]-[6].
However, coupling matrix synthesis for filters with inverse
frequency response has not yet been presented.
In this paper a novel method to synthesize a filter with
inverse (dual) frequency response of the same type from an
original filter response is presented. To this end, an original
filter is first synthesized using one of the available methods.
Fig. 1. An example of the proposed double conversion method for
synthesis of fourth order inverse Chebyshev lowpass prototype.
Then, a tandem combination of frequency inversion and
swapping S21 and S11 is utilized to generate the filtering
function of the inverse filter. This is followed by regular
coupling matrix extraction. At the end, an optional direct
coupling matrix scaling is performed to re-normalize the
response of the resultant filter based on the desired corner
frequency definition. This procedure for a fourth-order inverse
Chebyshev lowpass filter with minimum stopband attenuation
of 25 dB is shown in Fig. 1. The filter is generated from its
dual filter i.e. a fourth-order Chebyshev lowpass filter with
passband return loss of 25 dB. However, this method is quite
general and can be used to synthesize any filter type from its
inverse filter.
This method can be used to synthesize inverse Chebyshev
and elliptic filters only using generalized Chebyshev filter
synthesis without requiring the inverse Chebyshev and elliptic
filter synthesis. Elliptic filter synthesis requires a tedious
calculation involving elliptic integrals and the Jacobian elliptic
functions. Furthermore, this method can be used to prescribe
equi-ripple stopband levels without optimization or iteration
of the location of transmission zeros. In order to demonstrate
the proposed method, the coupling matrix of a third-order
inverse Chebyshev bandpass filter is generated from a thirdorder Chebyshev bandpass filter. The filters are fabricated
using microstrip square open-loop resonators and tested.
Overall, the presented double conversion method provides a
convenient synthesis tool for realizing complex filtering
functions.
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II. SYNTHESIS PROCEDURE
The transfer and reflection functions of a lossless two-port
network with the general Chebyshev filtering function, can be
written using characteristic polynomials P(s), F(s), and E(s) as
in [4]:
=
,
ε
(1)
and
=
,
(2)
where P(s) and F(s) contain the filter transmission and
reflection zeros, respectively. In addition, E(s) is a Hurwitz
polynomial and ε and εr are constant values used to normalize
the highest degree coefficients of the polynomials to one. The
polynomials can be synthesized using the recursive formulas
presented in [3], [4]. For a lossless network, the characteristic
polynomials are related to each other using conservation of
energy,
|
| =
|
|
|
|
ε
.
=
.
.
.
.
(3)
This relationship can be used to find any of the polynomials
when the other two are available. Having the polynomials and
the constants, the coupling matrix for the lowpass prototype
can be extracted [3], [4].
From general filter and network theory, a normalized
highpass filter can be designed from lowpass prototype using
frequency inversion
1
.
(4)
Applying the lowpass to highpass transformation, the order
of coefficients in the polynomials is reversed. Therefore, the
constants, ε and εr, are required to be recalculated to again
normalize the highest degree coefficients of the polynomials
to one. Through this transformation, the frequency response of
the filter is folded back from infinity to zero and vice versa, as
shown in the top-right of the Fig. 1.
It was proposed in [7] that the frequency inversion is
equivalent to coupling matrix inversion. Therefore, instead of
performing frequency inversion to generate a highpass
coupling matrix from an original lowpass prototype, one can
find the highpass coupling matrix by lowpass coupling matrix
inversion. However, it should be mentioned that the rank of
the (N+2)×(N+2) coupling matrix for any odd-order filter with
symmetric frequency response (like maximally-flat or
generalized Chebyshev with symmetric transmission zeros) is
(N+1) which is one less than the matrix dimension. Therefore,
the coupling matrix is singular and cannot be inverted. In
other words, the previously reported method is not general and
cannot be used in all cases; although for even-order filters and
non-symmetric odd-order filters, this method can expedite the
coupling matrix generation.
.
.
− .
.
.
.
.
.
− .
.
.
.
Fig. 2. A fifth-order elliptic filter with 45 dB equi-ripple stopband
attenuation and 54 dB equi-ripple passband return loss.
In the next step, in order to convert the highpass prototype
to a lowpass prototype, the new transfer and reflection
functions are swapped as proposed in [6] and [8] to design a
bandstop filter from an original bandpass filter. Swapping of
the transfer and reflection functions, converts the highpass
prototype into a new lowpass prototype as its reflection and
transfer responses become the new filter transfer and
reflection responses, respectively as shown in the bottomright of Fig. 1.
Through this synthesis procedure, the return loss
characteristic of the original filter in the passband, determines
attenuation level of the inverse filter in the stopband.
Therefore, the prescribed equi-ripple return loss level of an
original Chebyshev filter, for instance, becomes the minimum
(equi-ripple) attenuation level of the reverse filter in the
stopband. Similarly, the transmission characteristic of the
original filter in the stopband determines the return loss of the
inverse filter in the passband. For example, the monotonic
rejection response of a Chebyshev filter provides a reflection
response similar to that of maximally-flat filters for the
inverse filter with all reflection zeros at zero frequency. The
order of applying the lowpass to highpass transformation and
swapping the transfer and reflection functions can be switched
without any change in the final response. One of the properties
of the proposed method is that the resulting filter is always
fully-canonical when the original filter is a Chebyshev filter.
Finally, although the resulting filter has a normalized corner
frequency, if a new corner frequency is desired, the filter
needs to be re-normalized. This can be directly applied to the
final coupling matrix using the method described in [9]. Fig. 2
978-1-4799-8275-2/15/$31.00 ©2015 IEEE
=
.
.
.
Fig. 5. Measured responses of the fabricated filters.
.
− .
− .
.
.
.
=
.
.
.
.
− .
− .
.
.
.
Fig. 3. The synthesized third-order Chebyshev (RL=23 dB) and
inverse Chebyshev filters and their coupling matrices MORG and MINV,
respectively.
a)
b)
21mm
0.27 mm
0.15 mm
Fig. 4. Fabricated filters with inset for showing the smallest features.
a) Chebyshev, b) inverse Chebyshev.
illustrates the application of the proposed method to generate
the coupling matrix of a fifth-order, elliptic filter from an
original generalized Chebyshev filter. By changing the return
loss level of the original filter, a family of elliptic filters with
different attenuation levels in the stopband can be obtained.
III. FABRICATION AND MEASUREMENTS
A third-order inverse Chebyshev filter with equi-ripple
attenuation of 23 dB in the stopband was synthesized from an
original Chebyshev filter with return loss of 23 dB to
demonstrate the practical application of the proposed method.
The filters dimensions were then extracted using known
coupling mechanisms and implemented in a Rogers Corp.
RO3006® microwave substrate using microstrip square openloop resonators. Both filters were designed with a center
frequency of 1800 MHz. The bandwidth of the Chebyshev
filter was chosen to be 100 MHz. Since the inverse Chebyshev
filter has a different corner frequency definition than the
Chebyshev, its coupling matrix is scaled with ωc = 2.8 rad/s.
This results in two filters with the same 3-dB corner
frequency. The synthesized filters along with their coupling
matrices are shown in Fig. 3. A photo of the fabricated filters
is shown in Fig. 4.
The filters were measured using an Agilent N5225A PNA
calibrated using an electronic calibration kit. Fig. 5 shows the
measured response of the filters. Good agreement between the
results is observed. Both filters exhibit an insertion loss of
about 1.1 dB considering the fact that measurement includes
the connector losses. There is a minor shift in the center
frequency of both filters. This shift is also seen in the left
transmission zero of the inverse Chebyshev filter. These
discrepancies can be attributed to fabrication tolerances, along
with the parasitic coupling between resonators one and three
in the inverse Chebyshev filter.
IV. CONCLUSION
This paper presents an elegant method for synthesizing an
inverse filter response of the same type from an original filter.
The method is based on using lowpass-to-highpass transformation in tandem with swapping the transfer and reflection
functions or vice versa. Using only the generalized Chebyshev
filter synthesis method, inverse Chebyshev and elliptic filters
can be synthesized.
To demonstrate the capabilities of the proposed method, the
coupling matrix for a third-order order inverse Chebyshev
filter was synthesized from an original Chebyshev filter with
return loss of 23 dB. Both filters were made and measured.
Good agreement between simulated and measured results was
achieved. To the authors’ best knowledge, this is the first time
978-1-4799-8275-2/15/$31.00 ©2015 IEEE
that this double conversion method has been used to
synthesize an inverse filter response.
The proposed method can be used to investigate a wide
range of new filter responses for which polynomials may not
have been considered yet.
ACKNOWLEDGMENT
This work was supported by the Agency for Defense Development
(ADD) Daejeon, Republic of Korea (Contract Number:
UD120046FD).
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978-1-4799-8275-2/15/$31.00 ©2015 IEEE