IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 5, MAY 2004
1443
Implementation of First-Order and Second-Order
Microwave Differentiators
Ching-Wen Hsue, Senior Member, IEEE, Lin-Chuan Tsai, and Kuo-Lung Chen
Abstract—Simple and accurate formulations are employed
to represent discrete-time infinite impulse response processes of
both first- and second-order differentiators in the -domain.
These formulations, in conjunction with the representations of
transmission-line elements in the -domain, lead to transmission-line configurations that are eligible for wide-band microwave
differentiators. Both the first- and second-order differentiators
in microstrip circuits are implemented to verify this method.
The experimental results are in good agreement with simulation
values.
Index Terms—Equal-length line, microwave differentiator,
-transforms.
I. INTRODUCTION
T
HE differentiator is a very useful tool to determine and
estimate time derivatives of a signal. It has been used extensively in many areas, such as image processing, speech systems, and digital control. In radars, the velocity and acceleration of objects are computed from position measurements using
differentiators [1]. In biomedical engineering applications, it is
often necessary to compute higher order derivatives of biomedical data. The differentiators are mainly implemented in circuits for low-speed applications. Thus, the implementation of
differentiators for high-frequency applications has been largely
ignored.
Various methods have been developed to design both discrete
finite impulse response (FIR) and infinite impulse response
(IIR) differentiators [2]–[7]. Al-Alaoui [2] used Simpson’s rule
to develop a stable second-order recursive differentiator. Tseng
[3] studied a fractional-order FIR differentiator by solving
linear equations of Vandermonde form. In order to develop
a wide-band differentiator, Khan and Ohba [6] employed the
central difference approximations of the derivative of a function
to obtain a maximally linear differentiator. An important aspect
of the previous investigation is that the exploration focused on
the improvement of linearity over a wide frequency band.
Most of the differentiator studies thus far elaborated on
discrete-time signal processing (DSP) techniques for the
applications in low-frequency microchips. In particular, many
-domain formats of transfer functions have been obtained
to represent the characteristics of a differentiator. In this
Manuscript received September 23, 2003; revised January 6, 2004. This work
was supported by the National Science Council, R.O.C., under Grant NSC922213-EO11-012.
The authors are with the Department of Electronic Engineering, National
Taiwan University of Science and Technology, Taipei, Taiwan 106, R.O.C.
(e-mail: cwh@et.ntust.edu.tw).
Digital Object Identifier 10.1109/TMTT.2004.827015
paper, we present the scattering characteristics of equal electrical-length transmission lines in the -domain [8], [9]. As
a result, the transmission-line configuration can emulate the
characteristics of the differentiator developed in a DSP study,
and the operating frequency band of a differentiator is, thus,
extended further into the microwave range. Both first- and
second-order differentiators are implemented with microstrip
lines, of which the operating frequency is determined by the
physical length of each line section. It is, therefore, plausible
to fabricate differentiators having operating frequencies larger
than 10 GHz. The close agreement between theoretical values
and experimental results further validates the proposed scheme.
It is pertinent to point out that the transmission lines considered
here are assumed to be both lossless and dispersionless. In
particular, the dispersion effect between microstrip lines of
different widths over a wide bandwidth is neglected for the
current consideration.
II. DISCRETE-TIME DIFFERENTIATORS
It is well known that the operation of a time derivative of a
signal is represented by a complex-frequency variable in the
Laplace transform representation. Neglecting the loss factor, the
, where
complex-frequency variable is equal to , i.e.,
is the signal angular frequency. As a result, a differentiator
is a high-pass filter and the amplitude of its system function
increases linearly as the signal frequency increases. We consider
a transformation relating the complex-frequency variable and
the discrete-time variable in the -domain as follows:
(1)
is a normalization constant, is a real constant, and
where
represents a unit of time delay. Physically,
is the sampling time interval in the DSP study. If is set equal to one, the
transformation in (1) is called a bilinear transformation, which
is widely used in converting analog prototypes to discrete-time
prototypes [10]. When the frequency response of the differentiator is concerned, the parameter in (1) is replaced with the
following relation:
(2)
. The value
where is the frequency angle and
of strongly affects the linearity of the transformation in (1).
dictates the
On the other hand, the multiplication constant
amplitude response of (1). It is required that the amplitude response of (1) should be less than unity for the entire frequencies. Fig. 1 shows the amplitude response of (1) as a function
with different values of when the multiplication constant
0018-9480/04$20.00 © 2004 IEEE
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Fig. 1.
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 5, MAY 2004
Amplitude responses with different values of d.
Fig. 2.
is set equal to 0.417. Apparently, the transformation in (1)
has a good linearity in amplitude response when is set equal to
0.1658. The value of 0.417 is selected to assume that the maximum value of in (1) is unity for the entire frequencies when
. When is equal to one, the amplitude response of
. The bithe system function in (1) becomes infinite at
linear transformation, when is one, has a good linearity when
. Therefore, the bithe normalized frequency is less than
linear transformation is improper to be adopted as the system
function of a wide-band differentiator. Fig. 2. shows the relative error of the amplitude response of (1) for different values of
when they are compared to an ideal differentiator. The ideal
differentiator is assumed to have precisely linear amplitude re, the
sponse for all frequencies, as shown in Fig. 1. If
.
relative error is less than 1% (or 40 dB) when
For
, we, therefore, adopt (1) as the system function of the differentiator in a discrete-time IIR format and the
selected system function of the first-order differentiator is
Relative error of amplitude response for different values of d.
Fig. 3. Two-port device.
that their transfer functions are similar to the system functions
of differentiators.
III. IMPLEMENTATION OF DIFFERENTIATORS
A. First-Order Differentiator
For a two-port network shown in Fig. 3, the chain-scattering
(
) of a two-port
parameters ( or -parameters)
network are defined as follows:
(3)
(5)
If we implement a circuit with the system function shown
in (3), the differentiator is accurate for the operating frequency
up to 0.8 of the normalizing frequency. With a finite error tolerance, such a differentiator has a wider operating frequency
bandwidth than those previously reported [7]. In particular, the
concise mathematical expression will lead to a simple circuit
configuration of the differentiator.
For a second-order differentiator, the system function
is obtained by squaring (3), i.e.,
where
and
are, respectively, the incident and reflected
and
are, respectively, the incident
waves at port 1, and
wave and the reflected wave at port 2. In Fig. 3,
and
are
and
are independent varidependent variables, while
ables. Table I shows the matrices for two transmission-line
configurations [8], [9], namely, the serial transmission line and
, , and
shunt-short stub in the -domain, where
are the propagation constant, physical length, and characteristic
is the reference characterimpedance, respectively. Note that
istic impedance, which is assumed to be 50 , unless otherwise
mentioned.
It is assumed that all finite lines have the same electrical
, where is the propagation delay
length, i.e.,
time of finite lines. To obtain the matrices in the -domain,
.
we set
From (5), if the output port of a shunt-short stub is loaded with
), the transfer function of
a matched termination (i.e.,
(4)
After defining the discrete-time system functions, the
remaining task is to implement both first- and second-order
differentiators with equal electric-length transmission lines. In
other words, we synthesize the transmission-line circuits so
HSUE et al.: IMPLEMENTATION OF FIRST- AND SECOND-ORDER MICROWAVE DIFFERENTIATORS
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TABLE I
BASIC TRANSMISSION-LINE ELEMENT’S CHAIN
SCATTERING-PARAMETER MATRICES
Fig. 4.
Physical layout of microstrips for a first-order differentiator.
shunt-short stubs, and the term
represents the delay
serial transmission-line sections.
factor of
to approximate the system function
in
If we set
(4) and neglect the propagation delay factor, we obtain
the shunt-short stub
we obtain
is given by
. From Table I,
(6)
where
and
is the characteristic impedance
equal to
in (3), we get
of the shunt stub. If we set
and
. Notice that
is 17.86 if
is 50 . This reveals that a transmission line shunted with a
short-circuited stub can be employed to implement a first-order
microwave differentiator dictated by (3).
B. Second-Order Differentiator
serial secIf a transmission-line configuration consists of
tions and shunt-short stubs ( and are positive integers),
of such a
the overall chain-scattering parameter
circuit is obtained by the sequential multiplication of chain-scattering parameter matrices of all transmission-line elements [9].
The chain-scattering parameter matrix element
is given as
(7)
where all are real and are determined by the characteristic
is the reflecimpedances of all transmission-line elements.
tion coefficient defined in Table I. If the output of the transmission-line circuit is loaded with a matched termination, the
, is as
transfer function of the overall circuit, denoted as
follows:
(8)
where
is a function of the characteristic impedances of all shunted and serial transmission-line
in the numerator of (8) is due to
elements. The term
(9)
If we divide (9) with
, we get
(10)
The next step is to compare the coefficients of denominators
is as close to
as poson both sides of (10) so that
in (10) is determined by the characteristic
sible. Notice that
impedances of all transmission lines. Upon using the optimization method [9] in the sense of minimum square error for the
coefficients of the denominators in (10), we obtain the characteristic impedances of transmission lines.
To implement a differentiator with transmission lines, the
electrical length of each transmission-line section is set equal
, where
to 90 at the normalizing frequency. We have
represents the physical length of each transmission-line section
and
is the wavelength at the normalizing frequency.
IV. EXPERIMENTAL RESULTS
To construct a first-order microwave differentiator, we employ microstrips to emulate transmission lines. The microstrips
are assumed to be both lossless and dispersionless for the current consideration. Fig. 4 shows the physical layout of the microstrips, which is built on a Duroid substrate with a thickness
.
of 30 mil (0.762 mm) and relative dielectric constant
To implement the shunted transmission-line stub having a characteristic impedance of 17.86 , we use a parallel configuration, i.e., the equivalent microstrips are placed symmetrically
on both sides of the 50- line. The propagation delay time of
each shunted finite line is 20 ps, which corresponds to the normalizing (or maximum operating) frequency of 12.5 GHz. The
ground termination of shunted finite lines is implemented by
using multiple via-holes along the edges. Fig. 5 shows the magnitude responses of both simulated values and experimental reand reflection cosults of the transmission coefficient
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 5, MAY 2004
Fig. 5. Magnitude responses of both experimental results and theoretical
values of S (f ) and S (f ) of the first-order differentiator.
Fig. 7. Magnitude responses of both experimental results and theoretical
values of S (f ) and S (f ) of the second-order differentiator.
Fig. 6. Physical layout of microstrips for a second-order differentiator.
efficient
of the first-order differentiator for frequencies
extending from dc to 10 GHz. Notice that 10 GHz represents
0.8 of the full-band normalizing frequency. Measured
and
are in good agreement with the respective theoretincreases linearly as the frequency
ical values. Measured
increases.
We also use microstrips to construct a second-order microwave differentiator. Fig. 6 shows the physical layout of the
microstrip circuit, which is built on the same substrate as that
used for the first-order differentiator. The circuit consists of
) and three shunted stubs
seven-section serial lines (
). Of course, other configurations can be selected to
(
implement the differentiator provided that the condition
is met. The characteristic impedances of transmission lines are
obtained by using the optimization process [9] that involves
the comparison between the coefficients of the denominators
on both sides of (10). To assure the feasibility of microstrips,
the lower and upper bounds of the characteristic impedances
.
for the optimization process are set as
The characteristic impedances of serial lines from the leftto right-hand side are 54.19, 92.0, 75.54, 40, 40, 54.82, and
61.34 . We also use a parallel configuration to implement
the shunted stubs. The characteristic impedances of equivalent
shunted stubs from the left- to right-hand side are 49.91, 50.0,
and 40.0 .
Fig. 8.
Response of the first-order differentiator for ramp signal input.
Of course, the characteristic impedances of shunt stubs on
one side of the serial line in Fig. 6 are twice these values. The
propagation delay time of each finite line is 20 ps, which produces the normalizing frequency of 12.5 GHz. Once again, the
ground termination of shunted finite lines is implemented by
using multiple via-holes along the edges. The total length of the
differentiator excluding the reference 50- lines on both sides is
29.43 mm. Fig. 7 shows the experimental results, as well as the
and resimulated values of the transmission coefficient
flection coefficient
of the second-order differentiator for
frequencies ranging from dc to 10 GHz. As shown in this figure,
the measured frequency-domain results agree very well with the
theoretical values for frequencies up to 0.8 of the full-band normalizing frequency.
To examine the characteristics of the differentiators in the
time domain, we employ ramp signals as input signals to the devices. Fig. 8 shows the experimental results of the first-order differentiator when ramp signals with rise times of 100 and 150 ps
HSUE et al.: IMPLEMENTATION OF FIRST- AND SECOND-ORDER MICROWAVE DIFFERENTIATORS
1447
REFERENCES
Fig. 9.
Response of the second-order differentiator for ramp signal input.
are incident upon the device shown in Fig. 4. The 150-ps ramp
signal is turned into a square wave, while the 100-ps ramp signal
is transformed into a distorted pulse signal. The amplitudes of
output signals decrease in both cases. Little ripples appear on
both the rising and falling edges of output signals. Notice that
the rise time of output signals becomes 50 ps for two different
input signals. On the other hand, the output signals have a different fall time. The output signal associated with the 100-ps
rise-time input signal has a larger fall time. In Fig. 8, the theoretical results of output signals are shown to compare with the measured results, wherein the propagation delay time of transmission lines is taken into account. Fig. 9 shows the output signals
of the second-order differentiator when the same ramp signals
are incident upon the device shown in Fig. 6. Both output signals appear as distorted triangular waveforms. The output signal
associated with the input signal of 100-ps rise time has a larger
peak-to-peak value. In particular, two outputs decrease significantly, and the time duration of two outputs lasts longer than
that of output signals in the first-order differentiator. For convenience, the theoretical results of output signals are also shown
for comparison with the measured results.
[1] M. I. Skolink, Introduction to Radar Systems. New York: McGrawHill, 1980, pp. 399–408.
[2] M. A. Al-Alaoui, “Novel IIR differentiator from the Simpson rule,”
IEEE Trans. Circuits Syst. I, vol. 41, pp. 186–187, Feb. 1994.
[3] C.-C. Tseng, “Design of fractional order digital FIR differentiators,”
IEEE Signal Processing Lett., vol. 8, pp. 77–79, Mar. 2001.
[4] B. Kumar and S. C. Dutta-Roy, “Design of digital differentiators for
low-frequencies,” Proc. IEEE, vol. 76, pp. 287–289, Mar. 1988.
[5] S. C. Pei and J. J. Shyu, “Analytic closed-form matrix for designing
higher order digital differentiators using eigen-approach,” IEEE Trans.
Signal Processing, vol. 44, pp. 698–701, Mar. 1996.
[6] I. R. Khan and R. Ohba, “New design of full-band differentiators based
on Taylor series,” Proc. Inst. Elect. Eng.–Vis. Image Signal Processing,
vol. 146, no. 4, pp. 185–189, Aug. 1999.
[7] C.-W. Hsue, T.-R. Cheng, H.-M. Cheng, and H.-M. Chen, “A secondorder microwave differentiator,” IEEE Microwave Wireless Comp. Lett.,
vol. 13, pp. 137–139, Mar. 2003.
[8] T.-R. Cheng and C.-W. Hsue, “High-speed waveshaping using nonuniform lines and Z transform technique,” Proc. Inst. Elect. Eng., vol. 150,
pp. 77–83, Apr. 2003.
[9] D.-C. Chang and C.-W. Hsue, “Design and implementation of filters
using transfer functions in the Z domain,” IEEE Trans. Microwave
Theory Tech., vol. 49, pp. 979–985, May 2001.
[10] A. V. Oppenheim and R. W. Shafer, Discrete-Time Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1989.
[11] T. Edward, Foundations for Microstrip Circuit Design. New York:
Wiley, 1991.
Ching-Wen Hsue (S’85–M’85–SM’91) was born
in Tainan, Taiwan, R.O.C. He received the B.S. and
M.S. degrees in electrophysics and electronics from
the National Chiao-Tung University, Hsin-Chu,
Taiwan, R.O.C., in 1973 and 1975, respectively, and
the Ph.D. degree from the Polytechnic University
(formerly the Polytechnic Institute of Brooklyn),
Brooklyn, NY, in 1985.
From 1975 to 1980, he was a Research Engineer
with the Telecommunication Laboratories, Ministry
of Communication, Taiwan, R.O.C. From 1985 to
1993, he was with Bell Laboratories, Princeton, NJ, as a Member of Technical
Staff. In 1993, he joined the Department of Electronic Engineering, National
Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C., as
a Professor, and from August 1997 to July 1999, he was the Department
Chairman. His current interests are in pulse-signal propagation in lossless and
lossy transmission media, wave interactions between nonlinear elements and
transmission lines, photonics, high-power amplifiers, and electromagnetic
inverse scattering.
V. CONCLUSION
Simple and accurate formulations have been employed to represent both first- and second-order differentiators in the -domain. In particular, the -domain representations of scattering
characteristics of equal-length nonuniform transmission lines
facilitate the implementation of discrete-domain differentiators
in the microwave frequency range. These differentiators have
been implemented by using microstrip transmission lines. The
experimental results agreed very well with the simulated values.
It is possible that many other circuits developed in DSP studies
can also be implemented by using transmission lines for microwave applications.
Lin-Chuan Tsai was born in Taipei, Taiwan, R.O.C.,
in 1968. He received the M.S. degree in electronic
engineering from the National Taiwan University of
Science and Technology, Taipei, Taiwan, R.O.C., in
1998, and is currently working toward the Ph.D. degree in electronic engineering at the National Taiwan
University of Science and Technology.
He is currently a Project Engineer with the Mobile
Business Group, Chunghwa Telecom, Taipei,
Taiwan, R.O.C., where he is involved with the
wide-band code division multiple access (WCDMA)
network planning. His current interests are discrete time signal processing,
wireless communications, and microwave planar filter design and passive
circuit design.
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 5, MAY 2004
Kuo-Lung Chen was born in Keelung, Taiwan
R.O.C., in 1954. He received the B.S. degree in textile engineering from the National Taiwan University
of Science and Technology, Taipei, Taiwan, R.O.C.,
in 1980, the M.S. degree in computer science
and information engineering from the National
Chiao-Tung University, Hsin-Chu, Taiwan, R.O.C.,
in 1995, and is currently working toward the Ph.D.
degree in electronic engineering at the National
Taiwan University of Science and Technology.
From 1981 to 1996, he was an Engineer with
the Data Communications Institute, Ministry of Transportation and Communications (MOTC), Taiwan, R.O.C. From July 1996 to 1998, he was a
Section Chief with the Public Telecommunications Department, Directorate
General of Telecommunications (DGT). From October 1998 to July 2003,
he was a Station Director of the Northern Taiwan Regulatory Station, DGT,
MOTC. He is currently a Deputy Director of Public Telecommunications
Department, Directorate General of Telecommunications. His current interests
are discrete-time signal processing, wireless asynchronous transfer mode
(ATM), microwave planar filter design, and passive circuit design.