Filters and amplifiers can make a powerful combination, especially when the filter is used to equalize the amplifier. What follows is an approach to apply the complementary response characteristics of a bandpass filter to flatten the amplitude response of a transmit amplifier. The filter consists of a pair of asymmetric slow-wave open-loop resonators. It reduced ripple in a transmitter's passband from 3.3 dB before equalization to 0.5 dB after equalization.

High data rate performance is essential to modern wireless communications systems. High data rates can be achieved by improving the spectral efficiency (in terms of bits per second per Hertz of frequency bandwidth) by using multiple-input, multipleoutput (MIMO) techniques or orthogonal frequency-division multiplexing (OFDM) and by simply extending the operating bandwidth. Often, a combination of techniques can improve spectral efficiency. Future data rates are likely to exceed 1 Gb/s, with bandwidths as wide as 100 MHz and spectral efficiency past 10 b/s/Hz.

In systems with such high spectral efficiency, in-band amplitude flatness will be critical to achieving high performance, and equalizers will be needed. Such equalizers are widely used at present at baseband frequencies for channel equalization.^{1-3}

In addition, refs. 4 and 5 provide examples of microwave amplitude equalization. In ref. 6, an amplitude equalizer can be implemented in lumped-element or distributed form. The current report offers an approach based on equalizing a transmitter by means of the complementary characteristic of a bandpass filter. The filter consists of a pair of asymmetric miniature open-loop, slow-wave resonators. By optimizing the resonators, it is possible to achieve a ripple shape complementary to the transmitter's response, resulting in a flattening of the transmitter's amplitude (with a small loss in output power).

The passband response shapes of *Figs. 1(a)-1(c)* can generally be offset by the inverse ripple shapes of Figs. *1(d)-1(f)*. By cascading a bandpass filter with complementary response in the RF transmitter or receiver, the passband ripple will be compensated. The converse is also true for flattening the responses of *Figs. 1(d)-1(f)* by designing a bandpass filter with the respective shapes of *Figs. 1(a)-1(c)*. An approach in ref. 4 can be adopted to synthesize an amplitude equalizer with the ripple shapes of *Figs. 1(d)-1(f)* so the emphasis here will be to develop filters with the ripple shapes of *Figs. 1(a)-1(c*).

For experimentation, a bandpass filter was designed using microstrip open-loop structures. Numerous research works have been published on open-loop filters,^{7-9} with ref. ^{9} pointing out that the coupling effect of an open-loop resonator filter can both enhance and reduce stored energy.

When the coupled resonator circuits are overcoupled, two resonant peaks and one resonant trough associated with mode splitting become apparent in the filter's bandpass response. A novel architecture was used for the open-loop filter design (*Fig. 2*). With this configuration, two resonant peaks and one resonant trough can be adjusted by tuning the corresponding geometrical parameters of the structure.

Filter designs were simulated with the full-wave electromagnetic (EM) simulator IE3D from Zeland Software.^{10} The filter parameters included a dielectric substrate with permittivity, e_{r}, of 10.2 and thickness, h, of 25 mils. *Figure 3* shows the simulated frequency responses of filters with different coupling spacings (s). When the coupling spacing decreases, the two resonant peaks move outward and the trough in the middle of the passband deepens, which implies an increase in the amount of coupling.

*Figure 4* shows the simulated frequency response for different ratios of filter gap widths w1 and w2. As the ratio of w1/w2 increases, the passband moves toward lower frequencies and the stopband suppression improves. By exploiting the characteristics of the slow wave resonators, the filter can be modified for high spurious suppression while saving around 30 percent in size compared to common openloop- resonator bandpass filters.^{11}

What makes this filter's resonator structure different from conventional designs is the asymmetry of the two resonant loops. The size of dimension a does not have to equal dimension b, and the feed positions for the input and output ports may not be in line (that is, line l1 may not be of equal length as line l2). As *Fig. 5* shows, as the difference in size between dimensions a and b in the two loops change, the insertion loss of the two resonant show asymmetric characteristics. If dimension a is smaller than dimenion b, the lower resonant peak will be higher than the upper peak, and vice versa. By means of tuning, the difference between dimension a and dimension b can be manipulated to obtain any of the amplitude ripple shapes shown in *Fig. 1*.

*Figure 6* shows the effects of different line lengths for l1 and l2. The insertion loss and frequencies of the two resonant peaks and the trough can be controlled by tuning input position l1 and output position l2 independently. As l1 and l2 both become larger Fig. 6(a)>, the filter bandwidth changes very little, although the trough between the two resonant peaks becomes deeper. When l2 is fixed at 2 mm, and l1 is varied *Fig, 6(b)*>, the ripple shape of the bandpass filter varies with respect to l2. For a larger value of l2, the difference of the insertion losses between the first resonant peak and the trough remain almost the same whereas the insertion losses between the second resonant peak and the trough increases and the frequency of the resonant trough increases. When the value of l2 is fixed and the feeding position l1changes, the differences in insertion losses and resonant frequencies become functions of the changing l1. As l1 or l2 is larger, the feedline becomes closer to a virtual ground point for the resonator, resulting in a weaker coupling or a larger external quality factor (Q).^{13} Thus, the frequencies and insertion losses of the two resonant peaks and the middle trough can be adjusted by tuning the input and output positions of l1 and l2, respectively.

When the coupling spacing s decreases, a filter with wider bandwidth and deeper trough results. With l1 equal to l2, a filter with the response of Fig. 1(b) can be designed. When the dimension of a is much larger than that of b, a filter with the ripple shape of *Fig. 1(a*) can be designed. And when dimension a is much smaller than b, a filter with the ripple shape of *Fig. 1(c)* can be designed. The resonant peaks and troughs can be controlled by adjusting l1 and l2 independently.

Designing a microwave amplitude equalizer involves three steps: 1) measuring the passband ripple response without the equalizer, 2) designing a bandpass filter euqalizer with the inverse shape of the passband ripple of step 1, and 3) installing the equalizer in the microwave transmitter to improve the in-band amplitude flatness.

To verify this proposed equalization approach, a transmitter with 100-MHz bandwidth centered at 3.5 GHz was designed and fabricated, with the measured unequalized response (lower trace) and the required inverse response for equalization (upper trace) shown in *Fig. 7*. The measured RF channel passband response is denoted by a(Ω) and the desirable characteristic of the equalizer is denoted as b(Ω). The relationship between them is b(Ω) = a_{0} - a(Ω), where a_{0} is defined as being appreciably smaller than a_{min}.

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To verify the equalization approach, a transmit amplifier with large amount of ripple was chosen for the test. The ripple within the 100-MHz 50. bandwidth is as severe as 3.3 dB.

The filter (*Fig. 2*) was designed by choosing dimension a for the left loop to be larger than dimension b for the right loop, to realize ripple shape b(Ω). The passband of the filter was designed for coverage from 3.4 to 3.5 GHz and fabricated on RT/6010 laminate from Rogers Corp. (www. rogers.com) with 0.635-mm thickness and relative dielectric constant of er = 10.2. To compensate for fabrication tolerances, the simulated center frequency was lower than that of the desired filter. After elaborate tuning, the filter dimensions were: a = 3.01 mm, b = 2.98 mm, g = 0.48 mm, s = 0.65 mm, w1 = 1 mm, w2 = 0.2 mm, l1 = 2.0 mm, and l2 = 2.14 mm.

The filter had a pair of SMA connectors for evaluation with a commercial vector network analyzer (VNA). In a comparison of measured and simulated performance (*Fig. 8*), high insertion loss in the measured passband was likely due to conductor losses.

The amplitude equalizer consists of a buffer amplifier, shaping bandpass filter with an inverse passband response, and an attenuator. The filter is inserted between the amplifier and attenuator to meet the matching condition at the input and output ports because the shaping filter cannot be normally matched to 50 ohm due to the large ripple. The amplifier and attenuation block are used for solving this mismatch problem.4 After installing the filter-based amplitude equalizer into the RF transmitter, the resulting passband ripple was reduced to less than 0.5 dB compared to the original 3.3-dB transmitter passband ripple without the equalizer (*Fig. 9*).

REFERENCES

1. R. W. Lucky, H. R. Rudin, "Generalized automatic equalization for communication channels," Proceedings of the IEEE, Vol. 54, No. 3, March 1966, pp. 439-441.

2. P. Monsen, "Adaptive Equalization of the Slow Fading Channel," IEEE Trans. on Coms., Vol. 22, No. 3, August 1974, pp. 1064-1075.

3. P. Monsen, "Adaptive Equalization of the Slow Fading Channel," IEEE Trans. on Coms., Vol. 22, No. 3, August 1974, pp. 1064-1075.

4. H. Y. Hang and S. W. Yun, "A New Type of Amplitude Equalizer for In-Band Flatness Improvement, Microwave Journal, Vol. 45, Issue 2, February 2002.

5. J. Kampa and K. Petrus, "Microwave Amplitude Equalizer," Microwave, Radar, and Wireless Communications 2000, Proc. 13th Int. Conf., May 2000, Vol. 1, pp. 37-40.

6. M. Thornber, "Amplitude Equalizer Flattens Gain from 50 MHz to 7 GHz," Microwaves & RF, Vol. 46, No. 10, October 2007.

7. C-C Yu and Kai Chang, "Novel Compact Elliptic-Function Narrow-Band Bandpass Filters Using Microstrip Open-Loop Resonators with Coupled and Crossing lines," IEEE Trans. on MW Theory & Tech., Vol. 46, No. 7, July 1998.

8. A. Gorur, "A novel dual-mode bandpass filter with wide stopband using the properties of microstrip open-loop resonator," IEEE MW and Wireless Comps. Lett., Vol. 12, No. 10, 2002.

9. Jia-Sheng Hong and Michael J. Lancaster, "Couplings of Microstrip Square Open- Loop Resonators for Cross-Coupled Planar Microwave Filters," IEEE Trans. on MW Theory & Tech., Vol. 44, No. 12, December 1996.

10. Zeland Software,, IE3D Version10.1.

11. J.-S. Hong and M. J. Lancaster, "Microstrip slow-wave resonator filters," IEEE MTT-S Digest, December 1997, pp. 713-716.

12. J.-S. Hong and M. J. Lancaster, "Theory and experiment of novel microstrip slow-wave open-loop resonator filters," IEEE Trans. on MW Theory & Tech., Vol. MTT-45, No. 12, December 1997, pp. 2358-2365.

13. J. S Wong, "Microstrip tapped-line filter design," IEEE Transactions on Microwave Theory & Techniques, Vol. MTT-27, No. 1, January 1979, pp. 44-50.