Dual-Band RF Circuits and Components for Multi-Standard Software Defined Radios
The advent of multi-standard and multi-band software defined radio (SDR) applications has necessitated the design and deployment of dual-band RF components and circuits considering the numerous advantages of such designs over the traditional narrow band circuits and components. For example, a dual-band power amplifier (PA) not only simplifies the hardware complexity but also provides higher reconfigurability  and hence makes it a front runner for deployment in SDR architectures –. Furthermore, the evolution of communication technologies demands the use of dualband/ multi-band RF circuits so as to acc ommodate many standards simultaneously for facilitating and guaranteeing the backward compatibility of future standards (such as 4 G) based system for smooth network migration and upgrades. These technological requirements have also led to commercial introduction of dual-band base stations and repeaters –. Furthermore, the advancement in CMOS and other MMIC technologies, although, is challenging the transmission line based passive circuit techniques but the high power handling ability of transmission line based circuits are potentially very useful in applications such as the design of high power/high efficiency PAs and transmitters. This article elaborates on the techniques employed in the design of transmission line based dual-band RF components in the context of multi-band/multi-mode SDR architecture, highlighting the problems which need to be addressed during the design process.
ide-band circuits, although, are more generic and best suited for reconfigurable transceiver architecture but are severely limited in applications such as impedance matching [7–14]. Many pioneer researchers such as Bode , Fano , Youla , and Carlin et al. [12–13], have conducted their research to describe the limit of broadband impedance matching. Initial work by Bode was however focused on a load consisting of a resistance and a capacitor in parallel . Fano , Youla  and Carlin et al. [12–13] later developed the gain bandwidth restriction to an arbitrary load. In fact, the theory of Youla and Carlin is focused on the determination of the feasibility of matching an arbitrary load, so as to obtain a target reflection coefficient as a function of frequency .
For matching applications, the hypothesis of Fano establishes that arbitrary impedance cannot be matched to a pure resistance over the whole frequency spectrum or even at all frequencies within a finite frequency band . Instead the hypothesis suggests possibilities for obtaining matching at desired number of frequencies provided the required impedances have finite resistive components at those frequencies. Consequently the design of wide-band matching, which is a pre-requisite for obtaining optimum efficiency and linearity over the specified band from non-linear components such as PA, is not feasible especially for the situation when the intended bandwidth covers two distinct communication standards far apart in terms of their carrier frequencies. In such situations, dual-band/multi-band matching is highly useful as they provide optimum solution over a limited range of bandwidth around the chosen carrier frequencies of operation , .
The dual-band matching is also useful in receiver circuits such as diode based mixers. Such an application generally needs a broadband rat-race or branchline coupler with an octave bandwidth, which is difficult to fabricate and has limited performance –. In addition, there is always an issue of simultaneously matching the diodes to the respective optimum reflection coefficients at LO and RF especially when the LO and RF are far from each other. In practical designs, mismatch is allowed at LO as it has high power than RF –. A dual-band matching, however, can provide match at both LO and RF which will lead to eventual reduction in LO power consumption. Furthermore, other applications in passive RF circuit field mostly utilize cascading of several single sections for obtaining broad-band performances. This results into increased cost, size and acomplexity which consequently limit the usefulness of such circuits. It can thus be concluded that the dual-band/multi-band RF circuits have potential applications until there is some breakthrough research for catapulting the design of broad-band circuits.
In this article, the fundamental concept of dispersive circuits and their applications in dual-band circuitry is presented which is then followed by several examples of dual-band RF passive circuits and their associated design and miniaturization methodologies as well as dual-band impedance matching technique for high power applications. Finally, an example of dual-band wireless transmitter, suitable for SDR applications, is described that utilizes various dual-band RF circuits in conjunction with hybrid RF-digital pre-distortion (hybrid RF-DPD) setup realized on a field programmable gate array (FPGA) platform .
Multi-Band/Multi-Mode Software Defined Radio: An Architectural Overview
Software defined radio (SDR) is the radio terminal where the key parameters of the radios are defined in software and in which fundamental aspects of the radio’s operation can be reconfigured by upgrading the software . Although software plays an important role in SDR backend where the signal processing functionality is the key operation, yet this role is considered peripheral while hardware plays the dominant role in the SDR frontend. These roles of hardware and software are changing continuously over time, and with the advent of new technology the SDR architecture is moving towards software radio (SR) where receiver digitization will be at (or very near to) the antenna and software programmed in high-speed signal processing element will perform all the processing required for the radio –. Fig. 1 shows a typical block diagram of SDR terminal, where the analog to digital conversion and vice versa is performed at some stage downstream from the antenna typically after low noise amplification and down conversion to a lower frequency in subsequent stages. However, several variations have been proposed in the literature –, among which Fig. 1 is built on most common existing state of art in transceiver design. In such a configuration, the software can play a vital role in obtaining multi-mode capability but multi-band capability is mainly served by hardware in the RF front end.
Adopting multi-band architecture over wideband in implementing SDR front-end also reduces burden over the design of band-select filters after antenna without hampering the performance in presence of undesired interference. These band-select filters are used to select desired band of operation as per the intended standard of communication (e.g. WCDMA, WiMax etc.).
In the case of PA design, any mismatch from the optimal loads can severely affect efficiency and linearity of the transmitter. This, mismatch can occur due to (1) the bandwidth limitation of matching network as imposed by Bode, Fano, Youla and Carlin etc. [7–14] and (2) change in antenna input impedance. The latter case occurs due to change in antenna environment (e.g. mutual coupling of the antenna with the nearby object can cause detune of the antenna resonant frequency) , whereas, the former case is related to broad-band matching circuits, where, there is compromise in terms of level of mismatch tolerated as the bandwidth of operation increases [7–14]. With the mismatch at the PA output and between the antenna and PA, the efficiency and linearity of the transmitter reduces drastically –.
The efficiency is very important in concern to the transmitters as it primarily contributes towards power consumption in a typical wireless communication network. Moreover, to increase spectral efficiency, the modulation resulting into envelop varying signals having high PAPR (e.g. 64 QAM) are commonly used which poses constraints over the linearity of the PA in the transmitter, resulting into adjacent channel leakage and interference with the other standards operating in a nearby frequency range. Although, in order to meet spectral mask of 3G and 4G communication standards, a separate linearization technique must be adopted, still the PA designer should also seek for an optimum solution in terms of efficiency and linearity (IMD, ACLR etc.) through optimum matching condition. As an example, the power added efficiency (PAE) and IMD contours of a 1W GaN HEMT depicted in Fig. 2 clearly identifies that there is substantial degradation in PAE and linearity as the matching impedance moves away from the optimal solution. These contours provide an indication of the trade-off that can be adopted in the design of matching circuits for optimal PAE, Pout and IMD.
It is evident from Fig. 2(a) that the maximum Pout and PAE are 28 dBm and 21% respectively when the device is measured corresponding to the lower fundamental tone (ω1). The IMD contours in Fig. 2(b) demonstrates that the best matching, Γla depicted as the shaded overlapping contours of upper and lower IMDs, for relatively good linearity of 24 dBm results into a poor PAE of 5%. However the results also show that the matching region for the optimal PAE and Pout, shown as the shaded portion and depicted as Γla, provides a poor linearity. Therefore, depending on the design specifications the best matching conditions can be traded-off in order to achieve the best PAE, Pout, and IMD.
It should be, however, noted that even the choice of matching for the optimal PAE and linearity at a certain frequency may not result in best PAE and linearity over a band due to presence of mismatch occurring across the band [7–14]. It is due to this reason that multi-band/ dual-band circuits are employed, when the intended carrier frequencies are widely separated, for optimal operation of the radio.
The mismatch due to the change in the antenna input impedance is more difficult to manage and technology such as smart antenna where the adaptive variation in directivity and null-placement can reduce the effect of the object in the vicinity , are employed. Another approach adopted to overcome this problem is to design fully adaptive system that can adjust output power, load-line or bias with the change of the antenna impedance resulting into optimum performance in changing antenna environment .
Dispersive Circuits and Their Applications in Dual–Band Circuitry
The most common approach adopted in the design of dual-band circuits is to replace each transmission line sections in a conventional single-band design with a 2-port dispersive structure. These dispersive structures possess specific image impedance and phase constant which corresponds to the required values of the characteristic impedance and electric lengths at the respective two frequencies. For example, each section in a dual-band branch-line hybrid requires a phase shift of 90° or its odd-multiple with characteristic impedances, 50 Ω or 35.35 Ω, at the chosen two frequencies which is same as desired for the traditional single-band hybrid . For a standard transmission line segment, the image impedance is same as its characteristic impedance at two frequencies but the corresponding electric length is linear function of frequency. In other words, if n is the frequency ratio of the two chosen frequencies f1 and f2 then electric length at f2 will be n times the electric length at f1. It consequently leads to a simple premise that a circuit having controllable non-linear behavior of phase with respect to frequency can be customized to achieve a specific electric length or its multiple at any two frequencies of operation which are independent of the frequency ratio of the two frequencies. Fig. 3, which depicts the relationship between the frequency ratio and electric length, clearly conveys this idea that by controlling the non-linear phase characteristics through design parameters based on the type of circuit chosen, the desired electric lengths at the two frequencies can be achieved for any frequency ratio.
The circuits/2-port networks which have nonlinear phase characteristic with frequency are known as dispersive circuits and in principle the dispersion is responsible for frequency dependent phase/group velocity characteristic in such circuits. Stub loaded transmission line is the most common 2-port network exhibiting dispersive effect. In these structures the phase characteristic can be shaped by the stub and the parameters of the loaded line –. In addition, these circuits exhibit frequency dependent behavior of both image impedance as well as phase characteristics and therefore shaping of the phase characteristic also changes the image impedance and thereby necessitating simultaneous solutions for both these features during the design –. Alternatively these structures can also be thought as filters and studied through the image theory concept but this aspect is beyond the scope of this article.
It is worth discussing that, although s-parameters analysis is more rigorous and practical in analyzing the RF and Microwave circuits, image impedance and ABCD matrix approach can be used in much easier way to analyze and design of certain circuits where the transmission lines are involved as main design elements . Example of these circuits are hybrids , , , Wilkinson power divider , , , delay line ,  etc., where the key goal is to obtain desired performance using specific value of characteristic impedance and electrical length of the transmission line at the desired frequency.
In such cases, the theory of image filter design provides a direct relationship of the characteristic impedance and electric length of the emulated transmission line to the ABCD parameters to a 2-port network imitating the transmission line behavior. Indeed, due to this reason, this approach is sometimes preferred over network synthesis approach [7–8], while designing the dual-band circuits, where, a dispersive 2-port network is used to emulate the behavior of transmission line of certain characteristic impedance and electrical length at two frequencies of operation. Thus rather than shaping the input impedance (hence reflection coefficient) using a certain polynomial characteristics (such as Tchebyscheff, Butterworth etc.) –, the image impedance and image propagation constant characteristics are shaped . These characteristics to a certain limiting case can be considered equivalent to characteristic impedance and electrical length of a transmission line over a finite band and hence imitate the properties of the transmission line with required specifications.
Analysis of RF Passive Dual–Band Circuits
RF passive dual-band circuits such as power dividers, branch-line hybrid/rat-race baluns, phase-offset lines, delay lines etc. are designed using transmission lines with fix characteristic impedances and desired electric lengths at the specified frequencies. As a case study, design methodologies for realizing power divider and hybrids for dual-band operation as well as a common technique for miniaturizing these circuits are analysed in this section. However, it should be understood that these techniques are equally applicable for the design of any transmission line based dual-band circuits.
Stub-Loaded Transmission Line in Dual-Band Hybrid/Baluns Applications
The transmission line segments in branch-line/rat-race couplers are quarter wave length long whose ABCD parameters are given in equation (1) , 
To satisfy (1) at two distinct frequencies, a 2-port dispersive structure is required which can be achieved through the use of either lumped components  or transmission line sections –, –. The lumped components based designs are smaller in size but suffer from the limited bandwidth. In addition, these designs are dependent on the inductor models for achieving the required accuracy. The transmission line based designs, although, are larger in size but have the potential to provide superior performance if designed properly.
There have been two techniques to achieve dualband quarter-wave operation from a transmission line. First method is stub loading at the center of the transmission line  while the other is stub loading at the ends of the transmission line , –. Figs. 4(a) and 4(b) depict both these scenarios for the case of dual-band branch-line coupler. The transmission lines in these figures are dispersive and therefore capable of providing dual-band operation if their characteristics are shaped appropriately.
The center-tapped stub-loaded line (as shown in Fig. 4(c)) behaves as dual-band quarter-wave transformer if equations (2) and (3), derived from its respective ABCD matrix , are simultaneously valid and satisfy the ABCD parameters in (1).
where, ZS,T and uS,T are the characteristic impedance and electric length of unloaded line, ZT
is the overall image impedance of the loaded line, and BS,T is the susceptance of the stub loading the line in Fig. 4(c). A detail derivation of transforming a center-tapped stub-loaded dispersive structure in a dual-band quarter wave transformer is given in shaded block.
Similarly, a transmission line with edge-loading, as shown in Fig. 4 (d), behaves as dual-band quarter-wave transformer and emulates ABCD parameters given in (1) if expressions in equations (4) and (5) hold simultaneously .
where, ZS,PI and ΘS,PI are the characteristic impedance and electric length of unloaded line loaded with a stub possessing susceptance BS,PI and ZT is the overall image impedance of the loaded line in Fig. 4(d). A detail derivation of transforming a dispersive structure with stub loading at the edges in a dual-band quarter wave transformer is given in shaded block.
The electric lengths uS,T in (2)–(3) and ΘS,PI in (4)–(5) varies with frequency and for them to satisfy (1) simultaneously at two distinct frequencies the expression in (6) should hold true –.
where, n is the frequency ratio f2/f1 with f1 being the smaller frequency and p equals 1 for the principle argument of sine and tangent function of (2)–(5) corresponding to the smallest physical length of unloaded line in Figs. 4(c) and 4(d).
The negative and positive signs in (1) correspond to electric lengths 90° and 270° respectively. It can therefore be concluded that any odd multiple of electric length 90° provides same performance as quarter-wave transformer. For the dual-band situation, therefore, it has been practically found that there exist two solutions for (6) that provide the requisite dual-band quarter-wave transformers at two distinct frequencies –. First solution is achieved with electric length 90°, corresponding to the negative sign in (6), for both the frequencies. While the second solution is obtained with electric length 90° in first frequency and 270° in the second frequency and corresponds to the positive sign in (6). For smaller frequency ratios, the first case gives larger physical dimensions as compared to the second case –.
The most common design approach is to vary frequency ratio n in (6) for obtaining different values of electric length ΘS,T or ΘS,PI for which the structure presents a required image impedance ZT (and hence emulate a quarter-wave transformer with a required value of characteristic impedance ZT) with a realizable value of ZS,T or ZS,PI. The realizable value of ZS,T or ZS,PI depends on the minimum width of line that can be fabricated precisely with the fabrication facility. Therefore the minimum possible width of transmission line puts a constraint over the maximum value of ZS,T or ZS,PI. The standard design curves regulating the center-tapped and edge loaded structures for emulating dual-band quarter-wave transmission line of 50 Ω characteristic impedance are shown in Figs. 5(a) and 5(b) respectively.
Upon determining the values of ZS,T or ZS,PI for particular values of ΘS,T or ΘS,PI, n, and ZT, the corresponding values for the other values of ZT can easily be predicted by the linear relation given by (2) and (4) as described in . Once the conditions for ZS,T or ZS,PI and the associated ΘS,T or ΘS,PI are obtained such that the structures in Fig. 4 can emulate an image impedance of ±ZT at the two specified frequencies, the respective values of BS,T should satisfy (3) for the center-tapped loaded structure and BS,PI should satisfy (5) for the edge-loaded structure. In most practical designs the stubs are realized by using either an open circuit or a short circuit transmission line. In such a scenario, the stub parameters for center-tapped stub loading of a transmission line are given by expression in (7) .
where, ZP,T is the characteristic impedance and ΘP,T is the electric length of the stub. Similarly, for the edge-loaded structure the stub parameters are given by the expression in (8) .
where, ZP,PI is the characteristic impedance and ΘP,PI is the electric length of the stub.
For (7) and (8) to repeat at the two distinct frequencies, an expression given in (9) can be derived –.
where, q is the integer with value 1 for principle argument of the tangent function in (7) and (8) and n is the frequency ratio.
For each realizable value of ZS,T and ZS,PI obtained from Fig. 5(a) and Fig. 5(b) respectively, for a particular required value of ZT and n, the values of ΘP,T or ΘP,PI can be obtained using (9). These selected values of ΘP,T or ΘP,PI along with other design parameters can then be used to obtain value of ZP,T for center-tapped loaded line and ZP,PI for the edge-loaded using equation (7) or (8) respectively. It should however be kept in mind that the selection of the values of ZP,T or ZP,PI are dependent on the technology used for fabrication. It is also evident from (7) and (8) that for particular values of ZT, ZS,T or ZS,PI and ΘP,T or ΘP,PI the design parameter, ZP,T or ZP,PI, is linearly proportional to ZT. Therefore, once the design parameters for a particular n and ZT are known, the values of the corresponding ZP,T or ZP,PI for any other values of ZT for the same n can also be determined .
Fig. 6(a) presents a photograph of dual-band rat-race coupler operating at 1960 MHz and 3500 MHz, which has been designed using center-tapped stub loaded dispersive structure, whereas Figs. 6(b)–(d) depict the measured dual-band performance of this rat-race coupler . The insertion loss of less than 3.9 dB, return loss greater than 16 dB, and isolation better than 20 dB over the band of 80 MHz, centered on the two selected frequencies of 1960 and 3500 MHz, achieved from this design provides a simple proof of concept of the use of stub-loaded center-tapped dispersive structure in the design of dual-band rat-race coupler. The use of edge-loaded dispersive structure is shown in the design of a dual-band branch-line coupler operating at 1960 MHz and 3500 MHz whose photograph is shown in Fig. 7(a) and the corresponding measured results are given in Figs. 7(b)–(d). Once again the achieved results of 3.41 dB insertion loss, return loss of 20 dB, and isolation of 20 dB over the band of 80 MHz, provide a proof of concept of the edge-loaded dispersive structure in the design of dual-band branch-line coupler.
Stepped Impedance Transformer in Dual-Band Power Divider Application
The design of conventional Wilkinson power divider requires impedance transformer that can transform 50 V to 100 V. For dual-band operation, each branch of a conventional Wilkinson power divider to design dual-band Wilkinson power divider can be replaced by dispersive structures with 908 electric length possessing an image impedance of 70.7 V . In principle, for a Wilkinson power divider the main requirement is the impedance transformation , irrespective of whether it is achieved through the use of quarter-wave transformers or non quarter-wave transformers such as Chebyshev or Monzon multi-section transformers –. However the Monzon transformer, which is a cascade of transmission lines of two different characteristic impedances and same electric lengths, is simpler to design and therefore is the most commonly used topology for dual-band application –. Even and odd mode analysis of the Monzon type dual-band Wilkinson power divider results in utilization of a LC tank circuit placed in shunt with isolation resistor, as shown in Fig. 8(a), resonating at the center of the two frequencies in order to improve isolation over a band .
An alternative approach that employs a stub at the junction in place of LC tank circuit in shunt with isolation resistor has also been reported in . Once again the use of stubs turns the design into a dispersive circuit for the overall dual-band operation. As an example, schematic and photograph of the prototype of a Monzon transformer based dual-band power divider operating at 1960 MHz and 3500 MHz are shown Fig. 8 (a) and (b) respectively, while some important results depicted in Figs. 8(c)–(e) show extremely good performance of this prototype. This prototype provides insertion loss of less than 3.5 dB over the band of 100 MHz, centered on the two selected frequencies of 1960 and 3500 MHz. The corresponding return losses and isolations were better than 20 dB at all ports over this band.
Miniaturization of Dual-Band Circuits
It can be identified from Figs. 5(a)–(b) that the realizable values of loaded line impedances ZS,T or ZS,PI sometimes exist for higher values of electric lengths (denoted by values of p ? 1). A similar argument is equally valid for the characteristic impedance ZP,T or ZP,PI of the stubs loading the transmission line. Thus for some frequency ratios the circuit can only be realizable with longer lengths of transmission lines and therefore require miniaturization for use in practical applications. For example, Monzon transformer utilizes two cascaded transmission lines that results into a quite long circuit which occupies larger space on the board. Periodic loading of transmission line is one of the easiest and conventional methodologies used for size reduction of circuit structure . Fig. 9 shows miniaturized Monzon based dual-band power divider using periodic loaded slow wave structure. The periodic loading of transmission line results into slow-wave propagation which has lower phase velocity than the phase velocity of the substrate , . Such periodic loaded lines can achieve higher electric length with corresponding smaller physical length. These lines are also referred as artificial transmission lines which possess similar characteristic as normal transmission line such as linear relation of phase velocity and frequency, and constant characteristic impedance with a non-dispersive relation up to a range of certain frequency . This non-dispersive range of frequency is useful for miniaturization application without introducing higher order modes of the Bloch wave travelling through the structure . This frequency range is decided by the periodicity of the structure such that the distance between the consecutive stubs are small as compared to the wavelength of the highest frequency of operation. A very common phenomenon observed in periodic loading is the reduction of loaded impedance from the value of characteristic impedance of the unloaded line which directly relates the miniaturization to the ratio of the loaded impedance and the unloaded impedance which is referred as slow wave factor represented as ‘k’ in Fig. 9(c) .
Monzon transformer’s transmissions lines section are miniaturized individually by same periodicity, d, as shown in Fig. 9(a), with different stub lengths lP and different number of stubs, N, for each sections which results into different loaded impedances of sections 1 and 2. Since, the maximum size reduction is achieved with highest impedance possible; hence, the impedance of unloaded line, ZC in Fig. 9(c), and stubs are kept equal and have maximum value corresponding to the minimum width of the transmission line that can be realized by the given facility in fabrication. Due to dispersive nature, the frequency behavior of the periodic structure exhibits a frequency cut-off at which the propagation seizes, resulting into stop-band, which is decided by several other factors such as frequency behavior of loaded impedance, resonance of the stub and periodicity of the structure . It is important to note that the stub-loaded dispersive structure (of Fig. 4) can be seen as one specific case of periodic loading, where dispersion is used to obtain dual-band response with an assumption that in a very narrow range of band around the two carrier frequencies has linear-non-dispersive characteristics. For the special case of unequal power division in branch line hybrid miniaturization, an alternative methodology has been reported  in which stepped-impedance transformer is used in place of the loaded line in Figs. 4(b) and (d).
Dual–Band Matching Network For High Power Active Circuits Design
The most important high power active circuit is the PA,and its design consists of biasing and stabilizing the transistor, and designing the matching network. Fig. 10 shows a generic schematic of a dual-band power amplifier with dual-band biasing, stability and matching network. A simple biasing circuit for a radio frequency (RF) PA includes an inductor that feeds DC to the transistor, but blocks AC leakage from the RF path. However, these inductors suffer great losses, due to their low Q factors that are further lowered when high power handling capacity is required; therefore, the most conventional approach is to use a transmission line based 908 transformer short circuit at one of its ends, realizing an effective open circuit at the other end, connected to the RF path . This behaves as a lossless path for DC; however, effectively, no AC leaks through it. For a dual-band PA design, such a 90° transformer should be dual band with high characteristic impedance. This dual-band transformer is designed by using a T-type structure, which is a transmission line loaded by a stub at the center . Once the transistor is biased using this dual-band 90° transformer, a stability circuit is designed at the input. After stabilization, a dual-band matching network will be designed to present optimum load and source impedances to the transistor at two frequencies, when terminated by a 50 V load.
In general, matching circuit in any amplifier design can be seen as a 2-port network that provides desired complex input impedance when terminated by a standard load impedance of 50 V. This complex input impedance is the required value of impedance seen by the device in order to behave optimally in terms of performance such as gain, output power, efficiency etc. In most cases, the required complex loads seen by the device at different frequencies of operations can be different and can have arbitrary values. Thus, a dual-band matching network should provide different arbitrary impedances at the two frequencies, when terminated by a 50 V load. This hinders some early efforts reported – as they limit the performance of the designed PAs. Some designs which utilize transmission line based impedance transformers are suitable only for real impedance matching – and hence are not generic considering that most of the RF PA matching requires complex impedance matching network. Furthermore, an interesting effort has been reported for matching harmonics using the transmission line based matching technique . Such a matching network is quite good for matching high reactive impedances, which are usually required in harmonic terminations, but is not easy to employ for fundamental complex matching with high resistive components at two separate and uncorrelated frequencies simultaneously.
Matching techniques – which provide matching of two different complex loads to 50 V at two distinct frequencies have been reported but the methodologies have similar problems to stub loaded structures for obtaining realizable values of characteristic impedances of transmission lines. Moreover, methodology and the design equations reported in  are too complex and need higher order of numerical optimizations. As a consequence these techniques find convergence only for selected frequency ratios and complex input impedances.
Application of dispersive stub-loading circuit, reported recently , utilizes frequency varying image impedance of these circuits along with the dispersive characteristic to design dual-band/dual-characteristic impedance quarter-wave transformer that has two different characteristic impedances at two different frequencies with 908 or its odd multiple as electric length at two specified frequencies. The methodology reported here in is based on synthesizing a matching network that presents two different and desired complex input impedances when terminated by a 50 V load. The methodology is primarily based on the standard non 50 V quarter-wave transformer matching technique  with some modification, to realize two distinct reflection coefficients at the two chosen frequencies, in order to achieve dual-band operation. The key element of this network is a dual-band/dual-impedance quarter-wave transformer that simultaneously transforms the 50 V load seen at the reference plane (TT‘ in Fig. 11(a)) to the respective real conductance parts, G(f1) and G(f2), of the required complex impedances, Y(f1) and Y(f2), to be seen by the active device at frequencies f1 and f2.
The first step in such a design is to obtain the required values of the characteristic impedances for the transformer at the two frequencies. Once the required impedance transformer has been designed, the corresponding dual-band/dual-susceptance stub can be designed to realize an imaginary part of the required complex impedances, Y(f1) and Y(f2), at frequencies f1 and f2, as shown in Fig. 11(a). These dual-band/dual-susceptance stubs can be realized using the impedance buffer technique reported in , and are considered highly suitable for harmonic matching also.
As an example, Fig 11(b) shows a prototype of the matching technique  where the required values of reflection coefficients at two frequencies were obtained from load-pull data for a 10 W gallium nitride (GaN) based device (CGH40010 from Cree) with a goal of obtaining the maximum power-added efficiency (PAE) at the respective frequencies of operation . Fig. 11(c) shows the performance of the matching network in obtaining desired complex input impedances when terminated by 50 V.
One can also think of using such circuits to match the antenna and PA directly. In such a case, 50 V load is replaced by an input admittance of Antenna in Fig. 11(a) and an additional dual-band/dual-susceptance stub is needed to nullify the reactive components in the antenna input impedance as shown in Fig. 12(a). The resistive component which is in most case not 50 V, is then matched to required optimal impedance Y(f) to be seen by the transistor using dual-band/dual-impedance quarter- wave transformer. An alternate solution that does not utilize any additional stub is also shown in Fig. 12(b).
In this approach, the characteristic impedances of the transformer are synthesized in such a manner that when it is terminated by certain admittances with the real conductive part (which can be real part of antenna’s input admittance) GA, it will realize the desired input admittances, Y(f1) and Y(f2), at the two frequencies. Thus, if YJ(f1) and YJ(f2) are two such admittances that terminate the proposed transformer at reference JJ’ (in Fig. 12(b)), the first step in the design methodology involves the synthesis of the required values of the characteristic impedances, ZT(f1) and ZT(f2), and the corresponding values of the susceptances, jBJ(f1) and jBJ(f2), at reference JJ’ (Fig. 12(b)).
Once the values of jBJ(f1) and jBJ(f2) are known, a dual-band/dual-susceptance stub is designed to present these susceptances at reference JJ’, which in conjunction with a GA eventually realize the required terminating admittances, YJ(f1) and YJ(f2) at the input of matching network. However, to absorb the susceptance of antenna input admittance with the stub values BJ(f ), a stub with modified BT(f ) is actually used in Fig. 12(b). For a conventional impedance inverter, the following inverting relation holds:
Putting Y(f ) and YJ(f ) in their complex form in (10), and separating the real and imaginary parts, the susceptance value of the junction admittances (at reference JJ‘ in Fig. 12(b)) and the required characteristic impedances of the 90° impedance inverter can be synthesized as follows:
Once the values of ZT(f ) at frequencies f1 and f2 are known by (12), the dual-band/dual-impedance transformer can be used to realize it . Since, the dualband stub should also absorb the susceptive part of antenna admittance, hence, the required value of dual-band/dual-susceptance stub (i.e. BT(f1) and BT(f2) shown in Fig. 12(b)) can be obtained as
This dual-band stub of BT(f ) will provide required junction susceptance at reference JJ’ in Fig. 12(b).
In Fig. 12, it has been assumed that the real part of antenna input impedance, which is mainly comprised of radiation resistance and loss resistance, is independent of frequency to some extent. However, for dual-band operation when the two carrier frequencies are quite apart, this assumption may not be valid and the circuit should match between two different complex impedances with both having distinct real and imaginary components. In such case, the required values of characteristic impedances ZT(f1) and ZT(f2) should be different for Fig. 12(a) whereas, for Fig. 12(b) these values of impedances are synthesized differently and equations (10) to (13) should be modified accordingly.
Some efforts have already been done to match two arbitrary complex impedances , while utilizing step impedance transformers as shown in Fig. 13. This approach provides a closed form solution for the transmission line parameters i.e. the characteristic impedances and electric lengths of each section. Again, in Fig. 13(a), YA(f ) represents antenna input admittance and Y(f ) represents the required input admittance of the matching network which a transistor of PA should see in order to behave optimally in terms of efficiency and linearity. The key idea is to solve for the transmission line parameters ZA, ΘA, ZB, ΘB by enforcing a condition over the impedances Zin and Zout (in Fig. 13(a)) as given in Fig. 13(b) at two frequencies, where ‘Conj’ represents conjugate. If this conjugate condition is not applied, then the approach will result into the transcendental equations that can only be solved by optimization algorithm –. Details of this matching technique as shown in Fig. 13(a) are reported in .
Fig. 14 shows application of the dual-band matching network of Fig. 11 in the design of 10 W dual-band class-AB PA for code division multiple access (CDMA) and Worldwide Interoperability for Microwave Access (WiMAX) applications at 1960 and 3500 MHz respectively .
Figs. 14(b) and 14(c) shows the obtained efficiencies of 59.94% and 55.51% at 1,960 MHz and 3,500 MHz respectively which are in agreement with any of the reported results in literature.
As a concluding statement on the dual-band matching circuits, our experience show that there is no available optimal solution and the adopted technique depends on the type of applications. Therefore Table 1, which outlines the advantages and limitations of various available matching techniques, can be taken as a general guideline while designing matching circuits for dual-band applications.
It is also worth noting that apart from the matching application, these dual-band/dual-impedance quarter-wave transformers can also be used in other circuits/components such as applications requiring unequal power division in two distinct frequency bands. For example, branch-line and rat-race hybrid with unequal-power ratio where the transformer has different structure from the conventional stub loaded transformer and provide similar dual-band/dual-impedance characteristics at the two frequencies have been reported .
Proposed Wireless Transmitter Design Technique For SDR Applications
The research over multi-band/multimode wireless transmitter is mainly motivated by their application in SDR terminals. Such radio terminals are reconfigurable and can be used in Cognitive radio applications where optimally choosing an appropriate connectivity options can minimize path loss/shadowing hence necessary transmission power in consideration of predicted traffic loads and channel characteristics –. In terms of hardware infrastructure, broadband RF Front ends are the most desired solutions for these radios but as explained before a broader bandwidth minimizes the matching accuracy. This leads to the hypothesis that to cover distant RF carriers a considerable mismatch should be allowed which will reduce the efficiency of PA and as a consequence increase the energy consumption in transmitters and the entire networks. In addition, PA is the major contributor of the non-linearity in a wireless transmitter and therefore is primarily responsible for causing interference in the adjacent channels. In order to mitigate this efficiency and linearity problems, reconfigurable multi-band/multi-mode SDR systems must utilize PAs with high linearity and efficiency –.
Digital predistortion (DPD) is one of the most promising techniques used for linearizing the PAs for SDR transmitters –. The DPD is highly appreciated due to its accuracy as compared to their analog counterparts (APD). Furthermore, its simpler implementation in digital domain also provides it the requisite reconfigurable capability –. However, it is difficult to obtain wideband linearization with DPD due to limited speed of digital circuits involved in linearization . Keeping this aspect in perspective the hybrid RF–DPD architecture has been proposed as a compromise between APD and DPD techniques , –.
Hybrid RF–DPD achieves higher accuracy in linearization when compared to the analog predistortion (APD) architecture and the bandwidth is not limited by digital signal processing (DSP) computational speeds experienced by the conventional DPD. In the hybrid RF-DPD, the signal correction is directly done on the RF signal using a RF-vector multiplier under the control of the digital circuits  and therefore it can be easily customized for dual-band/multi-band applications. For dual-band operation, each of the components in the conventional RF-DPD can be replaced by the respective dual-band counterparts, described in this article, as shown in Fig. 15.
The delay line may not be critical as the delay offset between the RF and the digital path can be precisely compensated by a digital delay in FPGA. The present system is integrated with RF components, which are readily available , where the two different RF vector-multipliers are used at two different carrier frequencies and their replacement and recalibration is done while using system in different bands. However, one can think of utilizing some commercial broad-band RF-vector multiplier such as  depending on their operating frequency range.
In principle, after calibration as described in , this system works well for improving the linearity at both the specified frequencies as shown in Figs. 15(b) and (c). However, the main issue of utilizing predistortion as linearization tool is the overall efficiency reduction as the PA is operated at a certain average power backoff, which depends on the peak-to-average power ratio (PAPR) of the predistorted signal –. In order to enhance the efficiency in this back-off region the Doherty PA (DPA) is considered the most appropriate –. Considering all the technological limitations, dual-band/ multi-band transmitters with dual-band/multi-band DPA and a hybrid RF–DPD can be the most optimal solutions, in terms of high efficiency and linearity, for SDR applications. So far there have been some initial reports of the design of dual-band DPA –, but further investigation is required for optimizing those techniques and for extending those techniques to multi-band designs.
In this paper, a review of most commonly used transmission line based dual-band/multi-band circuits and systems have been presented. It has been shown that the dual-band/multi-band circuits can be realized through either the pi-type or the t-type transmission line structures. It has been demonstrated, through design examples, the ease and flexibility of pi-type over the t-type or vice versa depends on specific applications. It has also been demonstrated that the designs of dual-band/multiband circuits and components actually utilizes the dispersive characteristics of transmission line structures, and it is this dispersive feature which also enables the optimal miniaturization of dual-band/multi-band circuits and components. The realization of dual-band/ multi-band matching circuits is a tedious process and requires appropriate considerations while selecting the optimal design techniques. Finally, design technique utilizing RF-DPD predistorter for realizing highly linear and efficient modern wireless transmitter for SDR applications has been discussed. It has been proposed that the use of Doherty PAs can further enhance the performance of such transmitters. It is anticipated that this proposed technique can fulfill the future design needs of ever growing wireless applications and would eventually enable the development of spectrum and environmental friendly SDR and Cognitive Radio.
 E. McCune, “High-effi ciency, multi-mode, multi-band terminal power amplifiers,” IEEE Microwave, vol. 6, no. 1, pp. 44–55, Mar. 2005.
 F. M. Ghannouchi, “Power amplifi er and transmitter architectures for software defi ned radio systems,” IEEE Circuits Syst. Mag., vol. 10, no. 4, pp. 44–55, Dec. 2010.
 P. Kennington, in RF and Baseband Techniques for Software Defined Radio. Norwood, MA: Artech House, 2005, chs. 1 and 5, pp. 1–24, pp. 183–229.
 Powerwave, “Nexus RT 850/1900 MHz dual-band digital repeater,” Datasheet no. NP50B0-22111, pp. 1–2.
 Apple, “Apple MB763LL/A airPort extreme dual-band base station,” pt. MC340LL/A.
 Erricson, “RBS 2102 radio base station for GSM system,” AE/LZT 123 3066 R5, Ericsson Radio Systems AB 2001.
 G. L. Matthaei, L. Young, and E. M. T. Jones, in RF and Baseband Techniques for Software Defined Radio. Norwood, MA: Artech House, 2005, ch. 1, ch. 5, pp. 1–24, pp. 183–229.
 W. K. Chen, Theory and Design of Broadband Matching Networks. Pergamon International Library, 1976, ch. 4, pp. 217–311.
 H. W. Bode, Network Analysis and Feedback Amplifier Design. Princeton, NJ: Van Nostrand, 1945.
 R. M. Fano. (1948, Jan.). Theoretical limitations on the broadband matching of arbitrary impedances. Massachusetts Inst. Technol., Res. Lab. Electron., Rep. 41, pp. 1–48. [Online]. Available: http://hdl.handle.net/1721.1/5007
 D. Youla, “A new theory of broad-band matching,” IEEE Trans. Circuit Theory, vol. 11, no. 1, pp. 30–50, Jan. 1964.
 H. J. Carlin and P.J. Crepeau, “Theoretical limitations on the broadband matching of arbitrary impedances,” IRE Trans. Circuit Theory, vol. 8, no. 2, pp. 165–165, June 1961.
 H. J. Carlin and R. C. Shen, “Gain-bandwidth theory for optimizing transmission through a prescribed lossless two-port,” IEEE Trans. Circuit Theory, vol. 19, no. 1, pp. 98–100, Jan. 1972.
 Li Yiqun Li, B. Derat, D. Pasquet, and J. C. Bolomey, “Matching limits for a dual-band mobile phone antenna,” in Proc. IEEE 2007 Int. Symp. Microwave, Antenna, Propagation, and EMC Technologies for Wireless Communications, 2007, pp. 656 659.
 R. Negra, A. Sadeve, S. Bensmida, and F. M. Ghannouchi, “Concurrent dual-band class-F load coupling network for applications at 1.7 and 2.14 GHz,” IEEE Trans. Circuits Syst. II, vol. 55, no. 3, pp. 259–263, Mar. 2008.
 S. A. Mass, in Nonlinear Microwave Circuits. Norwood, MA: Artech House, 1988, ch. 5, pp. 15–64.
 Bahl and P. Bhartia, in Microwave Solid-State Circuit Design. New York: Wiley, 1998, pp. 540–598. <
 K. Rawat, M. Rawat, and F. M. Ghannouchi, “Compensating I-Q imperfections in hybrid RF/digital predistortion with adapted look up table implemented in FPGA,” IEEE Trans. Circuits Syst. II, vol. 57, no. 5, pp. 389–393, May 2010.
 F. Rivet, Y. Deval, J. B. Begueret, D. Dallet, P. Cathelin, and D. Belot “A disruptive receiver architecture dedicated to software-defi ned radio,” IEEE Trans. Circuits Syst. II, vol. 55, no. 4, pp. 344–348, Apr. 2008.
 J. M. M. Ferreras, R. G. García, and F. P. Martínez, “RF front-end concept and implementation for direct sampling of multiband signals,” IEEE Trans. Circuits Syst. II, vol. 58, no. 3, pp. 129–133, Mar. 2011.
 C. A. DeVries and R. D. Mason, “Subsampling architecture for low power receivers,” IEEE Trans. Circuits Syst. II, vol. 55, no. 4, pp. 304–308, Apr. 2008.
 V. K. Parikh, P. T. Balsara, and O. E. Eliezer, “All digital-quadraturemodulator based wideband wireless transmitters,” IEEE Trans. Circuits Syst. I, vol. 56, no. 11, pp. 2487–2497, Nov. 2009.
 P. K. Datta, X. Fan, and G. Fischer, “A transceiver front-end for ultra- wide-band applications,” IEEE Trans. Circuits Syst. II, vol. 54, no. 4, pp. 362–366, Apr. 2007.
 G. Hueber, R. Stuhlberger, and A. Springer, “An adaptive digital front-end for multimode wireless receivers,” IEEE Trans. Circuits Syst. II, vol. 55, no. 4, pp. 349–353, Apr. 2008.
 D. Chen, W. Wang, and T. Kwasniewski, “Highly integrated and tunable RF front ends for reconfi gurable multiband transceivers: A tutorial,” IEEE Trans. Circuits Syst. I, vol. 58, no. 9, pp. 2038–2050, Sept. 2011.
 V. Bezooijen, R. Mahmoudi, and A. H. M. V. Roermund, “Adaptive methods to preserve power amplifi er linearity under antenna mismatch conditions,”IEEE Trans. Circuits Syst. I, vol. 52, no. 10, pp. 2101–2108, Oct. 2005.
 Keerti and A. V. H. Pham, “RF characterization of SiGe HBT power amplifi ers under load mismatch,” IEEE Trans. Microwave Theory Tech., vol. 55, no. 2, pp. 207–214, Feb. 2007.
 S. E. Ramanujam, “Intermodulation distortion modelling and measurement techniques for GaN HEMT characterization,” Ph.D. dissertation, Kassel Univ., Germany, 2009.
 D. M. Pozar, in Microwave Engineering. New York: Wiley, 1998, chs. 7 and 8, pp. 351–414, pp. 422–493.
 K. Rawat, F. M. Ghannouchi, M. Rawat, and M. S. Hashmi, “Analysis of frequency selective impedance loading of transmission lines for dual band couplers,” Int. J. Microwave RF Comput. Aided Eng., vol. 21, no. 3, pp. 325–335, May 2011.
 K. K. M. Cheng and F. L. Wong, “A novel approach to the design and implementation of dual-band compact planar 90° branch-line coupler,” IEEE Trans. Microwave Theory Tech., vol. 52, no. 11, pp. 2458–2463, Nov. 2004.
 R. E. Collin, in Foundation for Microwave Engineering (IEEE Press Series on Electromagnetic Wave Theory). New York: IEEE, 1998, ch. 8, pp. 550–595.
 P. Peres, C. R. de Souza, and I. S. Bonatti, “ABCD Matrix: A unique tool for linear two-wire transmission line modelling,” Int. J. Elect. Eng. Educ., vol. 40, no. 3, pp. 220–229, July 2003.
 J. Reed and G. Wheeler, “A method of analysis of symmetrical fourport networks,” IRE Trans. Microwave Theory Tech., vol. 4, no. 4, pp. 246–252, Oct. 1956.
 E. J. Wilkinson, “A N-way hybrid power divider,” IRE Trans. Microwave Theory Tech., vol. 8, no. 1, pp. 116–118, Jan. 1960.
 H. Lin, M. D. Vincentis, C. Caloz, and T. Itoh, “Arbitrary dual-band components using composite right/left-handed transmission lines,” IEEE Trans. Microwave Theory Tech., vol. 52, pp. 1142–1148, Apr. 2004.
 Hsu, J. T. Kuo, and C. W. Chang, “Miniaturized dual-band hybrid couplers with arbitrary power division ratio,” IEEE Trans. Microwave Theory Tech., vol. 57, pp. 149–156, Jan. 2009.
 S. Dwari and S. Sanyal, “An arbitrary dual-band microstrip hybridring,” Wiley Microwave Opt. Lett., vol. 48, pp. 840–842, May 2006.
 K. Rawat and F. M. Ghannouchi, “A novel dual-band matching technique based on dual-characteristic impedance transformers for dualband power amplifi ers design,” IET Microwave Antenna Propagat., to be published.
 M. J. Park and B. Lee, “A dual-band Wilkinson power divider,” IEEE Microwave Wireless Comp. Lett., vol. 18, no. 2, pp. 85–87, Feb. 2008.
 G. Castaldi, V. Fiumara, and I. M. Pinto, “A dual-band Chebyshev impedance transformer,” Microwave Opt. Tech. Lett., vol. 39, no. 2, pp. 141–145, Oct. 2003.
 S. J. Orfanidis, “A two-section dual-band Chebyshev impedance transformer,” IEEE Microwave Wireless Comp Lett., vol. 13, no. 9, pp. 382–384, Sept. 2003.
 Monzon, “A small dual-frequency transformer in two sections,” IEEE Trans. Microwave Theory Tech., vol. 51, pp. 1157–1161, Apr. 2003.
 L. Wu, S. Sun, H. Yilmaz, and M. Berroth, “A dual-frequency Wilkinson power divider,” IEEE Trans. Microwave Theory Tech., vol. 54, no. 1, pp. 278–284, Jan. 2006.
 L. Wu, S. Sun, H. Yilmaz, and M. Berroth, “A new Wilkinson power divider design for dual band application,” IEEE Microwave Wireless Comp. Lett., vol. 17, no. 9, pp. 664–666, Sept. 2007.
 K. Rawat and F. M. Ghannouchi, “A design methodology for miniaturized power dividers using periodically loaded slow wave structure with dual-band applications,” IEEE Trans. Microwave Theory Tech., vol.
57, pp. 3330–3338, Dec. 2009.  G. Gonzalez, Microwave Transistor Amplifiers: Analysis and Design. Englewood Cliffs, NJ: Prentice-Hall, 1996.
 Y. Wu, Y. Liu, and S. Li, “A dual-frequency transformer for complex impedances with two unequal sections,” IEEE Microwave Wireless Comp. Lett., vol. 19, no. 2, pp. 77–79, Feb. 2009.
 M. J. Park and B. Lee, “Dual-band design of single-stub impedance matching networks with application to dual-band stubbed T-junctions,” Microwave Opt. Tech. Lett., vol. 52, no. 6, pp. 1359–136, July 2010.
 P. Colantonio, F. Giannini, and R. Giofrè, “A design technique for concurrent dual-band harmonic tuned power amplifi er,” IEEE Trans. Microwave Theory Tech., vol. 56, no. 11, pp. 2545–2555, Nov. 2008.
 M. L. Chuang, “Dual-band impedance transformer using two section shunt stub,” IEEE Trans. Microwave Theory Tech., vol. 58, no. 5, pp. 1257–1263, May 2010.
 Y. Wu, Y. Liu, S. Li, C. Yu, and X. Liu, “A generalized dual-frequency transformer for two arbitrary complex frequency-dependent impedances,” IEEE Microwave Wireless Comp. Lett., vol. 19, no. 12, pp. 792–794, Dec. 2009.
 X. Liu, Y. Liu, S. Li, F. Wu, and Y. Wu, “A three-section dual-band transformer for frequency-dependent complex load impedance,” IEEE Microwave Wireless Comp. Lett., vol. 19, no. 10, pp. 611–613, Oct. 2009.
 M. S. Hashmi, F. M. Ghannouchi, P. J. Tasker, and K. Rawat, “Highly reflective load-pull,” IEEE Microwave, vol. 12, no. 4, pp. 96–107, June2011.
 Grace, J. Chen, T. Jiang, and P. D. Mitchell, “Using cognitive radio to deliver green communication,” in Proc. 4th IEEE Int. Conf. Cognitive Radio Oriented Wireless Networks and Communications (CROWNCOM), Hannover, Germany, June 2009, pp. 1–4.
 C. Han, T. Harrold, S. Armour, I. Krikidis, S. Videv, P. M. Grant, H. Haas, J. S. Thompson, I. Ku, C.X. Wang, T. A. Le, M. R. Nakhai, J. Zhang, and L. Hanzo, “Green radio: Radio techniques to enable energy-effi cient wireless networks,” IEEE Commun. Mag., pp. 1–10, June 2011.
 G. Jeckeln, F. Beauregard, M. A. Sawan, and F. M. Ghannouchi, “Adaptive baseband/RF predistorter for power amplifiers through instantaneous AM-AM and AM-PM characterization using digital receivers,” in IEEE MTT-S Int. Microwave Symp. Dig., May 2000, vol. 1, pp. 489–492.
 W. Woo, M. D. Miller, and J. S. Kenney, “A hybrid digital/RF envelope predistortion linearization system for power amplifiers,”IEEE Trans. Microwave. Theory Tech., vol. 53, no. 1, pp. 229–237, Jan. 2005.
 K. Rawat, O. Hammi, and F. M. Ghannouchi, “Investigating effects of quadrature imperfection of vector multiplier in implementing RF/digital predistortion,” in Proc. 10th Annu. IEEE WAMICON, Clearwater, FL,Apr. 20–21, 2009, pp. 1–4.
 M. Rawat, K. Rawat, and F. M. Ghannouchi, “Adaptive digital predistortion of wireless power amplifi ers/transmitters using dynamic realvalued focused time-delay line neural networks,” IEEE Trans. Microwave Theory Tech., vol. 58, no. 1, pp. 95–104, Jan. 2010.
 “MAX2045/MAX2046/MAX2047 high gain vector multipliers datasheet,” Maxim Integr. Products, Inc., Sunnyvale, CA, Jan. 2003.
 “ADL5390 RF/IF vector multiplier,” Analog Devices, Norwood, MA, 2004.
 W. H. Doherty, “A new high effi ciency power amplifi er for modulated waves,” Proc. IRE, vol. 24, no. 9, pp. 1163–1182, Sept. 1936.
 B. Kim, J. Kim, I. Kim, and J. Cha, “The Doherty power amplifier,”IEEE Microwave, vol. 7, no. 5, pp. 42–50, Oct. 2006.
 P. Colantonio, F. Feudo, F. Giannini, R. Giofre, and L. Piazzon, “Design of a dual-band GaN Doherty amplifier,” in Proc. 18th Int. Conf. Microwave Radar and Wireless Communications (MIKON), 2010, pp. 1–4.
 X. Li, W. Chen, Z. Zhang, Z. Feng, X. Tang, and K. Mouthaan, “Aconcurrent dual-band Doherty power amplifier,” in Proc. Asia PacificMicrowave Conf., 2010, pp. 1–4.
Entry filed under: Electrical Engineering.