## Dual-Band RF Circuits and Components for Multi-Standard Software Defined Radios

*June 16, 2012 at 4:43 pm* *
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**Abstract
**

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 [1] and hence makes it a front runner for deployment in SDR architectures [2]–[3]. 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 [4]–[6]. 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.

**Introduction
**

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 [9], Fano [10], Youla [11], 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 [9]. Fano [10], Youla [11] 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 [14].

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 [10]. 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 [1], [15].

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 [16]–[17]. 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 [16]–[17]. 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 [18].

**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 [3]. 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 [19]–[22]. 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 [23]–[25], 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) [26], 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 [26]–[27].

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 (ω_{1}). The IMD contours in Fig. 2(b) demonstrates that the best matching, Γ_{la} 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 P_{out}, shown as the shaded portion and depicted as Γ_{la}, 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, P_{out}, 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 [3], 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 [26].

**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 [29]. 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 f_{1} and f_{2} then electric length at f_{2} will be n times the electric length at f_{1}. 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 [30]–[32]. 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 [30]–[31]. 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 [33]. Example of these circuits are hybrids [29], [32], [34], Wilkinson power divider [29], [32], [35], delay line [29], [32] 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.) [7]–[8], the image impedance and image propagation constant characteristics are shaped [30]. 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) [29], [32]

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 [36] or transmission line sections [30]–[31], [37]–[39]. 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 [30] while the other is stub loading at the ends of the transmission line [31], [37]–[39]. 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 [30], are simultaneously valid and satisfy the ABCD parameters in (1).

where, *Z _{S}*

_{,T }and u

_{S,T }are the characteristic impedance and electric length of unloaded line,

*Z*

is the overall image impedance of the loaded line, and

_{T}*B*

_{S}_{,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 [31].

where, Z_{S,PI} and Θ_{S,PI} are the characteristic impedance and electric length of unloaded line loaded with a stub possessing susceptance B_{S,PI} and Z_{T} 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 u_{S,T} in (2)–(3) and Θ_{S,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 [30]–[31].

where, *n *is the frequency ratio *f*_{2}/*f*_{1} with *f*_{1} 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 [30]–[31]. 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 [30]–[31].

The most common design approach is to vary frequency ratio n in (6) for obtaining different values of electric length Θ_{S,T }or Θ_{S,PI} for which the structure presents a required image impedance Z_{T} (and hence emulate a quarter-wave transformer with a required value of characteristic impedance Z_{T}) with a realizable value of Z_{S,T} or Z_{S,PI}. The realizable value of Z_{S,T} or Z_{S,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 Z_{S,T} or Z_{S,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 Z_{S,T} or Z_{S,PI} for particular values of Θ_{S,T} or Θ_{S,PI}, n, and Z_{T}, the corresponding values for the other values of Z_{T} can easily be predicted by the linear relation given by (2) and (4) as described in [30]. Once the conditions for Z_{S,T} or Z_{S,PI} and the associated Θ_{S,T} or Θ_{S,PI} are obtained such that the structures in Fig. 4 can emulate an image impedance of ±Z_{T} at the two specified frequencies, the respective values of B_{S,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) [39].

where, *Z _{P}*

_{,T}is the characteristic impedance and Θ

_{P,T}is the electric length of the stub. Similarly, for the edge-loaded structure the stub parameters are given by the expression in (8) [39].

where, *Z _{P}*

_{,PI}is the characteristic impedance and Θ

_{P,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 [30]–[31].

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 Z_{S,T} and Z_{S,PI} obtained from Fig. 5(a) and Fig. 5(b) respectively, for a particular required value of Z_{T} and n, the values of Θ_{P,T} or Θ_{P,PI} can be obtained using (9). These selected values of Θ_{P,T} or Θ_{P,PI} along with other design parameters can then be used to obtain value of Z_{P,T} for center-tapped loaded line and Z_{P,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 Z_{P,T} or Z_{P,PI} are dependent on the technology used for fabrication. It is also evident from (7) and (8) that for particular values of Z_{T}, Z_{S,T} or Z_{S,PI} and Θ_{P,T} or Θ_{P,PI} the design parameter, Z_{P,T} or Z_{P,PI}, is linearly proportional to Z_{T}. Therefore, once the design parameters for a particular n and Z_{T} are known, the values of the corresponding Z_{P,T }or Z_{P,PI} for any other values of Z_{T} for the same n can also be determined [30].

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 [30]. 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 [40]. In principle, for a Wilkinson power divider the main requirement is the impedance transformation [35], 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 [41]–[43]. 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 [44]–[46]. 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 [44].

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 [45]. 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 [46]. 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 [32], [46]. 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 [46]. This non-dispersive range of frequency is useful for miniaturization application without introducing higher order modes of the Bloch wave travelling through the structure [32]. 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) [46].

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 [46]. 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 [37] 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 [47]. 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 [30]. 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 [48]–[49] 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 [41]–[42] 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 [50]. 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 [51]–[53] 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 [51] 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 [39], 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 [47] 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*(*f*1) and *G*(*f*2), of the required complex impedances, *Y*(*f*1) and *Y*(*f*2), to be seen by the active device at frequencies *f*1 and *f*2.

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*(*f*1) and *Y*(*f*2), at frequencies *f*1 and *f*2, as shown in Fig. 11(a). These dual-band/dual-susceptance stubs can be realized using the impedance buffer technique reported in [50], and are considered highly suitable for harmonic matching also.

As an example, Fig 11(b) shows a prototype of the matching technique [39] 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 [54]. 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 Y_{J}(f1) and Y_{J}(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, Z_{T}(f1) and Z_{T}(f2), and the corresponding values of the susceptances, jB_{J}(f1) and jB_{J}(f2), at reference JJ’ (Fig. 12(b)).

Once the values of jB_{J}(f1) and jB_{J}(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, Y_{J}(f1) and Y_{J}(f2) at the input of matching network. However, to absorb the susceptance of antenna input admittance with the stub values B_{J}(f ), a stub with modified B_{T}(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 Z_{T}(f ) at frequencies f1 and f2 are known by (12), the dual-band/dual-impedance transformer can be used to realize it [39]. 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. B_{T}(f1) and B_{T}(f2) shown in Fig. 12(b)) can be obtained as

This dual-band stub of B_{T}(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 Z_{T}(f1) and Z_{T}(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 [52], 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), Y_{A}(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 Z_{A}, Θ_{A}, Z_{B}, Θ_{B} by enforcing a condition over the impedances Z_{in} and Z_{out} (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 [52]–[53]. Details of this matching technique as shown in Fig. 13(a) are reported in [52].

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 [39].

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 [37].

**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 [55]–[56]. 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 [1]–[3].

Digital predistortion (DPD) is one of the most promising techniques used for linearizing the PAs for SDR transmitters [57]–[60]. 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 [2]–[3]. However, it is difficult to obtain wideband linearization with DPD due to limited speed of digital circuits involved in linearization [57]. Keeping this aspect in perspective the hybrid RF–DPD architecture has been proposed as a compromise between APD and DPD techniques [18], [57]–[59].

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 [18] 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 [61], 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 [62] depending on their operating frequency range.

In principle, after calibration as described in [18], 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 [57]–[60]. In order to enhance the efficiency in this back-off region the Doherty PA (DPA) is considered the most appropriate [63]–[66]. 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 [65]–[66], but further investigation is required for optimizing those techniques and for extending those techniques to multi-band designs.

**Conclusion
**

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.

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