Design and characterization of downconversion mixers and the on-chip calibration techniques for monolithic direct conversion radio receivers

This thesis consists of eight publications and an overview of the research topic, which is also a summary of the work. The research described in this thesis is focused on the design of downconversion mixers and direct conversion radio receivers for UTRA/FDD WCDMA and GSM standards. The main interest of the work is in the 1-3 GHz frequency range and in the Silicon and Silicon-Germanium BiCMOS technologies. The RF front-end, and especially the mixer, limits the performance of direct conversion architecture. The most stringent problems are involved in the second-order distortion in mixers to which special attention has been given. The work introduces calibration techniques to overcome these problems. Some design considerations for front-end radio receivers are also given through a mixer-centric approach. The work summarizes the design of several downconversion mixers. Three of the implemented mixers are integrated as the downconversion stages of larger direct conversion receiver chips. One is realized together with the LNA as an RF front-end. Also, some stand-alone structures have been characterized. Two of the mixers that are integrated together with whole analog receivers include calibration structures to improve the second-order intermodulation rejection. A theoretical mismatch analysis of the second-order distortion in the mixers is also presented in this thesis. It gives a comprehensive illustration of the second-order distortion in mixers. It also gives the relationships between the dc-offsets and high IIP2. In addition, circuit and layout techniques to improve the LO-to-RF isolation are discussed. The presented work provides insight into how the mixer immunity against the second-order distortion can be improved. The implemented calibration structures show promising performance. On the basis of these results, several methods of detecting the distortion on-chip and the possibilities of integrating the automatic on-chip calibration procedures to produce a repeatable and well-predictable receiver IIP2 are presented.reviewe

Analog baseband circuits for WCDMA direct-conversion receivers

This thesis describes the design and implementation of analog baseband circuits for low-power single-chip WCDMA direct-conversion receivers. The reference radio system throughout the thesis is UTRA/FDD. The analog baseband circuit consists of two similar channels, which contain analog channel-select filters, programmable-gain amplifiers, and circuits that remove DC offsets. The direct-conversion architecture is described and the UTRA/FDD system characteristics are summarized. The UTRA/FDD specifications define the performance requirement for the whole receiver. Therefore, the specifications for the analog baseband circuit are obtained from the receiver requirements through calculations performed by hand. When the power dissipation of an UTRA/FDD direct-conversion receiver is minimized, the design parameters of an all-pole analog channel-select filter and the following Nyquist rate analog-to-digital converter must be considered simultaneously. In this thesis, it is shown that minimum power consumption is achieved with a fifth-order lowpass filter and a 15.36-MS/s Nyquist rate converter that has a 7- or 8-bit resolution. A fifth-order Chebyshev prototype with a passband ripple of 0.01 dB and a −3-dB frequency of 1.92-MHz is adopted in this thesis. The error-vector-magnitude can be significantly reduced by using a first-order 1.4-MHz allpass filter. The selected filter prototype fulfills all selectivity requirements in the analog domain. In this thesis, all the filter implementations use the opamp-RC technique to achieve insensitivity to parasitic capacitances and a high dynamic range. The adopted technique is analyzed in detail. The effect of the finite opamp unity-gain bandwidth on the filter frequency response can be compensated for by using passive methods. Compensation schemes that also track the process and temperature variations have been developed. The opamp-RC technique enables the implementation of low-voltage filters. The design and simulation results of a 1.5-V 2-MHz lowpass filter are discussed. The developed biasing scheme does not use any additional current to achieve the low-voltage operation, unlike the filter topology published previously elsewhere. Methods for removing DC offsets in UTRA/FDD direct-conversion receivers are presented. The minimum areas for cascaded AC couplings and DC-feedback loops are calculated. The distortion of the frequency response of a lowpass filter caused by a DC-feedback loop connected over the filter is calculated and a method for compensating for the distortion is developed. The time constant of an AC coupling can be increased using time-constant multipliers. This enables the implementation of AC couplings with a small silicon area. Novel time-constant multipliers suitable for systems that have a continuous reception, such as UTRA/FDD, are presented. The proposed time-constant multipliers only require one additional amplifier. In an UTRA/FDD direct-conversion receiver, the reception is continuous. In a low-power receiver, the programmable baseband gain must be changed during reception. This may produce large, slowly decaying transients that degrade the receiver performance. The thesis shows that AC-coupling networks and DC-feedback loops can be used to implement programmable-gain amplifiers, which do not produce significant transients when the gain is altered. The principles of operation, the design, and the practical implementation issues of these amplifiers are discussed. New PGA topologies suitable for continuously receiving systems have been developed. The behavior of these circuits in the presence of strong out-of-channel signals is analyzed. The interface between the downconversion mixers and the analog baseband circuit is discussed. The effect of the interface on the receiver noise figure and the trimming of mixer IIP2 are analyzed. The design and implementation of analog baseband circuits and channel-select filters for UTRA/FDD direct-conversion receivers are discussed in five application cases. The first case presents the analog baseband circuit for a chip-set receiver. A channel-select filter that has an improved dynamic range with a smaller supply current is presented next. The third and fifth application cases describe embedded analog baseband circuits for single-chip receivers. In the fifth case, the dual-mode analog baseband circuit of a quad-mode receiver designed for GSM900, DCS1800, PCS1900, and UTRA/FDD cellular systems is described. A new, highly linear low-power transconductor is presented in the fourth application case. The fourth application case also describes a channel-select filter. The filter achieves +99-dBV out-of-channel IIP2, +45-dBV out-of-channel IIP3 and 23-μVRMS input-referred noise with 2.6-mA current from a 2.7-V supply. In the fifth application case, a corresponding performance is achieved in UTRA/FDD mode. The out-of-channel IIP2 values of approximately +100 dBV achieved in this work are the best reported so far. This is also the case with the figure of merits for the analog channel-select filter and analog baseband circuit described in the fourth and fifth application cases, respectively. For equal power dissipation, bandwidth, and filter order, these circuits achieve approximately 10 dB and 15 dB higher spurious-free dynamic ranges, respectively, when compared to implementations that are published elsewhere and have the second best figure of merits.reviewe

CMOS radio frequency circuits for short-range direct-conversion receivers

The research described in this thesis is focused on the design and implementation of radio frequency (RF) circuits for direct-conversion receivers. The main interest is in RF front-end circuits, which contain low-noise amplifiers, downconversion mixers, and quadrature local oscillator signal generation circuits. Three RF front-end circuits were fabricated in a short-channel CMOS process and experimental results are presented. A low-noise amplifier (LNA) is typically the first amplifying block in the receiver. A large number of LNAs have been reported in the literature. In this thesis, wideband LNA structures are of particular interest. The most common and relevant LNA topologies are analyzed in detail in the frequency domain and theoretical limitations are found. New LNA structures are presented and a comparison to the ones found in the literature is made. In this work, LNAs are implemented with downconversion mixers as RF front-ends. The designed mixers are based on the commonly used Gilbert cell. Different mixer implementation alternatives are presented and the design of the interface between the LNA and the downconversion mixer is discussed. In this work, the quadrature local oscillator signal is generated either by using frequency dividers or polyphase filters (PPF). Different possibilities for implementing frequency dividers are briefly described. Polyphase filters were already introduced by the 1970s and integrated circuit (IC) realizations to generate quadrature signals have been published since the mid-1990s. Although several publications where the performance of the PPFs has been studied either by theoretical calculations or simulations can be found in the literature, none of them covers all the relevant design parameters. In this thesis, the theory behind the PPFs is developed such that all the relevant design parameters needed in the practical circuit design have been calculated and presented with closed-form equations whenever possible. Although the main focus was on twoand three-stage PPFs, which are the most common ones encountered in practical ICs, the presented calculation methods can be extended to analyze the performance of multistage PPFs as well. The main application targets of the circuits presented in this thesis are the short-range wireless sensor system and ultrawideband (UWB). Sensors are capable of monitoring temperature, pressure, humidity, or acceleration, for example. The amount of transferred data is typically small and therefore a modest bit rate, less than 1 Mbps, is adequate. The sensor system applied in this thesis operates at 2.4-GHz ISM band (Industrial, Scientific, and Medical). Since the sensors must be able to operate independently for several years, extremely low power consumption is required. In sensor radios, the receiver current consumption is dominated by the blocks and elements operating at the RF. Therefore, the target was to develop circuits that can offer satisfactory performance with a current consumption level that is small compared to other receivers targeted for common cellular systems. On the other hand, there is a growing need for applications that can offer an extremely high data rate. UWB is one example of such a system. At the moment, it can offer data rates of up to 480 Mbps. There is a frequency spectrum allocated for UWB systems between 3.1 and 10.6 GHz. The UWB band is further divided into several narrower band groups (BG), each occupying a bandwidth of approximately 1.6 GHz. In this work, a direct-conversion RF front-end is designed for a dual-band UWB receiver, which operates in band groups BG1 and BG3, i.e. at 3.1 – 4.8 GHz and 6.3 – 7.9 GHz frequency areas, respectively. Clearly, an extremely wide bandwidth combined with a high operational frequency poses challenges for circuit design. The operational bandwidths and the interfaces between the circuit blocks need to be optimized to cover the wanted frequency areas. In addition, the wideband functionality should be achieved without using a number of on-chip inductors in order to minimize the die area, and yet the power consumption should be kept as small as possible. The characteristics of the two main target applications are quite different from each other with regard to power consumption, bandwidth, and operational frequency requirements. A common factor for both is their short, i.e. less than 10 meters, range. Although the circuits presented in this thesis are targeted on the two main applications mentioned above, they can be utilized in other kind of wireless communication systems as well. The performance of three experimental circuits was verified with measurements and the results are presented in this work. Two of them have been a part of a whole receiver including baseband amplifiers and filters and analog-to-digital converters. Experimental circuits were fabricated in a 0.13-µm CMOS process. In addition, this thesis includes design examples where new circuit ideas and implementation possibilities are introduced by using 0.13-µm and 65-nm CMOS processes. Furthermore, part of the theory presented in this thesis is validated with design examples in which actual IC component models are used.Tässä väitöskirjassa esitetty tutkimus keskittyy suoramuunnosvastaanottimen radiotaajuudella (radio frequency, RF) toimivien piirien suunnitteluun ja toteuttamiseen. Työ keskittyy vähäkohinaiseen vahvistimeen (low-noise amplifier, LNA), alassekoittajaan ja kvadratuurisen paikallisoskillaattorisignaalin tuottavaan piiriin. Työssä toteutettiin kolme RF-etupäätä erittäin kapean viivanleveyden CMOS-prosessilla, ja niiden kokeelliset tulokset esitetään. Vähäkohinainen vahvistin on yleensä ensimmäinen vahvistava lohko vastaanottimessa. Useita erilaisia vähäkohinaisia vahvistimia on esitetty kirjallisuudessa. Tämän työn kohteena ovat eritoten laajakaistaiset LNA-rakenteet. Tässä työssä analysoidaan taajuustasossa yleisimmät ja oleellisimmat LNA-topologiat. Lisäksi uusia LNA-rakenteita on esitetty tässä työssä ja niitä on verrattu muihin kirjallisuudessa esitettyihin piireihin. Tässä työssä LNA:t on toteutettu yhdessä alassekoittimen kanssa muodostaen RF-etupään. Työssä suunnitellut alassekoittimet perustuvat yleisesti käytettyyn Gilbertin soluun. Erilaisia sekoittajan suunnitteluvaihtoehtoja ja LNA:n ja alassekoittimen välisen rajapinnan toteutustapoja on esitetty. Tässä työssä kvadratuurinen paikallisoskillaattorisignaali on muodostettu joko käyttämällä taajuusjakajia tai monivaihesuodattimia. Erilaisia taajuusjakajia ja niiden toteutustapoja käsitellään yleisellä tasolla. Monivaihesuodatinta, joka on alunperin kehitetty jo 1970-luvulla, on käytetty integroiduissa piireissä kvadratuurisignaalin tuottamiseen 1990-luvun puolivälistä lähtien. Kirjallisuudesta löytyy lukuisia artikkeleita, joissa monivaihesuodattimen toimintaa on käsitelty teoreettisesti laskien ja simuloinnein. Kuitenkaan kaikkia sen suunnitteluparametreja ei tähän mennessä ole käsitelty. Tässä työssä monivaihesuodattimen teoriaa on kehitetty edelleen siten, että käytännön piirisuunnittelussa tarvittavat oleelliset parametrit on analysoitu ja suunnitteluyhtälöt on esitetty suljetussa muodossa aina kuin mahdollista. Vaikka työssä on keskitytty yleisimpiin eli kaksi- ja kolmiasteisiin monivaihesuodattimiin, on työssä esitetty menetelmät, joilla laskentaa voidaan jatkaa aina useampiasteisiin suodattimiin asti. Työssä esiteltyjen piirien pääkohteina ovat lyhyen kantaman sensoriradio ja erittäin laajakaistainen järjestelmä (ultrawideband, UWB). Sensoreilla voidaan tarkkailla esimerkiksi ympäristön lämpötilaa, kosteutta, painetta tai kiihtyvyyttä. Siirrettävän tiedon määrä on tyypillisesti vähäistä, jolloin pieni tiedonsiirtonopeus, alle 1 megabitti sekunnissa, on välttävä. Tämän työn kohteena oleva sensoriradiojärjestelmä toimii kapealla kaistalla 2,4 gigahertsin ISM-taajuusalueella (Industrial, Scientific, and Medical). Koska sensorien tavoitteena on toimia itsenäisesti ilman pariston vaihtoa useita vuosia, täytyy niiden kuluttaman virran olla erittäin vähäistä. Sensoriradiossa vastaanottimen tehonkulutuksen kannalta määräävässä asemassa ovat radiotaajuudella toimivat piirit. Tavoitteena oli tutkia ja kehittää piirirakenteita, joilla päästään tyydyttävään suorituskykyyn tehonkulutuksella, joka on vähäinen verrattuna muiden tavallisten langattomien tiedonsiirtojärjestelmien radiovastaanottimiin. Toisaalta viime aikoina on kasvanut tarvetta myös järjestelmille, jotka kykenevät tarjoamaan erittäin korkean tiedonsiirtonopeuden. UWB on esimerkki tällaisesta järjestelmästä. Tällä hetkellä se tarjoaa tiedonsiirtonopeuksia aina 480 megabittiin sekunnissa. UWB:lle on varattu taajuusalueita 3,1 ja 10,6 gigahertsin taajuuksien välillä. Kyseinen kaista on edelleen jaettu pienempiin taajuusryhmiin (band group, BG), joiden kaistanleveys on noin 1,6 gigahertsiä. Tässä työssä on toteutettu RF-etupää radiovastaanottimeen, joka pystyy toimimaan BG1:llä ja BG3:lla eli taajuusalueilla 3,1 - 4,7 GHz ja 6,3 - 7,9 GHz. Erittäin suuri kaistanleveys yhdistettynä korkeaan toimintataajuuteen tekee radiotaajuuspiirien suunnittelusta haasteellista. Piirirakenteiden toimintakaistat ja piirien väliset rajapinnat tulee optimoida riittävän laajoiksi käyttämättä kuitenkaan liian montaa piille integroitua kelaa piirin pinta-alan minimoimiseksi, ja lisäksi piirit tulisi toteuttaa mahdollisimman alhaisella tehonkulutuksella. Työssä esiteltyjen piirien kaksi pääkohdetta ovat hyvin erityyppisiä, mitä tulee tehonkulutus-, kaistanleveys- ja toimintataajuusvaatimuksiin. Yhteistä molemmille on lyhyt, alle 10 metrin kantama. Vaikka tässä työssä esitellyt piirit onkin kohdennettu kahteen pääsovelluskohteeseen, voidaan esitettyjä piirejä käyttää myös muiden tiedonsiirtojärjestelmien piirien suunnitteluun. Tässä työssä esitetään mittaustuloksineen yhteensä kolme kokeellista piiriä yllämainittuihin järjestelmiin. Kaksi ensimmäistä kokeellista piiriä muodostaa kokonaisen radiovastaanottimen yhdessä analogisten kantataajuusosien ja analogia-digitaali-muuntimien kanssa. Esitetyt kokeelliset piirit on toteutettu käyttäen 0,13 µm:n viivanleveyden CMOS-tekniikkaa. Näiden lisäksi työ pitää sisällään piirisuunnitteluesimerkkejä, joissa esitetään ideoita ja mahdollisuuksia käyttäen 0,13 µm:n ja 65 nm:n viivanleveyden omaavia CMOS-tekniikoita. Lisäksi piirisuunnitteluesimerkein havainnollistetaan työssä esitetyn teorian paikkansapitävyyttä käyttämällä oikeita komponenttimalleja.reviewe

High Performance LNAs and Mixers for Direct Conversion Receivers in BiCMOS and CMOS Technologies

The trend in cellular chipset design today is to incorporate support for a larger number of frequency bands for each new chipset generation. If the chipset also supports receiver diversity two low noise amplifiers (LNAs) are required for each frequency band. This is however associated with an increase of off-chip components, i.e. matching components for the LNA inputs, as well as complex routing of the RF input signals. If balanced LNAs are implemented the routing complexity is further increased. The first presented work in this thesis is a novel multiband low noise single ended LNA and mixer architecture. The mixer has a novel feedback loop suppressing both second order distortion as well as DC-offset. The performance, verified by Monte Carlo simulations, is sufficient for a WCDMA application. The second presented work is a single ended multiband LNA with programmable integrated matching. The LNA is connected to an on-chip tunable balun generating differential RF signals for a differential mixer. The combination of the narrow band input matching and narrow band balun of the presented LNA is beneficial for suppressing third harmonic downconversion of a WLAN interferer. The single ended architecture has great advantages regarding PCB routing of the RF input signals but is on the other hand more sensitive to common mode interferers, e.g. ground, supply and substrate noise. An analysis of direct conversion receiver requirements is presented together with an overview of different LNA and mixer architectures in both BiCMOS and CMOS technology

Low-noise amplifiers for integrated multi-mode direct-conversion receivers

The evolution of wireless telecommunication systems during the last decade has been rapid. During this time the design driver has shifted towards fast data applications instead of speech. In addition, the different systems may have a limited coverage, for example, limited to urban areas only. Thus, it has become important for a mobile terminal to be able to use different wireless systems, depending on the application chosen and the location of the terminal. The choice of receiver architecture affects the performance, size, and cost of the receiver. The superheterodyne receiver has hitherto been the dominant radio architecture, because of its good sensitivity and selectivity. However, superheterodyne receivers require expensive filters, which, with the existing technologies, cannot be integrated on the same chip as the receiver. Therefore, architectures using a minimum number of external components, such as direct conversion, have become popular. In addition, compared to the superheterodyne architecture, the direct-conversion architecture has benefits when multi-mode receivers, which are described in this thesis, are being designed. In this thesis, the limitations placed on the analog receiver by different system specifications are introduced. The estimations for the LNA specifications are derived from these specifications. In addition, the limitations imposed by different types of receiver architectures are described. The inductively-degenerated LNA is the basis for all the experimental circuits. The different components for this configuration are analyzed and compared to other commonly-used configurations in order to justify the use of an inductively-degenerated LNA. Furthermore, the design issues concerning the LNA-mixer interface in direct-conversion receivers are analyzed. Without knowing these limitations, it becomes difficult to understand the choices made in the experimental circuits. One of the key parts of this thesis describes the design and implementation of a single-chip multi-mode LNA, which is one of the key blocks in multi-mode receivers. The multi-mode structures in this thesis were developed for a direct-conversion receiver where only one system is activated at a time. The LNA interfaces to a pre-select filter and mixers and the different LNA components are analyzed in detail. Furthermore, the design issues related to possible interference from additional systems on single-chip receivers are analyzed and demonstrated. A typical receiver includes variable gain, which can be implemented both in the analog baseband and/or in the RF. If the variable gain is implemented in the RF parts, it is typically placed in the LNA or in a separate gain control stage. Several methods that can be used to implement a variable gain in the LNA are introduced and compared to each other. Furthermore, several of these methods are included in the experimental circuits. The last part of this thesis concentrates on four experimental circuits, which are described in this thesis. The first two chips describe an RF front-end and a direct-conversion receiver for WCDMA applications. The whole receiver demonstrates that it is possible to implement A/D converters on the same chip as sensitive RF blocks without significantly degrading receiver performance. The other two chips describe an RF front-end for WCDMA and GSM900 applications and a direct-conversion receiver for GSM900, DCS1800, PCS1900 and WCDMA systems. These ICs demonstrate the usability of the circuit structure developed and presented in this thesis. The chip area in the last multi-mode receiver is not significantly increased compared to corresponding single-system receivers.reviewe

Digitally-Enhanced Software-Defined Radio Receiver Robust to Out-of-Band Interference

A software-defined radio (SDR) receiver with improved robustness to out-of-band interference (OBI) is presented. Two main challenges are identified for an OBI-robust SDR receiver: out-of-band nonlinearity and harmonic mixing. Voltage gain at RF is avoided, and instead realized at baseband in combination with low-pass filtering to mitigate blockers and improve out-of-band IIP3. Two alternative “iterative” harmonic-rejection (HR) techniques are presented to achieve high HR robust to mismatch: a) an analog two-stage polyphase HR concept, which enhances the HR to more than 60 dB; b) a digital adaptive interference cancelling (AIC) technique, which can suppress one dominating harmonic by at least 80 dB. An accurate multiphase clock generator is presented for a mismatch-robust HR. A proof-of-concept receiver is implemented in 65 nm CMOS. Measurements show 34 dB gain, 4 dB NF, and 3.5 dBm in-band IIP3 while the out-of-band IIP3 is + 16 dBm without fine tuning. The measured RF bandwidth is up to 6 GHz and the 8-phase LO works up to 0.9 GHz (master clock up to 7.2 GHz). At 0.8 GHz LO, the analog two-stage polyphase HR achieves a second to sixth order HR > dB over 40 chips, while the digital AIC technique achieves HR > 80 dB for the dominating harmonic. The total power consumption is 50 mA from a 1.2 V supply

Continuous-time low-pass filters for integrated wideband radio receivers

This thesis concentrates on the design and implementation of analog baseband continuous-time low-pass filters for integrated wideband radio receivers. A total of five experimental analog baseband low-pass filter circuits were designed and implemented as a part of five single-chip radio receivers in this work. After the motivation for the research work presented in this thesis has been introduced, an overview of analog baseband filters in radio receivers is given first. In addition, a review of the three receiver architectures and the three wireless applications that are adopted in the experimental work of this thesis is presented. The relationship between the integrator non-idealities and integrator Q-factor, as well as the effect of the integrator Q-factor on the filter frequency response, are thoroughly studied on the basis of a literature review. The theoretical study that is provided is essential for the gm-C filter synthesis with non-ideal lossy integrators that is presented after the introduction of different techniques to realize integrator-based continuous-time low-pass filters. The filter design approach proposed for gm-C filters is original work and one of the main points in this thesis, in addition to the experimental IC implementations. Two evolution versions of fourth-order 10-MHz opamp-RC low-pass filters designed and implemented for two multicarrier WCDMA base-station receivers in a 0.25-µm SiGe BiCMOS technology are presented, along with the experimental results of both the low-pass filters and the corresponding radio receivers. The circuit techniques that were used in the three gm-C filter implementations of this work are described and a common-mode induced even-order distortion in a pseudo-differential filter is analyzed. Two evolution versions of fifth-order 240-MHz gm-C low-pass filters that were designed and implemented for two single-chip WiMedia UWB direct-conversion receivers in a standard 0.13-µm and 65-nm CMOS technology, respectively, are presented, along with the experimental results of both the low-pass filters and the second receiver version. The second UWB filter design was also embedded with an ADC into the baseband of a 60-GHz 65-nm CMOS radio receiver. In addition, a third-order 1-GHz gm-C low-pass filter was designed, rather as a test structure, for the same receiver. The experimental results of the receiver and the third gm-C filter implementation are presented

Analysis of Non-Idealities on CMOS Passive Mixers

In the current state of the art, WiFi-alike standards require achieving a high Image Rejection Ratio (IRR) while having low power consumption. Thus, quadrature structures based on passive ring mixers offer an attractive and widely used solution, as they can achieve a high IRR while being a passive block. However, it is not easy for the designer to know when a simple quadrature scheme is enough and when they should aim for a double quadrature structure approach, as the latter can improve the performance at the cost of requiring more area and complexity. This study focuses on the IRR, which crucially depends on the symmetry between the I and Q branches. Non-idealities (component mismatches, parasitics, etc.) will degrade the ideal balance by affecting the mixer and/or following/previous stages. This paper analyses the effect of imbalances, providing the constraints for obtaining a 40 dB IRR in the case of a conversion from a one-hundred-megahertz signal to the five-gigahertz range (upconversion) and vice versa (downconversion) for simple and double quadrature schemes. All simulations were carried out with complete device models from 65 nm standard CMOS technology and also a post-layout Monte Carlo analysis was included for mismatch analysis. The final section includes guidelines to help designers choose the most adequate scheme for each case