Tunable High-Q N-Path Band-Pass Filters Modeling and Verification

Tunable High-Q N-Path Band-Pass Filters: Modeling and VerificationAmir Ghaffari,Student Member,IEEE,Eric A.M.Klumperink,Senior Member,IEEE, Michiel C.M.Soer,Student Member,IEEE,and Bram Nauta,Fellow,IEEEAbstract—A differential single-port switched-RC N-pathfilter with band-pass characteristic is proposed.The switching fre-quency defines the center frequency,while the RC-time and duty cycle of the clock define the bandwidth.This allows for high-Q highly tunablefilters which can for instance be useful for cognitive ing a linear periodically time-variant(LPTV)model, exact expressions for thefilter transfer function are derived.The behavior of the circuit including non-idealities such as maximum rejection,spectral aliasing,noise and effects due to mismatch in the paths is modeled and verified via measurements.A simple RLC equivalent circuit is provided,modeling bandwidth,quality factor and insertion loss of thefilter.A4-path architecture is realized in65nm CMOS.An off-chip transformer acts as a balun, improvesfilter-Q and realizes impedance matching.The differen-tial architecture reduces clock-leakage and suppresses selectivity around even harmonics of the clock.Thefilter has a constant 3dB bandwidth of35MHz and can be tuned from100MHz up to1GHz.Over the whole band,IIP3is better than14dBm, dBm and the noisefigure is3–5dB,while the power dissipation increases from2mW to16mW(only clocking power). Index Terms—Band-passfilter,CMOS,cognitive radio,commu-tated capacitors,frequency translatedfilter,high linearity,high-Q, inductorlessfilter,linear periodically time variant circuit,LPTV, N-pathfilter,software-defined radio,tunablefilter.I.I NTRODUCTIONI N software-defined radio(SDR)and cognitive radio trans-ceivers,programmability is not only desired in the digital back-end,but also for analog front-end functions.A major chal-lenge for such radios is the realization of an RF band-passfilter, with tunable center frequency over a wide frequency span.For cognitive radio applications in the TV-bands,relatively few and rather narrow spectral holes may exist between strong incum-bent TV transmitter signals[1].To reject the strong signals in order to avoid blocking of the receiver,such afilter should have very high linearity,high compression point but also very high Q(e.g.for10MHz bandwidth around500MHz). Although off-chip passivefilters provide these properties,in-tegrated CMOS alternatives are highly desired for reasons ofManuscript received September02,2010;revised January14,2011;accepted January21,2011.Date of publication April05,2011;date of current version April22,2011.This paper was approved by Guest Editor Yuhua Cheng.This work was supported by the Dutch Technology Foundation STW,Applied Sci-ence Division of NWO and the Technology Program of the Ministry of Eco-nomic Affairs(project08081).The authors are with the University of Twente,CTIT Institute,IC Design Group,7500AE Enschede,The Netherlands(e-mail:a.ghaffari@utwente.nl). Color versions of one or more ofthefigures in this paper are available online at .Digital Object Identifier10.1109/JSSC.2011.2117010Fig.1.Architecture of an N-pathfilter[5](p and q are the mixing functions and T is the period of the mixing frequency).size and cost.On-chip LCfilters can be implemented but with limited tuning range and low Q,especially below1GHz where the achievable Q of on-chip inductors is poor and coils take large area.Q-enhanced techniques[2]–[4]can improvefilter quality factor but degrade linearity and noise.Thus alternative tunable filters without inductors are highly wanted.Inductor-less tunablefilters based on periodically time variant networks have been addressed in literature under different names such as N-pathfilters,sampled datafilters, commutated capacitors,etc.[5]–[11].Discrete-time switched capacitor N-pathfilters are probably best known[8],but here we focus on their continuous-time predecessors.Fig.1shows a block diagram of an N-pathfilter composed of identical linear time-invariant(LTI)networks with impulse response and2frequency mixers(or modulators),driven by time/phase shifted versions of the clock and.The time shift between two successive paths is,where is the pe-riod of the mixer clock.If the LTI networks exhibit a low-pass characteristic around DC,the mixing results in a band-pass around the mixing frequency.Simply put,the input signal is down-converted to baseband,filtered by the LTI network and then up-converted again to the original band of.The center frequency is determined by the mixing frequency,insensitive tofilter component values.A high mixing frequency combined with a narrow low-passfilter bandwidth allows for a very high filter Q.While time-continuous N-pathfilters have been proposed for kHz operating frequencies in the1960s[6],they seem to have been largely forgotten until recently.CMOS technology now al-lows N-pathfilters to work at TV-band RF frequencies[9],[10] and even above1GHz[11].In[9]an8-path single-ended struc-ture is used,and in[10]we proposed a differential4-pathfilter0018-9200/$26.00©2011IEEEFig.2.(a)Switched-RC N-pathfilter.(b)Single port,single ended N-path filter.(c)Multiphase clocking.(d)Typical(in-band)input and output signal. combined with a broadband off-chip transformer.In[11]the differential4-pathfilters are applied in a quad-band SAW-less receiver.This paper aims to model and verify N-pathfilter ing linear periodically time-variant(LPTV)anal-ysis,exact expressions for the frequency response of differential N-pathfilters are derived.In[7]state-space analysis was used to derive the steady state and transient response for a single-ended N-pathfilter,which is however not directly applicable to our dif-ferential architecture.We will derive one set of equations that characterizesfiltering but also possible imperfections like har-monic folding,noise and the effects of the clock phase imbal-ance and mismatch.Moreover,an equivalent RLC tank circuit will be derived to approximate N-pathfilter behavior around the center frequency.Finally,we will verify the model via extended experimental results compared to[10].In Section II,we will derive the differential N-pathfilter architecture starting from Fig.1and then analyze its transfer function in Section III.Section IV presents basic characteristics of the differential N-pathfilter.Its chip implementation is discussed in Section V and Section VI covers the measurement and verification versus the model.II.N-P ATH F ILTERWe will now derive the differential N-pathfilter from Fig.1,where we aim for a high-linearity implementation using MOS-switches as passive mixers,and RC low-passfilters[see Fig.2(a)].Furthermore we will try to develop some intuitive understanding of thefilter behavior.A.Single Ended Switched RC N-Path FilterFig.2(c)shows a multi-phase clocking scheme for the switches with non-overlapping on-times.Thus no charge ex-change between capacitors can occur.For this reason and since a resistor is a memory-less element,it can be shared by all paths and shifted in front of them[Fig.2(b)].Moreover,if the clocks for thefirst and second set of switches are identical, thefirst set can also implement the function of the second set.becomes then available between the shared resistorand Fig.3.Single port differential4-pathfilter.switches.Fig.2(b)shows the resulting single-port single-ended N-pathfilter(is both input and output port).If we would use the capacitor voltages as outputs,the circuit behaves as a highly linear multiphase passive mixer[12],[13].To intuitively understand thefilter behavior of Fig.2(b)it is useful to model it as a two-step process:1)the input signal expe-riences down-conversion and low-passfiltering passing through the switches to the capacitors;2)the same switches up-convert thefiltered capacitor voltages to the output node.Another way to understand thefiltering is to realize that at any moment one and only one capacitor is connected to the output node.If we assume that the time constant,the output voltage will be the average of the input voltage over the time that the capacitor“looks at”.If the frequency of is equal to the switching frequency,a particular capacitor will periodically ob-serve the same part of the input waveform during every period. As each capacitor sees another part,the result is a staircase ap-proximation of,see Fig.2(d)(4-path example).In fact,the capacitors experience a steady DC voltage and,infirst order ap-proximation,conduct no current.If the input frequency deviates from the clock frequency the signal portion seen by a capacitor will“travel over the period”and the capacitors experience an AC voltage(with).Thus the capacitors conduct current while switches are on and the average voltages on the capacitors becomes closer to zero.Consequently signals at input frequen-cies below or above the switching frequency will be suppressed with an amount depending on the offset from the switching fre-quency,on-time of the switch and RC time.B.Differential Switched RC N-Path FilterIf we repeat the analysis for input signals around the har-monics of the clock frequency,we alsofind non-zero average values.Thisfits to the comb-like characteristic of N-pathfil-ters[5],i.e.,its repetitive selectivity around harmonics of the switching frequency.The differential architecture of Fig.3aims to cancel the even harmonic responses.Each path is differen-tial-in and differential-out,but contains one grounded capac-itor connected to two anti-phase driven switches.A4-phase 25%-duty-cycle clock provides all required clocks(see Fig.3). Now,for input signals around the even harmonics of the clockfrequency,no net charge is stored on the capacitors in steady state and no up-converted signal appears at the output.III.A NALYSISTo model the behavior of the N-pathfilter quantitatively,we will now apply LPTV state space analysis[14],[15]to derive the exact shape of the transfer function of a differential N-path filter.A.State Space Analysis of LPTV CircuitsFor an LPTV network which is periodic with the frequency ,the output spectrum is related to the input spectrum as[16](1) where represents the shifted version of the input spectrum to account for frequency translation(mixing),while describes the spectral shaping(filtering)properties of an LPTV network.To simplify analysis,we make two assumptions:1)the switches are ideal,i.e.,their off-impedance is infinite and on-impedance is zero;2)switching occurs instantaneously.The time interval is divided into portions(is the number of the states)and each portion identified by can be represented as,and(see Fig.4).During each interval there is no change in the state of the switches and the network turns to an LTI system.The state equations for interval can be written as(2) where is the input vector,the state vector andthe output vector.If we define and to be equal to the input and output respectively during the th interval, and zero otherwise,the state equations in(2)can be reformu-lated as[14](3) In(3)the Dirac’s delta function has been used to add the effects of the initial conditions to the equations at the beginning of an interval,while subtracting it at the interval’s end.Then the output is the sum of all responses from states in the system:(4) Since we are interested in the spectrum of the output of the system we need to take the Fourier transform from(4).It can be shown[15]that if we apply a complexexponentialas input,the output state at discrete moments can be Fig.4.Time intervals for the state space analysis.calculated by reforming the state equations to a set of difference equations at the switching moments.Consequently the relation between the input and the output will be of the form(5) The input spectrum can be represented as a summation of sinusoidal signals.As a result the infinite summation in(3)takes the form(6) By applying(6)to(3)and taking the Fourier transform,the spectrum of the state vector becomes(7) where is the length of the th time interval(see Fig.4)and is an identity matrix.B.Analysis of the Differential Single Port N-Path FilterWe will apply the analysis procedure described in the pre-vious part to derive the output spectrum of the differential N-pathfilter.Although Fig.3illustrates a4-path architecture, analysis is done for a general differential N-path system.At any moment two capacitors are connected to the differential output through two switches,which are activated with the same phase of the input clock.The resulting time-domain output signal is the superposition of the signals from different capacitorsFig.5.Differential single path circuit and the clock phases for the switches. at different moments without any overlap.Since there is no interaction between capacitor voltages,the analysis of the simple network illustrated in Fig.5suffices,where just one path is illustrated with its timing diagram for the switches.The state equations for this circuit are(8) where is the voltage on the capacitor in Fig.5.If one of the two switches is on,the output voltage will track the voltage on the capacitor.When switches are off,the voltage on the capac-itor will be held,but will not affect the output.Hence,the output spectrum contribution will be calculated in the track-mode.As the capacitor will deliver either(on)or( on),this contribution is.As a result,andin(7)should be defined from either thefirst or the third equation in(8),depending on which switch is on.As a result,according to(8)for and:,where is defined as the 3dB bandwidth of a single low-passfilter with resistor and capacitor(see Fig.5).Applying as the input in the state equations in(8)and also assuming that and,we canfind in(6)for as(9)Assuming the output voltage is following the voltage on ca-pacitor during interval,based on(8)the Fourier trans-form of the output can be found as(10) The output spectrum is the superposition of contributions for all paths,and taking into account the phase shifts between the contributions the complete output spectrum can be found as(11) where for each path(see Fig.5).According to(11)is composed of components generated by the paths.We will now derivefilter characteristics from(11).IV.C HARACTERISTICS OF A D IFFERENTIAL N-P ATH F ILTER A.Filtering and Harmonic Folding Back EffectsFor an ideal N-pathfilter we assume,where is the duty cycle of the multiphase clock.Analysis of(11)shows that is undefined for,but we can take the limit of when approaches con-tinuously to zero tofind.Moreover,in deriving(11)we assumed(non-overlapped switching).When ,there are periodic time intervals that all the switches are off and the output signal is tracking the input signal(instead of the voltages on the capacitors).The output spectrum contribu-tion generated due to this fact is not considered in(11).In order to include this effect in,the factor should beFig.6.Theoretical and simulated curve for at MHz for ,and pF.added to the part which is derived from(11)by taking the limit for to zero.Thefinal result becomes(12) Although(12)includes the clock duty cycle effectin the output spectrum,in an ideal N-pathfilter:.In(12)represents the desiredfiltering charac-teristic without any frequency translation.for a4-path filter with the values of,,pF and MHz is shown in Fig.6.In Fig.6the comparison between the theoretical transfer function from(12)and the sim-ulated results applying Spectre RF PSS-PAC is shown,whichfit completely on top of each other.Fig.6shows band-passfiltering around the switching frequency,for this case with1.8dB inser-tion loss.However,there are also response peaks around odd harmonics of the switching frequency,related to repetitive poles in the denominator of(12).Thanks to the differential architec-ture,peaking around even harmonics does not occur.From(11) we see that for whereand is zero for other values of.Thus thefilter not only shows a comb-filter characteristic,but also“folding back”from input frequencies around to the desired band around .The strongest terms which result in down-conversion in a 4-pathfilter are shown in Fig.7.For instance,renders non-zero,modeling folding from and to (both with frequency shift).For an8-path architecture the first folding back will happen from.In general,increasing the number of paths will increase the distance between and thefirst folded component around.Often,a low-pass pre-filter will be needed in front of the N-pathfilter to sufficiently suppress harmonic folding.For radio receiver applications,the pre-filter requirements depend on specific blocker scenarios.In fact the folded back power from blockers to the desired band and the amount of degradation in Fig.7.around the switching frequency for a4-pathfilter.This deter-mines the back from odd harmonics of the switching frequency MHz.Fig.8.N-pathfilter with switch resistance.the sensitivity of the receiver must be acceptable.Note that the implementation of a low-pass pre-filter is likely to be feasible, because it does not need to be tunable.In the proposed differ-ential N-pathfilter,second harmonic folding is cancelled which relaxes the pre-filter transition band.Moreover,increasing can relax the pre-filter transition band requirements.B.The Effect of the Switch ResistanceIn order to include the effect of the switch resistance in our analysis,we consider the model shown in Fig.8.Since in the architecture illustrated in Fig.3,at any moment just two of the switches are on,the model in Fig.8includes two switch resis-tances in front of an ideal N-pathfilter with zero switch resis-tances.Employing(11)with,the transfer function from to can be easily ing su-perposition of the and contributions wefind(13) If is set to zero in(13)it returns to the previous form of(11).Switch resistance can have strong impact on the max-imum achievable rejection in N-pathfilters.To understand this, consider the frequency transfer function including the switch re-sistance effect for:(14)Fig.9.Switch resistance effect on the maximum rejection of a4-pathfilter (,pF,MHz,).According to(14),close to the switching frequency the ef-fect of the switch resistance is not significant(term is close to1and dominates).But for frequencies further away from the switching frequency,where is close to zero,thefirst term often dominates and the output can be approximated as.Thus,the maximum filter rejection is limited by the switch resistance,as exemplified by Fig.9for the same4-pathfilter used for Fig.6but now with added.As a conclusion,in order to increase the maximum rejection of thefilter,the switch resistance should be very small with respect to the source resistance.C.Harmonic Selectivity in an N-Path FilterIn this section we will discuss the selectivity in the har-monics of the switching frequency both in an analytical and intuitive way.Aroundfilter response peaks thefilter becomes high-ohmic.To quantify the selectivity around the harmonics of the switching frequency,we approximate(12)for, for odd,resulting in(15) where and is the duty cycle of each clock phase.As an example,if we substitute and and,then,i.e.,1.8dB in-sertion loss in the pass-band,whichfits to Fig.6.According to (15)increasing N will reduce the insertion loss.As an example, for an8-path system the insertion loss becomes0.4dB.Equa-tion(12)was used to produce Fig.10(a),comparing a4-path and8-pathfilter with afixed total amount of capacitance.This figure suggests that thefilter attenuation around harmonics is reduced by increasing the number of paths.To understand this fact better,notice that the achievable rejection of thefilter at the odd harmonics of the switching frequency can be approxi-mated by(15).Assuming ideal clocking(substitutein(15))the suppression at the harmonics is solely defined by the number of paths.Fig.10(b)shows the calculated harmonic rejection applying(15).In most cases indeed harmonic suppres-sion decreases by increasing the number of paths,but there are a few exceptions[see Fig.10(b)].To get some intuitive insight how harmonic rejection and har-monic folding occurs,we added Fig.10(c).It showsthe time signals for a sine-wave with a frequency of applied to the 4-pathfilter of Fig.2(b).Each of the capacitors now Fig.10.(a)Harmonic selectivity comparing the4-pathfilter of Fig.6with an 8-pathfilter with the same total capacitance.(b)Odd harmonic selectivity for different number of paths and(eqn.(15)).(c)Waveforms intuitively illustrating3rd harmonic selectivity and3rd harmonic folding(down-mixing from to)for a4-pathfilter.“sees”the sine-wave for3/4of a period.The output after in-tegration is a square wave with frequency,which contains both the fundamental and the third harmonic,illustrating that both harmonic selectivity and harmonic folding occurs.Similar signals can be drawn for a larger number of paths.Doing so, we can observe that for more paths,i.e.,greater number of time slots in Fig.10(c),there is less time to cancel the positive parts of a signal with the negative parts during the integration taking place on each individual rger leads to less attenuation of harmonics at the output[Fig.10(b)],but less folding back since the staircase approximation in Fig.2(d)re-sembles the actual input signal better.D.Input Impedance of an N-Path FilterNow by applying(14)and(15)we can derive simple expres-sions for the input impedance of the N-pathfilter in Fig.8at the switching frequency,its odd harmonics and for frequencies far away from the peak points in Fig.9.It will be shown that these impedances are all resistive.The effect of the switch re-sistance can easily be taken into account by substituting(15)in (14)tofind an approximate equation for.Tofind the equivalent input impedance of the N-path systemwe define.As a result,.For a 4-path system(,)wefind(16) Equation(16)predicts that at the switching frequency the input impedance of the N-path system is resistive,which is sim-ilar to a tank circuit at the resonance frequency.For,(16) corresponds to the energy conservation based derivation in[12], but(16)can also be used for other values of.For frequencies far away from the switching frequency and its odd harmonics, using(14)we againfind purely resistive input impedance:(17)Fig.11.(a)Effects of increasing and decreasing of the clock duty cycle.(b)Insertion loss changes due to the duty cycle reduction.For an ideal system(17)reduces to.In-tuitively this can be understood because the capacitors act like a short circuit for.We can conclude that for the switching frequency,the N-path filter has high impedance(16)and for frequencies far away from the switching frequency,it renders small impedance(17).Later, in Section IV-F,we will apply derivations in(15)–(17)to derive a simple RLC model for the N-pathfilter.In Section V,(16)also will be applied to define the required conditions for input power matching.E.The Effect of the Duty Cycle of the ClockWhen the duty cycle of the multiphase clock is smaller than,all switches are off periodically for some time and the output signal of the N-pathfilter simply tracks the(unloaded) input signal.Applying(12)for a4-path architecture,thefilter shape for a duty cycle is illustrated in Fig.11(a). Comparing to the ideal case,notice from(17) that reducing the duty cycle from results in higher input impedance for the frequencies far away from the switching fre-quency and its odd harmonics,which translates to less rejection. On the other hand the reduced duty cycle decreases pass-band insertion loss,as predicted by(12)and also(15)[see Fig.11(b)]. Note that the degradation of the maximum rejection is much more than the reduction in the pass-band loss.As mentioned in Section III,our analysis does not include the case for which two switches can be on at the same time,resulting in undesired charge sharing between capacitors. Fig.11(a)also shows simulation results for two overlapping clock cases:1)and,resulting in com-plete destruction of thefilter shape;2),, where the switch resistance still limits the charge sharing be-tween capacitors and somefiltering remains,but with large in-sertion loss.F.RLC Model,Bandwidth and Quality FactorAround,thefilter transfer function as shown in Fig.6resembles that of a high-Q tank circuit.Now we want to quantify this similarity andfind an equivalent RLC model that predicts the quality factor and bandwidth for the N-path filter.Although(11)encompasses a repetitive pattern of poles and zeros,we are mostly interested in poles which occur close to theswitching frequency.Equating the denominator ofin(12)to zero to find the poles,wefind,Fig.12.Equivalent RLC circuit model for the N-pathfilter.,indeed odd harmonics of.In order to make a narrowband approximation,we just consider the poles close to the switching frequency,which areand set these poles equal to the poles of the transfer function of Fig.12,which is shown in(18):(18) As a result,and in the RLC model can be found as(19a)(19b) The value of in the RLC model is already derived in(16) for a4-path system with for the switching frequency. Note that and are independent of,in contrast to. However,the term in the denominator of(19b)can be non-negligible compared with.Thus,the maximum in the transfer function can be slightly shifted to higher frequencies, which alsofits with the N-pathfilter response.For this shift is negligible.To verify the validity of the RLC model,con-sider a4-pathfilter with and pF.Employing (16)and(19),wefind pF,nH,and.The comparison between the RLC model and the exact transfer function in Fig.13shows a nearly perfect match around the switching frequency.Applying the RLC model,we canfind the bandwidth as. Intuitively this can be understood considering that in Fig.5the resistor value charging the capacitor is and the capacitor sees this resistor value for a fraction of the period,i.e.,the “effective resistor”is.Therefore,the3dB bandwidth becomes times the low-passfilter bandwidth defined by. Finally,for a4-path system and.Thus,we see that the bandwidth not only depends on theparison of the transfer of the RLC model with the full N-pathfilter model(11).RC value for each path,but also on the duty cycle of the clock. Comparing a4-path and an8-pathfilterusing identical capacitors in each path,the8-pathfilter has half the bandwidth of the4-pathfilter.If we keep the total amount of capacitance equal,then the bandwidth will not differ by in-creasing the number of paths[see Fig.10(a)].In this paper,we exploit a1:2transformer to increase the source resistance,as seen by thefilter,and hence reduce the bandwidth and increase the Q of thefilter.A further increase in capacitor area can pro-vide even lower bandwidth and higher Q.Notice that as we discussed intuitively in Section II,the RC in each path of an N-pathfilter is performing integration on the input signal,when the corresponding switch is on.In contrast to a sampling system,there is no need for the voltage to settle during one on-time of the switch.Thus,the choice of the capac-itor size is independent of the switching frequency and can be solely determined based on the desired bandwidth.G.Imbalance Multiphase Clocking and Mismatch in the Paths Next,consider what happens if there is mismatch between paths or if clocking signals deviate from the ideal situation. With mismatch,we can expect that even-order terms are no longer perfectly cancelled and extra frequency components show up.As an example,,which was0for an ideal N-path system,can be non-zero around the desired band. According to(11),this renders an image response(e.g.,con-version from to)and also folding back from signals around to.As we use large valued integrated capacitors(10s of pF),good matching is possible and therefore we focus on quantifying clock phase errors.We model clock pulse width variations in the multiphase clocks and apply(11),considering unequal pulse widths.Fig.14shows the calculated image suppression and folding back from.According to Fig.14,one degree of phase error will result in42dB of image rejection and45dB suppression of second harmonic folding.H.NoiseMost output noise power is due to the thermal noise of the source and switch resistances,where noise folding from around harmonics of should be incorporated.As the switches do not carry DC current,theflicker noise of the switches can be neglected.Atany moment two switches are in the on-state(see Fig.3)and since noise contributions ofthe switches are not Fig.14.Image and second harmonic rejection with phase error in the driving clock phases.Fig.15.The model for noise calculation.(a)Source noise.(b)Switch resis-tance noise.correlated,we can use the model in Fig.15,similar to Fig.8.As the input impedance around according to(16)is significantly larger than,the noise voltage source has a much higher transfer to than has,which is beneficial for the noisefigure.Relation(13)between the input and output spectrum of an LPTV system can also be applied to random signals.For Fig.15(a),we canfind,the thermal noise due to:(20) Thefirst term accounts for the noise power,which appears at the output without any frequency translation,and the second part accounts for noise folding,where is the frequency-shifted version of the noise power generated by. Note that in(20)can be calculated by applyingin(11).To calculate the output noise power due to switch resis-tance,we consider Fig.15(b).Similar to the previous case, the frequency transfer from to can be calculated from(11)with.Then,by applying the procedure described in Section IV-B,the transfer function from to can be derived.Finally,for the circuit in Fig.15(b),wefind(21)。