Matrix Methods for the Dynamic Range Optimization of Continuous-TimeGm-CFilters

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008 2525 Matrix Methods for the Dynamic Range Optimization of Continuous-Time Gm - C Filters Juan F. Fernández-Bootello, Manuel Delgado-Restituto, Member, IEEE, and Angel Rodríguez-Vázquez, Fellow, IEEE Abstract—This paper presents a synthesis procedure for the optimization of the dynamic range of continuous-time fully differential - filters. Such procedure builds up on a general extended state-space system representation which provides simple matrix algebra mechanisms to evaluate the noise and distortion performances of filters, as well as, the effect of amplitude and impedance scaling operations. Using these methods, an analytical technique for the dynamic range optimization of weakly nonlinear - filters under power dissipation constraints is presented. The procedure is first explained for general filter structures and then illustrated with a simple biquadratic section. Index Terms— - filters, filter synthesis, dynamic range optimization, noise analysis, distortion analysis, low power. I. INTRODUCTION FULLY DIFFERENTIAL - circuits and techniques are widely employed to design integrated continuous-time filters. Over the years significant contributions have been made regarding the proposal of transconductor topologies with enhanced noise and distortion performance. Based on such transconductors, filters with enlarged dynamic range (DR)1can be built for different practical applications. However, despite the availability of high-performance transconductors, dynamic range optimization may be hampered due to inaccurate evaluation of the impact of noise and distortion on the overall filter performance. Regarding the impact of noise, one of the most significant early contributions was due to Groenewold [1]. He employed Manuscript received October 22, 2007; revised January 11, 2008. First published April 18, 2008; current version published October 29, 2008. This work was supported by the Spanish Ministry of Education & Science under Grant TEC2006-03022, and the Junta de Andalucía under Grant TIC-02818. This paper was recommended by Associate Editor T. B. Tarim. J. F. Fernández-Bootello and A. Rodríguez-Vázquez are with AnaFocus, Sevilla 41092, Spain. M. Delgado-Restituto is with the Instituto de MicroelectrÓnica de Sevilla, Centro Nacional de Microelectrónica, Consejo Superior de Investigaciones Científicas (CSIC), and Universidad de Sevilla, 41012 Sevilla, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSI.2008.921048 1DR is defined as the ratio between the maximum and minimum signal that can be processed by the filter. The latter is limited by the total integrated outputreferred noise power of the filter, U , and the former is limited by the maximum output distortion level which can be tolerated. Assuming a single-tone excitation and letting be the maximum signal level at the output of the filter, the dynamic range can be expressed as DR =(1 = 2) 1 (  =U )  P =U state-space descriptions to evaluate noise performance through simple matrix manipulations. Similar approaches to noise evaluation have been used in [2], [3]. In this paper, we adopt these noise evaluation techniques. Also, we adopt the general - structure proposed in [3]. Regarding the impact of distortion, different techniques have been reported; for instance, those presented in [4]–[9]. Proposals in [4], [5] are based on frequency-domain calculation using Volterra series which, although powerful, tend to be very cumbersome as the filter order increases. In [6], [7] nonlinearities of individual transconductors are propagated by means of partial transfer functions to the output, where the contributions are summed to estimate the overall distortion behavior of the filter. Finally, [8], [9] use time-domain analysis and state-space modelling for the evaluation of harmonic and intermodulation distortion in filters with no floating capacitors. None of these approaches provides the simple, general and systematic matrix manipulation techniques which are available for noise. Consequently, no systematic method is yet available to evaluate the impact of both noise and distortion on a general - filter structure, thereby limiting the ability of designers to maximize the dynamic range of practical filter implementations. This paper presents methods and techniques to evaluate the impact of both noise and distortion on the - filter structure of [3] through simple and systematic matrix algebra. For distortion evaluation we combine state-space and Volterra Series representations [10] in such a way that they can be easily embedded in matrix form. The paper also addresses the issue of scaling. The optimization of DR through modifications of the amplitude and impedance levels at the internal filter nodes is covered for both biquadratic sections and general - filters. In the case of biquadratic sections, the paper reports compact DR expressions which provide more accurate estimations of the influence of the quality factor, , than those in [1]. For general filters, the techniques presented in this paper overcome the lack of univocal solutions observed in the approach presented in [6]. The paper is structured as follows. Section II describes the state-space representation of general - filters, and provides relationships among the state-space matrix and a set of internal transfer functions which are essential for the foregoing analysis. Sections III and IV present the systematic techniques for the evaluation of noise and distortion, respectively. Section V deals with the scaling of - filters. Section VI addresses the dynamic range optimization of generic - filters and the procedure is illustrated in Section VII for biquad structures. Finally, Section VIII concludes the paper. 1549-8328/$25.00 © 2008 IEEE 2526 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008 Fig. 1. Simplified schematic of a fully balanced G - C filter. II. SPACE-STATE REPRESENTATION OF - FILTERS Fig. 1 shows the conceptual schematic of a generic continuous-time fully balanced - filter with integration nodes. We assume that each integration node can be connected to the input and to all the remaining integration nodes; we further assume that these connections can be realized with either capacitors or transconductors. It is illustrated in Fig. 1 for a generic integration node . Capacitive connections to other integration nodes are realized with capacitors ; transconductance connections to other integration nodes are realized with transconductors ; connections to the input are realized with capacitors and transconductors , respectively. This general diagram contemplates also the general case where the filter output is obtained as the linear combination of the input and an arbitrary number of internal node voltages.Using an extended state-space notation, the filter is described as [3] (1) which is actually a generalization of the representation given in [1]. In the expression above: •and are, respectively, the input and output voltages of the filter and is a state vector which gathers all the integration node voltages. Operator denotes matrix transpose. •is an matrix whose element represents the transconductance of the transconductor connected from node to node . •is an vector given by (2) where and represent, respectively, the transconductance and capacitance between the filter input node and the integration node ; is formed by transconductances and is composed by capacitances. •in (1) is a vector whose element denotes the voltage amplification between node and the output; realized by a transconductor with input and transconductance , loaded with a resistor of resistance implemented by a feedback transconductor [see Fig. 1(a)]. Parameter is the voltage gain of the forward path from the input to the output of the filter; realized by a transconductor with gain loaded by the same feedback transconductor as before.2 • Finally, is an matrix composed by capacitances with the following structure: (3) where the negative and positive components of the integration nodes are chosen such that all the out-of-diagonal entries are negative. In conventional - structures, the anti-diagonal entries of , namely and , coincide as they represent the same capacitor. However, there are practical topologies which lead to non-symmetrical matrices. This happens, for instance, in the so-called - opamp structures where the integrating capacitor is connected in feedback configuration around one opamp [11]. From the representation in (1) the input–output transfer function of the filter can be calculated as (4) In addition to this overall transfer function, other functions need to be defined. On the one hand, let be the transfer function from to the integration node . It can be shown that (5) where (6) 2For the sake of generality the mathematical formulation in the paper considers the general representation above. In many practical situations, there is no forward path from the input so that d =0 . Also, in many cases the output of the filter is taken from a single internal node so that only one entry of vector C is non-zero (i.e., C =[0 ; ... ;c ; ... ; 0] ). Moreover, if no output voltage amplification is required, then vector C is unitary (the only non-zero entry has unity value) and the output summing network can be suppressed. FERNÁNDEZ-BOOTELLO et al.: MATRIX METHODS FOR THE DYNAMIC RANGE OPTIMIZATION OF CONTINUOUS-TIME -FILTERS 2527 On the other hand, let be the transfer function from a current source connected to integration node (between the output terminals of transconductors with gain for )to the filter output for . If all the transfer functions , are collected into a vector (7) it is easy to show that (8) As will be shown afterwards, vectors and significantly simplify the evaluation of filter distortion. Also, they are useful to calculate the relative sensitivities of the filter transfer function with respect to the matrices in the representation (1). They can be easily obtained using the Hadamard product as3 (9) where the sensitivities are defined as . Up to now, it has been assumed that transconductances remain constant with frequency. In a more general case, the representation in (1) must be extended to also cope with frequency-dependent transconductances [7]. This can be done at the price of a more complex state-space matrix description. As an example, let us assume that transconductance exhibits a one-pole roll-off with time constant . This can be modelled by replacing transconductor by the network at the right of Fig. 2(a), where and the low-frequency transconductance is given by . Note that this model adds a new node to the topology (labelled ) per transconductor. Hence, matrices of the extended state-space representation must be rebuilt. This is indicated in Fig. 2(b) which shows the rows and columns that must be added to account for the new filter node, . For consistency, the low frequency behavior must be made equal to the original frequency-independent entry in (1). III. NOISE IN - FILTERS We assume noise is only contributed by transconductors and that frequency is high enough so that noise behavior is dominated by thermal contributions. Thermal noise contributions from different transconductors are assumed un-correlated (this is supported by the fact that transconductors are different physical entities), and are modelled by output current noise sources with double-sided power spectral density (PSD) where is the transconductance, is Boltzmann’s constant, 3The Hadamard product of two m 2 n matrices A and B , denoted by A  B , is an m 2 n matrix given by ( A  B ) = a b [4]. Fig. 2. (a) Circuit replacement for adding one-pole roll-off transconductance characteristics to the generic schematic of Fig. 1(a). (b) Required modifications on the state-space matrices. is the absolute temperature and is the noise excess factor of the transconductor—the value of this latter parameter depends on the actual transconductor implementation [12]. Using the transfer functions defined in the previous section [see (7) and (8)], the total output-referred noise voltage PSD of the filter can be expressed as (10) Here, the first term accounts for the noise contributions of all transconductors in the filter core. The second term corresponds to the output summing structure. From now on we will assume that this second term is either negligible (which occurs for large enough values of as it happens in practice), or null (which corresponds to the case where the filter output is simply taken from a single internal node). With this assumption, the total output noise of the filter is approximately given by (11) which represents an upper-limit value. In order to evaluate the integral in (11) it is worth noting that matrix , defined as [1], [13]4 w(12) can be algebraically obtained from the following generalized Lyapunov equation [14]: (13) 4 W is related to the observability grammian of the system, W ,as W = EWE [15]. 2528 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008 Fig. 3. Conceptual schematic at the i th integration node of the filter including nonlinearities. and, therefore, the total noise of the filter can be written as (14) where w, represent the elements at the diagonal of matrix is the sum of all the transconductances driving node . In the right-hand side of this equation, represents the noise contributed to the output from the th integration node. IV. DISTORTION IN - FILTERS We focus here on the impact of nonlinearities on the input–output voltage-to-current transformation. Assuming weakly nonlinear operation conditions and that transconductors are fully balanced; their input–output characteristic can be approximated by (15) where is the third-order nonlinearity coefficient and is the input voltage. This simplified model, where even-order nonlinear coefficients are null because of the balanced structure, suffices for most practical transconductors [11], [12]. Let us first consider that the output summing structure of Fig. 1(a) does not generate distortion; the influence of this structure will be computed in later on. Using (15), the extended state-space representation in (1) becomes (16) where is the Hadamard cube of x(it is obtained by three consecutive Hadamard products—see footnote 3). For illustration purposes, Fig. 3 shows a conceptual schematic (single-ended for simplicity of the drawing) which displays the components which contribute to the th integration node according to (16). In this figure, the linear and nonlinear components are clearly separated. Several approaches are found in literature for the analysis of the nonlinear behavior described by (16), [4]–[9]. Here we use Volterra’s series expansions [10]. This method consists in decomposing the internal nodes variables in operators, according to [16] (17) where is the input signal of the system, is an arbitrary amplitude scaling factor (18) is referred to as the th order Volterra operator and is the th Volterra kernel [10]. The Laplace transform of this multidimensional kernel is defined as (19) where is the -dimensional Laplace variable. Function describes in frequency-domain the th order distortion performance of the system. Hence, describes the linear behavior of the system, accounts for the third-order nonlinear behavior, and so on. The relevant feature of Volterra’s series expansions approach is that, for weakly nonlinear systems and low values of , series (17) rapidly converges and it can be approximated by the first few terms. Therefore, if the distortion behavior of a fully balanced - filter is dominated by the third-order nonlinearities of the transconductors, can be simply approximated by . Replacing this expression in (16) and grouping terms with the same power of , the following two linear systems in and are obtained (20) and (21) where . The first equations of these two systems can be mapped into the firstand third-order circuits shown in Fig. 4(a) and (b), respectively. Solving both circuits, the firstand third-order transformed kernels of the filter are respectively given by (22) and (23) From these expressions, the third-order harmonic distortion of the filter and its intermodulation performance can be estimated, with no transient analysis needed, by [16] (24) where is the amplitude of the input tones applied to the system. FERNÁNDEZ-BOOTELLO et al.: MATRIX METHODS FOR THE DYNAMIC RANGE OPTIMIZATION OF CONTINUOUS-TIME -FILTERS 2529 Fig. 4. (a) First-order and (b) third-order circuits at node i . Equation (24) can be related to vectors and , defined in (6) and (7), respectively, as shown in (25) at the bottom of the page. It reveals that the distortion evaluation of a - filter can be easily accomplished by simple matrix algebra. These closed-form expressions compare favourably to others in the literature, demanding a much more cumbersome formulation [4], [5]. Let us now consider the case where the outputsumming structure of Fig. 1(a) also contributes distortion. In this case, the conceptual schematic of the filter output takes the form in Fig. 5, similar to that in Fig. 3 for the filter core. As above, the analysis of this circuit encompasses decomposition into a firstand a third-order schematics (shown, respectively, in Fig. 6(a) and (b)) which are solved one after the other to give (26) Fig. 5. Conceptual schematic of the output structure including nonlinearities. Fig. 6. First-order (a) and (b) third-order (b) circuits to evaluate the distortion of the output stage. where and are obtained from (22) and (23). In these represents the linear part of the filter response, and the third-order nonlinear contribution. After some algebra, the third-order harmonic distortion of the filter and its intermodulation performance take the form in (27), shown at the bottom of the page, where terms and , defined in (25), are due to the core of the filter. As an illustration of the proposed distortion evaluation method, Fig. 7 shows in solid lines the calculated third-order intermodulation and harmonic distortion components of a fully balanced seventh-order low-pass Chebyshev filter with 10 MHz cut-off frequency. The filter uses a standard leap-frog structure (25) (27) 2530 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008 Fig. 7. Estimations of IM (2 ! 0 ! ) and HD ( ! ) . using transient analysis and the proposed method. and it is assumed that all transconductors exhibit a third-order nonlinear term . Calculations are made assuming that the filter is driven by input tones of 125 mV at different frequencies. The third-order intermodulation estimation assumes a 100 kHz offset frequency between the input tones. For comparison purposes, Fig. 7 includes also entries for the third-order intermodulation (asterisks) and harmonic distortion (circles) calculated at different input frequencies by means of conventional Fourier analysis of a transient response. As can be seen, both methods give approximately the same results, however, whereas the computational cost of the transient approach is 135 s cpu time, that of the proposed method is of only 0.6 s (data using a 1.8 GHz Pentium Mobile processor), i.e., more than two orders of magnitude lower. V. SCALING OF - FILTERS This section presents analytic expressions to account for the impact of scaling on filter noise and distortion. This is worth doing because scaling is customarily employed for filter design [6], [12], [17], [18]. We consider only scaling operations which retain the frequency response, , required by the application. This excludes frequency scaling operations which are easily implemented by multiplying all filter capacitances by the same factor [this makes and, thereafter, ] or by scaling all filter transconductances by [this makes and, thereafter, ]. In the foregoing analysis, filters are grouped into two categories. On the one hand, filters with all capacitors grounded excepting, perhaps, those connecting the integration nodes with the filter input (they are characterized by a diagonal matrix ). On the other hand, filters which include floating capacitors between integration nodes. Unless otherwise stated, if denotes a given variable in the prototype system, represents the corresponding transformed variable. A. Filters Without Floating Capacitors Table I summarizes results for the three types of scaling considered herein. The second column shows the basic and derived matrix equations for each type of scaling. The third column includes comments regarding the effects of corresponding transformation on the noise and distortion performance of the filter. A first scaling approach consists of multiplying each row of the state equation in (1) by a corresponding positive number, . This transformation, denoted as noise scaling in Table I, modifies the local impedance at each node of the filter without altering their voltage swings. Hence, it does not affect the distortion behavior of the filter-interesting property that will be exploited later on. The noise contributed to the output from the th integration node in the scaled filter becomes , whereas the sum of all transconductances driving the th node of the filter scales as . It means that to reduce the noise contribution at node by a factor , the total transconductance must be increased by the same factor. This increases the area occupation of the filter as well, because capacitances are also scaled by . For a given value of the total transconductance, , there is an optimum set of scaling coefficients , which minimizes the total noise contributed by the filter. After some calculations (detailed in Appendix I) it is found that such optimum set is obtained when all the diagonal elements of the transformed matrix are identical, i.e., w w w (28) which gives w w (29) In this case, the total noise of the filter can be expressed as (30) A special case of noise scaling is power scaling in which all the multiplying factors take on the same value and, hence, all capacitors and transconductors of the filter core are scaled by . In this case, the total output noise value of the filter is transformed according to , without affecting the distortion behavior. This fact will be used in Section VI to relate the total noise of the filter with its power consumption. Consider now that scaling is made by multiplying column entries instead of row entries. This transformation is labelled distortion scaling in Table I where scaling factors are called . It affects the amplitude level of the voltages at the internal nodes of the filter and, therefore, modifies its distortion behavior. Actually, distortion improves for scaling factors . However, this operation affects the noise behavior and defines a trade-off between noise and distortion. It is worth noting that matrix remains unaltered after distortion scaling and so w w . This fact will be exploited in the next section. FERNÁNDEZ-BOOTELLO et al.: MATRIX METHODS FOR THE DYNAMIC RANGE OPTIMIZATION OF CONTINUOUS-TIME -FILTERS 2531 TABLE I BASIC SCALING TRANSFORMATIONS A last option for filter scaling consists of applying a similarity transformation on the state-space representation in (1). A typical case, illustrated in Table I, corresponds to a diagonal transforming matrix with coefficients .Asin the previous case, this mapping carries on both noise and distortion modifications on the prototype filter, and a trade-off can be established as well: scaling factors decrease the noise contribution but worsen distortion. This type of transformation is typically used to equalize and maximize the peak voltages at the input of the transconductors in the filter—this is done by means of a diagonal matrix with coefficients , where is the maximum input signal range of transconductors and are the peak values of each defined in (5) (they usually occur near the passband of the filter) [17]. B. Filters With Floating Capacitors The presence of floating capacitors between the integration nodes makes the matrix non-diagonal. As a consequence, previous scaling operations cannot be applied in stand-alone manner. Otherwise the symmetry of [see (3)] would be lost, the feedback and forward paths of the floating connections would be different and, consequently, they would be unrealizable with simple capacitors. To overcome this situation, scaling operations should be pair-wise applied so that symmetry of is always restored after scaling.5 As an example, let us assume that a prototype filter with floating capacitors is scaled by a diagonal similarity transformation . The non-diagonal components of matrix change as , thus, breaking the symmetry for . In order to restore this property and, hence, allow the use of floating capacitors as in the prototype filter, one possibility is to noise scale the filter (see Table I), in such a way that the relationship is met for , what guarantees that . A similar procedure can be envi5Rigorously speaking, scaling could be performed in a single step by means of a similarity transformation with an orthogonal T matrix ( T = T ) . However, from a synthesis perspective, it is more simple and intuitive to use two consecutive scaling operations. 2532 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 55, NO. 9, OCTOBER 2008 sioned if, instead of noise scaling, a distortion scaling is applied for restoring symmetry. VI. DYNAMIC RANGE OPTIMIZATION OF GENERAL - FILTERS We have seen that for a given total transconductance the noise of a - filter can be minimized by making all diagonal terms of matrix identical. We have also seen that, by applying distortion scaling, the harmonic distortion and intermodulation performance can be improved while keeping the terms unaltered. Hence, an optimum noise-scaled filter will retain this feature after distortion scaling. This observation is at the core of a recently proposed procedure for dynamic range optimization of weakly nonlinear - filters under power dissipation constraints [6]. It consists of the consecutive application of noise and distortion optimizations on the prototype filter. However, this procedure presents two limitations. First, the noise and distortion contributions, as well as power consumption, of transconductors not accounted in matrix are neglected. Indeed, transconductors assessed in matrix are replaced by ideal current sources. This simplification precludes obtaining univocal solutions from the optimization procedure, as will be illustrated in the next section by means of an example. Second, distortion optimization is constrained by the condition that the noise generated at the filter core keeps unaltered. This condition reduces the design space of filter parameters and, consequently, may preclude that a global optimum is reached. These drawbacks are overcome by the procedure presented in this section. Without loss of generality and to keep mathematics as simple as possible, let us assume that all transconductors in the filter share the same topology and linear range, and exhibit the same current efficiency, ; where this latter parameter is defined as the ratio between the transconductance and the biasing current of the cell [18], [19]. In this case, the total power consumption of the filter, , is proportional to the sum of all its transconductances, ,as , where is the power supply voltage of the filter [18]. A practical way of fulfilling this assumption is by implementing all transconductance values through the parallel connection of unitary transconductors.6 Let us assume that a generic fully balanced - filter with no output network (see Fig. 1(a)) has been, first, optimally noise scaled and, then, distortion and power scaled. In this case, taking into account Table I and applying a power scaling factor , the total noise of the filter for a total power consumption becomes (31) 6This is, indeed, a common practice among integrated circuit designers. By using multiple instances of a given transconductor, the design complexity is notably reduced and the robustness of the filter against variations of the technological process is improved. Using this strategy in combination with proper layout techniques, matching between transconductors is largely favoured and the tunability of the filter, simplified [20], [21]. where and are the coefficients of the matrices and after optimum noise scaling, and is the sum of all the transconductances of the filter after distortion scaling. Useful for the foregoing analysis, can be also expressed as (32) where (respectively, ) is the sum of the transconductances of all the transconductors with input at the th node (respectively, filter input) of the optimum noise-scaled filter.7 Let us further assume, without loss of generality, that the distortion performance of the filter is evaluated by the third-order intermodulation. From Table I and assuming that input tones are close together , at can be expressed as (33) where , see definition in (7), is obtained after optimum noise scaling. Equation (33) can be also written as (34) where coefficients and , both independent of , are defined as (35) The maximum power at the output of the filter, , for a peak intermodulation distortion value, , can be obtained from (34) as (36) from where the dynamic range of the filter (see footnote 1) can be calculated, using (31), as (37) This expression can be recast as (38) where is an adimensional number, which depends on the particular transconductor implementation used in the filter, and depends on the filter structure. Parameter gives a measure on how large the dynamic range of a filter can be for a given power dissipation 7In this section and the following, if x denotes a variable or coefficient of the original prototype, ~ x refers to the corresponding variable or coefficient of the optimally noise scaled filter. FERNÁNDEZ-BOOTELLO et al.: MATRIX METHODS FOR THE DYNAMIC RANGE OPTIMIZATION OF CONTINUOUS-TIME -FILTERS 2533 and distortion performance. Hence, it can be used as a figure of merit for comparing transconductor implementations. Equation (38) also shows that for a given power consumption , maximum tolerated distortion and selected transconductor topology , the optimization of the filter dynamic range implies maximizing .As remains unaltered after scaling, the only scaling-dependent factor in is , which we define as a cost function in the dynamic range optimization procedure (39) Clearly, to optimize the dynamic range of a filter, must be minimized. The minimum of is calculated by setting , which after some algebra obtains the set of equations (40) where is the conjugate of and extracts the real part of the argument. Summing the left-hand terms of the above equations, on the one hand, and the right-hand terms, on the other, the following two relationships are obtained: (41) from where, equating the results and using (40), the optimum distortion-scaling coefficients can be calculated by recursively solving the expression (42) using the definition of in (34). With these values, the absolute minimum for becomes (43) Note that if input transconductors are not accounted for in the above analysis, which is the situation considered in [6], coefficients and are null and (43) becomes indeterminate.8 8By removing the effects of input transconductors on distortion and noise, they are assumed to perform as perfect voltage-to-current converters. In essence, these ideal converters play the same role as the input current-controlled current sources used in [6], i.e., to inject signal into a current-input filter. Assuming that B has no capacitive components, i.e., B is a null vector, the driving signal of the current-input filter takes in our model the form B v . In the case of [6], such driving signal is simply given by Bi . Obviously, the dimensionality of matrix B in both procedures is different but, from the point of view of optimizing the dynamic range of the current-input filter, defined by matrix A and E , this is irrelevant. Fig. 8. Fully differential G - C biquadratic section. This reveals that there is an infinite number of solutions that achieve the same dynamic range for a given power dissipation, and not a single solution as stated in [6]. It is to be understood that by including the effects of input transconductors on distortion and noise, the optimization algorithm finds the necessary constraint to achieve a single and univocal solution. It is also worth mentioning that the dynamic range optimization may result in a non-unity distortion-scaling coefficient for the output node of the filter, i.e., if the output is taken from node , coefficient will more likely be . This implies that vector becomes non-unitary, however, there is no need to add an amplification output stage to the filter (see Fig. 1) to guarantee an optimum dynamic range. Instead, the output of the filter could be directly taken from node . Note that any potential output network will ideally scale the output noise and desired signal power by the same factor while retaining the distortion performance of the filter core. Therefore, the dynamic range will remain unaltered after amplification. Finally, note that the optimization process described above is restricted to a single operating frequency. Therefore, such frequency must be carefully chosen so that it corresponds to the worst-case dynamic range of the filter. This can be done by a previous analysis on the noise and distortion dependence with frequency through (10) and (25). VII. CASE STUDY:BIQUADRATIC SECTIONS As a case of study, the procedure in the previous section is herein applied to the biquadratic section of Fig. 8. In the following analysis, transconductor-dependent parameters have been derived from a simple folded-cascode topology, designed in a 0.13 m CMOS technology at a power supply of 3.3 V. They are and giving . Using these transconductors, a low-pass filter with cut-off frequency at MHz has been designed. The power consumption of this filter is mW. Using the extended state-space representation of Section II, the biquad in Fig. 8 can be described by the matrices (44)