# A Platform-Based Methodology for System-Level Mixed-Signal Design

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## Abstract

The complexity of today's embedded electronic systems as well as their demanding performance and reliability requirements are such that their design can no longer be tackled with ad hoc techniques while still meeting tight time to-market constraints. In this paper, we present a system level design approach for electronic circuits, utilizing the platform-based design (PBD) paradigm as the natural framework for mixed-domain design formalization. In PBD, a meet-in-the-middle approach allows systematic exploration of the design space through a series of top-down mapping of system constraints onto component feasibility models in a *platform library*, which is based on bottom-up characterizations. In this framework, new designs can be assembled from the precharacterized library components, giving the highest priority to design reuse, correct assembly, and efficient design flow from specifications to implementation. We apply concepts from design centering to enforce robustness to modeling errors as well as process, voltage, and temperature variations, which are currently plaguing embedded system design in deep-submicron technologies. The effectiveness of our methodology is finally shown on the design of a pipeline A/D converter and two receiver front-ends for UMTS and UWB communications.

## Keywords

Robust Optimization Design Reuse Operational Transconductance Amplifier Robust Optimization Problem Architectural Constraint## 1. Introduction

Modern electronic systems are becoming increasingly complex and heterogeneous. Telecommunication and multimedia applications require highly integrated, high-performance systems, where analog, RF, and digital components must be efficiently packaged into a single chip. Emerging sensor and actuator swarm applications, as well, demand customized mixed-domain systems to be embedded into a myriad of extreme physical environments to provide a variety of personal or broad-use services. On the other side, manufacturing technology is evolving deeper into the nanometer era, where leakage power, increasing process variations, reducing supply voltage, and worsening signal integrity conditions make it daunting even to assess the required performance specifications. To build future integrated systems, designers need to face several challenges, at all levels of abstraction, from system conception to physical implementation. Design complexity is indeed rising while, at the same time, time-to-market constraints are becoming tighter, and dependable systems need to be built out of increasingly unreliable components. Addressing the above challenges requires innovative solutions not only in manufacturing technologies and circuit architectures, but also in design methodologies and tools.

A disciplined design style that reduces iterations in the flow should be based on a rigorous formalism leveraging accurate and robust *performance modeling* techniques to guarantee that performance variables of each component are correctly propagated across the design hierarchy. Moreover, *fast, global optimization* techniques need to be deployed to provide the best design options, for a given application, within a well-constrained and characterized search space. Finally, a practical framework should promote design reuse, and the *separation of design concerns* to reduce system complexity and boost designers' productivity.

In this paper, we present a system-level design methodology for mixed-signal electronic circuits, which is inspired by the above principles, and leverages the platform-based design paradigm (PBD) [1, 2] as the natural formalization framework. In PBD, a platform is expressed as a collection of components and composition rules. A design is obtained by composing components of the platform in a platform instance. The refinement process consists of mapping a functional description into a set of interconnected components. The design space is systematically explored through a meet-in-the-middle approach in which top-down design constraints of the system are mapped onto bottom-up performance characterizations of the components in the platform library. Based on this paradigm, we provide a unified framework to assist designers at all levels of abstractions. At the system level, a global optimization technique provides the best design options by leveraging tradeoffs among all components, rather than composing systems using locally optimized components. At the component level, designs in different domains (e.g., RF, analog, or digital) can be concurrently characterized to provide an interface that offers smooth system integration while, at the same time, hiding implementation details. This *orthogonalization of concerns* allows making design decisions at the system level, where system tradeoffs can be evaluated across all RF, analog, and digital components. Moreover, the design process can be significantly shortened, because of the hierarchical approach enabled by our methodology, which progressively reduces the number of design variables. To ensure that reliable systems are produced, accurate and robust circuit performance models are crucial in our methodology, since high-level models should directly correspond to feasible physical implementations. Designs should therefore be robust to both modeling errors and process, voltage, and temperature variations (PVT), increasingly important as process parameters (minimum channel length, device threshold, supply voltage, etc.) decrease. As presented in [3], we include into our formulation techniques from design centering, traditionally adopted for digital design. With respect to [3], we add details on our performance models, in comparison with other modeling approaches, as well as on the mathematical derivation of the performance margin evaluation algorithm used in our robust optimization. Moreover, we apply our methodology to an additional example.

This paper is organized as follows. Section 2 gives an overview of the PBD methodology applied to the analog and mixed-signal domains. In Sections 3 and 4, we discuss the robust system-level design problem and provide its mathematical formulation within the PBD paradigm. In Section 5, we illustrate our methodology using three case studies, namely, a pipeline A/D converter and two RF front-ends, for UMTS and UWB receivers. Finally, we draw some conclusions in Section 6.

## 2. Analog Platform-Based Design

Performing system-level design space exploration and optimization in a systematic way can have a great impact on system performance and cost. In a wireless receiver, for example, it allows distributing design requirements (e.g., gain, NF, linearity) among the chain building blocks, and early evaluation of several tradeoffs, such as preselect filter selectivity and power consumption versus front-end linearity, or base-band filter selectivity versus ADC resolution.

In traditional analog and mixed-signal design flows, experienced architects conduct system-level design, and system specifications are empirically partitioned among the various functional blocks that circuit designers have to implement. In fact, since an effective system-level optimization is not achievable without accurate knowledge of the achievable performance of the several building blocks, final system performance may largely deviate from the expected one, which can result in silicon respins. To simultaneously achieve high-quality system integration starting from accurate circuit characterizations, analog-PBD (APBD) has been proposed and formulated in [4, 5, 6, 7] as a meet-in-the-middle recursive process consisting in top-down optimization (platform mapping) and bottom-up characterizations, exporting the feasible design spaces of platform components to higher levels of abstraction.

Optimization is performed on behavioral models, that is, mathematical representations of electronic circuits, capturing their functionality as a function of a set of input, output, and configuration parameters. To allow information hiding and intellectual property (IP) protection, a feasible performance model is also provided for each circuit block, which exports the performance achievable by any available implementation of the block (in the platform library), without propagating implementation details. Performance models are built in a characterization process, as described in Section 2.1. Both models are accompanied by validity laws, that is, a set of constraints and inequalities delineating the validity regions of all component models and their compositions. An Analog Platform (AP) is therefore a library of components, each one decorated by the above set of models and laws. A design is a platform instance, that is, a correct composition of elements, implementing the desired function and, at the same time, optimizing a set of quality metrics.

### 2.1. Analog Performance Models

Performance models play a critical role in analog system-level design and particularly in platform-based design. Performance models are used to constrain the optimization process to achievable performances within the considered architecture space. Therefore, system-level design approaches have to consider the nature of performance models explicitly during system optimizations.

In the recent few years, a number of papers have appeared on the generation of performance models [8, 9] and even direct modeling of the feasibility region [10, 11]. The latter set of works aims at providing a classifier that separates feasible n-tuples of performances from unfeasible ones, without recurring to a regression-based approach. From the system-level perspective, feasibility models allow casting exploration problems in a more intuitive performance space rather than mapping down to implementation parameters. The number of variables in the optimization problems is consequently reduced (at least in nondegenerate cases) and architecture selection becomes readily available as different implementation topologies may share common performance spaces.

*generation schemes*, equation based and simulation based (Figure 1). The first approach requires deriving analytical expressions to estimate performance from configuration (regression case) or to model the performance space (classification case). The second approach is based on statistical approximation techniques, where a set of performance samples is evaluated and exploited to build a performance model approximation. In order to compare the different schemes, it is useful to introduce some figures of merit for performance models. The

*cost*of generating a performance model can be decomposed into different contributions: model setup, model generation, and model retargeting. In particular, the first two contributions are usually at odds and need to be traded off in real models. Another fundamental figure of merit is

*accuracy*. Accuracy is usually assessed through some function (e.g., average or maximum) of the estimation error. We can further distinguish between two different kinds of error, the error on the training data and the generalization error. The last figure of merit we consider is

*generality*of the approach, both in terms of classes of circuits and of the performance figures that can be captured. Analytical- and simulation-based models are at opposite ends of the spectrum of performance model schemes.

Analog platform performance models rely on Support Vector Machines (SVMs) as a way of approximating the classifier Open image in new window discriminating the feasible performance space. Given a set Open image in new window of simulated performance vectors (as detailed in [10]), SVM training selects a subset of vectors Open image in new window (support vectors) and corresponding weight coefficients Open image in new window so that the classifier function is obtained as

## 3. Robust System-Level Design

Robust design and optimization have traditionally been closely related subjects. In fact, it is almost impossible to consider an aggressive optimization scheme without considering the robustness of the achieved solutions. System-level design should embrace robust approaches for two separate reasons. From the system level, mixed-signal design has to cope with model inaccuracies that are intrinsic to the behavioral models exploited in design explorations. The more complex the system, the larger the hierarchical structure of the design and the higher the risk when performing nominal design optimizations. In fact, composition of high-level models may provide results whose accuracy is not easily bounded, so either a costly iterative scheme between top-down system-level design and bottom-up verification or relaxed (robust) constraint propagation is adopted. From the implementation level, any performance model is subject to two kinds of inaccuracies: intrinsic modeling errors and process variability. While some control is available on the former source (even if potentially very expensive or restrictive), the latter cannot be solved with deterministic approaches.

Early approaches to computer-aided design centering in an analog context date back to the early 80s [12, 13, 14]. All the approaches have a common dependency on the model used to estimate performance degradation on design parameters and, if yield is actually considered, on joint probability functions used to compute yield expectations. However, robust optimization for analog design has not been developed at the same level as nominal optimization. The largest obstacle on the way is represented by the complexity of the resulting optimization problem, which is usually captured as a semiinfinite programming problem. In [15], a circuit optimizer based on simulation is enriched with robust design features, showing significant improvements albeit constrained with scaling issues for complex circuits. The lesson learned from early attempts of including process variations and mismatch in automated circuit design is the tremendous complexity of the resulting problem.

Models generated with classic approaches based on Response Surface Methodology (RSM) [16] can become too expensive to build because of the number of primal parameters and the complexity of the necessary simulations. Instead, we propose an alternative approach, based on approximate models to be developed at the system level.

### 3.1. Previous Approaches

*flexibility functions*. Flexibility functions are introduced to capture the complexity of implementing a specific set of performances. Therefore, in place of optimizing for power or area, the optimization problems maximize the "flexibility" of achieving the optimum set of performances (i.e., minimize the "effort" of implementation). Albeit rigorously formulated, the methodology was rather limited in performing aggressive optimizations because of the halo inherently inserted by the heuristic flexibility functions.

More recently AMGIE [20] proposed to carry out hierarchical design via a set of optimization problems where, at each abstraction level, component performances are bounded to predefined ranges. A robust approach is achieved inserting margins Open image in new window on all performances, so as to compensate for modeling inaccuracies. However, Open image in new window has to be determined a priori so that its final value is not the result of an optimization problem. In particular, the cost of meeting the margin on performances is not traded off with the potential improvements in system performances, that is, the sensitivity of the goal function on Open image in new window is not evaluated at all, leaving a wide discretionality in determining performance margins.

Recent advances in convex optimization [21] have revitalized analytical approaches to analog design and, consequently, robust design. ROAD [22] introduces a robust optimization approach based on posynomial performance models. To improve accuracy, a simulator-in-the-loop approach is selected and local posynomial models are generated around design points. It is then possible to deal with nonconvex design spaces exploiting the possibility of exactly solving large-scale convex programs. OPERA [23] introduces a robust geometric optimization problem to maximize yield over statistical variations. Design process variations are captured with confidence ellipsoids and approximated to yield a convex problem. The robust design formulation computes optimal design parameters to meet a predetermined yield target. Convex optimization approaches, however, tend to limit designers in selecting cost function and formulating their problems. The efficiency achieved in actually solving the problem may be then counterbalanced by the effort required to model the system and validate the analytical expressions used to set the problem. Moreover, classic approaches to system design with convex optimization are based on generating a flat optimization problem, where all circuit topologies have been selected, thus setting a challenging problem as system complexity grows and mixed-signal designs are approached.

Recently, a hierarchical approach to robust system-level analog design has been presented [24]. Performance centering is sought through concurrent maximization of system-level flexibility based on behavioral models and implementation-level performance margins based on performance models. A possible limitation of the approach is the requirement of posynomial models to capture both system-level and implementation-level constraints. While this assumption is certainly acceptable for some classes of analog systems, it may be in practice a hard one to satisfy as it becomes increasingly difficult to guarantee (or even assess) model convexity as design hierarchy becomes deeper and high-level behavioral models are exploited in mixed-signal design space explorations.

In our framework, we extend the hierarchical approach by removing the posynomial constraints on design formulation. As in [23, 24], robustness is achieved through maximization of margins with respect to system specifications. Extending the approach to analog platforms, we obtain a two-fold advantage. First, very accurate performance models (not constrained to be convex, posynomial, or even in explicit form) can be exploited to estimate implementation margins. It is then possible to accurately weigh implementation margins since model inaccuracies are kept to minimum levels. Second, arbitrary system behavioral models and constraints can be used to formulate the optimization problem since analog platform-based design relies on global stochastic optimization approaches to find optimal implementations. Designers can then specify their systems without recurring to posynomial approximations and capturing arbitrary nonconvex constraints.

## 4. Mathematical Formulation

### 4.1. Nominal Optimization

In a nominal formulation, the optimization process mapping platform Open image in new window onto platform Open image in new window is mathematically captured as

### 4.2. Robust Optimization

To address this problem, an alternate formulation of the optimization problem is required. The sets of constraints Open image in new window and Open image in new window have to be satisfied with some margin so as to compensate for modeling inaccuracies. We can write the new set of constraints as Open image in new window and Open image in new window . Margins have an intuitive interpretation, defining a sphere (as defined by the norm adopted) Open image in new window for system constraints and Open image in new window for performance constraints around the optimal pair Open image in new window . The objective of the optimization problem is then changed so as to maximize margins Open image in new window and Open image in new window , which corresponds to the maximization of the volumes of the spheres around the optimum configuration and performance points. The original cost function is inserted as an added constraint with a dedicated Open image in new window . Given a minimum cost target Open image in new window , at optimum Open image in new window is maximized constrained on the other margin variables, so that a tradeoff is evaluated between cost value and robustness during the optimization. Therefore, problem (2) becomes

System-level constraints are usually available in explicit form; therefore Open image in new window can be immediately written as

and included in the optimization problem. Additional constraints may be inserted to set specific relations on Open image in new window , for example, Open image in new window . The problem is more involved with performance models, as analog platforms provide Open image in new window in implicit form with a nonlinear function Open image in new window . In this case, we interpret the margin Open image in new window in the following way. For a performance model Open image in new window , its frontier Open image in new window defines the boundary of the feasible region. Given a configuration point Open image in new window satisfying performance constraints Open image in new window , its margin Open image in new window can be obtained finding the closest configuration Open image in new window to Open image in new window and computing the norm of Open image in new window . If all components Open image in new window of Open image in new window have the same weight, then Open image in new window (the performance constraint Open image in new window is consistent with the formulation in (2) as it is equivalent to the argument of sgn in (1) being Open image in new window after a sign change). In this case, minimizing Open image in new window is equivalent to maximizing the volume of the sphere around Open image in new window that is enclosed in the feasible space (within its boundary Open image in new window ). The general case of different weights on different performance components can be immediately obtained adopting a different norm when computing Open image in new window . Since the different performances in the performance vectors used to generate Open image in new window can differ in orders of magnitude, they are all preconditioned to be normalized in the interval Open image in new window . In the following paragraph, we show how to compute Open image in new window based on the SVM representation of Open image in new window .

### 4.3. Performance Margin Evaluation

The problem of finding Open image in new window given Open image in new window and Open image in new window is analogous to the problem of finding the largest hyperellipsoid enclosed by Open image in new window . Initially we start solving the case of hypersphere enclosure, extending to the general case at the end of this paragraph. By definition, Open image in new window is the point on the boundary Open image in new window which shows minimum distance from Open image in new window . To simplify notation, we set Open image in new window and Open image in new window . Therefore, we can obtain Open image in new window solving the following optimization problem:

where Open image in new window is implicitly defined from (1) as

The optimization problem obtained substituting (6) into (5) is evidently nonlinear and can be interpreted as vector *projection* onto a nonconvex set. In fact, while the cost function in (5) is strictly convex, the equality constraint in (6) is nonlinear (and nonconvex). At optimum, the Karush-Kuhn-Tucker conditions require that

where Open image in new window is the Lagrange multiplier, the first equation states the feasibility condition for Open image in new window **,** and the other equations enforce that the gradient of the Lagrangian function Open image in new window vanishes at any optimal point. System (7) originates from the equivalent problem obtained by (5) after squaring the cost function. For each Open image in new window , the Open image in new window th component of Open image in new window can therefore be computed as follows:

By substituting (6) into (8), for each Open image in new window , we finally obtain the equations in (7).

The nonlinear system (7) can be solved with Newton-Raphson (NR) providing quadratic convergence if Open image in new window is "close" to Open image in new window . Open image in new window is therefore the radius of the largest hypersphere enclosed in Open image in new window . However, the nonlinear nature of (7) generates two problems. First, a multitude of solutions may exist, so we could achieve convergence on a point on Open image in new window which is not the closest to Open image in new window ; second, NR may not converge at all if a sufficiently good initial guess is not provided. To cope with the above problems we first adapt to our problem a more sophisticated implementation of the NR method, similar to the *damped* Newton's method [21], which tries to improve on basic NR poor global convergence. Then we add some ad hoc heuristics to generate a good initial guess.

Solving for Open image in new window one of the equations in (7) and substituting the result into the other equations, we obtain an Open image in new window -dimensional system in the unknown vector Open image in new window , which can be denoted as Open image in new window . We then combine NR method with the minimization of the function Open image in new window , in the sense that we accept the solution provided by each NR step only if the step considerably reduces Open image in new window . If this does not happen, we backtrack along the NR direction Open image in new window starting from the old point Open image in new window until we have an acceptable new point Open image in new window ( Open image in new window ). Since the NR step is a descent direction for Open image in new window , we are guaranteed to find an acceptable point by backtracking. The backtracking routine is based on the *line minimization* rule [25, 26] and consists in defining Open image in new window , as the restriction of Open image in new window along Open image in new window , and finding Open image in new window so as to minimize Open image in new window . To save on the number of function evaluations, a cubic approximation of Open image in new window is actually computed based on available information on Open image in new window and its derivative. Since the improved NR method can still occasionally fail converging on a local minimum of Open image in new window , we can try a new starting point according to the following heuristics:

(i)we compute the distance along reference axes in Open image in new window using bisection-based monodimensional methods. It is then possible to bound the distance of Open image in new window . We observed that in practical cases whenever this bound is smaller than some Open image in new window (whose actual value depends on normalization of Open image in new window ) convergence is always achieved and the correct Open image in new window is returned by Newton-Raphson,

(ii)we set Open image in new window to start iterations as we expect Open image in new window to define a relatively "thin" feasible space. Whenever the previous heuristics is not satisfied, we run Open image in new window NR iterations perturbing the initial point Open image in new window in the direction of the axis where the minimum distance has been found in the previous point (iterations are aborted after a predetermined number) until the minimum distance solution is reached. We observed that Open image in new window is generally sufficient to achieve convergence,

(iii)in case of nonconvergence, we return the bound computed in the first point. In practice, there is no consequence in doing this because it always happened for points deep in Open image in new window in our tests.

The above procedure can be extended to hyperellipsoids enclosure by scaling Open image in new window with a unitary matrix Open image in new window to obtain Open image in new window and extending the previous approach on Open image in new window . Margins found in this way need to be scaled back to the initial space through Open image in new window . This allows selecting different margins on different performances.

The overall algorithm complexity has been computed to be Open image in new window where Open image in new window is the number of performance figures in Open image in new window , Open image in new window is the number of performance vectors, and Open image in new window is the cost for evaluating the exponential function as in (1).

## 5. Examples

In this section we apply the previous results to the case studies reported in [4, 27, 28]. The original designs are reformulated according to (3). The selection of good cost functions is a crucial issue in system-level optimization, with implications that may become subtle when maximizing robustness. In our experiments, we used the following cost prototype:

A few considerations may help explain the form of (9). First, the volumes of the Open image in new window ellipsoid and the Open image in new window hypercube increase with number of dimensions for constant margin; therefore an overall normalization is achieved with the powers Open image in new window and Open image in new window of Open image in new window and Open image in new window products. As far as architecture margins are concerned, we can partition Open image in new window , where Open image in new window refers to the single platform component. Elements Open image in new window of Open image in new window are strongly related describing an ellipsoid embedded in Open image in new window . Therefore a single element is sufficient to describe the margin of the Open image in new window th component. If we consider that the composition of blocks is as robust as the weakest block, we can obtain a different cost function considering Open image in new window . The Open image in new window function is used to saturate the sensitivity on Open image in new window as margins too wide may cause degenerate robustness/performance tradeoffs. Finally, if we analyze the Pareto optimal curves as a function of Open image in new window and Open image in new window , we can easily obtain that the relative importance of two Open image in new window parameters is controlled by

so that Open image in new window sets the relative impact of variations of Open image in new window and Open image in new window . When Open image in new window and Open image in new window are considered, we obtain (for small Open image in new window )

which makes it clear how the parameter Open image in new window can be used to control sensitivity on Open image in new window without recurring to exponent ranges that may generate numerical issues during optimization. Equations (10) and (11) can be used as guidelines to set parameters in (9), as exemplified in the following case studies.

As a final remark, we notice that architecture performance margins are taken on lower-dimensional models than the corresponding platform ones. In fact, some parameters are simply "ancillary" parameters required for correct composition of platform models, and as such not related with the robustness of the solution. One other parameter, which we did not include when computing margins at the component (architectural) level, is power. Power may be considered as an annotation on circuit performances. In fact, in our case studies, if a given circuit exhibits a larger (or smaller) power consumption with respect to the estimated one, it does not affect circuit performances (which is obviously not true if gain is not met, for example). We remark that this is an arbitrary design choice and is not related to the presented methodology. On the other hand, in our examples we introduce margins on power at the system-level to trade the *global* power consumption with the robustness of the solution. Also, area has not been exploited as a robustness criterion, but this can be seamlessly introduced in the robust optimization scheme to export at the system level area penalties involved in topology selection.

### 5.1. Pipeline ADC

*PolyEstim*circuit. At the same time the distorted SHA characteristic is effectively inverted (rectified) by the

*PolyInv*circuit. As shown in Figure 7, the interstage residue amplifiers are fully differential switched capacitor systems (FB C in Figure 6) implemented with a telescopic Operational Transconductance Amplifier (OTA). Loading effects and switches nonidealities are also included in the model. The OTA optimization needs to be performed under the hypothesis of operation of digital calibration circuits, as detailed in [5]. In order to perform efficient high-level exploration across the analog/digital boundary while reducing the complexity of the problem we provided characterizations and feasible performance models for the main blocks, that is, the digital calibration logic and the first-stage residue amplifier. The remaining part of the converter was considered ideal. Indeed, the first stage in a multibit pipeline ADC is the most critical block since the accuracy required in terms of gain and linearity is maximum; the remaining stages have been lumped into one block in our macromodel. Since the first stage provides the first 4 bits, a nominal gain Open image in new window of 8 is required to the SHA. However, the presence of the digital correction circuit relaxes this constraint enabling power savings. In the nominal optimization, the cost function aims at minimizing power consumption Open image in new window of the overall ADC subject to performance models and minimum system requirements on DNL, INL, and the signal-to-noise ratio (SNR) due to thermal noise. The architectural space includes four correction algorithms to invert the polynomial nonlinearity corresponding to different accuracy and power consumption levels, based on [5]. Performances are evaluated through the behavioral model Open image in new window of the mixed-signal platform library, in which each component is embedded.

The extension to the robust approach of the optimization problem has been achieved through the following formulation, based on the cost template in (9):

where Open image in new window , Open image in new window , Open image in new window , and Open image in new window are system margins on power, DNL, INL, and SNR, respectively. Open image in new window , the architecture margin, is normalized in Open image in new window and is computed by exploiting an ellipsoid in which weight for the OTA bandwidth (BW) and open-loop gain ( Open image in new window ) is 2 times the other performance indices. The parameter Open image in new window controls cost function sensitivity on Open image in new window , hence the architecture margin on the optimum.

Performance of optimal ADC, OTA, and GDEC circuit for 3 different cost functions. Open image in new window denotes the system and architecture margins.

Performance | ||||||
---|---|---|---|---|---|---|

DNL (LSB) | 0.07 | 0.73 | 0.04 | 0.76 | 0.07 | 0.73 |

INL (LSB) | 0.43 | 0.57 | 0.04 | 0.96 | 0.45 | 0.55 |

SNR (dB) | 85.1 | 9.1 | 85.1 | 9.1 | 82.6 | 6.6 |

57.1 | 42.9 | 59.6 | 40.4 | 42.6 | 57.4 | |

52.8 | — | 55.3 | — | 37.8 | — | |

194 | 134 | 267 | 88.7 | 228 | 2.72 | |

BW (KHz) | 4269 | 1119 | 3768 | 739 | 2755 | 22.7 |

7.46 | — | 7.65 | — | 7.24 | — | |

4.2 | — | 4.2 | — | 4.8 | — |

We notice how in lower-power designs the system-level margin on SNR tends to decrease as well. Moreover, the unity gain frequency (and the bandwidth), which is the key parameter influencing the settling behavior of the SHA, tends to decrease thus impacting the accuracy of the system (i.e., INL, DNL, and Open image in new window ) and mandating more accurate and power expensive calibration circuits. We finally compare results in Table 1 with the optimal design reported in [4]. Using a nominal optimization technique, we obtained 52.5 mW ADC power consumption with approximately 9% margins. This implies that, in the nominal formulation, it was still possible to obtain reasonable architectural and system margins by acting both on optimization constraints and feasible performance model generation constraints as viable safety margin knobs. However, we had not chances to quantitatively explore and efficiently control the involved performance/margin tradeoffs as we have demonstrated here in the robust formulation.

### 5.2. UMTS Front-End

UMTS receiver robust optimization results as a function of Open image in new window . Larger values of Open image in new window decrease the sensitivity on Open image in new window in (9). Note that the LNA topology is also affected by robust optimizations.

nom. | 0 | 0.1 | 0.3 | |
---|---|---|---|---|

Gain (dB) | 30.8 | 29.9 | 31.2 | 27.7 |

Power (mW) | 10.9 | 14.4 | 12.9 | 11.7 |

Open image in new window (dBm) | −99.1 | −99.7 | −100.2 | −99.5 |

Open image in new window (dBm) | −98.7 | −99.5 | −100.3 | −99.7 |

− | 0.7/3.5 | 1.2/4.3 | 0.5/3.7 | |

NF (dB) | 6.8 | 6.9 | 6.4 | 6.9 |

18.2 | 15.4 | 17.3 | 15.7 | |

3.5 | 3.8 | 4.5 | 5.8 | |

5.7 | 7.3 | 6.1 | 4.2 | |

Topology | n | n | n | np |

Margin | 1.3% | 12.5% | 6.9% | 3.6% |

CG (dB) | 12.6 | 14.5 | 13.9 | 10 |

5.1 | 7.1 | 6.8 | 5.9 | |

Margin | 0.9 % | 15.8% | 6.1% | 4.8% |

### 5.3. UWB Front-End

In the nominal optimization, we aimed at minimizing power consumption ( Open image in new window ) of the RF front-end while meeting system constraints on Open image in new window , total gain Open image in new window , and noise figure ( Open image in new window ). Similar to (14), the robust optimization problem is formulated as follows:

where the system performance figures (power, Open image in new window , gain, and Open image in new window ) are calculated from the Open image in new window and Open image in new window using RF cascade equations, as follows:

The behavioral model Open image in new window is then built out of (16), with some additional *validity laws* enforcing correct block composition. As in the UMTS optimization problem, Open image in new window has been used to control the amount of margin on Open image in new window , hence the architecture margin at optimum. For this particular application, power, Open image in new window , and Open image in new window , the most critical performance parameters, have been given the same relative weight. Open image in new window dictates the power consumption margin, while robustness with respect to gain and second-order distortion variations is less of a concern for our UWB communication system. Open image in new window and Open image in new window set the system-level margins on noise figure and Open image in new window of the optimum system design. As in (14), the Open image in new window function saturates when margins become too large ( Open image in new window 15%).

UWB RF front-end receiver robust optimization results as a function of Open image in new window . Larger values of Open image in new window decrease the sensitivity on Open image in new window in (15).

nom. | 0 | 0.1 | 0.3 | |
---|---|---|---|---|

Gain (dB) | 18.32 | 19.84 | 17.72 | 19.64 |

Power (mW) | 7.95 | 8.59 | 8.33 | 10.28 |

4.16 | 4.16 | 3.99 | 3.75 | |

Open image in new window (dBm) | −24.07 | −24.68 | −21.00 | −18.65 |

0.75 | 0.80 | 0.46 | ||

0.84 | 1.01 | 1.25 | ||

5.32 | 9.00 | 11.35 | ||

19.42 | 21.57 | 20.30 | 21.56 | |

3.81 | 4.00 | 3.72 | 3.56 | |

Open image in new window (dBm) | −9.32 | −11.13 | −10.46 | −11.49 |

5.8 | 6.38 | 6.39 | 8.44 | |

Margin | 0.22% | 32.0% | 24.6% | 14.2% |

CG (dB) | −1.11 | −1.73 | −2.59 | −1.9 |

12.79 | 11.81 | 12.32 | 12.07 | |

Open image in new window (dBm) | −4.47 | −2.81 | −0.29 | 3.83 |

2.15 | 2.21 | 1.93 | 1.84 | |

Margin | 0.04 % | 17.13% | 0.1% | 0.1% |

As a final comment on the results, we could not perform a Monte Carlo analysis on the actual circuits for any design since the complexity of our systems rules out the possibility of performing any reasonable number of simulations to get meaningful results. In fact, this was an important motivation to introduce robustness early in the design cycle starting from the system level.

## 6. Conclusions

Platform-Based Design (PBD) is a promising methodology for embedded system design, aiming to improve design productivity by encouraging design reuse, orthogonalization of concerns, and system-level optimization. In this paper, we have illustrated the extension of PBD to mixed-signal systems. Furthermore, to ensure robustness with respect to both model and design uncertainties, we have proposed the application, within the PBD framework, of design-centering techniques. The proposed approach allows robust hierarchical design without any assumption on the mathematical properties of the system models, leading to a general formulation that can be used for robust automatic design-space exploration.

To demonstrate the effectiveness of the proposed design methods in different domains, we presented three case studies: a mixed-signal pipeline ADC and two RF front-ends, respectively, for UMTS and UWB receivers. In all cases, designs were efficiently composed from precharacterized components, as well as optimized at the system level, demonstrating the flexibility of the approach and significant improvements in terms of robustness.

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