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...ortest (weighted) path in a coefficient DDD, which is a DAG (direct acyclic graph), can be found by depth-first search in time O(V + E), where V is the number of DDD vertices and E is number of edges =-=[28]-=-. So it is O(V ) in DDDs. Once we find the shortest path from a DDD, we can subtract it from the DDD using SUBTRACT() operation [21], and then we can find the next shortest path in the resulting DDD. ...

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... GmbVbs testability analysis and yield enhancement [10]. Efficient symbolic techniques for linear(ized) analog circuit analysis are the basis of distortion analysis of weakly nonlinear circuits [11], =-=[12]-=-, symbolic modeling of time-varying systems such as mixers [13]. Previous attempts to generate interpretable expressions use various symbolic analysis methods to generate sum-of-product representation...

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...functions until errors are within the user-specified error bounds for the frequency range. M3 M4 M6 Vin1 Vin2 Cc M1 M2 Vbias M5 Fig. 1. A simplified two-stage CMOS opamp [8]. M7 VDD As illustrated in =-=[9]-=-, simple yet accurate symbolic expressions can be interpretable by analog designers to gain insight into the circuit behavior, performance and stability, and are important for many applications in cir...

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...ONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL 23, NO. 4, APRIL 2004 2 g Cgd GmVgs Cgs Cbg Fig. 2. MOSFET small signal model. d GmbVbs testability analysis and yield enhancement =-=[10]-=-. Efficient symbolic techniques for linear(ized) analog circuit analysis are the basis of distortion analysis of weakly nonlinear circuits [11], [12], symbolic modeling of time-varying systems such as...

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... generated in the final stage (as the model is recentered after every stage). Such nominal point approximation strategy used in this work is also adopted by most of symbolic approximation methods [1]–=-=[7]-=-. During the approximation, we monitor both magnitudes and phases of the transfer functions until errors are within the user-specified error bounds for the frequency range. M3 M4 M6 Vin1 Vin2 Cc M1 M2...

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...pts to generate interpretable expressions use various symbolic analysis methods to generate sum-of-product representations for network functions. This area has been studied extensively in 1960s-1980s =-=[14]-=-. The resulting approaches, however, are only feasible for very small circuits, since the number of expanded product terms grows exponentially with the size of a circuit, and the resulting expressions...

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...nt efficient algorithms of generating dominant terms for deriving interpretable symbolic expressions based on a compact determinant decision diagram (DDD) representation of circuit transfer functions =-=[21]-=-, [22]. We show that dominant term generation can be performed elegantly by DDD graph manipulations. Since we start with exact symbolic expressions in DDD representations, our new approximation method...

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... been developed. Approximation after generation is reliable but it requires the expansion of product terms first [6], [9], [15]. Some improvements based on nested expressions have been proposed [16], =-=[17]-=-. But they generally suffer symbolic term-cancellation and alignterm problems. Approximation during generation extracts only significant product terms [2], [5], [7]. It is very fast, but has two major...

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...functions. Other smallsignal characteristics such as sensitivities, symbolic poles and zeros, cannot be extracted in general. At the same time, several approximation before generation techniques [7], =-=[18]-=- were proposed in which the complexity of a circuit is simplified before symbolic analysis methods are applied. Recently, a signalflow graph based approximation before generation method was proposed [...

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...terpretable by analog designers. Recently, various approximation schemes have been developed. Approximation after generation is reliable but it requires the expansion of product terms first [6], [9], =-=[15]-=-. Some improvements based on nested expressions have been proposed [16], [17]. But they generally suffer symbolic term-cancellation and alignterm problems. Approximation during generation extracts onl...

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... nested expressions have been proposed [16], [17]. But they generally suffer symbolic term-cancellation and alignterm problems. Approximation during generation extracts only significant product terms =-=[2]-=-, [5], [7]. It is very fast, but has two major deficiencies: First, if accurate expressions are needed, the complexity of the approach becomes exponential. Second, it works only for transfer functions...

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...ability in approximation-before/after-generation for analyzing large analog circuits. Experimental results show that our algorithms outperform the best dominant term generation method reported so far =-=[4]-=-, [23]. Some preliminary results of this paper were presented s Go Cdb Csb b in [24], [25]. This paper is organized as follows. Section II reviews the concepts of DDDs and s-expanded DDDs. Section III...

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...es and phases of the transfer functions until errors are within the user-specified error bounds for the frequency range. M3 M4 M6 Vin1 Vin2 Cc M1 M2 Vbias M5 Fig. 1. A simplified two-stage CMOS opamp =-=[8]-=-. M7 VDD As illustrated in [9], simple yet accurate symbolic expressions can be interpretable by analog designers to gain insight into the circuit behavior, performance and stability, and are importan...

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...s have been developed. Approximation after generation is reliable but it requires the expansion of product terms first [6], [9], [15]. Some improvements based on nested expressions have been proposed =-=[16]-=-, [17]. But they generally suffer symbolic term-cancellation and alignterm problems. Approximation during generation extracts only significant product terms [2], [5], [7]. It is very fast, but has two...

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...he original DDD in which s is contained in some vertices is called complex DDD. The s-expanded DDD can be constructed from the complex DDD in linear time in the size of the original complex DDD [22], =-=[26]-=-. Before we generate the dominant terms, one problem we need to consider is symbolic cancellation. Symbolic canceling terms arise from the use of the MNA formulation in analog circuits. For instance, ...

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...ent symbolic techniques for linear(ized) analog circuit analysis are the basis of distortion analysis of weakly nonlinear circuits [11], [12], symbolic modeling of time-varying systems such as mixers =-=[13]-=-. Previous attempts to generate interpretable expressions use various symbolic analysis methods to generate sum-of-product representations for network functions. This area has been studied extensively...

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...n approach on the s-expanded DDDs before de-cancellation. IV. CONSECUTIVE K-SHORTEST PATH ALGORITHM FOR GENERATION OF DOMINANT TERMS We proposed an efficient algorithm for finding k dominant terms in =-=[27]-=-. The algorithm does not require DDDs to be native or to satisfy aforementioned graph theoretical property (Theorem 1), and thus can be applicable to any DDD graph. To differentiate this algorithm fro...

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...odel generated in the final stage (as the model is recentered after every stage). Such nominal point approximation strategy used in this work is also adopted by most of symbolic approximation methods =-=[1]-=-–[7]. During the approximation, we monitor both magnitudes and phases of the transfer functions until errors are within the user-specified error bounds for the frequency range. M3 M4 M6 Vin1 Vin2 Cc M...

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...del. d GmbVbs testability analysis and yield enhancement [10]. Efficient symbolic techniques for linear(ized) analog circuit analysis are the basis of distortion analysis of weakly nonlinear circuits =-=[11]-=-, [12], symbolic modeling of time-varying systems such as mixers [13]. Previous attempts to generate interpretable expressions use various symbolic analysis methods to generate sum-of-product represen...

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...] were proposed in which the complexity of a circuit is simplified before symbolic analysis methods are applied. Recently, a signalflow graph based approximation before generation method was proposed =-=[19]-=- and demonstrated successfully to symbolic pole and zero generation. Symbolic analysis based on the concept of signal paths in control theory was employed for pole and zero location analysis [3], [20]...

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...practical analog circuits. For each circuit, DC analysis is first carried out using SPICE and our program reads in small-signal element values from the SPICE output. The algorithms described in [21], =-=[22]-=- are used to construct complex DDDs and s-expanded DDDs. First we apply the proposed dominant term generation algorithms to derive interpretable symbolic expressions for transfer functions and poles f...

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...osed [19] and demonstrated successfully to symbolic pole and zero generation. Symbolic analysis based on the concept of signal paths in control theory was employed for pole and zero location analysis =-=[3]-=-, [20]. In this paper, we present efficient algorithms of generating dominant terms for deriving interpretable symbolic expressions based on a compact determinant decision diagram (DDD) representation...

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... longer interpretable by analog designers. Recently, various approximation schemes have been developed. Approximation after generation is reliable but it requires the expansion of product terms first =-=[6]-=-, [9], [15]. Some improvements based on nested expressions have been proposed [16], [17]. But they generally suffer symbolic term-cancellation and alignterm problems. Approximation during generation e...

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...[19] and demonstrated successfully to symbolic pole and zero generation. Symbolic analysis based on the concept of signal paths in control theory was employed for pole and zero location analysis [3], =-=[20]-=-. In this paper, we present efficient algorithms of generating dominant terms for deriving interpretable symbolic expressions based on a compact determinant decision diagram (DDD) representation of ci...

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...ty in approximation-before/after-generation for analyzing large analog circuits. Experimental results show that our algorithms outperform the best dominant term generation method reported so far [4], =-=[23]-=-. Some preliminary results of this paper were presented s Go Cdb Csb b in [24], [25]. This paper is organized as follows. Section II reviews the concepts of DDDs and s-expanded DDDs. Section III prese...

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...erimental results show that our algorithms outperform the best dominant term generation method reported so far [4], [23]. Some preliminary results of this paper were presented s Go Cdb Csb b in [24], =-=[25]-=-. This paper is organized as follows. Section II reviews the concepts of DDDs and s-expanded DDDs. Section III presents a dynamic programming based term generation algorithm based on the work of [4], ...