1. A Regularity-Based Hierarchical Symbolic Analysis
Alex Doboli
VLSI Systems Design Laboratory
State University of New York – Stony Brook, USA
This paper presents a novel hierarchical symbolic analysis method for representing the relationship between the parameters of an analog network
and its building blocks.
The originality of the method stems from exploiting regularity aspects. The regularity aspects that were identified are:
(1) structural regularity: majority of the network blocks are connected in identical templates
(2) symbolic parameter regularity: parameters for a connection template require similar sets of operations.
We mathematically show that the generated models are of polynomial size if the two kinds of regularity are exploited. The paper discusses the three components of the proposed symbolic analysis method:
(1) an efficient representation of symbolic expressions,
(2) an algorithm for construction of symbolic expressions and
(3) a decomposition technique for extracting the structural regularity of a network.
The described symbolic technique was implemented and used successfully for synthesis and optimization of different analog systems such as filters and communication systems.

2. Ranga Vemuri Method for Large-Scale Analog Networks
Digital Design Environments Laboratory University of Cincinnati, USA

3. What is Symbolic Analysis?
Symbolic analysis is the task of automatically deducing
relationships between network parameters and parameters of the
building blocks: C1 R1 R2 E1 - E2 + C2
Symbolic expression
for transfer function:
Symbolic analysis is the task of automatically deducing relationships between the network parameters and the parameters that correspond to the building blocks of the analog network.
For example, a symbolic expression can describe the AC transfer-function of a filter in terms of parameters of its building blocks i.e. op amps, resistors and capacitors.
Besides transfer-functions, symbolic formulas can also represent poles, zeros, resonant frequencies, sensitivities of system performances with respect to variations of building block parameters \cite {gielen91}. E2
m/R1R2C1C2 = E1
s2 + (1/R1C1 + 1/R2C1 + (1-m)/R2C2)s + 1/R1R2C1C2

4. Symbolic Analysis for Analog Synthesis
Signal flow and processing of a system
Performance Model Generation
Using Symbolic Analysis
Generic Performance Model of a system
Analog Basic Blocks
Architecture Generator
Analog system net-list
Models of
basic blocks
Update Performance Model
VASE synthesis methodology
Analog system net-list + performance models
System
Constraints
System-level Design Parameter Optimization
Analog system net-list + optimized design parameters

5. Related Work on Symbolic Analysis
Existing symbolic analysis methods include:
- Determinant-based methods [Gielen et al], SFG- based methods [Lin et al],
[Daems et al], two-graph methods [Wambacq et al] etc
Limitations of symbolic analysis methods:
Symbolic models are of exponential size
=> only small size applications can be addressed
Proposed solutions for coping with exponential model size:
Approximation of symbolic formulas [Yu et al], [Fernandez et al]
Hierarchical symbolic analysis methods [Starzyck et al], [Hassoun et al], [Shi et al]
Effective representation methods for symbolic formulas: DDD - [Shi et al]
The main challenge for any symbolic analysis method is the exponential growth of the produced symbolic expressions (1011 terms for an op amp).
Current research considers two ways of handling this limitation:
(1) Approximation of symbolic expressions and hierarchical methods.
Approximation methods retain only the significant terms of the symbolic expressions and eliminate the insignificant ones. The difficulty, however, lies in identifying what terms to eliminate and what the resulting approximation error could be.
(2) Hierarchical methods tackle the symbolic analysis problem in a divide-and-conquer manner. They consider only one part of the global network at a time and then recombine partial expressions for finding overall symbolic formulas.
Existing hierarchical methods have a fundamental limitation in that they require networks composed of loosely connected parts. Thus, such techniques are not feasible for addressing networks i.e. operational amplifiers that are built of tightly coupled blocks. This constitutes an important disadvantage as transistor-level optimization of analog circuits i.e. operational amplifiers is a primary aspect of analog synthesis.

6. Size Reduction of Symbolic Formula
Our Observations for
Size Reduction of Symbolic Formula
(1) Systems include structural regularity
(v1,i1)
* 100.0
(vo,io) +
* (-1.78)
* 1.0
(v2,i2)
The originality of our research stems from exploiting regularity aspects for addressing the exponential size of produced symbolic expressions.
Two kinds of regularity aspects were identified:
(1) Analog networks have structural regularity:
A filter can consist of multiple stages of blocks (adders, integrators)
that are linked in identical templates, that we call Generic Interconnection Templates (GIT). GITs can also be identified among the transistors of an operational amplifier.
If a generic symbolic expression describes a GIT then this expression can be re-used for all blocks connected in this template. This significantly reduces the size of our symbolic models. Moreover, exploiting structural regularity is not limited to networks built of loosely coupled stages.
The figure shows the block level structure of the telephone receiver system. All blocks are connected with a unique GIT.
=> Generic symbolic formulas can be shared for
- Blocks interconnected in similar connection patterns

7. Size Reduction of Symbolic Formula
Our Observations for
Size Reduction of Symbolic Formula
Symbolic formulas for different parameters have similar
operation structure that enables further formula sharing:
E11 = S11 + S13 * S31* T11/(1 - S33 * T11)
E12 = S12 + S13 * S32* T11/(1 - S33 * T11)
E21 = S21 + S12 * S32* T11/(1 - S33 * T11)
E22 = S22 + S23 * S32* T11/(1 - S33 * T11)
(2) Symbolic expression regularity:
Symbolic parameters of a GIT are expressed by applying the same set of operations to distinct variables. The uniform set of operations can be encapsulated in a generic symbolic function and re-used for expressing all symbolic parameters of the template.
Note the similar operations for the expressions of E11, E12, E21, E22.
Section 5 argues that symbolic models are of polynomial size if generic symbolic functions are employed for analysis.

8. Symbolic Expression Size
Symbolic expressions are polynomial size:
Model Size = O (mn2b3 + nb2 + b(m3 + n3))
where:
m - Maximum number of connection wires for any pair of blocks
n - Maximum number of ports for a block
b - Number of blocks in a net-list
Lemma:
Representing all symbolic parameters of the composed block that corresponds to a block with n1 ports linked to a block with n2 ports via m wires requires a number of APT nodes of the order
O [m(n12 + n22) + m2 (n1+n2) + m3].
Assume a net-list of b interconnected blocks. Two blocks are connected through maximum m wires. Besides the m wires, a block has at most n other ports. The number of APT nodes required for representing all parameters of the net-list is of the order
O [mn2b3 + nb2 + b(m3 + n3)].

9. Constituting Elements of the Proposed Symbolic Analysis Method
Effective representation of symbolic expressions
Technique for building symbolic models:
Library of block interconnection patterns
Algorithm for building symbolic models
This paper discusses a novel hierarchical method for top-down symbolic analysis of large analog networks. For a network parameter, the technique produces a computational tree that describes how it relates to parameters of the building blocks of the network.
The proposed technique includes three specific components:
(1) An efficient representation of symbolic expressions:
The representation Analog Performance Tree (APT) explicitly targets the expression of hierarchy and regularity aspects of symbolic expressions. It is an un-interpreted variant of the closed-form, symbolic expressions that are created by traditional methods.
(2) An algorithm for constructing APTs for symbolic expressions:
The method traverses top-down the hierarchy of a network and builds symbolic expressions depending on the interconnection templates that were found, without explicitly solving the nodal equations. The algorithm exploits the hierarchy and regularity structure of a network for producing compact symbolic expressions.
(3) A new technique for extracting the structural regularity of a network:
We propose a decomposition method that identifies frequently used GITs. This is different from other partitioning approaches for hierarchical symbolic analysis in that ours can handle tightly connected networks.

10. Analog Performance Trees –
Proposed Representation for Symbolic Expressions
Symbolic function for parameter E11:
Symbolic function for parameter E21: + / A
E21
S21
S12
S32
T11
S33
E11
S11
S13
S31
T11
S33
Linked to A - * B B * 1 * C
APTs are set-up to share the structural similarity of symbolic expressions:
a common symbolic function is shared by all symbolic parameters
actual parameters are the parameters of the building blocks. These parameters are linked to the formal parameters of the symbolic function (I.e. S11, S12, S13 etc)
each symbolic function calculates a value that describes the interconnection involving the building blocks described by parameters Sij C D F G
D,F G

11. Algorithm for Symbolic Analysis
Block1
Block7
Block6
Block5
Block4
Block3
Block2
Block8
Stage 1
Stage 2
Step 1:
Block123
Step 2:
Overall System
The suggested symbolic analysis technique includes following steps:
(1) The first step is regularity extraction for the input network. This step aims to identify uniform stages in a network and block pairs connected with identical Generic Interconnection Templates (GITs). The paper discusses in Section 4 a decomposition technique for performing this task.
(2) As a second step, our algorithm sets-up the symbolic expressions for network parameters.
The algorithm traverses top-down, in pre-order the decomposition hierarchy. At each step, the algorithm computes only parameters that correspond to composed blocks that represent clusters of two or more blocks. Parameters are calculated applying a decomposition formula that is discussed in Section 5.
The method starts by generating symbolic expressions for the parameters of the overall system that is formed of stages Stage1 and Stage2. In Step2, Stage 1 is decomposed into blocks Block123 and Block4. Because Block4 is a basic blocks its parameters are not further expanded. Note that Stage 1 - Stage 2 and Block123 and Block4 are connected using the same connection style. Therefore, symbolic formulas for this connection style can be re-used.

12. Algorithm for Symbolic Analysis
Step 3:
Block1
Block3
Block2
Block12
Step 4:
Block123
Step 5:
Share symbolic tree for Stage 2 by linking corresponding parameters
Next, only the parameters of block Block123 required by the previous step are calculated. Step 3 shows how these parameters are computed. The process continues in a top-down fashion until basic building blocks Block 1 and Block 2 are reached in Step 4.
Step 5 calculates the parameters for Stage 2. As Stage 2 has exactly the same block structure as Stage 1, the top down expansion for Stage 1 is re-used. Only difference is that parameters for the building blocks of Stage 2 are linked instead of the parameters for the building blocks of Stage 1.

13. Library of Interconnection Patterns
General case:
. . .
n wires
p wires
m wires
Particular cases:
Connections between any pair of blocks can be characterized using the general case consisting of a block with n ports connected through p wires with another block with m wires. All possible connection are obtained by instantiating values for m, n and p.
We defined a library of interconnection styles to allow further optimizations of the symbolic formulas describing block interconnections. Some popular block interconnections are shown in the bottom part of the Figure.

14. Size with proposed Method
Experimental Results
Size with Two-graph Method
Size with proposed Method
Example
Network 36
624
CMOS Miller
OTA
346
531
BiCMOS Miller
OTA
By comparing Columns 2 and 3 in Table, we can notice that the model size
for large networks i.e. BiCMOS Miller OTA and BiCMOS fully-differential OTA is much less than the model sizes produced by the two-graph method.
The reason is that for our technique, due to the two kinds of sharing, symbolic models increase at a polynomial rate with the network size (number of nodes and edges in the graph).
The two-graph method generates models that increase at an exponential rate.
For small networks, however, the model size is less for the two-graph method than for the proposed technique. The reason is that symbolic expressions produced by the two-graph technique have all possible simplifications performed i.e. additions of constant 0 or multiplications with constant 1.
The APT trees generated by our method represent these operations, as they are not further reducible. If shared solving functions were replicated and all computations were symbolically performed then further reductions in the size of the APTs are possible.
3.55 x 105
1289
BiCMOS fully
Different. OTA
2.96 x 1011
4431

15. Filter Synthesis Using the Proposed Symbolic Analysis Method
Fifth order filter
Second order filter
The described symbolic technique was used successfully for synthesis and optimization of different analog systems such as filters and communication systems.
The figures present the optimization of second and fifth order filters. Symbolic models created by the presented method were employed for performance evaluation during filter optimization.

16. Conclusions Automated Performance Model Generation:
Developed a decomposition-based method for symbolic analysis of large systems
Method exploits structural regularity for blocks connected in similar patterns
Size of symbolic formulas is polynomial in the general case
Technique can be used for any system or circuit
(other hierarchical symbolic analysis methods require systems
built of loosely connected stages)
The originality of our method is that it exploits regularity aspects for reduction in the complexity of symbolic expressions.
Two kinds of regularity aspects are used by the proposed technique:
structural regularity: network blocks are connected in identical connection templates and
(2) regularity of symbolic expressions: distinct symbolic parameters are described by applying the same set of operations to distinct variables.
We mathematically showed that produced models are of polynomial size if the two kinds of regularity are considered.
For large networks, the size of the models produced by our symbolic analysis method is much less than the size of the models produced by other methods i.e. the two-graph method. Size reduction is mainly a consequence of exploiting the regularity of
symbolic expressions.
Method can be applied to any system/circuit network. Other methods are restricted to networks of a particular kind.