Outline

Tile "An Efficient Substrate Coupling 
Noise Analysis with Block Structural-Model-Reduction and Partition"

Section I Introduction
Define the problem of model reduction and its application in substrate noise analysis.

Review of the traditional model reduction approach by PRIMA 
and explain its two major limitations: 
(i) the reduced model loses structure such that hard to be realized; 
(ii) the reduced model is not compact; 
and (iii) the reduced model is dense and hard to analyze with large number of ports.

Review recent approaches: 
(i) the structural model reduction [Freund:ICCAD'04] preserve structure; 
(ii) the TBR (truncated balanced realization) [Philips:TCAD'05] 
and structural model reduction improve the compactness, i.e., 
the low order can achieve higher accuracy; 
and (iii) hierarchical decomposition during reduction  
[Freidman:ICCAD'04] can alleviate the complexity introduced by ports...

Their limitations will be discussed. In this paper, we propose 
an nxn structural model reduction to further 
improve the compactness of the reduced model. 
We then use a hierarchical decomposition of dense macro-model 
into a number blocks with a subset of ports, 
and apply the decomposed macro-model to verify if the voltage bounce 
is under a specified target based on a linear programming 
based robustness verification. The major contributions:

(1) Model Reduction Compactness: under the same accuracy, 
this approach converges XX times faster than PRIMA 
and results in a realized model XX times smaller model size.

(2) Ability to Handle Large Number of Ports: 
to show what is the largest port number that can 
be handled when using hierarchical decomposition and not; 
and what is maximum speedup in both cases.

Section II nxn Structural Model Reduction

Present the theory and derivation of nxn structural model reduction. 
Give the theorem about preservation of the passivity. 
(1) Try to find the explanation or mathematic derivation showing 
that the structural model reduction not only preserves 
the structure but also improves the accuracy. 
(2) Due to the preservation of structure and reciprocality, 
a realization procedure of the macro-model is presented. 

Section III Hierarchical Decomposing of Macro-model

Present the theory and derivation of hierarchical decomposing of macro-model. 
Show the reason why the overall matrix is sparse; 
discuss the cost of decomposing 
and how to reduce the cost decomposing by approximation.

Section IV Robustness Verification of Substrate Noise

Present the observation of monotony of substrate RC network; 
the definition of the robustness in the substrate noise problem; 
and the formulation of robust verification in linear programming. 

Section V Experiment

(1) Impact of nxn structural model reduction
I will compare our nxn structural-mode-reduction with PRIMA 
in following aspects by increasing (1) circuit size (same port#); 
(2) port number (same circuit size): 

Accuracy: under the same model order, this approach is about XX% accurate than PRIMA;
Efficiency: under the same accuracy, this approach converges XX times faster 
than PRIMA and results in a realized model XX times smaller model size.

Discuss the impact of partition size. 

(2) Impact of hierarchical decomposition
I will compare our hierarchical partition of ports with original model 
and flat macro-model in the linear programming based robustness 
verification by changing (1) circuit size and port number; (2) partition size:

Ability to handle large sized circuit: to show what is the largest circuit 
that can be handled when using reduction and hierarchical decomposition 
and without using reduction and hierarchical decomposition; 
and what is maximum speedup in both cases. 

Ability to handle large number of ports: 
to show what is the largest port number that 
can be handled when using hierarchical decomposition 
and without using hierarchical decomposition; and what 
is maximum speedup in both cases. 

Discuss the impact of partition size. 

(3) Application of the above flow to substrate contact placement 
Discuss the setting of experiment. Present the generated noise map 
at low and high frequency. Show the impact of guard ring. 
Compare the output waveform in both time/frequency domains 
for two different locations by the result of the noise map. 

Section VI Abstract/Conclusion