By Mitchell Aaron Thornton
Spectral ideas in VLSI CAD became a topic of renewed curiosity within the layout automation group as a result of emergence of recent and effective equipment for the computation of discrete functionality spectra. some time past, spectral computations for electronic good judgment have been too complicated for useful implementation. using choice diagrams for spectral computations has vastly decreased this quandary taking into consideration the advance of recent and precious spectral concepts for VLSI synthesis and verification. numerous new algorithms for the computation of the Walsh, Reed-Muller, mathematics and Haar spectra are defined. The relation of those computational ways to conventional ones is additionally provided.
Spectral thoughts in VLSI CAD offers a unified formalism of the illustration of bit-level and word-level discrete services within the spectral area and as determination diagrams. an alternate and unifying interpretation of selection diagram representations is gifted because it is proven that a number of the diversified ordinary sorts of choice diagrams are simply graphical representations of assorted discrete functionality spectra. Viewing a variety of choice diagrams as being defined via particular units of transformation capabilities not just illustrates the connection among graphical and spectral representations of discrete features, but in addition supplies perception into how a number of choice diagram kinds are related.
Spectral innovations in VLSI CAD describes a number of new functions of spectral ideas in discrete functionality manipulation together with selection diagram minimization, common sense functionality synthesis, expertise mapping and equivalence checking. using linear modifications in determination diagram dimension aid is defined and the connection to the operation referred to as spectral translation is defined. a number of tools for synthesizing electronic common sense circuits in accordance with a subset of spectral coefficients are defined. An equivalence checking process for sensible verification is defined dependent upon using matching pairs of Haar spectral coefficients.
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Extra resources for Spectral Techniques in VLSI CAD
1 illustrates the computation of the Rademacher-Walsh spectrum by matrix multiplication. In general, this approach require 22n multiplications and 22n additions. 1 Computation of the Walsh spectrum of a 3-variable function using R-encoding. 23 The Spectral Domain 3 -1 -1 -1 1 1 1 -3 1 1 1 1 1 1 1 1 = 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 -1 1 1 -1 0 1 1 0 0 0 0 1 An alternate formulation represents the function by the vector Z in which logic In this case the spectrum is given by S=WnZ ois coded as +1 and logic 1 is coded as -1, which we term S-encoding.
N},a f:. 2) so the information content of the R and the S spectral coefficients is the same. 1, the fact that (WI )-1 = ~ WI and the fact that scalar multipliers can be factored out of a Kronecker product. S. E. Muller  which led to considerable interest in the ReedMuller (AND-XOR) expansion of Boolean functions. e. integer addition is replaced with summation (mod 2). 3 Computation of the Reed-Muller spectrum of a :i-variable function. 3 (Mn)-l = M n over GF(2). 1 and the fact (Ml}-l = Ml over 0 GF(2).
The presentation here and the development throughout the rest of this book are directed toward binary systems. e. a function whose inputs and whose output assume p distinct values with p;::: 2). 4 DECISION DIAGRAMS This chapter focuses on the use of various forms of Decision Diagrams (DDs) for the representation of the spectra of Boolean functions. Several DD variants will be described with their unique characteristics discussed followed by a description of their usage in representing Boolean functions in the binary as well as the spectral domains.
Spectral Techniques in VLSI CAD by Mitchell Aaron Thornton