Doctor of Philosophy in Electrical Engineering and Computer Science University of California at Berkeley

Professor Edward A. Lee, Chair

## Abstract

In this thesis, we present various scheduling techniques for programs expressed as Synchronous Dataflow (SDF) and Multidimensional Synchronous dataflow graphs. Synchronous dataflow has proven efficient as a specification model for block-diagram based programming environments for signal processing. Two key reasons for its popularity are that a) static schedules can be constructed at compile time, thus eliminating the overhead due to dynamic scheduling, and b) it models multirate signal processing application very naturally and intuitively. The first property is particularly important in environments where code-synthesis for embedded processors is desirable, since embedded signal processing applications have rigid throughput requirements in order to meet hard- real-time constraints. Multidimensional Synchronous Dataflow (MDSDF) is an extension of SDF to multiple dimensions; in this model, applications such as image and video signal processing, which operate on multidimensional signal spaces, are more naturally modeled and specified than in SDF.The amount of on-chip memory available on embedded processors is often severely limited. Adding off-chip memory is usually not an option because it entails a speed penalty, it increases overall system cost and size, and increases power requirements. Thus, when generating software implementations from SDF specifications for a uniprocessor, the scheduling problem of minimizing code size and buffer memory size becomes crucial. If the scheduling is not done carefully, the blowup in the size of the implementation can preclude implementation on the processor. Previous techniques have focused on scheduling to minimize code size. However, this neglected the buffer memory usage of the schedule; in this thesis, we develop techniques that jointly minimize for code size and buffer-memory size of the software implementation. We give three polynomial-time algorithms: a dynamic programming algorithm that is used as a post-optimization step, and two heuristics that use different approaches to constructing these schedules. The general problem is shown to be NP-complete, thus justifying the use of heuristics. An extensive experimental study is given to show the efficacy of all these techniques.

We extend these scheduling results to MDSDF specifications. We then tackle a problem of a different type. A multidimensional signal can be sampled in many different ways. A straightforward extension of one-dimensional sampling results in the so-called rectangular sampling structure, where the samples lie on a rectangular grid. However, a more general sampling structure is a geometrical lattice; sampling lattices that are not rectangular can have many advantages in certain applications. For example, a signal sampled on a non-rectangular lattice can have a lower sampling density than one sampled on an equivalent rectangular lattice. For real-time processing of multidimensional signals, a lower sampling density means fewer samples to process in a given time interval. The standard MDSDF model suffers from the inability to model multidimensional systems sampled on arbitrary sampling lattices; hence, we give an extension of MDSDF that is capable of modeling such systems. The model we give preserves the property of static, compile-time schedulability. However, constructing such schedules requires the solution to some challenging problems. In particular, we show that an augmented set of balance equations have to be solved simultaneously in the extended model. The additional equations are quite different from the usual balance equations in SDF and MDSDF; they involve computing so-called "integer volumes" of parallelepipeds. This computation turns out to be an interesting number-theoretic problem, and we present several approaches for solving it. Finally, we present a practical example of a video sampling structure conversion system to show the usefulness of the generalized MDSDF model.

Citation:

Memorandum No. UCB/ERL M96/79, Electronics Research Laboratory,
College of Engineering, University of California, Berkeley, Ca 94720,
December 10, 1996.