Tsung-Hao Chen and Kuang-Ching Wang May 10 2002 Spring 2002 ECE 734 Course Project An Integrated Loop Transformation and Optimization Tool for FIR/IIR filters Tsung-Hao Chen and Kuang-Ching Wang May 10 2002

Introduction Single-input single output FIR/IIR filters Generalized z-transformed polynomial representation Integrated tool suite for loop retiming, pipelining, unfolding, and look ahead transformation. GUI for configuration and presentation.

Project Scope

Direct Form II IIR/FIR Filters Matrix representation for weighted (delays) directed graph

Algorithms Loop retiming Pipelining Unfolding transformation [Parhi] Critical path analysis – modified Bellman Ford algorithm for longest path search. Iteration bound analysis – Longest Path Matrix (LPM) algorithm and DFG transformation. Retiming algorithm [Parhi] with Floyd Warshall shortest path algorithm Pipelining Delay injection and retiming Retiming algorithm’s [Parhi] limitation: allows only forward retiming. Unfolding transformation [Parhi] Look-ahead transformation Direct look-ahead transformation [Parhi]

Delay Injection and Retiming Feed forward cutsets on right leg only. Cutset pipelining can be done with delay injection in all entering branches of output node. Delays are expected to be pushed back (away from output). Retiming algorithm [Parhi] has limitation; delays can only be pushed toward output.

Retiming algorithm limitation Shortest path lengths must be less than or equal to zero. [Parhi] convention: r(i) is retiming value for node I, denoting the number of delays pushed backward. R(i) is never positive implies: solution space for retiming is not fully exploited.

Graphical User Interface A 2nd order IIR example.

Retiming Retiming reduces critical path to as little as iteration bound. In this example, critical path reduces from 7 to 4.

Pipelining for IIR Critical path limited by feedback loops (4). Feed-forward cutset retiming applicable on right leg only. Pipelining provides no improvement for IIR

Pipelining for FIR Critical path reduced to minimum operation latency.

Unfolding Filter DFG as shown. As stated in [Parhi], critical path can be reduced to iteration bound with unfolding alone. Observation suggests: not true. Retiming must be applied to achieve iteration bound. For IIR/FIR, retiming alone achieves iteration bound.

Look-ahead Transformation Filter DFG as shown. Filter coefficients calculated for reference.

Thank you.