Iteration Bound. Lan-Da Van ( 倫 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C.

Transcription

1 Iteration Bound Lan-Da Van ( 倫 ) Ph. D. Department of Computer Science National Chiao Tung University Taiwan R.O.C. Spring 27 ldvan@cs.nctu.edu.tw

2 Outline Introduction Data Flow Graph (DFG) Representations Loop Bound and Iteration Bound Compute the Iteration Bound Longest Path Matrix Algorithm (LPM) Minimum Cycle Mean Method (MCM) Conclusion Lan-Da Van VLSI-DSP-2-2

3 Introduction Iteration bound maximum loop bound OR maximum non-loop bound Clock period critical path period cycle time /clock rate Sample rate throughput rate Impossible to achieve an iteration bound less than the theoretical iteration bound with infinite processors Lan-Da Van VLSI-DSP-2-3

4 Outline Introduction Data Flow Graph (DFG) Representations Loop Bound and Iteration Bound Compute the Iteration Bound Longest Path Matrix Algorithm (LPM) Minimum Cycle Mean Method (MCM) Conclusion Lan-Da Van VLSI-DSP-2-

5 for n DSP Program: Example to y ( n) ay( n ) + x( n) Intra - Iteration Period : Ak B k Inter - Iteration Period : B k A k + Data Flow Graph Lan-Da Van VLSI-DSP-2-

6 Critical Path: The path with the longest computation time among all paths that contain zero delays Loop: Directed path that begins and ends at the same node Loop Boundtl/wl where tl is the loop computation time and wl is the number of delays in the loop Critical Loop: the loops in which has maximum loop bound. Iteration Bound: maximum loop bound / non-loop bound i.e. a fundamental limit for recursive / non-recursive algorithms Loop Bounds in DFG Loop bounds : /2 u.t. (max) /3 u.t. / u.t. Lan-Da Van VLSI-DSP-2-6

7 Definition: T Iteration bound t max{ l l L w l } T (a) Iteration bound 6/2 3 u.t. (b) Iteration bound Max { 6/2 / } u.t. Lan-Da Van VLSI-DSP-2-7

8 Outline Introduction Data Flow Graph (DFG) Representations Loop Bound and Iteration Bound Compute the Iteration Bound Longest Path Matrix Algorithm (LPM) Minimum Cycle Mean Method (MCM) Conclusion Lan-Da Van VLSI-DSP-2-8

9 Longest Path Matrix Algorithm (LPM) A series of matrix is constructed and the iteration bound is found by examining the diagonal elements of the matrices. d : # of delays Compute L~Lm m 2 3 d ( m) l :The longest computation time of all paths from delay i j element d i to delay element d j that pass through exactly m- delays where the delay d i and delay d j are not included for m- delays. If no path exists then the value of l equals -. Lan-Da Van VLSI-DSP-2-9

10 Lan-Da Van VLSI-DSP-2- A DFG with Three Loops Using LPM (/3) () L

11 Lan-Da Van VLSI-DSP-2- A DFG with Three Loops Using LPM (2/3) ) max( ) ( () ) ( m j k k i K k m j i l l l + + (2) L (2) L

12 Lan-Da Van VLSI-DSP-2-2 A DFG with Three Loops Using LPM (3/3) (3) L () L } { max ) ( } {2... m l T m i i d m i 2 } { max } {2... d m i T

13 A Filter Using LPM () L T 8 8 (2) L max{ } Lan-Da Van VLSI-DSP-2-3

14 Minimum Cycle Mean Method (MCM). Construct the new graph G d (DFG G) transform from DFG decide the weight of each edge 2. Compute the maximum cycle mean construct the series of d+ vectors f (m) m 2 d An arbitrary reference node is chosen in Gd (called this node s). The initial vector f () is formed by setting f () (s) and setting the remaining nodes of f () to infinity. find the max cycle mean 3. Find the min cycle mean between each cycle Lan-Da Van VLSI-DSP-2-

15 A DFG with Three Loops Using MCM (/) VLSI Digital Signal Processing Systems delay > node longest path length (computation time) >weight w(ij) If no zero-delay path exists from delay d i to delay d j then the edge I -> j does not exist in Gd. Lan-Da Van VLSI-DSP-2-

16 A DFG with Three Loops Using MCM (2/) VLSI Digital Signal Processing Systems Longest path length path which pass through no delays longest : two loops that contain D a and Db max { 6 } 6 cycle mean 6/23 Lan-Da Van VLSI-DSP-2-6

17 A DFG with Three Loops Using MCM (3/) Cycle mean Average length of the edge in c (Cycle Loop ) Compute the cycle mean Lan-Da Van VLSI-DSP-2-7

18 Lan-Da Van VLSI-DSP-2-8 A DFG with Three Loops Using MCM (/) we will find d+ vectors f (m) m d (2) f (3) f 8 () f () f () f )) ( ) ( min( ) ( ) ( ) ( j w i i f j f m I i m +

19 A DFG with Three Loops Using MCM (/) T min ( max ( ( d ) ( i) f i { 2... d} m {2... d } d m f ( m) ( i) )) T min{ 2 2 } 2 Lan-Da Van VLSI-DSP-2-9

20 A Filter Using MCM T f () min{ 8 6} 8 f Lan-Da Van VLSI-DSP-2-2 f () (2) 2 2

21 Conclusion When the DFG is recursive the iteration bound is the fundamental limit on the minimum sample period of a hardware implementation of the DSP program. Two algorithms to compute iteration bound LPM and MCM were explored. Lan-Da Van VLSI-DSP-2-2

22 Self-Test Exercises STE: For the DFG shown in Fig. 2.2 the computation times of the nodes are shown in parentheses. Compute the iteration bound of this DFG using (a) the LPM algorithm and (b) the MCM algorithm. (Also see Problem 2. of the text book.) STE2: Repeat STE for the DFG shown in Fig. 2. assuming that addition and multiplication require and 2 u.t. respectively. (Also see Problem 2. of the text book.) Lan-Da Van VLSI-DSP-2-22