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Chapter 3 Algorithm Representation & Iteration Bound

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Representations of DSP Algorithms

Mathematical formulationsBehavioral description languages

Applicative language Represents a set of equations satisfied by the variables, e.g. Silage

Perspective language Explicitly specify the order of assignment, e.g. C and other HLLs

Descriptive language Represents the structure of a DSP system, e.g. VHDL, Verilog

Graphical representationsBlock diagramsSignal flow graph (SFG)Data flow graph (DFG)Dependence graph (DG)

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Block Diagrams (1)

Consists of functional blocks connected with directed edges

Functional block, e.g. Add, Mult

Unit delay element

Directed edge representing the data flow between blocks

Basic blocks

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Block Diagrams (2)

3-tap FIR example

Alternative block diagram with data broadcast

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Signal Flow Graph (1)

A collection of nodes and directed edgesNode: computation or taskDirected edge (j,k)

a linear transformation from node j to node k Usually as constant gain multiplier or delay elements

Widely used in digital filter structures

Flow graph reversal (transposition)A transform to obtain equivalent structureApplicable to single-input single output systemReverse the directions of all edgesExchange the input output nodeRetain the edge gain and edge delay

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Signal Flow Graph (2)

SFG of a 3-tap FIR filter

Original SFG

Transposed SFG

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Signal Flow Graph (3)

Limitations of transpositioncan be applied to MIMO systems described by symmetric transform matrices

More on SFGApplicable to linear network

Cannot be used to described multi-rate system

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Data Flow Graph (1)

DFGNode: computation (function or subtask)

Directed edge: data path or communication between nodes

Associated edge delay: non-negative

Associated node delay: execution time of each node

Block diagram Conventional DFG Synchronous DFG

add

mpy

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Data Flow Graph (2)

Applications: high level synthesis

Firing rulesA node can fire whenever all the input data are available

Concurrency: multiple nodes can be fired simultaneously

Data driven (implicit) scheduling

Precedence constraintIntra-iteration: imposed by edge with no delay

Inter-iteration: imposed by edge with delay

fine-grain (atomic) v.s. coarse grain DFG

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Data Flow Graph (3)

3-tap FIR filter example

Direct form

Transpose form

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Data Flow Graph (4)

Synchronous DFGNumber of data samples produced or consumed by each node is specified a priori

Single rate system

Multi-rate system: different nodes working on different frequencies

Multi-rate system can be represented by a single rate system via unfolding (unrolling)

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Introduction to Iteration bound

DSP algorithms often contain feedback loopsImpose an inherent lower bound on the achievable iteration or sample period

Iteration bound

Impossible to achieve an iteration period less than the iteration bound even with infinite HW

Iteration kIteration k-1

Iteration k+1Iteration k+2

t

Iteration period

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Data Flow Graph Representations

For n = 0 to ∞y(n) = ay(n-1) + x(n)

Iteration – execution of each DFG node oncePrecedence constraints

Intra-iteration – no delay on edgeInter-iteration – at least one delay on edge

Execution time of a

node

Inter-iteration

Intra-iteration

Critical pathAB

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Critical Path

Critical path of a DFGThe path with the longest computation time among all paths containing zero delaysThe minimum computation time for one iteration of the DFG6→3→2→15→3→2→1Iteration period = 5 u.t.

Iteration boundRecursive DFG has a lowerbound on the shortestiteration period

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Loop bound and iteration bound (1)

Loop boundMinimum time to execute one loop in the DFG

tl / wl: tl = loop computation time, wl = number of delays in the loop

(a) loop bound = (4+2)/2 = 3

(b) loop bound 1 = (4+2)/2 = 3

(b) loop bound 2 = (2+4+5)/1 = 11

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Loop bound and iteration bound (2)

In (a), two independent sets of computing threadsTwo iterations in every 6 u.t. iteration period = 3 u.t.

A0→B0 A2→B2 A4→B4 A6→…

A1→B1 A3→B3 A5→B5 A7→…

In (b)Loop 1: A→B→ALoop 2: A→B→C→A (critical loop)

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Loop bound and iteration bound (3)

Loop bound of the critical loop iteration bound of the DSP algorithm

Algorithms to find T∞

Longest path matrix algorithm

Minimum cycle mean algorithm

Negative cycle detection algorithm

u.t. 111

11,

2

6max

max

Ll

l

l

Ll

T

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tT