Chapter 3
Algorithm Representation &
Iteration Bound
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Representations of DSP Algorithms
Mathematical formulations
Behavioral 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 representations
Block diagrams
Signal 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 edges
Node: computation or task
Directed 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 structure
Applicable to single-input single output system
Reverse the directions of all edges
Exchange the input output node
Retain 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 transposition
can be applied to MIMO systems described by
symmetric transform matrices
More on SFG
Applicable to linear network
Cannot be used to described multi-rate system
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Data Flow Graph (1)
DFG
Node: 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
add
mpy
Block diagram
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Conventional DFG
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Synchronous DFG
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Data Flow Graph (2)
Applications: high level synthesis
Firing rules
A node can fire whenever all the input data are available
Concurrency: multiple nodes can be fired simultaneously
Data driven (implicit) scheduling
Precedence constraint
Intra-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 DFG
Number 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 loops
Impose 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
t
Iteration k-1
Iteration k
Iteration k+1
Iteration k+2
Iteration period
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Data Flow Graph Representations
For n = 0 to ∞
y(n) = ay(n-1) + x(n)
Intraiteration
Execution
time of a
node
Critical path
AB
Interiteration
Iteration – execution of each DFG node once
Precedence constraints
Intra-iteration – no delay on edge
Inter-iteration – at least one delay on edge
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Critical Path
Critical path of a DFG
The path with the longest computation time among all
paths containing zero delays
The minimum computation time for one iteration of the
DFG
6→3→2→1
5→3→2→1
Iteration period = 5 u.t.
Iteration bound
Recursive DFG has a lower
bound on the shortest
iteration period
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Loop bound and iteration bound (1)
Loop bound
Minimum 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 threads
Two 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→A
Loop 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
T 
max
 tl 


 wl 
max
 6 11 
 ,   11 u.t.
2 1 
l L
T 
l L
Algorithms to find T∞
Longest path matrix algorithm
Minimum cycle mean algorithm
Negative cycle detection algorithm
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