SCHEDULING SCHEDULING SOURCES- Mark Manwaring Kia Bazargan Giovanni De Micheli Gupta Youn-Long Lin M. Balakrishnan M. Balakrishnan Camposano, Camposano, J. Hofstede, J. Hofstede, Knapp, Knapp, MacMillen MacMillen Lin Lin
Dec 21, 2015
SCHEDULINGSCHEDULINGSOURCES- Mark ManwaringKia BazarganGiovanni De Micheli GuptaYoun-Long Lin
M. BalakrishnanM. BalakrishnanCamposano, Camposano, J. Hofstede, J. Hofstede, Knapp,Knapp,MacMillenMacMillenLinLin
Overview of Hardware SynthesisOverview of Hardware Synthesis
assign times to operations under given constraints
reduce the amount of hardware, optimize the design in general.
May be done with the consideration of additional constraints.
assign operations to physical resources under given constraints
Outline of schedulingOutline of scheduling• The scheduling problem.
1. Scheduling without constraints.
2. Scheduling under timing constraints.• Relative scheduling.
3. Scheduling under resource constraints.• The ILP model (Integer Linear Programming).• Heuristic methods (graph coloring, etc).
Timing constraints versus resource constraintsresource constraints
Example of scheduling:Example of scheduling:ASAPASAP
This is As Soon as Possible Scheduling (ASAP).
It can be used as a bound in other methods like ILP or when latency only is important, not area.
What is necessary to solve the What is necessary to solve the scheduling problem?scheduling problem?
• Circuit model:• Sequencing graph.• Cycle-time is given.• Operation delays expressed in cycles.
• Scheduling:• Determine the start times for the operations.• Satisfying all the sequencing (timing and resource) constraints.
• Goal:• Determine area/latencyarea/latency trade-off.
Do you remember what is latency?
•Scheduling affects
•Area: maximum number of concurrent operations of same type is a lower bound on required hardware resources.
•Performance: concurrency of resulting implementation.
Taxonomy of scheduling Taxonomy of scheduling problems to solveproblems to solve
1. Unconstrained scheduling.
2. Scheduling with timing constraints:• Latency.• Detailed timing constraints.
3. Scheduling with resource constraints.
• Related problems:• Chaining. What is chaining?• Synchronization. What is synchronization?• Pipeline scheduling.
• Simplest scheduling model
• All operations have bounded delays.
• All delays are in cycles.
• Cycle-time is given.
• No constraints - no bounds on area.
• Goal• Minimize latency.
Scheduling 2Scheduling 2
Scheduling problems are NP-hard, so all kind of heuristics are used
• ASAP – As soon as possible• ALAP• List scheduling – Resource Constrained algorithms• Force directed algorithms – time constrained• Path based• Percolation algorithms• Simulated annealing• Tabu search and other heuristics• Simulated evolution• Linear Programming• Integer Linear Programming - time constrained
What are types of Scheduling What are types of Scheduling Algorithms?Algorithms?
• Time & Resource Tradeoff
• Scheduling is temporal binding
Types of Scheduling ProcessesTypes of Scheduling Processes
Assignment of operations to time (control Assignment of operations to time (control steps) within given constraints and steps) within given constraints and
minimizing a cost functionminimizing a cost function
• Time-constrained• Resource-constrained
• With or withoutor without control (conditions)• With or withoutor without iterations (infinite loops)
• Constructive• Iterative
Simplest model of Simplest model of schedulingscheduling
• All operations have bounded delays.
• All delays are expressed in numbers of cycles of a single one-phase clock.• Cycle-time is given.
• No constraints - no bounds on area.
• Goal:• Minimize latency.
• Unconstrained scheduling used when
• Dedicated resources are used.
• Operations differ in type.
• Operations cost is marginal when compared to that of steering logic, registers, wiring, and control logic.
• Binding is done before scheduling: resource conflicts solved by serializing operations sharing same resource.
• Deriving bounds on latency for constrained problems.
Minimum Latency Unconstrained Minimum Latency Unconstrained Scheduling ProblemScheduling Problem
Minimum-latency Minimum-latency unconstrainedunconstrained
scheduling problemscheduling problem
• Given a set of operations V with set of corresponding integer delays D and a partial order on the operations E:
• Find an integer labeling of the operations
: V --> Z : V --> Z ++ , such that:
• t i = (v i ),
• t i t j + d j i, j such that (v j , v i ) E
• and tn is minimum.
di
dj
(v j , v i ) t i t j + d j
t j
t i
Input to di must be stable
Example of using Example of using mobilitymobility
• Operations with zero mobility:
• {v 1, v 2, v 3, v 4, v 5 }.
• They are on the critical path.
• Operations with mobility one:
• {v 6 , v 7 }.
• Operations with mobility two:
• {v 8 , v 9 , v 10 , v 11 }.
mobility two:
1. Start from ALAP
2.Use mobility to improve
Classes of scheduling algorithmsClasses of scheduling algorithms
Operation Scheduling FormalisationOperation Scheduling Formalisation
• ALAP solves a latency-constrained problem.
• Latency bound can be set to latency computed by ASAP algorithm.
• Mobility• Defined for each operation.• Difference between ALAP and ASAP schedule.• Zero mobility implies that an operation can be started
only at one given time step.• Mobility greater than 0 measures span of time interval in
which an operation may start.
• Slack on the start time.
Latency Constrained Latency Constrained SchedulingScheduling
• Motivation• Interface design.• Control over operation start time.
• Constraints• Upper/lower bounds on start-time difference of any operation pair.
• Minimum timing constraints between two operations• An operation follows another by at least a number of prescribed time
steps• lij 0 requires tj ti + lij
• Maximum timing constraints between two operations• An operation follows another by at most a number of prescribed time
steps• uij 0 requires tj ti + uij
Scheduling under Detailed Scheduling under Detailed Timing ConstraintsTiming Constraints
• Example• Circuit reads data from a bus, performs computation, writes result
back on the bus.
• Bus interface constraint: data written three cycles after read.
• Minimum and maximum constraint of 3 cycles between read and write operations.
• Example• Two circuits required to communicate simultaneously to external
circuits.
• Cycle in which data available is irrelevant.
• Minimum and maximum timing constraint of zero cycles between two write operations.
Scheduling under Detailed Scheduling under Detailed Timing ConstraintsTiming Constraints
Scheduling under Scheduling under detailed detailed timing timing constraintsconstraints
• Motivation:• Interface design.• Control over
operation start timeoperation start time.
• Constraints:• Upper/lower bounds on start-time
difference of any operation pair.
• Feasibility of a solution.
Constraint graph modelConstraint graph model• Start from a sequencing graph.
• Model delays as weights on edges.
• Add forward edges for minimum constraints.
• Edge (vi , vj) with weight lij => t j t i +lij .
• Add backward edges for maximum constraints.• Edge (vi , vj) with weight:
• - u ij => t j t i + uij
• because t j t i + uij => t i t j - uij
Add this edge for max constraint
Add this edge for min constraint
• Unbounded delays• Synchronization.• Unbounded-delay operations (e.g.
loops).
• Anchors.• Unbounded-delay operations.
• Relative scheduling• Schedule ops w.r. to the anchors.• Combine schedules.
Method for Scheduling with Method for Scheduling with Unbounded-Delay OperationsUnbounded-Delay Operations
• Presence of maximum timing constraints may prevent existence of a consistent schedule.
• Required upper bound on time distance between operations may be inconsistent with first operation execution time.
• Minimum timing constraints may conflict with maximum timing constraints.
• A criterion to determine existence of a schedule:
• For each maximum timing constraint uij
• Longest weighted path between vi and vj must be uij
Method for Scheduling Under Method for Scheduling Under Detailed Timing ConstraintsDetailed Timing Constraints
Example of using Example of using constraint graph with constraint graph with minimum and maximum minimum and maximum constraintsconstraints
Sequencing graph Constraint graph
explain
So now we can calculate this table from
sequencing graph
• Weight of longest path from source to a vertex is the minimum start time of a vertex.
• Bellman-Ford or Lia-Wong algorithm provides the schedule.
• A necessary condition for existence of a schedule is constraint graph has no positive cycles.
Method for Scheduling Under Method for Scheduling Under Detailed Timing ConstraintsDetailed Timing Constraints
Methods for Scheduling Methods for Scheduling Under Detailed Timing Under Detailed Timing
ConstraintsConstraints
Shown in last slide
Will follow
Methods for schedulingMethods for schedulingunder under detailed timingdetailed timing constraints constraints• Start from the Sequencing Graph. Assumption:
• All delays are fixed and known.
• Set of linear inequalities.
• Longest path problem.
• Algorithms for the longest path problemlongest path problem were discussed in Chapter 2 of De Micheli:• Bellman-Ford,Bellman-Ford,• Liao-Wong.Liao-Wong.
Bellman-Ford’s Bellman-Ford’s algorithmalgorithm
BELLMAN_FORD(G(V, E, W))BELLMAN_FORD(G(V, E, W))
{
s 1 0 = 0;
for (i = 1 to N)
s 1 i =w 0, i ;
for (j =1 to N){
for (i =1 to N){
s j+1 i = min { s j i , (s j k +w q, i )},
}
if (s j+1 i == s j i i ) return (TRUE) ;
}
return (FALSE)
}
ki
LongestLongest path problem path problem
• Use shortest path algorithms:Use shortest path algorithms:• by reversing signs on weights.
• Modify algorithmsModify algorithms:• by changing min with max.
• Remarks:Remarks:• Dijkstra’s algorithm is not relevant.• Inconsistent problem:
• Positive-weighted cycles.
Example – Bellman-FordExample – Bellman-Ford
• Iteration 1: l 0 =0, l 1 =3, l 2 = 1, l 3 =.
• Iteration 2: l 0 =0, l 1 =3, l 2 =2, l 3 =5.
• Iteration 3: l 0 =0, l 1 =3, l 2 =2, l 3 =6.
Use shortest path algorithms: by reversing signs on weights.
source
3 -1=2
1+4=5
2+4=6
Liao-Wong’s Liao-Wong’s algorithmalgorithm
LIAO WONG(G( V, E F, W))
{
for ( i = 1 to N)
l 1 i = 0;
for ( j = 1 to |F|+ 1) {{
foreach vertex v i
l j+1 i = longest path in G( V, E,W E ) ;
flag = TRUE;
foreach edge ( v p, v q) F {
if ( l j+1 q < l j+ 1
p + w p,q ){
flag = FALSE;
E = E ( v 0 , v q) with weight ( l j+ 1 p + w p,q)
}
}
if ( flag ) return (TRUE) ;
}}
return (FALSE)
}
adjust
Example Example – – Liao-WongLiao-Wong
• Iteration 1:Iteration 1: l 0 = 0, l 1 = 3, l 2 = 1, l 3 = 5.
• Adjust: add edge (v 0 , v 2 ) with weight 2.
• Iteration 2:Iteration 2: l 0 = 0, l 1 = 3, l 2 = 2, l 3 = 6.
Only positive edges from (a)
(b) adjusted by adding longest path from node 0 to node 2
Looking for longest path from node 0 to node 3
Method for schedulingMethod for schedulingwith with unbounded-delayunbounded-delay operations operations• Unbounded delays:
• Synchronization.• Unbounded-delay operations (e.g. loops).
• Anchors.• Unbounded-delay operations.
• Relative scheduling:• Schedule operations with respect to the anchors.• Combine schedules.
Relative scheduling Relative scheduling methodmethod
• For each vertexFor each vertex:• Determine relevant anchor set R(.).R(.).• Anchors affecting start time.• Determine time offset from anchors.
• Start-time:• Expressed by:
• Computed only at run-time because delays of anchors are unknown.
Relative scheduling under Relative scheduling under timingtimingconstraintsconstraints
• Problem definition:Problem definition:• Detailed timing constraints.• Unbounded delay operations.
• Solution:Solution:• May or may not exist.• Problem may be ill-specified.
Relative scheduling Relative scheduling under timingunder timingconstraintsconstraints
• Feasible problem:Feasible problem:• A solution exists when unknown delays are zero.
• Well-posed problem:Well-posed problem:• A solution exists for any value of the unknown delays.
• Theorem:Theorem:• A constraint graph can be made well-posed iff there
are no cycles with unbounded weights.
(a) & (b) Ill-posed constraint (c) well-posed constraint
Example ofExample of Relative scheduling under timing constraintsRelative scheduling under timing constraints
Relative scheduling approachRelative scheduling approach
1. Analyze graph:1. Detect anchors.
2. Well-posedness test.
3. Determine dependencies from anchors.
2. Schedule operations with respect to relevant anchors:• Bellman-Ford, Liao-Wong, Ku algorithms.
3. Combine schedules to determine start timesstart times:
Example of control-unit synthesized for Example of control-unit synthesized for Relative scheduling Relative scheduling
Scheduling Scheduling ProblemProblemScheduling Problem Scheduling Problem
formalizationformalization
ASAP scheduling algorithmASAP scheduling algorithmASAP ( Gs(V, E)){
Schedule v0 by setting t S 0 = 1;
repeat { Select a vertex vi whose predecessors are all
scheduled;
Schedule vi by setting t S i = max t S j + dj ;
}
until (vn is scheduled) ;
return (t S );}
j:(vj,vi)E
Similar to breadth-first search
ALAP scheduling algorithmALAP scheduling algorithm
• As Late as Possible - ALAP• Similar to depth-first search
Remarks on ALAP and mobilityRemarks on ALAP and mobility• ALAP solves a latency-constrained problem.
• Latency bound can be set to latency computed by ASAP algorithm. <-- using bounds, also in other approaches
• Mobility:• Mobility is defined for each operation.• Difference between ALAP and ASAP schedule.
• What is mobility?number of cycles that I can move upwards or downwards the operation
• Slack on the start time.
Scheduling under Scheduling under
resource constraintsresource constraints• Classical Classical scheduling problem.
• Fix area bound - minimize latency.
• The amount of available resources affects the achievable latency.
• DualDual problem:• Fix latency bound - minimize resources.
• Assumption:• All delays bounded and known.
Minimum latencyMinimum latency resource-constrained resource-constrainedscheduling problemscheduling problem
• Given a set of operations V with integer delays D a partial order on the operations E, and upper bounds {ak ; k = 1, 2,…,nres}:
• Find an integer labeling of the operations : V --> Z+ such that :• t i = '(v i ),• t i t j +d j 8 i; j s:t: (v j ; v i ) 2 E,• jfv i jT (v i ) = k and t i l < t i +d i gj a k• and tn is minimum.
Scheduling under Scheduling under resource resource constraintsconstraints
• Intractable problem.
• Algorithms:• Exact:
• Integer linear program.• Hu (restrictive assumptions).
• Approximate:• List scheduling.• Force-directed scheduling.
Resource Constraint SchedulingResource Constraint SchedulingML-RCSML-RCS: :
minimize latency, minimize latency, bound on resourcesbound on resources
MR-LCSMR-LCS: : minimize resources, minimize resources,
bound on latencybound on latency
Algorithm of Hu for Algorithm of Hu for Resource Constraint SchedulingResource Constraint Scheduling
ML-RCSML-RCS: : minimize latency, minimize latency,
bound on resourcesbound on resources
Example of using Example of using Hu’s algorithmHu’s algorithm
• Assumptions:• One resource type only.• All operations have unit delay.
Distance from sink
• Additional assumptions:• Graph is a forest.
• Algorithm:• Label vertices with distance
from sink.• Greedy strategy.• Exact solution.
Hu's Hu's algorithmalgorithm
Algorithm Algorithm
Hu's schedule with Hu's schedule with aa resources resources• Set step, l = 1.• Repeat until all operations are
scheduled:• Select s a resources with:
• All predecessors scheduled.• Maximal labels.
• Schedule the s operations at step l.• Increment step l = l +1.
•All operations have unit delay.
•All operations have the same type.
• Minimum latency with a = 3 resources.• Step 1:
• Select {v 1 , v 2 , v 6 }.
• Step 2: • Select {v 3 , v 7 , v 8 }.
• Step 3: • Select {v 4 , v 9 , v 10 }.
• Step 4: • Select {v 5 , v 11 }.
Algorithm Algorithm
Hu's schedule with Hu's schedule with aa resources resources
We always select 3 resources
• Minimum latency with a = 3 resources.• Step 1:
• Select {v 1 , v 2 , v 6 }.
• Step 2: • Select {v 3 , v 7 , v 8 }.
• Step 3: • Select {v 4 , v 9 , v 10 }.
• Step 4: • Select {v 5 , v 11 }.
Algorithm Algorithm
Hu's schedule with Hu's schedule with aa resources resources
We always select 3 resources
Exactness of Hu's algorithmExactness of Hu's algorithm• Theorem1:
• Given a DAG with operations of the same type.
• a is a lower bound on the number of resources to complete a schedule with latency .
• is a positive integer.• Theorem2:
•Hu's algorithm applied to a tree with unit-cycle resources achieves latency with a resources.
• Corollary:• Since a is a lower bound on the number of resources for achieving , then is minimum.
• Assumptions• a1 = 2 multipliers with delay
1.• a2 = 2 ALUs with delay 1.
• First Step• U1,1 = {v1, v2, v6, v8}• Select {v1, v2}• U1,2 = {v10}; selected
• Second step• U2,1 = {v3, v6, v8}• select {v3, v6}• U2,2 = {v11}; selected
• Third step• U3,1 = {v7, v8}• Select {v7, v8}• U3,2 = {v4}; selected
• Fourth step• U4,2 = {v5, v9}; selected
•Priority list heuristics.•Assign a weight to each vertex indicating its scheduling priority•Longest path to sink.•Longest path to timing constraint.
List Scheduling List Scheduling AlgorithmsAlgorithms
List schedulingList scheduling algorithmsalgorithms
• Heuristic method for:
• 1. Minimum latency subject to resource bound.
• 2. Minimum resource subject to latency bound.
• Greedy strategy (like Hu's).
• General graphs (unlike Hu's).
List scheduling algorithmList scheduling algorithmfor minimum resource Usagefor minimum resource Usage
• Candidate Operations Ul,k
• Operations of type k whose predecessors are scheduled and completed at time step before l
• Unfinished operations Tl,k are operations of type k that started at earlier cycles and whose execution is not finished at time l
• Note that when execution delays are 1, Tl,k is empty.
}),(:)(:{, Evvjl dtkvΤypeVvU ijjjiikl and
}),(:)(:{, Evvjl dtkvΤypeVvT ijjjiikl and
List scheduling algorithmList scheduling algorithmfor minimum latency for resource boundfor minimum latency for resource bound
List scheduling algorithmList scheduling algorithmfor minimum latency for resource boundfor minimum latency for resource bound
LIST_L ( G(V, E), a ) {l = 1;repeat {
for each resource type k = 1, 2,…,nres {
Determine candidate operations U l,k ;
Determine unfinished operations T l,k ;
Select Sk U l,k vertices, such that |Sk| + |T l,k| a k ;
Schedule the S k operations at step l;
}l = l +1;}until (vn is scheduled) ;
return (t );
}
• Assumptions:• a 1 = 3 multipliers with
delay 2.
• a 2 = 1 ALUs with delay 1.
List scheduling algorithmList scheduling algorithmfor minimum latencyfor minimum latency
1. List scheduling algorithm1. List scheduling algorithmfor minimum latency for resource boundfor minimum latency for resource bound
Now we needtwo time units for multiplier
• Assume =4• Let a = [1, 1]T
• First Step• U1,1 = {v1, v2, v6, v8}• Operations with zero slack {v1, v2}• a = [2, 1]T
• U1,2 = {v10}• Second step
• U2,1 = {v3, v6, v8}• Operations with zero slack {v3, v6}• U2,2 = {v11}
• Third step• U3,1 = {v7, v8}• Operations with zero slack {v7, v8}• U3,2 = {v4}
• Fourth step• U4,2 = {v5, v9}• Both have zero slack; a = [2, 2]T
• Assumptions:• a 1 = 3 multipliers with delay 2.
• a 2 = 1 ALUs with delay 1.
1. 1. List schedulingList scheduling algorithm algorithmfor for minimum latencyminimum latency for for resource boundresource bound
• Solution• 3 multipliers as assumed• 1 ALU as assumed• LATENCY 7
Compute the latest possible start times tL by ALAP ( G(V,E), );
2.2. List schedulingList scheduling algorithm algorithm for for minimum resource usageminimum resource usage
• overlap
2.2. Example: List scheduling algorithm for Example: List scheduling algorithm for minimum resource usageminimum resource usage
From ALAP
We can move them
• Solution• 2 multipliers • 1 ALU • LATENCY 4
Now we assume the same time of each operation
List Scheduling Algorithm example List Scheduling Algorithm example ML-RCSML-RCS
** * *
* *
-
-
+
+
<
Assume 3 multipliers Assume 1
ALU
t=0mul 3m 1 ALU
t=1 1 ALU
t=2 3mul
t=3 1 ALU
t=4 1 ALU
t=5 1 ALUNow we assume the same time of
each operation
Other way of explanation of the same as in last
slide
List Scheduling ExampleList Scheduling Example• The scheduled DFG• DFG with mobility labeling (inside <>)
• ready operation list/resource constraint
Static-List SchedulingStatic-List Scheduling
• DFG
• Partial schedule of five nodes
• Priority list
The final schedule
Scheduling with chainingScheduling with chaining• Consider propagation delays of resources not in terms of cycles.
• Use scheduling to chain multiple operations in the same control step.
• Useful technique to explore effect of cycle-time on area/latency trade-off.
• Algorithms:• ILP, • ALAP/ASAP, • List scheduling.
SummarySummary
• Scheduling determines area/latency trade-off.• Intractable problem in general:
• Heuristic algorithms.• ILP formulation (small-case problems).
• Chaining:• Incorporate cycle-time considerations.
Finite Impulse Finite Impulse Response FilterResponse Filter
D
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IN[i]
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c0 c3c2c1 c7c6c5c4 c10c9c8
A0
A0
A3A2A1 A6A5A4 A8A7 A9
B0 B1
B1
B4B3B2 B5 B7B6 B8 B9
IN[i-1] IN[i-2] IN[i-6]IN[i-5]IN[i-4]IN[i-3] IN[i-8]IN[i-7] IN[i-10]IN[i-9]
This can be directly used for synthesis with 11 multipliers, 10 adders and 10 registers. But the latency would be 1 multiplier + 10 adders
Example 1
FIR FIR SchedulingScheduling+
+
+
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+
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10
9
8
7
6
5
4
3
2
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c0 c1
c2
c3
c4
c5
c6
c7
c8
c9
c10
A0
A4
A3
A2
A1
A5
A6
A7
A8
A9
B0
B3
B2
B1
B4
B6
B5
B9
B8
B7
IN[i] IN[i-1]
IN[i-6]
IN[i-5]
IN[i-4]
IN[i-3]
IN[i-2]
IN[i-8]
IN[i-7]
IN[i-9]
IN[i-10]
OUT
1 adder, 1 multiplier
2 multipliers
This is also bad as we use both
multipliers only in stage 1
There are many ways to solve this problem,
transform the tree, schedule, allocate,
pipeline, retime
Example: Cascade Filter Example: Cascade Filter optimizationoptimization
IN[i] OUT+ +
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*
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A0[i]
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c0c1
c2
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C0[i-1]
C0[i-2]c5
c4 c3
C1
C5
C4
C2
C3
Look to ci ci coefficients and compare with two previous slides.
Example 2
This is not FIR,
six coefficients
Cascade Cascade Filter Filter
Scheduling, Scheduling, cont example cont example
22
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+
*********
+
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1
2
3
4
5
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c1
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A5
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c0[i] C3
C1
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A4
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c0c1
c2
BC0[i]
C0[i-1]
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C1
C5
C4
C2
C3
2 mul, 2 add
Infinite Impulse Infinite Impulse Response FilterResponse Filter
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D
+ +**
OUT
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A2 C1
C3
B
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C4
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D
D1D2
C5C6A5A6
Example 3
IIR Filter, 10 coefficients
IIR Filter Scheduling continuedIIR Filter Scheduling continued
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1
7
3
4
6
5
2
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c4
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C0[i-2]C0[i-1]
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D0[i-1]
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D1
D2
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B
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A4
A6
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OUT
IN
IIR Filter, 10 coefficients
Two adders two multipliers