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Embedded Computer Architecture 2. (BOCA). Bijzondere Onderwerpen Computer Architectuur Block B Index Transformations. The convolution algorithm.

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Boca

Embedded Computer Architecture 2

(BOCA)

Bijzondere Onderwerpen

Computer Architectuur

Block B

Index Transformations


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The convolution algorithm

The behavior of a linear time-invariant system can be fully described by its impulse response h, i.e. the response on the output to a single unit pulse on the input.

The response y on the output to an input stream x then follows from:

or


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The convolution algorithm

continue

with j = z – i, we obtain:

and if the impulse response h is finite (bounded), i.e.

we get


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x

y

F

The dependency graph of the convolution

We will investigate the dependency graph for both the convolution formulas

and


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x

y

F

The DG of the convolution

In which h is the impulse response defining the system F.

Recurrent relations:


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x

y

F

The DG of the convolution

In which h is the impulse response defining the system F.

Recurrent relations:

Let x be a bounded sequence of length NX, i.e.:

for

and

And let hbe a bounded sequence of length NH, i.e.:

for

and


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hence

The convolution

and

for

and

and

for

z-i<0

NX=NH= 5

i

-1

0

1

2

3

4

5

-1

0

1

2

or

3

4

5

z

6

7

8

i>4

or

i<0

9

z-i>4


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and

Hence, with

we obtain

for

so

and with

we obtain

for

so

so

can be replaced by

The convolution

Recall:


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implies

for all i

Notice that

i.e. z < 0 or z > NX + NH-2

implies

for all i, and thus yz= 0

The convolution

Recall:

Hence, yzis a bounded sequence too.

Consequently the recurrent relations become:

for


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global variables

xi

hz-i

hz-i

xi

x

sz,i-1

sz,i

sz,i-1

sz,i

Fz,i

+

local variables

The DG of the convolution

Implementation basic cell

Basic cell


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x0

h0

x1

y0=s0,0

0

h1

x2

y1=s1,1

0

x3

h2

y2=s2,2

0

h3

y3=s3,3

0

h4

y4=s4,3

0

y5=s5,3

0

y6=s6,3

0

0

y7=s7,3

The DG of the convolution

Dependency graph

(globally recursive)

i

z

NX = 4

NH = 5


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global dependencies (broadcasts)

Transformation from a globally recursive to a locally recursive DG

Recall the recurrent relations of the globally recursive graph:

for

for


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h0

x0

y0=so,o

0

h1

x1

y1=s1,1

0

h2

x2

y2=s2,2

0

h3

x3

y3=s3,3

0

h4

y4=s4,3

0

y5=s5,3

0

y6=s6,3

0

y7=s7,3

0

The DG of the convolution

Dependency graph

(locally recursive)

NX = 4

NH = 5

Easier:

First make the graph and then the recurrent relations


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The DG of the convolution

Alternatives for the locally recursive dependency graph

global local

for xi : 2 alternatives

total : 8 alternatives

for hi : 2 alternatives

for  : 2 alternatives


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h0

0

h1

0

h2

0

h3

0

h4

0

The DG of the convolution

One of the alternative

locally recursive graphs

y0=so,o

0

y1=s1,0

0

y2=s2,0

0

y3=s3,0

y4=s4,0

x0

y5=s5,1

x1

NX = 4

NH = 5

y6=s6,2

x2

y7=s7,3

x3


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x

y

F

The dependency graph of the convolution

We will investigate the dependency graph for both the convolution formulas

and

We have investigated this one

This one can be investigated

similarly


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xz-i

hi

sz,i-1

sz,i

Fz,i

The DG of the convolution

h0

x0

y0=so,o

0

h1

x1

y1=s1,1

0

h2

x2

y2=s2,2

0

x3

h3

y3=s3,3

0

h4

y4=s4,3

0

y5=s5,3

0

NX = 4

NH = 5

Basic cell

y6=s6,3

0

y7=s7,3

0


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The DG of the convolution

x0

The locally recursive DG

h0

y0=s0,0

0

x1

h1

y1=s1,1

0

x2

h2

y2=s2,2

0

x3

h3

y3=s3,3

0

h4

y4=s4,3

0

y5=s5,3

0

NX = 4

NH = 5

y6=s6,3

0

y7=s7,3

0


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The DG of

versus

x0

h0

h0

y0=s0,0

x0

0

x1

y0=s0,0

0

h1

h1

y1=s1,1

x1

0

x2

y1=s1,1

0

h2

h2

x2

y2=s2,2

0

x3

y2=s2,2

0

h3

h3

x3

y3=s3,3

0

x4

y3=s3,3

0

h4

h4

y4=s4,3

x4

0

x5

y4=s4,4

0

y5=s5,3

x5

0

x6

y5=s5,5

0

x6

y6=s6,3

0

y6=s6,6

0

y7=s7,3

x7

y7=s7,7

0

0


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Algebraic description of a DG

Node description

Each node in an N-dimensional DG can be described by a vector in an N-dimensional index space.

1

2

3

4

5

6

7

8

(6,3)


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our choice

Algebraic description of a DG

Edge description

Alternative 1

Alternative 2

3

3

2

2

source

destination

source

destination

6

6

(0,1)

(6,2)

(6,2)

(6,3)


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Algebraic description of a DG

A dependency graph is fully described by:

  • The set of nodes

  • The set of intermediate edges

  • The set of input edges

  • The set of output edges

virtual nodes:

in which


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Algebraic description of a DG

0,0

1,0

1,1

nodes: Vn= {(0,0),(1,0),(1,1)}

intermediate edges: Eint= {((0,0),(1,0)),((1,0),(0,1))}

input edges: Ein= {((0,-1),(0,1)),((1,-1),(0,1)),((-1,0),(1,0)),((0,1),(1,0))}

output edges: Eout= {((0,0),(0,1)),((1,1 ),(0,1)),((1,0),(1,0)),((1,1),(1,0))}


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Shifting the origin = adding a vector

is vector in the new description

is vector in the old description

causes a shift of the origin

is a non-singular matrix

Index transformations

An N-dimensional dependency graph can be linearly transformed by:

Just cosmetic.

Does not give new designs

Chosing a different base with the same origin

= multiplying with a non-singular matrix A

So

In which:


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Index transformations

Shifting the origin with a vector is just cosmetic and does not lead to any new designs. So we refrain from this operation.

Hence,

Because A is non-singular, the inverse tranform is always possible.

All vectors in the algebraic description of the DG have to be

multiplied with A.


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0,0

0,1

0,2

0,0

1,1

1,2

1,0

0,0

1,1

2,2

1,-1

2,0

2,1

2,2

2,0

2,-2

3,-1

4,0

Examples of index transformations

Transformation

Transformation

An edge consists of two vectors

Both are multiplied with A


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The DG of

versus

x0

h0

h0

x0

y0=s0,0

0

y0=s0,0

0

x1

h1

h1

x1

y1=s1,1

y1=s1,1

0

0

x2

h2

h2

x2

y2=s2,2

0

y2=s2,2

0

x3

h3

h3

x3

y3=s3,3

y3=s3,3

0

0

x4

h4

h4

x4

y4=s4,4

y4=s4,4

0

0

x5

x5

y5=s5,4

y5=s5,5

0

0

x6

x6

y6=s6,6

y6=s6,4

0

0

0

x7

y7=s7,4

0

0


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The DG of

versus

Applying the transformation

h0

(0,0) → (0,0)

x0

y0=s0,0

0

h1

x1

(1,0) → (1,1)

y1=s1,0

0

h2

(1,1) → (1,0)

x2

y2=s2,0

0

h3

x3

(2,0) → (2,2)

y3=s3,0

0

h4

(2,1) → (2,1)

x4

0

y4=s4,0

(2,2) → (2,0)

x5

0

y5=s5,1

x6

0

y6=s6,2


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The DG of

versus

Applying the transformation

x0

h0

h0

x0

y0=s0,0

y0=s0,0

0

0

x1

h1

h1

x1

y1=s1,0

y1=s1,1

0

0

x2

h2

h2

x2

y2=s2,0

0

y2=s2,2

0

x3

h3

h3

x3

y3=s3,0

y3=s3,3

0

0

x4

h4

h4

x4

y4=s4,4

0

0

y4=s4,0

x5

x5

y5=s5,4

0

0

x6

y5=s5,1

x6

y6=s6,4

0

0

y6=s6,2

y7=s7,4


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Examples of index transformations

The previous transformation elucidates that index transformations on a formula are identical to the transformations in the N-dimensional index space.

Hence, transformation in the N-dimensional index space might support index transformation on a formula.


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