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Determining Fixed-Point Formats using the Worst-Case Peak Gain measure

Anastasia Volkova, Thibault Hilaire, Christoph Lauter

Sorbonne Universit´ es, University Pierre and Marie Curie, LIP6, Paris, France

ASILOMAR 49 November 10, 2015

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Motivation

0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 1

Filter implementation flow:

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Motivation

0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 1

Filter implementation flow: Transfer function generation

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Motivation

0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 1

Filter implementation flow: Transfer function generation State-space, DFI, DFII, . . .

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Motivation

0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 1

Filter implementation flow: Transfer function generation State-space, DFI, DFII, . . . Software or Hardware implementation

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Motivation

0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 1

Filter implementation flow: Transfer function generation

! Coefficient quantization

State-space, DFI, DFII, . . . Software or Hardware implementation

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Motivation

0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 1

Filter implementation flow: Transfer function generation

! Coefficient quantization

State-space, DFI, DFII, . . .

! Large variety of structures with no common quality criteria

Software or Hardware implementation

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Motivation

0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 1

Filter implementation flow: Transfer function generation

! Coefficient quantization

State-space, DFI, DFII, . . .

! Large variety of structures with no common quality criteria

Software or Hardware implementation

! Constraints: power consumption, area, error, speed, etc.

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Motivation: Automatized filter implementation flow

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Motivation: Automatized filter implementation flow

Focus on Fixed-Point realization.

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Motivation: Automatized filter implementation flow

Focus on Fixed-Point realization. Optimization of wordlengths: Take more - pay more Take less - overflow risk What we want in the end: Rigorous algorithm for Fixed-Point Formats (FxPF) determination Integration into automatized code generator for filters Multiple wordlength paradigm

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LTI filters

H := (A, B, C, D) is a Bounded Input Bounded Output LTI filter in state-space representation: H x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k) with q inputs, n states and p outputs and state matrices

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Basic brick: interval through filter

The Worst-Case Peak Gain theorem

Time Amplitude

8k, |u(k)| ≤ ¯ u Input u(k)

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Basic brick: interval through filter

The Worst-Case Peak Gain theorem

H

y(k) u(k)

amplification/attenuation

Time Amplitude

8k, |u(k)| ≤ ¯ u Input u(k)

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Basic brick: interval through filter

The Worst-Case Peak Gain theorem

Time Amplitude

Output y(k) H

y(k) u(k)

amplification/attenuation

Time Amplitude

8k, |u(k)| ≤ ¯ u Input u(k)

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Basic brick: interval through filter

The Worst-Case Peak Gain theorem

Time Amplitude

Output y(k) 8k, |y(k)| ≤ hhHii ¯ u H

y(k) u(k)

amplification/attenuation

hhHii = |D| + P∞

k=0 |CAkB|

Worst-Case Peak Gain

Time Amplitude

8k, |u(k)| ≤ ¯ u Input u(k)

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Two’s complement Fixed-Point arithmetic

m + 1 −ℓ s w −2m 20 2−1 2m−1 2ℓ

t = −2mtm +

m−1

• i=ℓ

2iti Wordlength: w Most Significant Bit position: m Least Significant Bit position: ℓ := m − w + 1

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Two’s complement Fixed-Point arithmetic

m + 1 −ℓ s w −2m 20 2−1 2m−1 2ℓ

t = −2mtm +

m−1

• i=ℓ

2iti quantization step: 2ℓ t represented by integer T = t · 2ℓ T ∈ [−2m; 2m − 2ℓ] ∩ Z

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Two’s complement Fixed-Point arithmetic

m + 1 −ℓ s w −2m 20 2−1 2m−1 2ℓ

t = −2mtm +

m−1

• i=ℓ

2iti y(k) ∈ R wordlength w bits minimal Fixed-Point Format (FPF) is the least m: ∀k, y(k) ∈ [−2m; 2m − 2m−w+1]

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Problem statement

Let H := (A, B, C, D) be a LTI filter: H x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k) Given wordlength constraints wx and wy for each state and output variable input domain ¯ u we need to determine the minimal FPF for all variables of filter H, i.e. find the least mx and my such that ∀k, xi(k) ∈ [−2mxi; 2mxi − 2mxi−wxi+1] ∀k, yi(k) ∈ [−2myi; 2myi − 2myi−wyi+1].

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Modification of H

Let ζ(k) := x(k) y(k)

• be a vertical concatenation of state and
• utput vectors.

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Modification of H

Let ζ(k) := x(k) y(k)

• be a vertical concatenation of state and
• utput vectors.

Then the state-space takes the following form: Hζ    x(k + 1) = Ax(k) + Bu(k) ζ(k) = I C

• x(k)

+ D

• u(k)

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Modification of H

Let ζ(k) := x(k) y(k)

• be a vertical concatenation of state and
• utput vectors.

Then the state-space takes the following form: Hζ    x(k + 1) = Ax(k) + Bu(k) ζ(k) = I C

• x(k)

+ D

• u(k)

We seek to determine the least mζ such that ∀k, |ζi(k)| 2mζi − 2mζi−wi+1.

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Computing the MSB using the WCPG theorem

Applying the WCPG theorem on filter Hζ gives ∀k, |ζi(k)| ( Hζ ¯ u)i

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Computing the MSB using the WCPG theorem

Applying the WCPG theorem on filter Hζ gives ∀k, |ζi(k)| ( Hζ ¯ u)i Therefore, the smallest mζi satisfying ( Hζ ¯ u)i 2mζi − 2mζi−wi+1, will satisfy the wordlength constraints.

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Computing the MSB using the WCPG theorem

Applying the WCPG theorem on filter Hζ gives ∀k, |ζi(k)| ( Hζ ¯ u)i Therefore, the smallest mζi satisfying ( Hζ ¯ u)i 2mζi − 2mζi−wi+1, will satisfy the wordlength constraints. We can compute mζi with mζi =

• log2 (

Hζ ¯ u)i − log2

• 1 − 21−wζi

.

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Taking the quantization errors into account

The exact filter Hζ is: Hζ    x (k + 1) = Ax (k) + Bu(k) ζ (k) = I C

• x

+ D

• u(k)

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Taking the quantization errors into account

The actually implemented filter H♦

ζ is:

Hζ♦    x♦(k + 1) = ♦mx

• Ax♦(k) + Bu(k)
• ζ♦(k)

= ♦mζ I C

• x♦(k) +

D

• u(k)
• where ♦m is some operator ensuring faithful rounding:

|♦m(x) − x| 2m−w+1.

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Taking the quantization errors into account

The actually implemented filter H♦

ζ is:

Hζ♦    x♦(k + 1) = ♦mx

• Ax♦(k) + Bu(k)
• ζ♦(k)

= ♦mζ I C

• x♦(k) +

D

• u(k)
• where ♦m is some operator ensuring faithful rounding:

|♦m(x) − x| 2m−w+1. It holds H♦

ζ

   x♦(k + 1) = Ax♦(k) + Bu(k) + εx(k) ζ♦(k) = I C

• x♦(k) +

D

• u(k) +

εx(k) εy(k)

• with

|εx(k)| 2mx−wx+1 and |εy(k)| 2my−wy+1.

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Implemented filter decomposition

ζ♦(k) u(k)

H♦

ζ 11/18

Implemented filter decomposition

ζ♦(k) u(k)

H♦

ζ 11/18

Implemented filter decomposition

ζ♦(k) u(k)

H♦

ζ

q−1 X(k + 1) X(k) A

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Implemented filter decomposition

ζ♦(k) u(k)

H♦

ζ

q−1 X(k + 1) X(k) A

Hζ H∆

ζ♦(k) u(k) ζ(k) ∆ζ(k)

• εx(k)

εy(k)

• 11/18

Implemented filter decomposition

ζ♦(k) u(k)

H♦

ζ

q−1 X(k + 1) X(k) A

Hζ H∆

ζ♦(k) u(k) ζ(k) ∆ζ(k)

• εx(k)

εy(k)

• mζ

! Filter H∆ depends on the MSBs of filter Hζ

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Implemented filter decomposition

ζ♦(k) u(k)

H♦

ζ

q−1 X(k + 1) X(k) A

Hζ H∆

ζ♦(k) u(k) ζ(k) ∆ζ(k)

• εx(k)

εy(k)

• mζ

! Filter H∆ depends on the MSBs of filter Hζ

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Implemented filter decomposition

ζ♦(k) u(k)

H♦

ζ

q−1 X(k + 1) X(k) A

Hζ H∆

ζ♦(k) u(k) ζ(k) ∆ζ(k)

• εx(k)

εy(k)

• mζ

! Filter H∆ depends on the MSBs of filter Hζ

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Two step approach

Step 1 Determine the MSBs mζ for the exact filter Hζ, applying the WCPG theorem; Step 2 Compute the error-filter, induced by the format mζ and deduce the FPF of the implemented filter H♦

ζ Hζ H∆

ζ♦(k) u(k) ζ(k) ∆ζ(k)

εx(k) εy(k)

• mζ

Hζ H∆

ζ♦(k) u(k) ζ(k) ∆ζ(k)

εx(k) εy(k)

• mζ

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MSB computation error analysis

mζi =

• log2 (h

hHζi i ¯ u)i + log2

• 1 − 21−wi

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MSB computation error analysis

mζi =

• log2 (h

hHζi i ¯ u)i + log2

• 1 − 21−wi

| {z } \ hhHζii + εW CP G

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MSB computation error analysis

mζi =

• log2 (h

hHζi i ¯ u)i + log2

• 1 − 21−wi

| {z } \ hhHζii + εW CP G c mζi =

• log2 (h

hHζi i ¯ u)i + log2

• 1 − 21−wi

+ log2

• 1 + (εW CP G · ¯

u)i (h hHζi i ¯ u)i

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MSB computation error analysis

mζi =

• log2 (h

hHζi i ¯ u)i + log2

• 1 − 21−wi

| {z } \ hhHζii + εW CP G c mζi =

• log2 (h

hHζi i ¯ u)i + log2

• 1 − 21−wi

+ log2

• 1 + (εW CP G · ¯

u)i (h hHζi i ¯ u)i

• = mζi + d. . .e

|{z}

∈{0,1}

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MSB computation error analysis

mζi =

• log2 (h

hHζi i ¯ u)i + log2

• 1 − 21−wi

| {z } \ hhHζii + εW CP G c mζi =

• log2 (h

hHζi i ¯ u)i + log2

• 1 − 21−wi

+ log2

• 1 + (εW CP G · ¯

u)i (h hHζi i ¯ u)i

• = mζi + d. . .e

|{z}

∈{0,1}

Adjust error term εWCPG in order to be at most off by one.

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Algorithm

Step 1 Determine the MSBs mζ for the exact filter Hζ, applying the WCPG theorem; Step 2 Compute the error-filter H∆, induced by the format mζ and deduce the MSB m♦

ζ of the

implemented filter; Step 3 If m♦

ζi == mζi then return m♦ ζi

• therwise mζi ← mζi + 1 and go

to Step 2.

Hζ H∆

ζ♦(k) u(k) ζ(k) ∆ζ(k)

• εx(k)

εy(k)

• mζ

Hζ H∆

ζ♦(k) u(k) ζ(k) ∆ζ(k)

εx(k) εy(k)

• mζ

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Numerical examples

Example: Random filter with 6 states, 1 input, 3 outputs ¯ u = 3.7776, wordlengths set to 7 bits

mζ = (4, 4, 4, 4, 2, 3, 6, 5, 5) m♦

ζ = (4, 5, 4, 4, 2, 3, 6, 5, 5)

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Numerical examples

Example: Random filter with 6 states, 1 input, 3 outputs ¯ u = 3.7776, wordlengths set to 7 bits

mζ = (4, 4, 4, 4, 2, 3, 6, 5, 5) m♦

ζ = (4, 5, 4, 4, 2, 3, 6, 5, 5)

m♦

ζ = (4, 5, 5, 4, 3, 4, 6, 5, 5)

After 3 iterations:

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Numerical examples

¯ x2 x♦

2 (k)

x2(k) ¯ x♦

2

After 3 iterations:

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Conclusion and Perspectives

Conclusion Rigorous procedure for Fixed-Point Formats determination Filter computation errors are taken into account, ensuring that no overflow occurs Multiple-wordlength paradigm Perspectives Integrate into optimization procedures in automatic workflow Solve off-by-one problem

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Thank you! Questions?

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