Formal High-Level Model of a Radar Signal Processing System

George Ungureanu1 Anders Åhlander2 Timmy Sundström3 Ingo Sander1 Ingemar Söderquist3

1School of EECS, KTH Royal Institute of Technology, Stockholm, Sweden2Business Area Surveillance, Saab AB, Gothenburg, Sweden2Business Area Aeronautics, Saab AB, Linköping, Sweden

October 9, 2019

The Context of this Work

NFFP7: CORRECTI Saab AB & KTH

I current industrial design flows do not havea clear path from the functionalspecification to product

I information on security, safety, behaviorare important aspects to be expressed andassessed

I identified by the industryI emphasized by certification authorities

I aim to decrease the need for late stagetesting and related development costs.

1 / 26

Overview

Modeling with ForSyDe

I push orthogonalization in modeling one step further

I harmoniously combine different DSLs capturing different aspectsLanguage 1

Abstraction1

Language 2

Abstraction2

Language 3

Abstraction3

α

αS2

αS2S3

The AESA Case Study

I Active Electronically Scanned Array Antenna

I typical high-performance component in avionicsunder real-time constraints

I extremely costly for civil applications today

2 / 26

Why We Use Models?

To a scientist, the value of a model lies in how well its properties match those of atarget (...). But to an engineer the value of an object lies in how well its propertiesmatch a model. A scientist asks, “Can I make a model for this thing?” An engineerasks, “Can I make a thing for this model?”1

...to understand systems v(t) = v(0) + 1m

∫ t0

F (τ)dτ

...to design systems

...to create systems

3 / 26

Why We Use Models?

To a scientist, the value of a model lies in how well its properties match those of atarget (...). But to an engineer the value of an object lies in how well its propertiesmatch a model. A scientist asks, “Can I make a model for this thing?” An engineerasks, “Can I make a thing for this model?”1

...to understand systems v(t) = v(0) + 1m

∫ t0

F (τ)dτ

...to design systems

...to create systems

1Edward Ashford Lee. Plato and the Nerd: The Creative Partnership of Humans and Technology. MIT Press, 2017.

3 / 26

Layered Languages

Each layer consists of:

I a set of structured types= encode properties

I a set of functions over these types= transformations, rules

I a set of higher order functions (HOF)= conduits between layers Lan

guage1

Ab

stra

ctio

n1

Language 2

Ab

stra

ctio

n2

Language 3

Ab

stra

ctio

n3

α

αS2

αS2S3

f : α→ α

f : S2(∗)→S2(∗)

hof : (∗ →∗)→ S2(∗)

→ S2(∗)

f : S3(∗)→S3(∗)

hof : (∗ →∗)→ S3(∗)

→ S3(∗)

4 / 26

A Language for Timed Behavior

+

SY.comb

SY process

4,3,2,1

3,2,1

SignalsI encode temporal information

I define tag systems2

ProcessesI act according to MoC semantics

I created with process constructors (HOF)

2E.A. Lee and A. Sangiovanni-Vincentelli. A framework for comparing models of computation. Dec. 1998.

5 / 26

A Language for Timed Behavior

+

SY.comb

SY process

4,3,2

3,22

SignalsI encode temporal information

I define tag systems2

ProcessesI act according to MoC semantics

I created with process constructors (HOF)

2Lee and Sangiovanni-Vincentelli, A framework for comparing models of computation.

5 / 26

A Language for Timed Behavior

+

SY.comb

SY process

4,3

3

4,2

SignalsI encode temporal information

I define tag systems2

ProcessesI act according to MoC semantics

I created with process constructors (HOF)

2Lee and Sangiovanni-Vincentelli, A framework for comparing models of computation.

5 / 26

A Language for Timed Behavior

+

SY.comb

SY process

46,4,2

SignalsI encode temporal information

I define tag systems2

ProcessesI act according to MoC semantics

I created with process constructors (HOF)

2Lee and Sangiovanni-Vincentelli, A framework for comparing models of computation.

5 / 26

A Language for Timed Behavior

×

CT.comb

CT process

2 4 6 8 100

0.5

1

0(t)

1(t)

2 4 6 8 10

−1−0.5

0

0.5

1

sin(t) −1−0.5

0

0.5

1

SignalsI encode temporal information

I define tag systems2

ProcessesI act according to MoC semantics

I created with process constructors (HOF)

2Lee and Sangiovanni-Vincentelli, A framework for comparing models of computation.

5 / 26

A Language for Timed Behavior

×

CT.comb

CT process

6 8 100

0.5

1

0(t)

1(t)

6 8 10

−1−0.5

0

0.5

1

sin(t)

2 4

−1−0.5

0

0.5

1

(0× sin)(t)

SignalsI encode temporal information

I define tag systems2

ProcessesI act according to MoC semantics

I created with process constructors (HOF)

2Lee and Sangiovanni-Vincentelli, A framework for comparing models of computation.

5 / 26

A Language for Timed Behavior

×

CT.comb

CT process

−1−0.5

0

0.5

1

−1−0.5

0

0.5

12 4 6 8 10

−1−0.5

0

0.5

1

(0× sin)(t)(1× sin)(t)

SignalsI encode temporal information

I define tag systems2

ProcessesI act according to MoC semantics

I created with process constructors (HOF)

2Lee and Sangiovanni-Vincentelli, A framework for comparing models of computation.

5 / 26

A Language for Structured Parallelism

f

f

f

f

· · ·

(a) farm

g

g

g

· · ·

(b) reduce

f

f

f

· · ·

(c) pipe

f

f

f

· · ·

(d) recur

f

f

f

· · ·

(e) prefix

Regular Structures (e.g. Vectors)

I encode spatial information

I enable catamorphisms3

Parallel PatternsI functional relations between elements

I potential for parallel distribution

I created with skeletons (HOF)

3David B Skillicorn. Foundations of parallel programming. 6. Cambridge University Press, 2005.

6 / 26

A Language for Testing Properties

pre-condition⇒ statement

Example:

∀a ∈ List(α) ⇒ reverse(reverse a) = a

GeneratorsI express pre-conditions

I abstract random data generators

I “smartened” by algebraic properties4

Properties & CombinatorsI test truth statements

I systematic composition of generators

4John Hughes. “QuickCheck testing for fun and profit”. In: International Symposium on Practical Aspects of Declarative Languages. Springer. 2007.

7 / 26

The AESA Radar

I typically thousands of antennaelements

I hundreds of GOPS

I when part of control loop, real-timeaspect becomes critical

rangebins

antennas

pulse windowbeam

NA

Nb

NFFT

AESA

video indata

DBF PC

DFB

DFB

CFAR

CFAR

INT

corner turn

8 / 26

The AESA Signal Processing Chain as a Layered Model

DBF

coef

.

SY

SDF PC

fir(coef )

SDF.comb

transpose

SDF.comb

transpose

SDF.comb%

DFB

fDFB

SDF.comb

DFB

fDFB

SDF.comb

CFAR

fCFAR

SDF.comb

CFAR

fCFAR

SDF.comb

++

SDF.comb

SDF

SY INT

coef

.

CTfarm

{⊥}(NbinNFFT /2)

c1 p1c2

p2

c2

p2

c3

p3

c3

p3

c4

p4

c4

p4

c5

c5

2

c1 = p1 = Nb; c2 = NFFT ×Nb; p2 = c4 = p4 = c5 = Nb ×NFFT ; c3 = p3 = NFFT ;

4ForSyDe-Atom code available at https://github.com/forsyde/aesa-radar

9 / 26

Digital Beamforming (DBF)

I forms “listening directions”

I vector-matrix multiplication

rangebins

antennas

pulses

pulse

NA

Nb

NFFT

...

· · ·

b1

...

· · ·

b2

...

· · ·

bNB

farmreduce reduce reduce

farm

a1fanout · · ·

a2fanout · · ·

aNAfanout · · ·

beamSigs

beamMatrix

antennaSigs

×α11 ×α12 ×α1NB

×α21 ×α22 ×α2NB

×αNA1 ×αNA2 ×αNANB

+

+

+

+

+

+

〈〈αij〉〉

dbf :: Vector (SY.Signal (Complex Float)) -> Vector (SY.Signal (Complex Float))

dbf antennaSigs = beamSigs

where beamSigs = reduce (farm21 (comb21 (+))) beamMatrix

beamMatrix = farm21 (\c -> comb11 (*c)) beamConsts sigMatrix

sigMatrix = farm11 fanout antennaSigs

10 / 26

Pulse Compression (PC)

I echo is passed through amatched filter to decodeits modulation

I basically a movingaverage/FIR filter

rangebins

bea

ms

pulse

pulse

NB

Nb

NFFT

procPC

mav(coefs)

SDF.comb

Nb NbSY

SDF

pcfarm

〈bdbf1 ...bdbfNB 〉 〈bpc1 ...b

pcNB

〉

11 / 26

Corner Turn (CT) & Doppler Filter Bank (DFB)rangebins

bea

ms

pulse windows

direction of consumption

NA

Nb

NFFT

rangebins

bea

ms

pulse windows

direction of production

NA

Nb

NFFT

⇒

bea

ms

rangebins

rangebins

pulse windows

pulse windows

left channel

right channel

NFFT /2

NFFT

12 / 26

Constant False Alarm Ratio (CFAR)

rangebins

bea

ms

Doppler windowsbeam

NA

Nb

NFFT

CFAR(aij ) = 2(5+log2 aij )−max(EMV (aij ),LMV (aij ),MD(aij ))

EMV (aij ) =1

N

N−1∑k=0

log2 1

4

3∑l=0

a(i−2−4k−l)j

LMV (aij ) =

1

N

N−1∑k=0

log2 1

4

3∑l=0

a(i+2+4k+l)j

MD(aij ) = log2

(N

mink=1

(aik )

)

∀i ∈ [eb, lb], j ∈ [1,N] where

N = NFFT

eb = NFFT + 1

lb = Nb − NFFT − 1fCFAR :: RealFloat r => Range (Window r) -> Range (Window r)

fCFAR rbins = V.farm41 (\ m -> V.farm31 (normCfa m)) md rbins lmv emv

where

md = V.farm11 (logBase 2 . V.reduce min) rbins

emv = (V.farm11 aritMean . V.stencil nFFT) rbins

lmv = (V.drop 2 . V.farm11 aritMean . V.stencil nFFT) rbins

normCfa m a l e = 2 ** (5 + logBase 2 a - maximum [l,e,m])

aritMean = V.farm11 (/n) . V.reduce (+) . V.farm11 geomMean . V.group 4

geomMean = V.farm11 (logBase 2 . (/4)) . V.reduce (+)

13 / 26

Why Have We Done This?

...to understand systems v(t) = v(0) + 1m

∫ t0

F (τ)dτ

...to design systems

...to create systems

14 / 26

...To Understand the AESA Signal Processing System

0 50 100150200250

0

200

400

600

800

1000

antenna 0

0 50 100150200250

antenna 1

0 50 100150200250

antenna 2

0 50 100150200250

antenna 3

0 50 100150200250

antenna 4

0 50 100150200250

antenna 5

0 50 100150200250

antenna 6

0 50 100150200250

antenna 7

0 50 100150200250

antenna 8

0 50 100150200250

antenna 9

0 50 100150200250

antenna 10

0 50 100150200250

antenna 11

0 50 100150200250

antenna 12

0 50 100150200250

antenna 13

0 50 100150200250

antenna 14

50100150200

antenna 15

72

64

56

48

40

32

24

16

8

0 50 100 150 200 250

0

200

400

600

800

1000

6.36.7

60.691.064.8

14.523.3

17.820.7

86.037.518.6

beam 0

0 50 100 150 200 250

12.012.2

126.7189.2135.8

30.947.7

38.444.8

180.287.742.5

beam 1

0 50 100 150 200 250

6.513.2

13.4144.3215.4156.2

35.253.1

43.751.4

205.3103.049.7

beam 2

0 50 100 150 200 250

12.712.9

139.9208.1152.1

34.351.0

42.350.0

199.0100.748.7

beam 3

0 50 100 150 200 250

12.712.9

139.9208.1152.1

34.351.0

42.350.0

199.0100.748.7

beam 4

0 50 100 150 200 250

6.513.2

13.4144.3215.4156.2

35.253.1

43.751.4

205.3103.049.7

beam 5

0 50 100 150 200 250

12.012.2

126.7189.2135.8

30.947.7

38.444.8

180.287.742.5

beam 6

0 50 100 150 200 250

6.36.7

60.691.064.8

14.523.3

17.820.7

86.037.518.6

beam 7

15 / 26

Why Have We Done This?

...to understand systems X v(t) = v(0) + 1m

∫ t0

F (τ)dτ

...to design systems

...to create systems

16 / 26

...To Design the AESA Signal Processing System

I system structure

I behavior particularities

I tested properties

I validated the designDBF

coef

.

SY

SDF PC

fir(coef )

SDF.comb

transpose

SDF.comb

transpose

SDF.comb%

DFB

fDFB

SDF.comb

DFB

fDFB

SDF.comb

CFAR

fCFAR

SDF.comb

CFAR

fCFAR

SDF.comb

++

SDF.comb

SDF

SY INT

coef

.

CTfarm

{⊥}(NbinNFFT /2)

c1 p1c2

p2

c2

p2

c3

p3

c3

p3

c4

p4

c4

p4

c5

c5

2

c1 = p1 = Nb; c2 = NFFT ×Nb; p2 = c4 = p4 = c5 = Nb ×NFFT ; c3 = p3 = NFFT ;

17 / 26

Why Have We Done This?

...to understand systems X v(t) = v(0) + 1m

∫ t0

F (τ)dτ

...to design systems X

...to create systems

18 / 26

Synthesizing the PC component to FPGA

rangebins

bea

ms

pulse

pulse

NB

Nb

NFFT

procPC

mav(coefs)

SDF.comb

Nb NbSY

SDF

pcfarm

〈bdbf1 ...bdbfNB 〉 〈bpc1 ...b

pcNB

〉

Given Specification

“For each antenna the data arrives pulse by pulse, and each pulse arrives range bin by range bin. Thishappens for all antennas in parallel, and all complex samples are synchronized with the same samplingrate, e.g. of the A/D converter.(...)The goal of the pulse compression is to collect all received energy from one target into a single rangebin. The received echo of the modulated pulse is passed through a matched filter. Here, thematched filtering is done digitally. ”

19 / 26

A Closer Look at the MAV Operation

reduce( comb(+)++ ) ◦ farm( reduce′(+) ◦ farm(×) creduce′(+) ◦ farm(×) c ) ◦ recur( delay(0)tail )

recur

farm

reduce

farm(×)〈1, 2, 1〉

reduce′(+)

tail

++

farm(×)〈1, 2, 1〉

reduce′(+)

tail

++

farm(×)〈1, 2, 1〉

reduce′(+)

tail

++

farm(×)〈1, 2, 1〉

reduce′(+)

〈0, 0, 0, 1〉

〈0, 0, 0, 1〉 〈0, 0, 1〉 〈0, 1〉 〈1〉

〈0, 0, 0〉 〈0, 0, 1〉 〈0, 2〉 〈1〉

〈0〉 〈1〉 〈2〉 〈1〉〈0, 1〉 〈0, 1, 2〉 〈0, 1, 2, 1〉

NFarm = NInput × NCoefficients

%

%

recur

farm

reduce

{0, 0, 0, 1}

{0, 1, 2, 1}

×1

0

×2

+

0

×1

+

NFarm = NCoefficients

20 / 26

A Closer Look at the MAV Operation

reduce( comb(+)++ ) ◦ farm( reduce′(+) ◦ farm(×) c ) ◦ recur( delay(0) )comb(+) c 7→ comb(×c) delay(0)

recur

farm

reduce

farm(×)〈1, 2, 1〉

reduce′(+)

tail

++

farm(×)〈1, 2, 1〉

reduce′(+)

tail

++

farm(×)〈1, 2, 1〉

reduce′(+)

tail

++

farm(×)〈1, 2, 1〉

reduce′(+)

〈0, 0, 0, 1〉

〈0, 0, 0, 1〉 〈0, 0, 1〉 〈0, 1〉 〈1〉

〈0, 0, 0〉 〈0, 0, 1〉 〈0, 2〉 〈1〉

〈0〉 〈1〉 〈2〉 〈1〉〈0, 1〉 〈0, 1, 2〉 〈0, 1, 2, 1〉

NFarm = NInput × NCoefficients

%

%

recur

farm

reduce

{0, 0, 0, 1}

{0, 1, 2, 1}

×1

0

×2

+

0

×1

+

NFarm = NCoefficients

20 / 26

Refining PC into a FIR Process Network

Fu

nction

Layer

Function

Skeleton

Layer

VectorFunction

MoC Layer

Process v

vvv

vvv

tvvv

t

vvv

tvvv

tvvv

tvvv

tvvv

tvvv

t

procPC

mav(coefs)

SDF.comb

Nb NbSY

SDF

pcfarm

〈bdbf1 ...bdbfNB 〉 〈bpc1 ...b

pcNB

〉

Fu

nction

Layer

Function

MoCLayer

Process

Skeleton Layer

Process

Network

v

vt

vt

vt

vt

%

%

%

recur

farm

reduce

pcfarm

· · ·

· · ·

· · ·

×c1

0

×c2

+

0

×c3

+

0

×cN

+

21 / 26

Floating Point to Fixed Point Number Representation

Specification

“After the DBF stage the numbers are represented withsufficient accuracy using 20 bits.”

Fu

nction

Layer

Function

MoCLayer

Process

Skeleton Layer

Process

Network

v

vt

vt

vt

vt

pc :: Vector (SY.Signal (Complex Float)) -> Vector (SY.Signal (Complex Float))

pc :: Vector (SY.Signal (Complex Fixed20)) -> Vector (SY.Signal (Complex Fixed20))

22 / 26

Floating Point to Fixed Point Number Representation

Specification

“After the DBF stage the numbers are represented withsufficient accuracy using 20 bits.”

Fu

nction

Layer

Function

MoCLayer

Process

Skeleton Layer

Process

Network

v

vt

vt

vt

vt

pc :: Vector (SY.Signal (Complex Float)) -> Vector (SY.Signal (Complex Float))

pc :: Vector (SY.Signal (Complex Fixed20)) -> Vector (SY.Signal (Complex Fixed20))

22 / 26

Load Balancing the FIR Reduction

Function

LayerFunction

MoCLayer

Process

Skeleton Layer

Process

Network

v

vt

vt

vt

vt

%

%

%

recur

farm

reduce

· · ·

· · ·

· · ·

. . ....

×c1

0

×c2

0

×c3

0

×cN

+ + +

+

23 / 26

RTL Synthesis

0 100 200

0

200

400

600

800

1000

59.688.663.8

16.121.4

18.621.8

86.139.420.2

beam 0

0 100 200

8.69.8

126.2188.5135.1

32.745.8

38.944.7

181.988.743.9

beam 1

0 100 200

7.49.2

11.4144.5216.6153.6

37.252.5

44.050.7

207.2104.050.9

beam 2

0 100 200

7.69.0

11.0139.6209.5148.3

36.051.0

42.348.9

200.1102.449.8

beam 3

0 100 200

7.69.0

11.0139.6209.5148.3

36.051.0

42.348.9

200.1102.449.8

beam 4

0 100 200

7.49.2

11.4144.5216.6153.6

37.252.5

44.050.7

207.2104.050.9

beam 5

0 100 200

8.69.8

126.2188.5135.1

32.745.8

38.944.7

181.988.743.9

beam 6

0 100 200

59.688.663.8

16.121.4

18.621.8

86.139.420.2

beam 7

24 / 26

Why Have We Done This?

...to understand systems X v(t) = v(0) + 1m

∫ t0

F (τ)dτ

...to design systems X

...to create systems X

25 / 26

Conclusions & Future work

A Framework for Layered Languages

I orthogonalize aspects of CPS modeling

I isolates design problems

Demonstration on the AESA System

I using ForSyDe ecosystem (https://forsyde.github.io)

I open-source (https://github.com/forsyde/aesa-radar)

I technical report

Future WorkI explore other modeling paradigms (e.g. contracts)

I HW/SW transformation-based synthesis flow

Design of Sensor SignalProcessing with ForSyDeModeling, Validation and Synthesis

George UngureanuSchool of EECSKTH Royal Institute of Technologyugeorge@kth.se

Timmy SundströmBusiness Area AeronauticsSaab AB

Anders ÅhlanderBusiness Area SurveillanceSaab AB

Ingo SanderSchool of EECSKTH Royal Institute of Technology

Ingemar SöderquistBusiness Area AeronauticsSaab ABVersion 0.2

26 / 26

Thank you!

26 / 26