A Reconfigurable Architecture for the FFT Operator in a SoftWare Radio Context

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INSTITUT D’ÉLECTRONIQUE ET DE TÉLÉCOMMUNICATIONS DE RENNES

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A Reconfigurable Architecture for the FFT Operator in a SoftWare

Radio Context

Ali AL GHOUWAYEL - Yves LOUËT IETR – Supélec, Campus de Rennes

Équipe SCEE

Séminaire de recherche TAMCIC ENST Bretagne

Jeudi 8 Juin 2006

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The SoftWare Radio (SWR) Technology Channel coding in the SWR context

Arithmetic elements

FFT: a Common and reconfigurable Operator Summary and outlook

Outline

Outline Outline

Outline

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Outline Outline Outline Outline

Arithmetic elements

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The SoftWare Radio The SoftWare Radio The SoftWare Radio

The SoftWare Radio TechnologyTechnologyTechnology (1/4)Technology (1/4)(1/4)(1/4)

Principal motivations :

The SWR aims to :

To

Provide a reprogrammable or reconfigurable radios. In other words, to get a communication system able to support several communications standards.

Replace the analog radio systems by digital ones and emphasizes digital signal processing

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An important technique:

The parameterization

The parameterization aims to:

Decrease the runtime of the software reconfiguration Optimize the sharing between the software and the

hardware of the execution platform

This essential part of SWR is based on two approaches:

The Common Function (CF) approach The Common Operator (CO) approach

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Our study is focused on the CO approach:

FFT (Fast Fourier Transform) is the concerned operator [2]

[1] A. Rhiemeier, ’’Benefits and Limits of Parameterized Channel Coding for Software-Radio’’, 2^ndKarlsrhue Workshop on Software Radios, Germany, March 2002.

[2] J. Palicot, C. Roland, ’’FFT: a basic Function for a reconfigurable Receiver’’, ICT’2003, February 2003, Papeete, Tahiti

The CF approach can be regarded as:

The trick’s of research of the function that is used by a predefined set

of standards and therefore, establish the generic one.

Example: Channel coding function [1]

The CO approach, can be regarded as:

For a given level of granularity, the research of the operator that is used by the maximum number of functions

Examples: FFT [2], MulDiv [3]

The CF and the CO

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Channel Coding in Frequency domain with the FFT

The several operations already performed with FFT operator are:

Filtering Function

Channel estimation and Equalizer Despreading and Rake

Multicarrier (de)modulation Channelization

Our problematic

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Outline Outline Outline Outline

Arithmetic elements

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Channel Channel Channel

Channel CodingCodingCoding in Coding in in thein thethe SWR the SWR SWR contextSWR contextcontext (1/9)context (1/9)(1/9)(1/9)

Our channel coding study is restricted on the cyclic codes: The Reed-Solomon (RS) code

Two ways to perform the RS coding:

1. In Time domain

2. In Frequency domain Encoding in time domain:

g₁

b₀

g₂

b₀ b₀

g_n-k-1

Code word Switch

b₀

Information sequence

) ) (

( ) ) (

( M x

x g

x M x X

C

k n

+

=

− C(x) : code word

M(x): information sequence

g(x): generator polynomial g(x)=(x−α ^j⁰)(x−α ^j⁰⁺¹)...(x−α ^j⁰⁺²^t⁻¹)

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M₀ M₁… M_k-2 M_k-1

Pad obligatory symbols

M₀ M₁…M_j0-10 0…0M_j0+2t-1 M_n-1 ^C(f) _IFFT

A nonsystematic code word

{

t 2

Encoding in frequency domain: The encoding process consists to constrain certain spectral component to zero and load the information symbols into convenient component, the inverse Fourier transform

produces the code word.

c⁽n)=c₀ c₁…c_n-1

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•

Decoding process in time and frequency domain

Code word

Decoding process

Decoding in time domain

Decoding in

frequency domain

FFT

Berlekamp-

Massey algorithm Chien-Search Forney algorithm Berlekamp-

Massey algorithm in time domain Error correction

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In the SWR context, the objectif of this study is to use the FFT operator to perform the channel coding

(Encoding/Decoding), then:

1- The Frequency channel coding will be treated

2- The FFT operator should be reconfigured in a such way to support the computations over the Galois Field (GF)

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Decoding in frequency domain

^:

Let us consider a received word r(x), in frequency domain the decoding process consists to:

1. Compute R(f)=FFT(r(x))

2. Solve the linear recursion by Berlekamp-Massey algorithm :

R(f)=[R₀ R₁… R_j0…R_J0+2t-1…R_n-1], t : power correcting code

and is the error locator polynomial 3. Find the root of by the « Chien Search »

4. Compute the error value : Forney algorithm

t t

k S

S

t j

j k j

k 1,...,2

1

∑

=

− = +

Λ

−

=

0−1

= +

j j

j R

S Λ(x)

) (x Λ

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Channel CodingCodingCodingCoding thethethethe in SWR in SWR in SWR in SWR contextcontextcontextcontext (6/9)(6/9)(6/9)(6/9)

Phase 3

Syndrome computation

Phase 1

Chien search

Forney algorithm Berlekamp

algorithm

Phase 2

N cycles 8t cycles + 2t cycles ^{N cycles} + ^{3 cycles}

n computatio

x x), ( ) ( Λ^' Γ

The idea is to perform the two most long-time stage

(Syndrome computation and Chien Search) for RS codes with FFT.

The different stages of RS decoding in the frequency domain can be gathered in three main phases as following :

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[4] A. Al Ghouwayel, Yves Louët, J. Palicot, ’’A Reconfigurable Architecture for the FFT operator in a Software Radio Context’’, IEEE International Symposium on Circuits and Systems (ISCAS' 2006), Island of

Phase 3

Syndrome Computation

with FFT

Phase 1

Chien Search with FFT

algorithm

Phase 2

Log N cycles 8t cycles + 2t cycles Log N cycles+ 3 cycles n

computatio x x), ( ) ( Λ^' Γ

Phase 3

Syndrome computation

Phase 1

Chien search

algorithm

Phase 2

N cycles 8t cycles + 2t cycles ^{N cycles} + ^{3 cycles}

n computatio

x x), ( ) ( Λ^' Γ

With FFT

Runing-time gain with FFT [4]:

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Channel CodingCodingCodingCoding in in in thein thethe SWR the SWR SWR contextSWR contextcontext (7/9)context (7/9)(7/9)(7/9)

Our Objective : Use the same FFT architecture for RS channel coding and classical operations,

To be able to use the most effecient FFT algorithm, it is necessary to have the following caracteristics:

1. The Transform length is an even number 2. The properties of symetry

( : The primitive element of Galois Field (GF), N: transform block length )

k k N

k N k

N α α and α α

α = = =−

+ +2

, 1

Channel CodingCodingCodingCoding in in in thein thethe SWR the SWR SWR contextSWR contextcontext (8/9)context (8/9)(8/9)(8/9)

α

We need to find the RS codes wich adapted with the FFT

 But : The classical RS codes do not correspond to the FFT architecture

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The candidate codes are the RS codes defined over GF(F_t),

RS codes defined over GF(F_t) :

is the Fermat number,

F₀, F₁, F₂, F₃, F₄ are the only Fermat prime numbers,

The encoding and decoding principles are the same ones as that of codes defined over GF(2ⁿ)

The arithmetic operations in this field are modulo (F_t) operations

This code was the recommended coding scheme for use on spacecraft-to- ground telemetry channels that has the compatibility with operational

requirements of the European Space Agency (ESA) [6].

1 2² +

= ^t Ft

[6] M.R. Best, H. F. A Roefs, « Technical assistance telemetry channel coding investigation »,

Contract no. 4184/79/NL/HP, Final report 1981. National Aerospace Laboratory, Amesterdam, The Netherlands.

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Bit Error Rate (BER) of RS(16,12) defined over GF(17) :

Performances of RS code Performances of RS code Performances of RS code

Performances of RS code defineddefineddefineddefined over

over over

over GF(FGF(FGF(FGF(F_tttt))))

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Time and frequency domain decoding

Frequential processing

Reduced runing time: n(t+1)clock cycles

Complex Hardware*

Temporal processing

longer runing-time: n²clock cycles

Easily designed Hardware

Performances comparison

* Additional point: The FFT is often already implemented in many standards by the mean of the OFDM modulation (DVB-T, ADSL, …), then, the computations of RS coding in frequency domain do not increase the Hardware complexity.

This remark justifies the advantage of the common operator approach

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Towards FNT (Fermat Number Transform)

As previously mentionned, the idea is to reconfigure the complex FFT in a such way to obtain an FFT over GF(F_t) called FNT. To do this, it is necessary to reconfigure the arithmetic operator (adder, multiplier).

Once the modulo (F_t) operators are reconfigured, we can define the reconfigurable Butterfly, and consequently

The global architecture of the reconfigurable FFT operator will be defined.

Practical realization

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First or intuitive idea : Compute the FFT and

then reduce the result mod(F_t)

Second idea :

Reconfigure the FFT architecture

in such way to perform the arithmetic operation mod (F_t) [5].

As previously mentionned, the idea is to reconfigure the complex FFT in a such way to perform the modulo (Ft) computation :

Two ways to perform the FFT over GF(Ft) :

Practical realization

[5] A. Al Ghouwayel, Yves Louët, J. Palicot, ’’A Reconfigurable Butterfly Architecture for Fourier and

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The SoftWare Radio (SWR) Technology Channel Coding in the SWR context

Arithmetic elements

Outline

Outline Outline

Outline

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Complex Butterfly:

Butterfly contains:

1 complex addition 1 complex subtraction

1 complex, constant multiply

Arithmetic elements (1/4)

u= a + j b v= c + j d

x= u+ v^W_N^r y= u- WN^r v

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Our proposed Modulo (F

_t

) Multiplier [5] :

Arithmetic elements (2/4)

[5] A. Al Ghouwayel, Yves Louët, J. Palicot, ’’A Reconfigurable Butterfly Architecture for Fourier and

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Beuchat modulo (F

_t

) Adders [6] :

n bits

n+1 bits

x y

n+1 bits

(b)

n bits

n+1 bits

x y

n+1 bits

n+1 bits 2ⁿ

1 0

(a)

Most significantbit

[6] Jean-Luc Beuchat, ’’Some Modular Adders and Multipliers for Field Programmable Gate Arrays’’,

Arithmetic elements (3/4)

(x+y+1)mod (2ⁿ+1)

x+y +1 (not x+y)

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Our proposed Modulo (F

_t

) operators [5]

n+1 bits (x+y)mod (2ⁿ+1)

x y

n+1 bits

2ⁿ

1 0

n bits

1 1 ,...,

0 n+

s

2 1 ,...,

0 n+

s

Arithmetic elements (4/4)

[5] A. Al Ghouwayel, Yves Louët, J. Palicot, ’’A Reconfigurable Butterfly Architecture for Fourier and n bits

n+1 bits (x-y)mod (2ⁿ+1)

x y

n+1 bits

2ⁿ

1 0

1

1 ,...,

0 n+

s

Modulo (F_t) Adder Modulo (F_t) Subtracter

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The SoftWare Radio (SWR) Technology Channel Coding in SWR context

Arithmetic elements

Outline

Outline Outline

Outline

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s₀ s₄ s₂

s₆ s₁ s₅

s₃ s₇

0

w2

0

w2 0

w2

0

w2

0

w4

1

w4

0

w4

1

w4

0

w8

1

w8

2

w8

3

w8

S₀ S₁ S₂ S₃ S₄

S₅ S₆ S₇

kn N kn+N /2

FFT : a Common and reconfigurable Operator (1/4)

The FFT operator over the complex field:

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s₀ s₁

s₂ s₃ s₄ s₅

s₆ s₇

α0

α2

α0

α2

α0

α1

α2

α3

S₀ S₁ S₂ S₃ S₄

S₅ S₆ S₇

k k N

k N

k

N α α α α

α = = = −

+ + 2

, 1

number Fermat

F F

N = = 2²^t + 1, :

FFT : a Common and reconfigurable Operator (2/4)

The FFT operator over the Galois field:

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The architecture of the Butterfly over GF(Ft):

FFT : a Common and reconfigurable

Operator (3/4)

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FFT : a Common and reconfigurable Operator (4/4)

The architecture of the FNT operator

Architecture simulated by Software

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The SoftWare Radio (SWR) Technology Channel Coding in the SWR context

Arithmetic elements

FFT : a Common and reconfigurable Operator Summary and outlook

Outline

Outline Outline

Outline

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Summary and outlook

SWR

Parametrization: CO approach CF approach

Channel coding : Frequency domain Time domain

CO approach: FFT

Frequency domain Channel coding with the Common Operator FFT

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Summary and outlook

We have getting out a subclass of RS code (RS over GF(F_t))

Demonstration of the efficient use of the FFT in encoding and decoding process

Definition of the structure of FFT over GF(F_t) Definition of the reconfigurable Butterfly

Designed of a reconfigurable operator that operates over two different field (C and GF)

Current work: Implementation of this operator with FPGAs to study the performances in term of slice and delay

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Thank you for your attention !

Questions ?

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Coordonnées Coordonnées Coordonnées Coordonnées

Ali AL GHOUWAYEL

IETR / Supélec, Campus de Rennes SCEE Team

[email protected] +33 2.99.84.45.38