Symmetric Radial Basis Function Network Equaliser

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NEWCOM-ACoRN Joint Workshop 2006

Symmetric Radial Basis Function Network

Equaliser

S. Chen†, A. Wolfgang†, S. Benedetto‡, P. Duhanel§ and L. Hanzo†

†

School of ECS, University of Southampton, UK

‡

Dipartimento di Elettronica, Politecnico di Torino, Italy

§

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Outline

❏

_{Channel equalisation revisit}

Symmetric property of Bayesian solution

❏

_{Symmetric RBF equaliser}

Exploit prior knowledge (symmetry)

❏

_{Adaptive equalisation algorithms}

✰

_{Kernel-type algorithm (Off-line, block data)}

✰

_{Nonlinear LBER algorithm (On-line, recursive)}

✰

_{Clustering algorithm (On-line, recursive)}

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Equaliser Model

❏ _{Channel model}_{: received signal}

x(k) =

n_c−₁

X

i=0

cib(k − i) + n(k)

❏ _{Signal model}_{: equaliser input} x(k) = [x(k) x(k ₋ 1)_{· · ·}x(k ₋ ne + 1)]T

x(k) = Cb(k) + n(k) = ¯x(k) + n(k)

❏ _Equaliser_{: uses information contained in} x(k) to detect symbol b(k ₋ τ)

ci, 0 ≤ i ≤ nc − 1, are channel taps, with channel length nc

C is ne × L channel convolution matrix, with L = nc + ne − 1

ne is equaliser order and τ decision delay

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Bayesian Equaliser

❏ _Assume _BPSK_: _b₍_k₎ _{∈ {−}₁_,₊₁_}_{, denote} _N_b _{= 2}L _legitimate

combina-tions of b(k) as bq, 1 ≤ q ≤ Nb, and (τ + 1)-th element of bq as bq,τ

❏ _{Noiseless channel output ¯}x(k) takes values from signal state set

X = _{x¯q = Cbq, 1 ≤ q ≤ Nb}

❏ _Optimal _{Bayesian equaliser}

yBay(k) = N_b

X

q=1

sgn(bq,τ)βqe

−kx(k)−x¯qk2

2σ2_n

where βq = _N 1

b(2πσ2_n)L/2, with decision

ˆ_b₍_k ₋ _τ_{) = sgn(}_y_Bay₍_k_{)) =}

 



+1, yBay(k) ≥ 0

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Symmetry Property

❏ _{Signal state set} _X _{can be divided into two subsets conditioned on} _b₍_k₋_τ₎

X(±) = _{x¯i ∈ X,1 ≤ i ≤ Nsb : b(k − τ) = ±1}

Sizes of _X(+) and _X(−) are both Nsb = Nb/2

❏ _{Symmetry property}_{: for any ¯}x(+)_i _{∈ X}(+) there exists a ¯x(_i−) _{∈ X}(−) such that ¯x(_i−) = ₋x¯(+)_i

❏ _{With this} symmetry property, Bayesian equaliser can be re-arranged as

yBay(k) = N_sb

X

q=1

βq e

−k

x(k)−x¯(+) q k2

2σ2_n

− e−

kx(k)+¯x(+) q k2

2σ2_n

!

where ¯x(+)q ∈ X(+), Bayesian solution has an inherent odd symmetry

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Symmetric RBF Equaliser

❏ RBF network equaliser

ˆ_b₍_k ₋ _τ_{) = sgn(}_y_RBF₍_k_{)) with} _y_RBF₍_k_{) =}

M

X

i=1

θiφi(x(k))

θi are RBF weights and M number of RBF units

❏ _{Symmetric RBF node}

φi(x) = ϕ(x;µ_i, ρ_i2) − ϕ(x;−µ_i, ρ2_i)

µ_i _{∈ R}ne is RBF centre, ρ2

i RBF variance, and ϕ(•) usual RBF function

❏ _{Gaussian RBF function}

ϕ(x;µ_i, ρ2) = e−

kx−µ ik2

2ρ2

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Block-Based Kernel Algorithm

❏ _{Given training set} _D_K ₌ _{x(k), d(k) = b(k ₋ τ)_}K_k₌₁, set M = K and

µ_i = x(i) for 1 _≤ i _≤ K, and fix all RBF variances to ρ2

❏ _Define _ε₍_i_{) =} _d₍_i₎ ₋ _y_RBF ₍_i_{). Then we arrive} _{regression model}

d = Φθ + ε

where d = [d(1) d(2)_{· · ·}d(K)]T, ε = [ε(1) ε(2)_{· · ·}ε(K)]T,

Φ = [φ₁ φ₂ _{· · ·}φ_M] _{∈ R}K×M

with φ_i = [φi(x(1)) φi(x(2))· · ·φi(x(K))]T, and θ = [θ1 θ2 · · ·θM]T

❏ _{Sparse kernel modelling problem} _⇒ _{select sparse} _M_spa_{-term subset model}

❏ OFS-FRCSM:

Orthogonal Forward Selection based on incrementally maximising Fisher

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Bit Error Rate of RBF Equaliser

❏ _Generic _{symmetric RBF equaliser}

yRBF(k;w) = M

X

i=1

θi ϕ(x(k);µ_i, ρ2_i) − ϕ(x(k);−µ_i, ρ2_i)

where w includes all RBF centres µ_i, variances ρ2_i and weights θi

❏ _Define _{probability density function} _{(PDF) of signed decision variable}

ys(k) = sgn(b(k − τ))yRBF (k;w) as py(ys)

❏ _{Error probability} _{or BER of RBF equaliser}

PE(w) = Prob{ys(k) < 0} =

Z 0

−∞

py(ys)dys

❏ _Nonlinear _{minimum bit error rate} _{(MBER) solution}

w_MBER = arg min

w

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Approximate Minimum Bit Error Rate

❏ _Unknown _p_y₍_y_s₎ _⇒ _{Parzen window estimate}

˜

py(ys) =

1

K√2πσ

K

X

k=1

e−

(_ys−sgn(b(k−τ))_yRBF(k;w))2

2σ2

where σ2 is kernel variance

❏ _{With this estimated PDF, estimated or} _approximate _{BER is given by}

˜

PE(w) =

Z 0

−∞

˜

py(ys)dys =

1

K

X

k=1

Q(˜gk(w))

where Q(_•) is the usual Gaussian error function and

˜

gk(w) =

sgn(b(k ₋ τ))yRBF(k;w)

σ

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Nonlinear Least Bit Error Rate

❏ _{To derive a sample-by-sample adaptive algorithm, consider} _{single-sample} PDF “estimate” of py(ys)

˜

py(ys, k) =

1

√

2πσ e

−(ys−sgn(b(k−τ))yRBF(k;w))2

2σ2

❏ _{Conceptually, with this} _{instantaneous} _{PDF “estimate” we have}

one-sample BER “estimate” ˜PE(w, k) = Q(˜gk(w))

❏ _Using _{instantaneous} _or _stochastic _gradient _∇_P˜_E₍w, k) gives rise to

w(k) = w(k₋1)+_√ξ

2πσe

−y

2

RBF(k;w(k−1))

2σ2 sgn(b(k−τ))∂yRBF (k;w(k − 1))

∂w

which we refer to as Nonlinear Least Bit Error Rate (NLBER)

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Clustering Algorithm

❏ _Set _M ₌ _N_sb_, _θ_i ₌ _{β >} _{0 and} _ρ2

i = ˆσn2 ∀i, where ˆσn2 is estimated σn2

❏ _Then _{symmetric RBF} _{equaliser realises exactly} _Bayesian _{equaliser by} placing its centres µ_i = x(+)_i _{∈ X}(+)

❏ _{Denote legitimate combinations of} b(k) corresponding to b(k ₋ τ) = +1 as b(+)_i , 1 _≤ i _≤ Nsb

❏ _{During training,} _supervised _clustering

if (b(k) == b(+)_i )

ˇ

x(k) = x(k);

else if (b(k) == ₋b(+)_i )

ˇ

x(k) = ₋ x(k);

µ_i(k) = µ_i(k ₋ 1) + µ(ˇx(k) ₋ µ_i(k ₋ 1)) ;

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A Simulation Example

❏ _{Three-tap channel}

x(k) = 0.3 b(k) + 0.8 b(k ₋ 1) + 0.3 b(k ₋ 2) + n(k)

❏ _{Equaliser order} _n_e _{= 4}

x(k) = [x(k) x(k ₋ 1) x(k ₋ 2) x(k ₋ 3)]T

and decision delay τ = 2

❏ _{Size of} _X(+) _is _N

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OFS-FRCSM Results

Bit error rate comparison of optimal Bayesian equaliser and symmetric RBF equaliser based on OFS-FRCSM algorithm

❏ _{Symmetric RBF size} _M _{= 30}

❏ _{Data block length} _K _{= 600}

❏ _{RBF variance} _ρ2 ₌ _σ2

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NLBER Results

❏ _{SNR= 13 dB, and average over 10 runs}

❏ _First _M _{data points as initial centres} µ_i(0), θi(0) = 0.01 and ρ2_i(0) = 4σ_n2

❏ _{Step size} _µ _{= 0}_._{1 and kernel width} _σ ₌ _σ_n _{for NLBER}

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NLBER Results (continue)

Bit error rate comparison of optimal Bayesian equaliser and symmetric RBF equaliser based on NLBER algorithm

❏ _{First 30 data as initial centres}

µ_i(0), initial weights θi(0) = 0.01

and initial variances ρ2_i(0) = 4σ_n2

❏ _{Step size} _µ _{= 0}_._{1 and kernel width}

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Clustering Results

❏ _{SNR= 10 dB, symmetric RBF size} _M _{= 32 and RBF variance} _ρ2 ₌ _σ2

n

❏ _{Learning curve shows Euclidean distance between set of RBF centres and} set of channel states

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Clustering Results (continue)

Bit error rate comparison of optimal Bayesian equaliser and symmetric RBF equaliser based on clustering algorithm

Symmetric Radial Basis Function Network Equaliser