Adaptive Minimum Bit Error Rate Beamforming Assisted QPSK Receiver

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Adaptive Minimum Bit Error Rate Beamforming Assisted

QPSK Receiver

S. Chen, L. Hanzo, N.N. Ahmad and A. Wolfgang

School of Electronics and Computer Science University of Southampton

Southampton SO17 1BJ, U.K.

E-mails: {sqc,lh,nna00r,aw03r}@ecs.soton.ac.uk

Presented at IEEE International Conference on Communications Paris, France, June 20-24, 2004

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Overview

Adaptive beamforming assisted multiuser detection for multiple receive antennas aided SDMA systems with QPSK modulation scheme

Motivation for minimum bit error rate design

System model and standard minimum mean square error solution

Minimum bit error rate beamforming solution

Adaptive implementation of minimum bit error rate design

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Motivation

128-subcarriers OFDM 4-receive-antennas aided SDMA, observing user 1 BER with increasing number of users:

-10 -5 0 5 10 15 20 25 30 35 40

Average SNR (dB) 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 A v er ag e B E R

P=1, 1 user AWGN 11 Users 10 Users 9 Users 8 Users 7 Users 6 Users 5 Users 4 Users 3 Users 2 Users 1 User

-10 -5 0 5 10 15 20 25 30 35 40

Average SNR (dB) 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 A v er ag e B E R

P=1, 1 user AWGN 11 Users 10 Users 9 Users 8 Users 7 Users 6 Users 5 Users 4 Users 3 Users 2 Users 1 User

(a) MMSE (b) MBER

Given number of antennas, capacity is fixed. But changing design from MMSE to MBER ⇒

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System Model

L receive antennas and M users,

point-source model with narrow band channels Ai for 1 ≤ i ≤ M

Received signal model:

x(k) = ¯x(k) + n(k) = Pb(k) + n(k)

where n(k) = [n₁(k)· · ·n_L(k)]T is system noise vector, user QPSK symbol

user M user 1 user 2

θ₁

...

l=1

l=2

l=L

λ/2

vector b(k) = [b1(k)· · ·bM(k)]T, and system matrix

P = [α1A1s1 α2A2s2 · · ·αMAMsM]

with si, 1 ≤ i ≤ M, denoting steering vectors and α2_i transmitted signal powers.

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Beamforming

Linear beamformer:

y(k) = wHx(k) = wH(¯x(k) + n(k)) = ¯y(k) + e(k)

with beamformer weight vector w = [w₁ w₂ · · ·w_L]T, and the decision for b₁(k):

ˆ_b

1(k) = sgn(yR(k)) + jsgn(yI(k))

Let wHP = wH[p1 p2 · · ·pM] = [c1 c2 · · ·cM]. Then

y(k) = c1b1(k) +

M

X

i=2

cibi(k) + e(k)

To make sure c1 being real and positive, weight vector rotation:

wnew = c

old 1

|cold₁ |w

old

Minimum mean square solution: wMMSE = PPH + σ_n2IL

−1

p1, σ_n2 being system noise

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Bit Error Rate

Conditional PDF of y(k) given b1(k) = +1 + j:

p(y| + 1 + j) = 1

Nsb

X

¯

y(q)∈Y₊_,₊

1

2πσ2

nwHw

exp −|y − y¯

(q)_|2

2σ2

nwHw

!

where y¯(q) ∈ Y+,+ are points of y¯(k) conditioned on b1(k) = +1 + j and Nsb is number of

points in Y+,+

BER:

P_E(w) = 1

2 PER(w) + PEI(w)

with

P_ER(w) = 1

N_sb

X

¯

y(q)∈Y₊_,₊

Q g_R(q)(w) P_EI(w) = 1

N_sb

X

¯

y(q)∈Y₊_,₊

Qg(_Iq)(w)

g_R(q)(w) = sgn(b

(q)

R,1)¯y (q)

R

σn

√

wH_w g

(q)

I (w) =

sgn(b(_I,q₁))¯y_I(q)

σn

√

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Minimum Bit Error Rate

MBER solution:

wMBER = arg min

w PE(w)

No closed-from solution, but it can be obtained via gradient-based optimization, with gradient for normalized w given by

∇PE(w) = 1

2 ∇PER(w) + ∇PEI(w)

∇P_ER(w) = 1 2Nsb

√

2πσn

X

¯

y(q)∈Y₊_,₊

exp   − ¯

y_R(q)

2

2σ2

n





sgn

b_R,(q)₁ y¯_R(q)w − ¯x(q)

∇P_EI(w) = 1 2Nsb

√

2πσn

X

¯

y(q)∈Y₊_,₊

exp   − ¯

y_I(q)

2

2σ2

n





sgn

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Adaptive Implementation

Given a block of training data {x(k), b1(k)}K_k₌₁, a Parzen window estimate for

the PDF of y(k), p(y), is given by

ˆ

p(y) = 1

K2πρ2

n K

X

k=1

exp −|y − y(k)|

2

2ρ2

n

!

where ρn is kernel width

From the estimated PDF pˆ(y), one obtains the estimated BER PˆE(w)

Block-data based adaptive MBER solution: minimizing PˆE(w) using a

gradient-based optimization

To derive sample-by-sample adaptation, consider one-sample PDF “estimate”:

ˆ

p(y, k) = 1

2πρ2

n

exp −|y − y(k)|

2

2ρ2

n

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Least Bit Error Rate

Conceptually, from one-sample estimate pˆ(y, k), one has instantaneous BER

ˆ

PE(w, k)

Minimizing this instantaneous BER with stochastic gradient

∇Pˆ_E(w, k) =

−sgn(bR,1(k)) exp

−y

2

R(k)

2ρ2_n

+ jsgn(bI,1(k)) exp

−y

2

I(k)

2ρ2_n

4√2πρ_n x(k)

leads to the LBER:

w(k + 1) = w(k) + µ−∇PˆE(w(k), k)

c1 = wH(k + 1)p1

w(k + 1) = c1

|c1|

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Comparison with Least Mean Square

Compared with LMS:

w(k + 1) = w(k) + µ(b1(k) − y(k))

∗

x(k)

c₁ = wH(k + 1)p₁

w(k + 1) = c1

|c1|

w(k + 1)

LBER has a similarly low complexity:

algorithm multiplications additions square root exp(•)

LMS 16 × L + 6 14 × L − 2 1 –

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Simulation (Fixed Channels)

3-element antenna array, 4 users, and fixed channel conditions:

Ai = 1 + j0 for 1 ≤ i ≤ 4

Desired user signal to noise ratio:

SNR = 1

σ2

n

15o 30o

λ/2

1

desired

45o

source

source 3 interferer source 2

interferer

70o

source interferer

4

Desired user signal to interferer i ratio:

SIRi =

α2₁

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Comparison of BERs

(a) desired user and all three interfering users had equal power: SIRi = 0 dB, i = 2,3,4

(b) desired user and interfering users 2 and 3 had equal power but interfering user 4 had 6 dB more power than desired user: SIR2 = 0 dB, SIR3 = 0 dB, SIR4 = −6 dB

(c) all three interfering users had 2 dB more power than desired user: SIRi = −2 dB, i = 2,3,4

-10 -8 -6 -4 -2 0

0 5 10 15 20 25

log10(Bit Error Rate)

SNR (dB) MMSE

MBER

-10 -8 -6 -4 -2 0

0 5 10 15 20 25 30

SNR (dB) MMSE

MBER

-10 -8 -6 -4 -2 0

0 5 10 15 20 25 30

SNR (dB) MMSE

MBER

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Comparison of PDFs

Case (a) with SNR= 15 dB: conditional PDFs, marginal conditional PDFs, and

signal subsets Y₊_,₊

−1 0

1 2

3 4

−1 0 1 2 3 4 0 0.1 0.2 0.3 0.4 0.5 0.6

Re[y] Im[y]

pdf

−1 0

1 2

3 4

−1 0 1 2 3 4 0 0.2 0.4 0.6 0.8

Re[y] Im[y]

pdf

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Comparison of PDFs

Case (c) with SNR= 20 dB: conditional PDFs, marginal conditional PDFs, and

signal subsets Y₊_,₊

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Learning Curves of LBER

Case (b) with SNR= 17 dB: (i) w(0) = wMMSE, and

(ii) w(0) = [0.0 + j0.1 0.1 + j0.0 0.1 + j0.0]T

DD: decision-directed adaptation with ˆb1(k) substituting b1(k)

1e-6 1e-5 1e-4 1e-3 1e-2

0 200 400 600 800 1000

Bit Error Rate

sample

MMSE DD training MBER

1e-6 1e-5 1e-4 1e-3 1e-2 1e-1

0 200 400 600 800 1000

Bit Error Rate

sample

MMSE DD at k=140 training MBER

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Simulation (Fading Channels)

Same 3-element antenna array

and 4 users, but magnitudes of

channels Ai, 1 ≤ i ≤ 4, were

Rayleigh processes, each with root mean power √0.5 + j√0.5

Fading was continuous, yielding different channel magnitude and phase for each transmitted symbol

Fading is slow at normalized

Doppler frequency 10−6

Frame structure: 40 training

symbols followed by 400 data symbols

-4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

0 10 20 30 40 50 60

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Conclusions

• An adaptive beamforming assisted multiuser detection scheme based on the

minimum bit error rate design has been derived for multiple receive antennas aided SDMA systems

• The minimum bit error rate design provides better performance and improves

system capacity, compared with the standard minimum mean square error design

• Sample-by-sample adaptation has been realized using the least bit error rate

algorithm, which has a similarly low complexity as the least mean square algorithm, for the QPSK modulation

• Our approach can be extended to space-time multiuser detection scheme for