Presented by:

Dr. Shafayat Abrar

Contents

Introduction

Existing Solutions for Blind Beamforming

Minimum Entropy Deconvolution (MED) Criteria

Proposed Algorithms

Simulation Results

Conclusion

Q/A Session

Adaptive Filters

Definition

A filter that self-adjusts its coefficients

according to an algorithm driven by an error

signal.

Types

There are two types of adaptive filters

Blind Adaptive Filters

Non-Blind Adaptive Filters

Non-Blind Adaptive Filters

Adaptive Filters that require a desired signal

for their operation. Such filters try to

minimize the error between the filter output

and the desired signal.

Examples

○ Least Mean Square (LMS) Algorithm

○ Recursive Least Squares (RLS) Algorithm

○ Least Mean Fourth (LMF) Algorithm

Blind Adaptive Filters

Adaptive Filters that do not require a desired

signal for their operation. Such Algorithms try

to restore certain signal properties hence they

rely on signal statistics.

Examples

○ Constant Modulus Algorithm (CMA)

○ Multiple Signal Classification (MUSIC) Algorithm

○ Multi-Modulus Algorithm (MMA)

Beamforming

Definition

A signal processing technique used in sensor

arrays for directional signal transmission or

reception.

Adaptive Beamforming

Transmission or reception of signals in

different directions without having to

mechanically steer the array

Blind Adaptive Beamforming

Adaptive Beamforming realized with the help

of blind adaptive filters is called Blind Adaptive

Beamforming.

Narrowband Signals

For purpose of beamforming ‘narrowband’

means that the bandwidth of the impinging

signal should be narrow enough to make sure

that the signals received by the opposite ends

of the array are still correlated with each other.

Adaptive Beamforming System

Constant Modulus Algorithm (CMA)

CMA/Godard Algorithm Forces output to have

a constant Modulus

CMA has the following cost function

In special case of CMA has the

following weight update equation

cma {(| | ) }p q

nJ E y R

p q 2

2

1 ( | | )n n n n nw w y R y x

: step size parameter, : output, : Dispersion Constant, : Regressorn ny R x

Multi Modulus Algorithm (MMA)

MMA utilizes the dispersion of real and

imaginary parts separately

The cost function of MMA is given as

The weight update equation of MMA is given

as

2 2 2 2 2 2

mma , ,[( ) ( ) ]n R R n I IJ E y R y R

2 2

1 , , , ,[ ( ) ( )]n n n R R n R n I I n I nw w y R y y R y x

, ,, : Real and Imaginary Part of Output

, : Real and Imaginary Part of Dispersion Constant

n R n I

R I

y y

R R

One of the earliest principle for designing blind

cost functions

Proposed by Wiggins in 1977

He suggested to maximize the following cost

for seismic data (Super Gaussian)

4

1

1

2

2

1

1

1| |

1| |

B

n b

b

B

n b

b

yB

yB

: Number of Equalized SamplesB

Gray generalized Wiggins idea to two degrees

of freedom in 1979 as follows

Donoho then developed general rules for

designing MED type estimators

Several cases of MED have appeared in

context of blind deconvolution of seismic data

have appeared in literature

1( , ) 1med

1

1

1| |

J

1| |

Bp

n bp q b

pB q

q

n b

b

yB

yB

Designing Blind Cost Function

We use following in the design of

cost function for Advanced Phase

Shift Keying (APSK)

constellations

MED principle

The probability density function

(PDF) of transmitted (APSK)

PDF of noisy received signal

PDF of

Continous APSK

Gaussian PDF

(Received Signal)

Using MED principle along with PDFs of APSK

constellation and noisy received signal results

in the following cost function

Maximizing the above cost can be interpreted

as finding weights that

Drive the distribution of away from Gaussian

towards uniform

This results in removal of interference from received

APSK signal

2

†

2arg max =

max

n

w

n

E yw

y

ny

Stchochastic Gradient Based implementation of

the equation

requires inclusion of a differentiable constraint:

one possibility is given below

† 2arg max | | s.t. max | |n n aw

w E y y R

† 2arg max | | s.t. fmax( ,| |)n a n aw

w E y R y R

†w

By optimizing the given expression for the

following update equations are obtained

Where

The given algorithm is called β-CMA

22 / ( ) 1L a aMM P R

: Total Number of Signal Alphabet

: Alphabets on Modulus

: Average Signal Energy

: Outermost Modulus

L a

a

a

M s

M R

P

R

1 f ( ) ,

1, if  | |f ( )

, if  | | .

n n n n n

n a

n

n a

w w y y x

y Ry

y R

To obtain an adaptive blind beamforming

algorithm for Square-QAM we note that

In-phase and quadrature components of square-

QAM are statistically independent of each other

Exploiting this independence and applying MED we

get the following cost function

Optimization of the given equation yields the

following algorithm which is termed β-MMA

2

, ,max , s.t. max maxn R n I nw

E y y y R

1 , ,

,

,

f f

1, if  | |f

, if  | |

n n R R n I I n n

L n

L

L n

w w y y x

y R

y R

2

, ,( 2) / (3 ), max maxR n I nR R R a a

Signal to Interference

Plus Noise Ratio (SINR)

We Define

2( ) ( )SINR

H H

n k k k nk H

n k n

w h h w

w R w

th

2 th

( ) : Steering Vector for k source

: Energy of the k source

: True autocorrelation of the interference

k

k

k

h

R

Simulation Parameters for β-CMA

Signal to Noise Ratio (SNR)=30dB

Signal to Interference Ratio (SIR)=10dB

Interference1 = 16 APSK (8,8), impinging angle

Interference2 = 16 APSK (4,12), impinging angle

Desired Signal = 32 APSK, impinging angle

Antenna Array Elements =9

Distance between antenna elements=λ/2 where λ is

wavelength of signal

1 5

2 50

40d

16 APSK (8,8)

32 APSK

16 APSK (4,12)

Simulation Results

Simulation Parameters for β-MMA

Interference1 = 4 QAM (8,8),

impinging angle

Interference2 = 16 QAM (4,12),

impinging angle

Desired Signal = 64 QAM,

impinging angle

Antenna Array elements, Inter element spacing,

SNR and SIR values are same as for β-CMA

1 0

2 45

45d

Simulation Results

Design of cost functions using MED is

discussed

Two algorithms named β-CMA and β-MMA are

proposed for APSK and QAM constellations

respectively

Superior performance of proposed schemes is

shown with the help of SINR comparison with

conventional schemes