Presented by: Dr. Shafayat Abraranumali.com/files/Minimum_Entropy_Beamforming_Pres.pdf ·...
Transcript of Presented by: Dr. Shafayat Abraranumali.com/files/Minimum_Entropy_Beamforming_Pres.pdf ·...
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