TEMPORAL BF - GUC · directly at each antenna element at RF frequencies In modern beam steered array antennas; SA arrays , the pattern is shaped using signal processing to perform

ADAPTIVE ANTENNAS. PROF. A.M.ALLAM

1 1

TEMPORAL BF

ADAPTIVE

ANTENNAS


2

1-Introduction

In traditional (conventional, past ) array antennas the main beam is

steered to directions of SOI, it is called beam steered arrays, phased

arrays, or scanned arrays

In traditional array antennas, electronic beam steering the beam is steered

via phase shifters where the change of the phase of the current is carried out

directly at each antenna element at RF frequencies

In modern beam steered array antennas; SA arrays , the pattern is shaped using

signal processing to perform certain optimum criteria

Digital beam formed antennas arrays (DBF) or AA arrays are modern beam

steered arrays when adaptive algorithms are employed

The digital implementation of these algorithms requires that the array outputs

be digitized through the use of A/D converters

This digitization can be performed at either IF or baseband frequencies


3

The robust design of an adaptive array system is

a multi disciplinary process,

Signal processing

Transceiver design

Array design

Antenna element design

Signal propagation characteristics

It includes:


4

Adaptive beamforming (ABF)-2

Application on angle

of arrival (AOA)

A base-station with

circular adaptive

antenna arrays

Mobile communication


5

Temporal reference BF

-It uses the temporal signal properties, such as embedded training sequences,

to form a reference. This signal is known to both the TX and RX and is sent

from the TX to RX during the training period

-It uses the Weiner Hoph solution (equation ) to get the weight vector from

the reference and the received signal

XXX pRW1

)()(1

1

kXkXK

R HK

k

RXX is an MxMcorrelation matrix of array antenna output X(m)

p is an Mx1 cross correlation vector between X(m) and the reference signal d(m)

)()( *

1

kdkXpK

k

Then it uses the optimization adaptation algorithm to continuously update

the weight vector W such that the BF can receive a signal arriving from a

several directions and attenuate signals from other directions

RXX= ARSSAH + Rnn

)]().([ kXkXER H

XX

)]().([ kSkSER H

SS

)]()([ * kdkXEpX

OR


6

Example of reference signal:

CDMA: spreading code of the user

GSM: Training sequence in each frame

0 7

The first frame is 8 time slots

Includes 26 bits training sequence

Generally, the process of generating the reference signal is very specific

and system dependence

The reference signal does not need to be an exact replica of the desired signal,

just be correlated with the desired and uncorrelated with the interference

In general to generate a reference signal you need to know simple information

like the frequency and type of modulation of the desired signal


7

Weiner and steepest descent methods

-In non blind algorithms , the BF in

the RX uses the information of the

training signal to compute the

optimal weight vector, Wopt

-The optimum weight vector; Wopt is

determined on the base of minimizing the

mean squared error between the desired

signal d(t) and the array output X(k) or X(t)

Output y(t)

X weights

Desired d(t)

-There are two methods of calculating

Wopt before performing the adaptive

algorithms, Weiner Hoph method (solution

or equation) and Steepest descent method

i.e., Weiner and steepest descent methods are based on minimizing the

mean squared error and are explained in the following:

x2(t)

xM(t)

x1(t)


8

oWeiner solution (method, equation)

-Let y(k) and d(k) denote the sampled

signal of y(t) and d(t) at time instant k

, respectively. Then the error signal is

given by

-The mean squared error is defined

by the cost function

)()()( kykdke (1)

])([2

keEJ (2)

-Substituting eqn(1) in eqn(2) given y(k) as

, one gets )()( kXWky H

y(k), d(t), e(t) are single

element matrix [ ]1x1

2)()([ kykdEJ

])}()()}{()([{ *kykdkykdE

])}()()}{()([{ *kXWkdkXWkdE HH

])()()(*)()()()([2

WkXkXWkdkXWWkXkdkdE HHHH

WRWpWWpkdE XX

H

X

HH

X ])([2

Output y

X weights

x2(t)

xM(t)

x1(t)

e(t) Desired d(t)


RX is the MxM autocorrelation matrix of the array output vector X(k),

)()(*..)()(*)()(*{)]()([ 21 kdkxkdkxkdkxEkXkdEP M

HH

X

pX is the M x1 cross-correlation vector between the array output vector

X(k) and the desired signal d(k)

-The cost function J attains its minimum when all the elements of its gradient vector

are simultaneously zero, i.e., when the mean squared error J is minimized, the

gradient vector will be equal to an M x 1 null vector, i.e.,

-The gradient vector of J, is defined by where denotes the

conjugate derivative with respect to the complex vector W

)(J *2)(

W

JJ

*W

)]()([ kXkXER H

XX

)(*)({

.

.

)(*)({

)(*)({

2

1

kdkxE

kdkxE

kdkxE

M

)]()([ * kdkXEPX


10

optXXX WRp XXXopt pRW

1 (5)

WRpW

JJ XXX 22002)(

*

(3)

-At which the cost function attains its minimum as

022)( optXXXWWRpJ

opt(4) (4)

optWWJJmin optXX

H

optX

H

optopt

H

Xd WRWpWWp 2

Substitute for Wopt and px (5)

(6) optXX

H

optd WRWJ 2

min

2

d is the power of the desired signal

optXX

H

opt WRW is the output power of array at Wopt

WRWpWWpkdEJ XX

H

X

HH

X ])([2


oSteepest descent method

-It overcomes the matrix inversion which is computationally intensive and can lead

to an unstable result

-Wopt is calculated in a recursive way. It begins with an initial value W(0) for the

weight vector, which is chosen arbitrarily, typically, W(0) is set equal to a column

vector of an identity matrix

1

0

0

X weights

x2(t)

xM(t)

x1(t)

e(t)

Desired d(t)

-The gradient vector of the cost

function is computed

at time k , i.e., the kth iteration

)))((( kWJ

W(0)

)(WJ

-The next guess of the weight vector is

computed by making a change in the

initial or present guess in a direction

opposite to that of the gradient vector

Select

μ W(k+1)

)))(((2

1)()1( kWJkWkW (7)

where μ is a positive real-valued constant

that is referred to as the step size parameter

or weighting constant


12 12 12

-Substitute for from eqn(3) in eqn(7) we get ))(( WJ

)]([)()1( kWRpkWkW XXX (8)

)]()([)()1( * kekXEkWkW (9)

-The iteration process is repeated until the algorithm converges onto the optimal value

of the weight vector Wopt , which would satisfy

J(Wopt) ≤ J(W) for all W (11)

-The stability of this method depends on μ & RX . For stability and convergence of

this algorithm

max

20

(12)

where λmax, is the largest eigen value of RX . It would indeed converge to

the optimum Wiener solution

)}()]()([)](*)([{)()1( kWkXkXEkdkXEkWkW H

)}]()()(*){([)()1( kWkXkdkXEkWkW H

Substitute (8),(9 )in (3)

)](*)([2/)]()1([2)))((( kekXEkWkWkWJ (10)

WRpW

JJ XXX 22002)(

*

)))(((

2

1)()1( kWJkWkW ( ) (3) (7)


13 13 13

-Assuming there is a single minima, it is logical to consider that successive

corrections to the weight vector in the direction of the negative of the gradient

vector should eventually lead to the minimum mean squared error Jmin, at which

the weight vector assumes its optimum value Wopt

In reality, however, exact measurements of the gradient vector are not possible since

this would require prior knowledge of RXX & pX , consequently, the gradient vector

must be estimated from the available data,

i.e., the weight vector is updated in accordance with an

adaptive algorithm that adapts to the incoming data

EX: two element array


14 14 14

Least Mean Squares Algorithm (LMS)

A significant of the LMS algorithm is its simplicity. It does not require measurements

of the relevant correlation functions, nor does it require matrix inversion

-It uses the steepest descent method but substitute for the expected values of (10)

)]()([2))(( * kekXEWJ by the instantaneous estimate )()(2))(( * kekXWJ

-Start from eqn (7) )))]((([2

1)()1( kWJkWkW

)](*)(2[2

1)( kekXkW

)]()()()[()()1( * kWkXkdkXkWkW H

)](*)([)()1( kekXkWkW (13)


15

-Hence the algorithm is

)()()()1( * kekXkWkW

)()()( kykdke

)()( kXWky H

μ

Output signal y(k)

W(k)

∑ X X

∑

∑ Desired signal

e(k) Error signal

X(k)

Array output vector

Array

antenna

d(k) + -

delay

W(k-1)

μX(k)e*(k)


16

The response of LMS algorithm is determined by three principle factors μ,

number of weights and eigen values of R of the input data

-The disadvantages of LMS algorithm are : •It must go through many iterations before satisfactory convergence is achieved (slower

convergence )

•It may not track the desired signal in a satisfactory manner if the signal characteristics are

rapidly changed

•μ is too small:

Effect of step size parameter μ

-Convergence is slow and we will have the overdamped case

-If the convergence is slower than the changing angles of arrival, it is possible that the adaptive

array cannot acquire the signal of interest fast enough to track the changing signal

-The LMS algorithm will overshoot the optimum weights of interest and we have the

underdamped case. The weights will oscillate about the optimum weights but

will not accurately track the solution desired

•μ is too large:

It is therefore imperative to choose a step size in a range that insures convergence


17 17

Example: array of 8 elements θo=30o, θinterference = -60o


18


19


20


21

Direct Matrix Inversion (DMI) or Sample Matrix Inversion (SMI) Algorithm

It uses K samples X(k), k = 1,2, . . . , K of the array signals to get estimate of R by

means of a simple averaging scheme

where denotes the estimate at the kth instant of time and X(k) denotes the

array signal sample, also known as the array snapshot, at the kth instant of time

)(kRX

)()(1

)(1

iXiXk

kR Hk

i

X

(14)

1

)1()1()()1(

k

kXkXkRkkR

H

XX

(15)

The process of estimating the , may be combined to update the inverse

from array signal samples using the matrix inversion lemma as follows

)(kRX

)(

1kRX

XX pkRkW )1()1(1

(16)

)()()( *

1

idiXkpk

i

X

(17)

)()1()(1

)1()()()1()1()(

1

1111

kXkRkX

kRkXkXkRkRkR

X

H

X

H

XXX

optXX WkWRkRk

)()(1

IRo

X

1)0(

1


22 22 4/10/2017 LECTURES 22

-The advantages of DMI(SM{) algorithm are :

•Faster than LMS because the time taken by snapshots or blocks are less than the convergence

time of LMS

-The disadvantages of DMI(SM) algorithm are :

•R may be ill conditioned which make problems when converted

•In case of large arrays there is a challenge in matrix inversion

• The sample matrix of DMI algorithm is the average estimate of the correlation matrix of the

array using K time samples (snapshots)

• Each snapshot ( time sample) is called a block , k is the block number , K is the block length

• Since we use K length block of data, DMI is called block adaptive approach. The adapting of

the weights are done block by block


23 23 4/10/2017 LECTURES 23

The SMI pattern is similar to the LMS pattern and was generated with no iterations. The time

taken by the total number of snapshots K is less than the time to convergence for the LMS

algorithm

Example: array of 8 elements θo=30o, θinterference =-60o


Since the signal sources can change or slowly move with time, we might want to

deemphasize the earliest data samples and emphasize the most recent ones. This can

be accomplished by modifying the estimate correlation matrix R and vector p as:

Recursive Least Squares algorithm (RLS)

)()()()()(1

1

1 kXkXiXiXkR HHk

i

ik

X

)()()()()(ˆ **1

1

1 kXkdiXidkpk

i

ik

)()()1(ˆ kXkXkR H (18)

)()()1(ˆ * kXkdkp (19)

)()()(1

1

1 iXiXkR Hk

i

k

X

)()()(ˆ *1

1

1 iXidkpk

i

k

This is called a weighted estimate and α is called the forgetting factor

It is a positive number between 0,1, we restore the ordinary least squares algorithm α = 1

Let us break up the summation into two terms: the summation for values up to i =

k−1 and last term for i = k

Thus, future values for R and p estimates can be found using previous values


25 25 4/10/2017 LECTURES 25

-The RLS algorithm can be described by the following equations

-Its advantage over LMS is that it is faster than the simple LMS

- Generally it is easy updating the inverse of the correlation matrix

)]1()()(*)[()1()( kWkXkdkGkWkW H

)()()( 1 kXkRkG

))()1()(

)1()()()1()1((

1)(

1

1111

kXkRkX

kRkXkXkRkRkR

X

H

X

H

XXX


26

-It uses the method of least squares to adjust the weight vector W(k)

-We choose the weight vector W(k), so as to minimize a cost function that consists of

the sum of squared errors over a time window

-In the exponentially weighted RLS algorithm, at time k, the weight vector is chosen

to minimize the cost function

where

and λ is a positive constant close to, but less than one

k is the iteration number, i.e., the observation time of a sample snapshot of vector X(k)

-Notice that , with the arrival of new data samples X(k), estimates are updated

recursively, introduce a weighting factor λk-1 to the sum-of-error-squares in eqn (18)

-Also the earlier samples are weakened by the weighting factor ( forgetting factor)

-The RLS algorithm is obtained from minimizing eqn(18) by expanding the

magnitude squared and applying the matrix inversion lemma

Recursive Least Squares algorithm (RLS) “ANOTHER APPROACH”

2

1

1 )()( iekJk

i

k

(18)

)()()()()()( iXkWidiyidie H


27

-The RLS algorithm can be described by the following equations

where I is the M x M identity matrix, and δ is a small positive constant

called the regularization parameter, which is assigned with a small value

for high SNR and a large value for low SNR

-Its advantage is , it is faster than the simple LMS algorithm. This

improvement in performance, however, is achieved at the expense of a

large increase in computational complexity

)1()()()( kWkXkdkH

)()1()(1

)()1()(

1

1

kXkPkX

kXkPkK

H

IP 1)0(

)1()()()1()( 11 kPkXkKkPkP H

)()()1()( kkKkWkW

Documents

TEMPORAL BF - GUC · directly at each antenna element at RF frequencies In modern beam steered array antennas; SA arrays , the pattern is shaped using signal processing to perform