study of broadband postbeamformer interference canceler antenna array processor using orthogonal...

Study of Broadband Postbeamformer InterferenceCanceler Antenna Array Processor using

Orthogonal Interference Beamformer

Lal C. Godara and Presila Israt

The School of Information Technology and Electrical Engineering,

The University of New South Wales,

Australian Defence Force Academy,

Canberra ACT 2600, Australia

Email : l.godara@adfa.edu.au, p.israt@student.adfa.edu.au

Outline

• Introduction

• Preliminary Review

• Proposed Method

• Examples and Discussion

• Conclusion

Introduction

• Processing of narrowband signals induced on an array of

sensors is normally carried out by multiplying these signals by

complex weights and summing to produce the array output.

Output y(t)

x1(t)

x(t)

xL (t)

+

w l*

w L*

w1*

Postbeamformer Interference Canceller structure

• PIC Processor is a two channel processor which cancels

directional interference by forming two beams using an

antenna array.

Two Beam Processor

PIC Output

– One beam referred to as the signal beam is steered

in the signal direction,

– the second beam referred to as the interference

beam is steered away from the signal direction and

– the weighted output of the second beam is

subtracted from the first beam to form the output of

the processor.

Narrowband Signals

• In the case of narrowband signals the beam steering is

achieved by complex weighting and no physical

delays are required.

• This scheme of complex weighting to steer beams is

not possible to process broadband directional signals.

Broadband Signals

• For this case a processor normally requires a tapped

delay line filter (TDL) and physical delays between

the TDL filter and the sensors are used to steer a

beam.

• These delays are not only cumbersome and

expensive to use but also deteriorate the array

performance due to implementation errors.

Aim of this paper

• This paper presents a technique to process broadband

signals without using steering delays and

• studies the performance of PIC processor using this

method in the presence of broadband directional

signals.

Preliminary Review

• Consider an array of L elements in the presence of

two uncorrelated broadband directional sources and

uncorrelated noise.

• One source is a signal source and the other one is an

unwanted interference.

Array Signals

• Let the induced voltage on the array elements be

sampled at a sampling frequency fs and an NL

dimensional vector X(t) denotes the N samples, that

is,

TTTT )T)(N(tx,T),(tx(t),x(t)X 1

Correlation matrix

• Define an NL × NL dimensional matrix R as

TtXtXER )()(

))1(())1(()( TntxTmtxER jimnij

Fourier transform

• Define a correlation matrix at the kth bin as

)(~)(~)(~

kxkxEkR H

),())(()(~

keRkekR Hli

Hli Lil ,,1,

.

1

1)(

)1()2

(

)2

(

kNN

j

mkN

j

e

eN

ke

Proposed Method

PIC at kth bin

• Figure shows the structure of the PIC processor at the

kth bin. The output q(k) is given by

)(~)()(~)()( kxkUwkxkVkq HH

Output Power

)()()( kqkqEkP

)()(~

)()()(~

)( kUkRkUwwkVkRkV HH

)()(~

)()()(~

)( kVkRkUwkUkRkVw HH

1

0

).(N

k

kPP

Optimal weights

• The optimal weight which minimizes P(k) is given by

.)()(

~)(

)()(~

)(ˆ

kUkRkU

kUkRkVw H

H

Signal and Interference Beams

• Signal Beam

• Interference Beam )()()( kSkPkU

L

kSkV

)()( 0

Steering Vector

L

kSkSIkP

H )()()( 00

)(2))((

lN

kj

l ekS

Where P(k) is a projection matrix and is given by

The steering vector in direction θ is defined as

Algorithm summary

The following summarizes the algorithm to process

broadband signals assuming that the frequency bin

k = k1, k1 + 1, . . . , k2

corresponding to the desired frequency band.

Algorithm

)()()( kSkPkU

• Estimate R using

• Calculate R(k) using

• Select V(k) and U(k) using

• Calculate the optimal weight using

.)()(

~)(

)()(~

)(ˆ

kUkRkU

kUkRkVw H

H

L

kSkV

)()( 0

),())(()(~

keRkekR Hli

Hli Lil ,,1,

TtXtXER )()(

Power

The mean output power at each bin is given by

)()()( kqkqEkP

)()(~

)()()(~

)( kUkRkUwwkVkRkV HH

)()(~

)()()(~

)( kVkRkUwkUkRkVw HH

Signal to Noise ratio (SNR)

• Substituting ( ) the correlation matrix for𝑅 𝑘 signal and noise,

the expressions for the output signal power and the

noise power can be obtained. Then the output SNR( ) 𝑘

is given by

))(ˆ( kwPS

))(ˆ( kwPN

))(ˆ(

))(ˆ()(

kwP

kwPkSNR

N

S

Theoretical Results

Let Ω( ) denote the output SNR at the centre frequency of the 𝑘

k 𝑡ℎbin. From the analysis of narrowband PIC it follows that

)()(1)(

)()(

)(

)()(

0

02 kakk

kak

k

kLpk

n

S

)(

)()(

2

kLp

kka

I

n

)()(

)()()()()(0

kUkUL

kUkSkSkUk H

HII

H

Theoretical Result

)()(

1

)()(

)(

)(

)()( 2

2

2

kLpk

kLpk

k

k

kLpk

I

n

I

n

n

S

200 )()()()(

1)(L

kSkSkSkSk

HII

H

When the interference beam is formed with 𝑆𝜃 ( ) = 𝑘 𝑆 𝐼 ( ), 𝑘

then 𝛾 0( ) = ( ) and 𝑘 𝑘𝜌 Ω(k) becomes

Value of ρ for 4 element array with N=125

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Interference angle in degree

a

t diffe

rent bin

k = 28

k = 52

Comments

• One observes from figure that when θs is close to θI , ρ(k) is very small. As

θs moves away from θI , ρ(k) varies monotonically with k whereas when θs

is far away from θI , ρ(k) is between about 0.9 and 1.0 and variation with k

is not monotonical.

• Ω(k) depends on ρ(k) which in turn depends on array geometry and relative

positions of the two directions. For example when interference is about

1400, ρ(k) increases monotonically between 0.5 to 0.95 whereas when

interference is at about 600 the increase with bins is not monotonical and the

variation between about 0.92 and 1.0. This variation reflects on Ω(k).

Example

• A linear array of 4 elements

• Number of samples N = 125

• Desired normalised frequency 0.22 to 0.42

• Interference direction variable

• Signal direction =110 degree

• Signal power =1

• Interference power =1

• Noise =1

Output SNR (Theory vs. Simulation)

30 35 40 45 50

0

2

4

6

Different bin k

Sig

nal

to n

ois

e ra

tio

s in

dB

SNR(k)(k)

I = 105

I = 70

I = 95

Comments

• One observes from Figure that SNR( ) 𝑘 and Ω( ) 𝑘 in all cases are almost the

same. Furthermore for interference source, making angles of 1050 and 950

with the line of the array, both SNR( ) 𝑘 and Ω( ) 𝑘 increase with 𝑘 similar to

the increase in ( ) whereas for interference angle of 70𝜌 𝑘 0, the signal to

noise ratio increases with first and then drops after reaching to a 𝑘

maximum as expected from the variation in ( ) 𝜌 𝑘 shown in previous

Figure. Thus confirming the analytical study that the output SNR varies

with k and the variation depends upon the relative positions of signal and

interference sources.

Signal to noise ratios (effect of large no. of samples)

250 300 350 400-2

0

2

4

6

8

Different bin k

Sig

nal to

nois

e ra

tios in

dB

SNR(k)(k)

I = 95

I = 70

I = 105

Comments

• Figure shows the same results for = 1000 to show that as 𝑁 𝑁

increases SNR( ) approaches Ω( ).𝑘 𝑘

Output signal to noise ratios (for low noise)

30 35 40 45 505

10

15

20

25

30

Different bin k

Sig

nal to

noise

ratios in

dB

SNR(k)(k)

I = 105

I = 95

I = 70

Comments

• Figure shows the effect of low background noise 0.01 and 100

respectively. One notes that for a very low noise Ω( ) 𝑘 ∝ (𝑘) 𝑘

which is evident from the figure . One also observe from the

figure that the effect of finite bandwidth is clearly evident in

this case. For the case of high noise, it follows that SNR is

very small in all cases which is evident from the next figure.

Example(s): Signal to noise ratios (for large noise)

30 35 40 45 50-15

-10

-5

0

Different bin k

Sig

nal

to n

ois

e ra

tio i

n d

B

I = 70

I = 95

I = 105

Example(s): Output interference Power

30 35 40 45 50-30

-25

-20

-15

-10

-5

0

Different bin k

Ou

tpu

t In

terf

eren

ce P

ow

er i

n d

B

I = 70

I = 95

I = 105

Comments

• Figure shows the variation in 𝑝𝐼 ( ) as a function of . One 𝑘 𝑘

observes from the figure that the process cancels less

interference when the interference is closer to the signal

source.

Conclusion

• We proposed OIB method for PIC processor which effectively cancels

interference without suppressing the signal.

• Analytical expression for SNR was derived to study the performance

of OIB method.

• Analytical results show that the output SNR depends on the relative

positions of the sources, input signal power, input interference power

and background noise.

• Simulation results were presented to support the analytical expression.

QUESTION