Array Fundamentals and
Conventional Spatial Filtering:
beamforming
Director: Yufeng Zhang
Reporter: Cuiqin Zhao
No:1200500973
Contents






1 Array Fundamentals
1.1 Spatial Signals
1.2 Array Signal Model
1.3 Spatial Sampling
2 Conventional Spatial Filtering: beamforming
- Beam response
2.1 Spatial Matched Filter
-Element spacing
-Array aperture and resolution

3 Conclusions
1 Array Fundamentals


Much as a frequency-selective
filter emphasizes signals at a
certain frequency ,we can choose
to focus on signals from a
particular direction. It has the
ability to spatially discriminate,
and passes signals from the
certain directions while rejected
those from other directions
Sensor array: spatial sampling of
a spatially propagating signal.
1.1 Spatial Signals(1)

Spatial signals
are signals that
propagate
through space.
A propagating wave emanating from a source located at
r0 :
s (t , r ) 
c
Wavelength:  
Fc
A
r  r0
2
e
 r  r0 
j 2 Fc  t 

c


(2)
(1)
1.1 Spatial Signals(2)

Cone angle ambiguity surface for a uniform linear
array (ULA)

The difference in propagation distance between
neighboring elements:
d x  r sin az cos el  r sin 
(3)
1.2 Array Signal Model(1)

Block diagram of propagating signal arriving at a
sensor with receive

The discrete-time signal form a ULA
x(n)  [ x1 (n)
x2 (n)
T
xM (n)]
(4)
 The discrete-time model is
xm (n)  H m ( Fc ,s )s0 (n)e
 j 2 Fc m
 wm (n)
(5)
1.2 Array Signal Model (2)

The full-array discrete-time model
x(n)  M v(s ) s (n)  w(n) (6)
 Where the array response vector
1
1 e j 2 Fc 2 ( )
v( ) 
M


e
 j 2 Fc M ( ) T

The delay to the mth element with respect to the first
element in the array  m ( )  (m  1) d sin 
c
T
So
 d sin   
 d sin   

 j 2 
 j 2 ( M 1) 
1 
 
 


v( ) 
e
1 e

(7)
M 

1.3 Spatial sampling





To avoid spatial alisaing
1
(8)
Spatial sampling frequency: U s 
d
Frequency of propagating signal: U  sin  (9)

Normalized spatial frequency:
U d sin 
(10)
u

Using u :
US

1
1 e j 2
v( ) 
M
e
 j 2 ( M 1) 

T
(11)
Since normalized frequency are unambiguous for
 1  u  1 and the full range of possible   900
2
2
unambiguous angles is ,the sensor spacing must be
d 
(12)

2
2 Conventional Spatial Filtering:
beamforming


Definition: to
combine the signals
from all the sensors in
a manner, that is, with
a certain weighting, so
as to examine signals
arriving from a
specific angle.
The output is
M
y (n)   cm* xm (n)  c H x(n)
m 1
(13)
Beam response
for a given weight vector c: c( )  c v( )
2
 Beam-pattern:
c( )

 Convert spatial frequency to angle as:   arcsin 
H
 Response
d
0
-5
Power Response (dB)
-10
-15
-20
-25
-30
-35
-40
-45
-50
-90 -80 -70 -60 -50 -40 -30 -20 -10
0
10 20 30 40 50 60 70 80
Angle (deg)
90
2.1 Spatial Matched Filter
2.5
2

Array response vector: the beamforming weight vector from
direction s
1.5
1
0.5
cmf (s )  v(s )
spatial matched filter: steering vector
beam-former.
The output of the spatial matched
filter: y(n)  c H ( ) x(n)  v H ( ) x(n)
0
0
20
40
60
80
60
80
100
120
140
160
180
200
9
8
7
6
5
mf
s
 M s(n)  v H (s ) w(n)
s
4
3
2
1
0
0
20
40
100
120
140
160
180
200
Element spacing d(1)

0
The beampatterns of spatial matched filtersswith
for ULA with element spacing of  / 4 ,  / 2 , 
and 2
0
-5
-10
-10
Power Response (dB)
Power Response (dB)
0
-20
-30
-40
-15
-20
-25
-30
-35
-40
-45
-50
-80 -65 -50 -35 -20 -5 10 25 40 55 70
Angle (deg)
-50
-80 -70 -60 -50 -40 -30 -20 -10
0
10 20 30 40 50 60 70 80
Angle (deg)
0
0
-5
-5
-10
-10
Power Response (dB)
Power Response (dB)
Element spacing(2)
-15
-20
-25
-30
-35
-15
-20
-25
-30
-35
-40
-40
-45
-45
-50
-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80
-50
-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80
Angle (deg)
Angle (deg)
Array aperture and beamforming
resolution
The aperture is the distance between the first and last
elements.
 Beam-forming resolution: the angular extent between
the nulls of the mainbeam nn or the half-power points
of the mainbeam (-3dB) 3dB

3dB 

L
0
-5
-5
-10
-10
Power Response (dB)
Power Response (dB)
0
-15
-20
-25
-30
-35
-40
-15
-20
-25
-30
-35
-40
-45
-45
-50
-80 -70 -60 -50 -40 -30 -20 -10
0
10 20 30 40 50 60 70 80
Angle (deg)
-50
-80 -70 -60 -50 -40 -30 -20 -10
0
10 20 30 40 50 60 70 80
Angle (deg)
0
0
-5
-5
-10
-15
Power Response (dB)
Power Response (dB)
-10
-20
-25
-30
-35
-40
-15
-20
-25
-30
-35
-40
-45
-45
-50
-80 -70 -60 -50 -40 -30 -20 -10
0
10 20 30 40 50 60 70 80
Angle (deg)
-50
-80 -70 -60 -50 -40 -30 -20 -10
0
10 20 30 40 50 60 70 80
Angle (deg)
Conclusions


A brief background in some array fundamentals,
including spatially propagating signals and array signal
model.
Introduce the concept of beam-forming, that is ,the
spatial discrimination or filtering of signals collected
with a sensor array ,we look at conventional, that is,
non-adaptive, beam-forming and touch upon many of
the common considerations for an array that affect its
performance, for example, element spacing,
resolution, and side-lobe levels.
thanks