Frequency-Difference Beamforming
Shima H. Abadi
Frequency-difference beamforming stems from the following ray-path approximation for the
sound channel's impulse response:
L
 
G(rj , rs ,)   Alj expirlj c .
(1)
l 1
Here L is the number of ray paths between the sound source and receiving array, 1 ≤ l ≤ L, Alj is
an amplitude for each ray to each receiver, rlj is the effective length of each ray path to each
receiver, and c is an appropriate average sound speed. In general, Alj is a complex number and
may depend on frequency but such dependence is neglected here. An equivalent formulation based
on
a modal sum, instead of (1), is likely possible but is not discussed here. Then, normalized
received signal will be:
~
Pj ( ) 
Pj ( )

N
j 1
Pj ( )
2

 r 
eis ( )
L
Alj exp i lj  .

l

1
2
 
N
 c 
 j 1 G(rj , rs , )
(2)
This equation explicitly shows how the frequency  influences phase, even though the path
amplitudes Alj, path lengths rlj, and average sound speed c are unknown. Eq. (2) can be developed
into an expression that includes a frequency difference that is small enough for plane-wave
beamforming: evaluate (2) at two differentfrequencies 2 > 1, complex conjugate the 1evaluation, and form the normalized field product,
~
~
Pj* (1) Pj (2 ) 
L L
 ( r  1rl j) 
expis (2 )  s (1 )
Al*jAmj expi 2 mj
.

2
2
 
 
N
N
c
l 1 m 1


G
(
r
,
r
,

)
G
(
r
,
r
,

)
 j 1 j s 1  j 1 j s 2
(3)
The phase relationship embodied in (3) is of interest because the source phase difference, s(2) –
s(1), appears on the right side, and because the exponential phase inside the double sum is
proportional to 2 – 1 when m = l. In (3) the square-root factors are real functions, so equating
phases in (3) leads to:


~
~
arg Pj* (1 ) Pj (2 )   s (2 )   s (1 ) 
L
 (  1 )rlj  L , L *
 (2 rmj  1rlj ) 
arg  Alj* Alj exp i 2
   Alj Amj exp i
.
c
c
l

1
l

m





(4)
Here, the double sum over ray paths has been separated into diagonal (l = m) and off-diagonal (l ≠
m) terms. The diagonal terms in (4) explicitly include the frequency difference 2 – 1 and take
the following form:
L
 (  1 )rlj 
diagonal terms =  Blj exp i 2
,
c
l 1


(5)
where Blj  Alj* Alj (no sum implied). Interestingly, (5) is functionally the same as (1) with 
replaced by 2 – 1. In both (1) and (5) the rlj correspond to L signal-paths having arrival angles
l at the receiving array. Thus, conventional delay-and-sum beamforming of the field product
~
~
Pj* (1 ) Pj (2 ) at the difference frequency 2 – 1 may yield a useful estimate of the signal phase
difference  s (2 )   s (1 ) when the beam steering angle is equal to l, and the off-diagonal terms
in (4) are unimportant as will be described below.
In the current investigation, such a signal-phase-difference estimate is developed from

N ~
~
b( , 1 , 2 )   j 1 Pj* (1 ) Pj (2 ) exp i(2  1 ) ( , rj ),
(6)
where b is the frequency-difference beamforming output,  is the time delay, and  is the beam
steering angle defined with respect to broadside ( = 0). If P˜ j* (1  0) is a non-zero constant that

is independent of j, then (6) reduces to conventional delay-and-sum beamforming in the limit
1  0 ,

N ~
lim b( , 1 , 2 )  b( ,0, 2 )   j 1 Pj (2 ) exp i2 ( , rj ).
1  0

(7)
For evenly spaced elements along a linear vertical array (the array geometry of interest here), the
time delays in an isospeed sound channel are simply related to the beam steering angle,

 ( , rj )   ( , z j )  ( j  1)d c sin  ,
(8)
where d is the distance between array elements. When there is significant vertical variation in the
channel's sound speed c(z), the time delays (, zj) can be selected in accordance with ray group
velocities to account for the curvature of the incoming wavefronts (see Dzieciuch et al. 2001, Roux
et al. 2008).
Structure of the Field Product
The mathematical structure of the field product in (6) can be illustrated by using (1) in the
simple case of two ray paths (L = 2) when the Alj are real coefficients. First use (1) with L = 2, to
find:
 
Pj ( )  S ( )G(rj , rs ,  )  S ( )A1 j expi r1 j c  A2 j expi r2 j c .
(9)
Thus, the field product becomes:
Pj* (1 ) Pj (2 )  S * (1 ) S (2 ) 
*

 i r 
 i r  
 A1 j exp  1 1 j   A2 j exp  1 2 j  


 c 
 c 

which, after some algebra, reduces to

 i r 
 i r   (10)
 A1 j exp  2 1 j   A2 j exp  2 2 j  ,


 c 
 c 

 2

  
  
 A1 j exp i

r1 j   A22 j exp i
r2 j  
c
c






Pj* (1 ) Pj (2 )  S * (1 ) S (2 )
,
1
 2 A A cos  1  2  r  r  exp i   r1 j  r2 j  



 2 j ij
 1j 2 j
c
c  2  









(11)
where  = 2 – 1. The first two terms inside the big parentheses on the right side of (13) are
the diagonal terms of the field product. They follow the form of (5) and their phases only depend
on , c , and the two ray path lengths. When Pj* (1 ) Pj (2 ) from (11) is beamformed at the
difference frequency, , these diagonal terms will make a contribution to b(,1,2) that does

not depend on 1.
On the other hand, the third term inside the big parentheses on the right side of (11) results
from combining the two off-diagonal terms of the field product. It depends on , 1, c , and the
sum and difference of the two ray path lengths. Thus, when Pj* (1 ) Pj (2 ) from (11) is beamformed

at the difference frequency, , this term will change as 1 is varied. In general, when L ray paths
connect the source and the receiving array, the number of desired 1-independent diagonal terms
(signal) increases like L while the number of undesired 1-dependent off-diagonal-term
contributions (noise) increases like L(L – 1)/2. Thus, for an arbitrary L, there may be an inherent
limit to frequency-difference beamforming's utility since its signal-to-noise ratio may decrease like
(L – 1)–1 with increasing L.
References
Shima H. Abadi, Hee Chun Song, David R. Dowling: "Broadband sparse-array blind
deconvolution using frequency-difference beamforming", Journal of the Acoustical Society of
America, Vol. 132, Issue 5.
Shima H. Abadi, Hee Chun Song, David R. Dowling: "Frequency-Difference Beamforming with
Sparse Arrays", Journal of the Acoustical Society of America, Vol. 132, Issue 3.
Shima H. Abadi, Hee Chun Song, David R. Dowling: " Broadband Sparse-Array Blind
Deconvolution Using Unconventional Beamforming", Journal of the Acoustical Society of
America, Vol. 132, Issue 3.