Frequency Sum Beamforming
Shima H. Abadi
Frequency-sum beamforming is an unconventional beamforming technique that manufactures higher
frequency signal information by summing frequencies from lower-frequency signal components. It is intended for
acoustic environments where a free-space propagation model is expected to be useful but perhaps slightly imperfect.
For the near-field acoustic imaging geometry shown in Fig. 1, the environment’s Green’s function is approximately:


 
exp i r j  rs c
 
,
(1)
G (r j , rs ,  ) 
 
4 r j  rs


where rs is the source location, r j is a receiver location, and c is an appropriate average sound speed since
inhomogeneities may cause mild variations in sound speed.
Sound Source at rs
y
Receiving Array Elements at rj
•
j=1
•
•
j=2 …
•
•
x
•
•
•
j=N–1 j=N
FIGURE 1. This is the generic geometry. A linear recording array receives signals broadcast by a near-field sound source. The
origin of coordinates coincides with the center of the array and the array elements lie along the x-axis.
Thus, the temporal Fourier transform,
Pj ( ) , of the signal recorded at the jth receiver (1 ≤ j ≤ N), can be
modeled as:


 
exp i r j  rs c
 
Pj ( )  S ( )G (r j , rs ,  )  S ( )
,
 
4 r j  rs
(2)
where S() is the Fourier transform of the broadcast signal. The conventional narrowband near-field delay-and-sum

beamforming output B1 ( r ,  ) at temporal frequency  is

1
B1 (r ,  ) 
1

 
 Pj ( ) exp  i r  rj c
N
j 1

2
,
(3)
where

r is the search location, and 1 is a normalization factor that can be chosen in variety of ways. The resolution
(or transverse spot size) of such conventional beamforming is proportional to c
L , where L is the dimension of the
array perpendicular to the average source-array direction. Thus, higher frequencies hold the promise of higher
resolution acoustic imaging.
Frequency-sum beamforming increases the resolution of B1 from (3) by manufacturing a higher frequency
from a quadratic or higher field product that is used in place of
Pj ( ) in (3). For example, consider two nearby
frequencies, 1 = 0 +   and 2 = 0 –  , that lie in the signal’s bandwidth, and form the quadratic product:


 
exp i 20 rj  rs c
 
 
Pj (1 ) Pj (2 )  S (1 ) S (2 )G (rj , rs , 1 )G (rj , rs , 2 )  S (1 ) S (2 )
 2 .
16 2 rj  rs
(4)
The phase of the final form in (4) depends on the sum frequency 2 0. Thus, the quadratic field product
Pj (1 ) Pj (2 ) = Pj (0   ) Pj (0   ) can be used for beamforming at this higher frequency,

1
B2 (r , 0 ) 
2

 
 Pj (0   ) Pj (0   ) exp  i 20 r  rj c
N
j 1
2
,
(5)
in the hope of obtaining an acoustic image of the source with twice the resolution of B1 from (3) evaluated at 0. Here,
 might be zero or it might be the frequency increment between neighboring complex amplitudes calculated from a
fast-Fourier transform (FFT) of the array-recorded signals. In general,  should be chosen to optimize the
beamformed output, but such an optimization effort is not considered here.
The quadratic nonlinearity that leads to B2 is readily extended to higher powers of the recorded field. For
example, a fourth-order nonlinear field product can be constructed as follows:

1
B4 (r , 0 ) 
4


2
 
 Pj (0  2 ) Pj (0   ) Pj (0   ) Pj (0  2 ) exp  i 40 r  rj c .
N
j 1
(6)
This construction should have four times the resolution of B1 from (3) evaluated at 0 since the sum frequency
manufactured by the nonlinear product is 4 0. Again,   can be chosen to be zero or set to another appropriate value.
The subscripts 1, 2, and 4 in (3), (5), and (6), respectively, denote the number of the complex field amplitudes used in
the beamforming.
Frequency sum beamforming increases the beamforming resolution in free space. However, it cannot be used
to increase beamforming resolution in a multipath environment without the appearance of a fictitious peak in the
beamformed output. The cross terms produced by the quadratic product in frequency-sum beamforming cannot be
suppressed by averaging through the signal bandwidth. The possibility of removing the fictitious peaks in the
frequency sum beamforming output in multipath environments is currently being investigated.
References
Abadi, S. H., Van Overloop, M. J., Dowling, D. R., (2013) “Frequency-sum beamforming in an inhomogeneous
environment”, Proceedings of Meetings on Acoustics, Vol. 19, 055080.