One-Channel Inverse Filter: spatio-temporal control of a complex
wave-field from a single point
Supplement material
About the virtual sources
On the Figure SM1a is given the schematic illustration of the image-sources decomposition used to
modelize the contribution of reflections in a reverberated rectangular cavity. Thus, based on this type
of decomposition, an illustration of the definition of “virtual sources” is given on the Figures SM1b
for more complex shapes. Considering any reverberating cavity, the impulse response between a
source S and a receiver R can be decomposed in several virtual impulse responses (Figure SM1b1).
Represented in the image-sources space, each of these virtual impulse responses corresponds to a
ring-shape source (the “virtual source”) that includes an ensemble of image-sources (Figure SM1b2).
In this illustration, the image-space is conjectured to have a random distribution of image-sources,
represented as white dot. It might be unrealistic since the density might not be uniform and is likely
to be peace-wise continuous in some regions. The purpose here is purely pedagogic.
Ampl. [a.u]
b1)
Time [a.u]
b2)
S1
S 3
S2
S
R
a)
Image-sources
space
Reverberating
cavity
Fig. SM1: a) Illustration of the image-sources theorem: every reflexion can be seen as the result of
the direct path between an image S’ of the initial source S located in the image plane. b1) An impulse
response measured between a source S and a receiver R is decomposed into different time windows.
b2) Illustration of the concept of “virtual sources” in the image-sources space. Each time window of
b1) corresponds to an ensemble of image-sources distributed in a ring surrounding the cavity. The
image-sources are schematically represented by the white spots. The rings in the image-sources
plane are becoming progressively wider as the time windows are later in the impulse response.
Comparison between One-Channel Inverse Filter and Spatio-Temporal
Inverse Filter
The virtual propagation operator is the core of the One-Channel Inverse Filter focusing technique. It
was inspired by the propagation operator of the Spatio-Temporal Inverse Filter and the link between
those « virtual » and « non-virtual » propagation operators is highlighted on the figure SM2. For the
Spatio-Temporal Inverse Filter (Figure SM2a), the propagation operator is built column after column
from the impulse responses recorded on the M receivers (or control plane) after the successive
emission of impulses from the different emitters. While for the One-Channel Inverse Filter (see figure
SM2b which is the same as the figure 3 of the article), there is only one set of impulse responses left
(e.g one column in the propagation operator of the Spatio-Temporal Inverse Filter). This lack of
emitters is thus filled by using the decomposition of the impulse responses into virtual sources as
already described in the article.
a)
h1j(t)
Emitters
Receivers
1
1
m
hmj(t)
j
J
h
mj
(t )1 j  J ,1 m M
M
(t )
(t )
hh11(t)
h1j(t)
hh12
1J(t)
 11




 
h12
(t )
 
mj(t)





hMM11 ((t)
hMj(t)
hh12
t)
t )
MJ((t)
Reverberating
cavity
hMj(t)
b)
Emitter
h1(t)
Receivers
1
hm(t)
h
v
mj
M
Reverberating
cavity

(t ) 1 j  J ,1 m M
v (t ) 
(t )
hhv11(t)
h1jv (t)
hh12
1J(t)
 11




v (t )

 

h12
(t)
mj





v
v
hvMM11 (t)
hMj
(t)
hh12
(t )
t )
MJ((t)
hM(t)
FIG. SM2: construction of the propagation operator. Comparison between the multi-channel
technique of the Spatio-Temporal Inverse Filter (a) and the single-channel one of the One-Channel
Inverse Filter (b). The latter figure is the same as the figure 3 of the article.
As a result, the Figure SM3 gives the comparison in terms of directivity pattern of the focusing
obtained with both techniques. Thus, the One-Channel Inverse Filter, while using only one emitter,
gives a similar result to the Spatio-Temporal Inverse Filter which involves 32 emitters on this
particular time window (from -25 µs to 25 µs).
-200
0
200 [s]
Intensity [dB]
0
OCIF20
STIF
-10
-20
-30
-40
-20
-10
0
10
Position [mm]
20
FIG. SM3: Comparison of the directivity pattern obtained with the Spatio-Temporal Inverse Filter
(dashed line) and the One-Channel Inverse Filter (with 20 virtual sources). The focus obtained with
only one emitter is similar to the one obtained with the multi-channel of the Spatio-Temporal Inverse
Filter for this time window ranging from -25 µs to 25 µs.