Speech Transcript - AN ANALYTICAL APPROACH TO LOCAL SOUND FIELD SYNTHESIS USING LINEAR ARRAYS OF LOUDSPEAKERS

0:00:13	alright right sorry for the delay
0:00:15	and
0:00:16	so uh that that some of it's and as a as we use it
0:00:20	uh it may be described as follows and a sum of the elements of a some services
0:00:24	that's out the such
0:00:26	that uh
0:00:26	sounds that would give an design a physical properties
0:00:29	you also over it extended region
0:00:32	oh and and not the for such a a and of the elementary sound was is is this is the
0:00:38	fifty six to a lot the care that is installed
0:00:41	no all of the power
0:00:42	and a because a of a circle of uh
0:00:45	two uh one point five meters that ideas
0:00:49	and so it practice these uh uh elementary sound sources what
0:00:52	uh that a lot because of the we you rather speak about the secondary sources because
0:00:57	is certain locations you assume properties that are are difficult to to source it to a
0:01:03	a discrete a lot to get what spatial extent
0:01:06	and
0:01:07	and if you are familiar with a work you might to be where how much appreciate the state each and
0:01:13	even more important the transparency of an analytical approach you
0:01:17	the most uh a uh double double but that's not a weighted and this was proposed by that backup
0:01:21	then that one variant of the ambisonics sonics approach
0:01:24	uh
0:01:25	according to that in
0:01:26	and also the spectral of in like that that the the but which is a
0:01:30	and extension of the so said set to a lower in there a race
0:01:34	and i i was be a particularly about but local sound it's and
0:01:38	and we use that a local of that to this is a good and you as as you have already
0:01:42	learned than the previous talks
0:01:43	um a practical implementation of some fields and that this system don't work perfectly but perfectly
0:01:49	physical physical
0:01:50	but um you can achieve
0:01:53	a a a a low the increase of the course of the physical occurs
0:01:56	this is what we
0:01:57	so a local self
0:02:00	so quick uh
0:02:02	introduction to the mathematical formulation
0:02:05	so these on the analytical approach is they typically assume a and and and an acoustically transparent
0:02:10	continuous distribution
0:02:12	of these elementary sources which are referred to as a second or so
0:02:17	and then you can establish what we don't listen to a this is
0:02:20	equation
0:02:21	um which
0:02:22	at that was the following
0:02:24	we assume a a continuous distribution
0:02:26	uh of taking small system the
0:02:29	you you all maybe a
0:02:30	and the the led to the function G represents the spatial that the transfer function of these second or sources
0:02:36	so for example a one who or a similar or
0:02:39	like
0:02:39	so so a school
0:02:41	a a a a a a or six so of the complex
0:02:44	a directivity T
0:02:45	and X not um um represents a position on the secondary source distribution X positions space
0:02:53	and and you that the for this transfer function because it may be interpreted as a a green function
0:02:59	and and then
0:03:00	the a the of the crime stick for each of which is individual for a a a source and if
0:03:05	few integrate great with this continuous distribution
0:03:08	the result of is
0:03:09	the is see it
0:03:11	and S
0:03:12	so examples for um such as as a to the things are
0:03:17	these are the the most different want on the left side you have a cycle
0:03:20	distribution and closing the target model
0:03:23	and a not on the right hand side you have an example for uh
0:03:27	for the situation when only since is a a a like a typical be or result of plane is designed
0:03:32	as a a a a a circular distribution
0:03:35	and a to uh the problem and so a a uh uh for the moment is that usually you do
0:03:39	not want how to how what's some you most if you drive you all speakers or the secondary sources and
0:03:44	the specific
0:03:46	you were rather want to
0:03:47	to know how to drive the second or sources and all those that that is is a six of the
0:03:51	walls so you will want to a educate S
0:03:55	and calculate that of the
0:03:57	there are two methods to do this
0:03:59	one is the X is it yeah yeah solutions to this one but um so you you can transform this
0:04:05	a i put it into a the um the spatial frequency domain
0:04:09	which is a which is the domain exactly that is on the geometry of the of the second or sources
0:04:14	you shows good at the start from knicks expansion or wave number maybe or a similar
0:04:19	and then this in one turns into uh a multiplication which can use of these uh for the driving signal
0:04:25	than inverse transform in order to the a
0:04:29	this signal that have to implement
0:04:30	i this is what a it is a specific or something formation
0:04:34	a does it also has that it is
0:04:37	but
0:04:37	so
0:04:38	and and i i have a i'm having a technical problem
0:04:48	and of it and that seems to be a problem with this design it
0:05:17	can
0:05:35	right
0:05:36	number
0:05:36	and and
0:05:39	oh
0:05:40	or
0:05:43	great
0:05:44	we
0:05:47	sorry again
0:05:49	and
0:05:51	um
0:05:53	should
0:05:56	i
0:05:56	sure
0:06:00	most
0:06:10	yeah hmmm
0:06:13	really
0:06:14	and a
0:06:15	maybe re
0:06:17	i
0:06:37	okay
0:06:52	what
0:06:59	and sorry yeah right can i have it my computer i'm quite sure that i
0:07:04	like
0:07:11	i
0:07:15	i
0:07:26	oh
0:07:29	i
0:07:31	oh
0:07:32	and
0:07:33	vol
0:07:34	oh
0:07:45	that
0:07:47	or
0:07:49	i
0:07:49	oh
0:07:54	that's
0:07:54	i
0:07:55	oh
0:08:02	oh
0:08:11	i
0:08:11	or
0:08:12	yes
0:08:13	okay
0:08:14	i don't
0:08:17	i'm very sorry
0:08:29	thank you for pitch
0:08:30	so
0:08:31	me and the are trying to to these include it is explicit solution is an implicit missing
0:08:36	we a you want um exploit the has
0:08:39	but i
0:08:40	i i i have
0:08:42	that's
0:08:43	and where you exploit the physical relationship between the sound in of volume
0:08:47	and uh and this something at the ball all real this volume which is for example described by the case
0:08:53	of and once my ready to goes
0:08:55	in order to to a derive of the time function from a description of the desired sound
0:09:00	it's you don't to explicitly so
0:09:02	a
0:09:03	equation
0:09:04	and this is done with fit in this is and all this that was also a sort of a
0:09:09	accelerated
0:09:10	so for this in a and we want to uh this is it with a linear a speaker rate is
0:09:15	explicit solution and then as and this is equation
0:09:18	i
0:09:18	given as i think about from minus infinity to T
0:09:22	yeah although i don't we assume a a secondary sources
0:09:26	a a yeah X axis
0:09:28	and then um then you can into a an interpret this is this equation as a convolution along a
0:09:35	X
0:09:36	a X is
0:09:36	and so for this someone was that the was zero
0:09:40	which relates to a more quantities and wave number domain
0:09:43	which you can then you use the um it were in this as a a a a a a with
0:09:47	respect to X
0:09:48	which you can use a rewrite
0:09:51	and and uh to to to solve for the by signal and then you of for both
0:09:55	had transform over to
0:09:57	finally obtain a
0:09:58	but then we can i think of obviously he's of abilities of such a linear a are um
0:10:03	are are limited and you have to reference
0:10:05	this thing is that's all
0:10:07	to to a line
0:10:08	and this is and you want look at where it is correct
0:10:10	and you
0:10:12	and that that is a that that's probably in most cases
0:10:16	so that looks as follows
0:10:18	if we want to read we're guess fact way which is shown
0:10:21	and
0:10:22	yeah the source located at
0:10:24	as you cross section two or something
0:10:27	we are and one right hand side we assume pointing is distribution of uh uh model
0:10:32	is just look the next section
0:10:34	and we synthesized this um
0:10:36	and and then it what's quite but but for any frequency you can imagine
0:10:42	so that's why point to the from what was on that data of course and feeling in a be we
0:10:46	we can assume a continuous distribution but in practice
0:10:49	it will run like this
0:10:51	so we have
0:10:52	uh uh uh discrete distribution of of a finite number of uh of speakers
0:10:57	and
0:10:58	and uh and you have a before that this to two
0:11:02	in a very see
0:11:02	for example they anything less
0:11:05	and what we do in order to describe
0:11:08	we don't assume a discrete distribution of secondary sources but we assume a continuous distribution that is excited at is
0:11:15	me points that need to be some a
0:11:17	not the the secondary so
0:11:19	but the driving function
0:11:21	yeah a that is that all these i go back
0:11:24	on be into the relations we have a uh we have a
0:11:28	exploit in be continuous case stay still value
0:11:32	and a and B for i want to um
0:11:34	describe the the consequence of the spatial sampling of what to work with excursion and to you be something of
0:11:40	a time domain signal which are
0:11:42	a probably all familiar with
0:11:43	yeah yeah that is a a a a a a a week because it i do this i can i
0:11:47	Z are emphasized a relation
0:11:50	a relations which uh
0:11:52	"'cause" it
0:11:54	so let us to having a a a a a a a a a a time of my think and
0:11:58	the with um and uh
0:12:00	symmetric uh um a spectrum
0:12:03	and so typically of like a lot to uh and the only think that and used on this something use
0:12:09	i'm of a signal
0:12:10	a repetitions of the spectrum of and this is
0:12:13	a period of repetitions
0:12:16	depends on on the vol
0:12:18	and able to reconstruct a
0:12:20	this is a you know i i i i don't have to to in order to
0:12:23	to extract the based
0:12:26	all
0:12:26	and is a discrete uh signal
0:12:29	which contains a than for you one only if there um
0:12:32	the
0:12:33	the you need
0:12:34	signal is it's
0:12:37	and with a a a a a a a a and you can
0:12:39	indeed achieve a perfect
0:12:41	that
0:12:42	two
0:12:42	i see
0:12:45	so
0:12:45	i i had uh emphasise that in high domain mean yeah it's and time and something we used to be
0:12:50	the repetitions time
0:12:52	it away
0:12:53	and a no pass filter is then used in order to interpolate these discrete
0:12:57	time a signal
0:12:59	it back to continuous time
0:13:01	in of something this is that the different because space is more nation and time
0:13:05	so
0:13:06	can can a proof that station something need to repetition
0:13:10	as base frequency domain
0:13:11	but in which for print is a representation of space frequency domain that depends on the geometry of the a
0:13:17	second or so
0:13:19	is to using you have some examples of me i don't know the are
0:13:22	a a a a a a listed here
0:13:24	and we will uh and that is a uh uh and i'm a a a range and then uh obtained
0:13:30	from digits wave that will be
0:13:31	and these are of just cannot get what it is is the way
0:13:34	to circle
0:13:36	and then in in a in a with a spatial or something such than the is data transfer functions as
0:13:41	a secondary sources as they use of a is discrete signal into
0:13:45	continuous space
0:13:46	and we have a a a a a a a is a sum
0:13:49	yeah investigations of one
0:13:51	a for example how a lot bigger
0:13:53	he's a in order to record to the uh
0:13:56	to derive right a a a a and type to go
0:14:00	and a like
0:14:01	so
0:14:01	no
0:14:02	the continuous time function in this is a specific wave number of uh the way
0:14:07	looks like this and if me
0:14:09	some
0:14:10	we obtain a reputation
0:14:12	which
0:14:12	it do it at a over there
0:14:15	so we now consider a and an or you use a lot of a uh uh uh you know
0:14:22	spatial frequencies because the higher one are then
0:14:25	and and in here by G
0:14:27	directive given of because
0:14:28	and i and you get out of a
0:14:30	and and and read only send
0:14:32	so we see an of C
0:14:34	there is no and interference of the different different spectrum to show that the above
0:14:39	a wrong
0:14:40	nine hundred hertz
0:14:41	these uh rubber this winter fee
0:14:44	so if we look at the sum but the synthesized
0:14:46	it looks as follows on that and side of the continuous distribution
0:14:49	and on the right hand side of discrete distribution
0:14:52	with a lot because they single
0:14:53	twenty sent
0:14:54	at a time but
0:14:56	you don't see any considerable difference
0:14:58	between two
0:14:59	if we go higher are two nine hundred for we see
0:15:02	some some
0:15:03	on the way
0:15:04	and we go even higher
0:15:06	then we have a
0:15:07	indeed double
0:15:08	a a sort of the from
0:15:10	no at this and not my uh and
0:15:14	uh
0:15:15	we are we're going to back to it the later
0:15:17	now the uh uh we are of course not forced to to use some stuff continued it got up to
0:15:22	five five
0:15:23	we could E
0:15:25	a a just a a a a a with the meeting
0:15:27	i i and by
0:15:28	a i all the unwanted a specific or to zero of that's just one line because of an implementation
0:15:35	and if we then a sum of this timing function
0:15:37	these repetitions do not over that
0:15:39	and leave
0:15:40	our based and uh i'm for all
0:15:43	and if we then look at the uh you want something
0:15:46	again this is the design a result the perfect we've got in B C and D
0:15:50	that in the set of of a a of this uh a a and B Y axes
0:15:55	there are indeed very soon
0:15:57	but uh uh of course a a a a a a to the locations of this far away from the
0:16:01	right
0:16:02	we uh uh we have to uh
0:16:05	we have a
0:16:06	well one uh
0:16:08	the of it i
0:16:11	uh a result
0:16:12	and since we achieve a low which is in a a curve as C we this is one of the
0:16:17	so
0:16:18	of course he's not increase of a course come at the cost of a
0:16:22	for a an increase of the duration as well
0:16:25	of course you not forced to to do this band but an an imitation a symmetric you can do also
0:16:30	a the method pretty good
0:16:31	to the we can see
0:16:33	then the
0:16:34	the uh uh uh a as it do not overlap i
0:16:38	and then
0:16:39	these the region to region of increase the C can be you and was the city
0:16:44	direct
0:16:45	and my might yeah i want to mention that all is not that that i i uh a uh a
0:16:50	a a lot of the shown
0:16:51	are what equipment to like
0:16:53	to
0:16:56	to that one so we we consider this situation
0:16:58	and try to improve it
0:17:00	so i quickly compute
0:17:02	in order not to uh use it too much or with the you
0:17:05	the problem of that this can be elegantly for data by a to the great
0:17:10	as it and i'm very transparent the formation in terms of their limitations
0:17:15	no this cannot be implemented in practice we have to use a discrete
0:17:18	um a discrete arrangement of secondary sources which
0:17:21	can to but are different
0:17:23	a local increase of for um
0:17:26	uh a C can be achieved by appropriate
0:17:28	spatial and
0:17:29	limitation
0:17:30	and this is and what we do
0:17:33	as a result
0:17:34	and we have presented a similar to and not exactly the same but that compared them for um
0:17:40	for a circular arrangements
0:17:42	a secondary source
0:17:43	and uh i can you know
0:17:45	the only thing or or or or do you think that the working one at the moment and this is
0:17:49	something i want to do
0:17:51	size
0:17:51	is that
0:17:53	you cannot be used
0:17:54	the of but there's a big difference uh uh two in the way a some it looks like in a
0:17:59	simulation
0:18:00	and the way a it sounds like a when when you it
0:18:03	so you cannot be to come simulation or something so like
0:18:06	and this is what we are what
0:18:08	at the moment to see you to exploit a static still
0:18:11	see if it's really
0:18:13	three for or or not
0:18:15	thank you much
0:18:18	i
0:18:20	i
0:18:21	i
0:18:22	oh
0:18:25	yeah
0:18:33	and
0:18:34	and
0:18:35	i
0:18:38	oh
0:18:40	i
0:18:41	yeah
0:18:43	oh
0:18:44	oh
0:18:46	so
0:18:49	yeah yeah some an and some relationship between uh the a a uh
0:18:55	space
0:18:55	spectrum of us all and the the
0:18:59	lee or the properties of
0:19:02	the space
0:19:03	i guess go back to this frequency domain illustration
0:19:07	i
0:19:07	right
0:19:08	is this one
0:19:10	so uh in that case
0:19:11	the E
0:19:13	um
0:19:14	zero you know uh everything
0:19:16	a the energy of around zero a frequency corresponds to a components of the sound for travelling a regular
0:19:23	to this uh to the secondary source
0:19:25	your from that means if quickly
0:19:26	i go back again
0:19:28	if we can the energy of the percent
0:19:30	there is a a is a sum that
0:19:32	yeah um
0:19:33	properties
0:19:34	almost at a particular
0:19:36	to of the second source distribution and a re
0:19:39	yeah
0:19:41	you and not in a a a a a a shift so like change the region
0:19:45	what we the energy in the space we can to domain is
0:19:48	wrote
0:19:49	the synthesized so
0:19:52	i
0:19:53	we see a
0:20:01	one
0:20:01	you
0:20:03	me
0:20:04	i
0:20:07	oh
0:20:09	and
0:20:11	oh
0:20:18	a
0:20:19	oh
0:20:20	where
0:20:25	um
0:20:29	a a good point
0:20:30	yeah
0:20:31	but
0:20:32	i would assume that a a a uh is complex valued it to as a couple
0:20:37	so
0:20:40	yeah
0:20:40	so far i but at uh if a if it is this a correct
0:20:44	you agree
0:20:45	no make is that something maybe michael or more quickly uh uh uh uh is up the G
0:20:56	a
0:20:56	oh
0:21:05	hmmm
0:21:07	yeah
0:21:09	yeah
0:21:11	and
0:21:13	i
0:21:15	i
0:21:17	i

AN ANALYTICAL APPROACH TO LOCAL SOUND FIELD SYNTHESIS USING LINEAR ARRAYS OF LOUDSPEAKERS

Spatial and Multichannel Signal Processing

Presented by: Jens Ahrens, Author(s): Jens Ahrens, Sascha Spors, Deutsche Telekom Laboratories, Germany