my
oh
oh
close one can everyone hear me
okay uh
good to all if you my name is uh probably in
and from university of michigan
a this is work is with mad wise set and then killed but and also joint work with profits a
mike flynn and and to and shows that that are
and as a that such just some gonna talk about a a little or compressive sampling at C
that we called as in and P P M
so before explain what random B P a is i need to explain uh what the P P M stands
for
so it stands for a pulse position modulation and and as you can probably and the stand from the name
that information that they basically
is present in the position of the pulses that it but uses
uh
for going to the D is i have to mention that the P P M at C falls and to
the general category of a C is that
oh less time encoders
that is because they can what that would that information into time delays and then digitise their time information using
efficient
D C time to digital can like this
the advantage of that is that it low was that out of the C because you can now replace all
the and the looks at you
in the classical it C with the uh digital parts
and that reduces the power
and also uh
but the current scaling down trends in the is but the chip at a are going down and the gate
delay so going down and
the power supply what is are going down it is much easier to get a finer time resolution
then um is a leash and so making use of all these uh uh selecting all this five
uh this P P M at C was uh proposed a a in those and nine by not uh eight
et cetera
and uh well not going into the details the at sequence consists the P P made is it consists of
uh friends ramp or signal
the am signal is P R T and uh the input signal is compared with this different than friends signal
you continuously
and the points where a the signal uh the um
and just say
a those are the points which are recorded
and uh the because the starting point of the rams are known
and the slope of the times so known
uh the delta that one and L delta out to the
position of the pulses
a gives you the information about the whole data show of the signal at that point as well
so effectively that what they T C at it's is uh
nonuniform signal dependent kind of sampling
and uh classically be uh because of uh a non-uniform it sure you can note the the roots difficult to
use linear reconstruction so
a that a uh some classical classically some reconstruction techniques for use not only trick get things
oh we are going to be did this one to mention that those algorithms need that the
the signal to be sampled at about board uh one point two times the nyquist rate
and below the nyquist rate those on returns i direct
so our goal here is to take this P P a at C and can what in didn't do or
compressive sampling at C
but which we mean that we need a uh sampler in the at see that of the signal X sub
nyquist rate
in the time domain and then of we also on construction reconstruction inside at see that the
will be fast enough and accurate
i need to reconstruct the signal and the frequency domain of course to use that compressive sensing design
we need to assume that does signal is sparse
uh
so i in but because you made a sparse in the frequency domain and it is as sparse which means
it has only S dominant frequencies
so if you saw the coefficients uh i'm bit you'd it has to dig get faster and they should be
only has uh dominant one
and of course a straightforward way to can what the P P M midi seen to uh
compressive sampling it is is to just a at the
uh sub nyquist rate and then of course uh use uh the new reconstruction techniques
we use uh a a a a a a a matching pose a kind of reconstruction technique
um
after that one way
and as we will see that that is much inferior to the and the P P M design what it
what the random and that is just introduced random it's into a system appropriately
uh specifically we can make this starting points of the rams random
and uh
we have tried different a random distributions but the uniform distribution into a uh to the best
so the simulations of the presentations is for that
and uh uh again on so the
sampling is no random um non-uniform
and it but results all the proper is and that by is of the P M design
and we used the uh
uh uh the use to different algorithms to reconstruct the signal both of them are
uh three matching pose a kind of all returns but does since i have time a little time i
this "'cause" only the first target it
uh so uh
warming the measurement matrix of a system before a going to that i just one explain a little but of
how would the measurement matrix is going to be different
from the a classical you compress sensing matrix
so usually in the compressed sensing if we assume there are uh F is you input signal and hand why
are you measurements
we take random linear measurements and from this matrix five
and uh of of if as you assume if you assume that the uh the in a measure and you
the measurements and nothing but the random on grid sampling
and the matrix for is just a collection of random rows from the identity matrix
and if S is sparse in the frequency domain
where site use a dft matrix and X is the representation of S in the frequency domain and we assume
that X is sparse
i want to reconstruct a
so an night measurement matrix is just a multiplication of sight
by time side
which would be a a a sub separate dft matrix
but in the case of P P em five is not a make it's like this but it's an interpolation
matrix now
so be is not going to be a separate it uh idea of team
so if you wanna look at what be is
uh we go to that the and then step it that wasn't the labs iteration
so if you as you that the signal has only one and C S not
uh that is the signal is equal to to the G to by not be and if you if it's
given that there the signal is sampled at time points to you want to take K
then we know that the measurements are have to look like that
so uh and my uh K is the number of measurements that that a C takes
and N is the number of measurements if we sampled at nyquist rates of course we want
keep K much fess than and
so much a matrix is a cake cross in matrix
and using this observation be
uh if you are looking for them
at the frequency F and then they're looking for measurements in this manner
and if you're looking for assume we're looking for measurement in this manner and so on
we can fill up the damage of a matrix and normalize it appropriately
and uh
point to note here is that the measurement matrix is a random and the randomness comes from this points uh
time points do you want to get the signal was sampled
and uh these since these points are uh you non-uniform and they don't lie on any nyquist grid
uh because of that the matrix B uh i does not have to necessarily satisfy any a typical
so we don't check for a right B and rather we
uh make do with something much weaker
we look at the correlations relations between the different columns and they'll expose bounds on them
and use those bonds to for their give a at a and is what or you got it
so uh uh and uh
so reconstruction algorithm
uh it's so it's just
so uh is similar to a uh any matching and really was it only them and stuff going into the
details of this one mention that
it has to blocks the frequency identification block and the coefficient estimation collection
and the most intense step in this plot is that least squares
which we to do with the digits it's next iteration
and uh the the coefficient estimations step is present inside the iterations
and the most intensive part here is the might it an of uh be transpose times a
the residual
and uh
because of the special structure of a matrix B we can formulate the be transposed are
uh
uh inverse and you have fifty
and we can use some existing algorithms for attending this item a hundred of um
out of N log N
and so if you number of iterations but um is i
the uh average run-time is order of i and log in
so do not just to have it be for a look at why and how the algorithm works
the be if you look at this to be transposed are
initially i is nothing but they that measurement Y
so that is nothing but be transposed be times et
and we can prove that if you have a and F number of measurements K when K is big enough
for as is the sparsity of the signal
it's lance is some constant
we can show that the a diagonal elements of for this guy matrix is uh are a small enough
and they can so the prove that
the estimate does that we get
their expected values uh quite close to the original value you don't that variance is bounded by the
a energy of the signal and
a brief sketch of the proof
uh uh prove has more of a a kind of leading to it
uh so we can for the prove that Y is uh if like
the be it's is i U but as i sit it or
and when gaze big enough we can prove that would probably be one minus or of that's learn squared
this kind of a and T
a X S on the uh at is is the
uh best test "'em" presentation of a signal X and the uh a constant at
for is kind of a signal dependent constant because
uh a it is a constant that separates the dominant frequency components from the non dominant ones for example in
this figure
uh they are five frequency components but we are in to sort only the three dominant once
and this as five kind of a threshold which shows separates is uh a company and stuff that is there
so which goes into this that's pressure
so uh i a set four
in the first iteration uh that estimate as
i quite close to the actual value but the variance is quite high because the a signal is left
i to be estimated
and that is indicated but this long that and so here
and they can prove that the probability of one man sort of epsilon square at good fraction of the use
of a a if i correctly
a ones those that i in fact is to make at the
so their contribution can be subtracted from the signal and you that's it you can be up to
and
so because a good fraction of that at fight in the first iteration
the variance not drops down because uh
the amount of energy in the signal also goes down
and we can prove that with a similar probability get good fraction of those uh is which switch not identified
properly in the first tradition
but lower be identified T
so the net number of frequencies are i it got goes up
and
so that the the result of that is that the variance skip going down and down from iteration to iteration
eventually
uh after sufficient number of iterations the at is small enough that
all the estimate this can be identified correctly
and then estimate only if we could just can be identified T and then estimate
oh are going to uh some
sort of a target it them
so we have a series of uh results that support the at
and uh the first one is way really construct might be don't signals with the on with them one that
just discussed
and that might be don't signals that just mean on signals with the
oh a linear combination of sinusoids and the sign so it's have random phases and frequencies
and they have a a you
and uh of this kind of signal is that take in and then we add additional white gaussian noise to
it
and reconstructed it the different input as a non levels the signal is sampled with both the regular P P
and design it is the with the
no than a P P M
and also with their and then P M and then reconstruct
the line line points to the regular P M and the blankets language to the and them one and the
black line response to over
benchmark that we colours as the estimate quest
which is nothing but the input signal sampled at the nyquist rate
the same quantization level as the as that of these at he's and then truncated to people only the estimate
in the frequent
i min
so that truncation actually improves the performance of the benchmarks so this is uh
and good benchmark
and uh uh uh uh a and at the point to note here is even when the input as and
i mean noise is like gaussian noise is added
because of the fine at the time resolution as and team at C there is some more
uh a quantization it that already present a signal
and a looking at the results as we can see the adding random to the system definitely improves the performance
of the at C
is is uh a a a that and P P and performs much better than this one at all snr
levels and
is separating the benchmark a much closer to it
that is going to the back correlation properties of the as measure
and then that point to what is that as we increase the number of tones in the signal at is
we make the signal less sparse
the
the a good but is a lot of degradation in the performance of the constant be but the regular P
M bit as and the and a P P is
the could be an affected compared to the benchmark
so i same number of measurements the than the B M is or
we can skate less but signals much better than the regular P B
in a second experiment is also just a proof of concept it's movement meant where with a steak a simple
one don't signal and then we construct a reading number of uh measurements and rating the input as an a
and i'll explain the poor out and the tight first but what as an hour is they X axis on
the Y y-axis is given by the percent at sampling needed for sets as
but it posted at sampling it simply this uh issue okay oh what and
and uh sets as is some criteria we define as an example for this to the
a particular experiment
and was it down sampling the it was as is the least uh as the used number of samples
that you need to succeed uh using this terms
so can as you can see in that class
but i and a P B M needs a much like a much less measurements just that C
and these and and an increases this
quickly dropped down to about three percent and stays about the same and the gap but also increases as as
and not increases
and the have on the left is the case when there is no arts about motion was and only the
measurement noise
even in this case the
the regular P P i'm kind of big it's
once you go as than twenty for about twenty percent of the measurements
but as the
the you the trying to be B can go as low as the three or two
uh you know next experiment i would this one a mentioned that uh
uh a of via a have are dealt with the on with frequencies in these two experiments in this that's
them and read look at an off peak frequency and how they with them forms
oh a frequency i mean the frequency which is lies
all the nyquist read that we searching on
and we know that that "'cause" this the spectral leakage and adversely affects effects a sparsity
and V uh it prior to encounter that using the hamming window approach we multiply the signal with a hamming
window uh or the sampling so the sampling process is an effective
and since the having window is uh a reversible
it's nonzero at all times so we can we can be words it after the reconstruction
uh i as you can see uh the performance with the many are definitely improves the performance of the system
are close close it to the benchmark
i i i as an at low snr reversing the hamming window to the noise so it to doesn't work
well
and have an plot at the regular people "'em" here because the old them doesn't come at all
uh in our next experiment we look at uh something of a bad because signal at is an fm signal
and we amplitude thirty three percent the nyquist rate and we have a similar results that the hamming window the
performance of improves
a log
and same for the him signal
and uh going into the all with them do i just one mention here that the
we not get them to you "'em" some action
conditions on the signal and because of that very able to reduce the number of iterations from i to just
one
and so it is computationally very less it's than a read them one
and uh uh its performance is the compare will go on with them one at high snrs
i it low an it actually does for much better than of them one
so if you know that the if you know the actually conditions on the input signal that side place and
then
or also if there is an is a little all with them to was back
and uh we have a a similar results for the uh practical signals the and it from signals
and for the for the T that the all with them to i would ask you to refer to the
people
and uh
so i in confusion so we have a
compressive sampling a a them
P P M at C
uh that
that keeps all the advantages of the regular B "'em" at C and also takes all the methods of the
compressive a sampling technique
since that's they is that mike which it can handle uh
and signals that the less parts
and signals knows with of at frequencies and
the reconstruction algorithms are are simple in and can be made simpler
uh for practical hardware in
should
uh
so that they can my presentation thank i attention
i i mean just at then you slightly showed uh of core frequency estimation
so you all the significant eating image always a sick the signal last a which to performance right
um
would you consider oh have you tried using uh uh
a fine else
speaker the fact um frequency grid for construction
a a frequency they definitely improves performance the again it increases the competition
and uh
the you want to implement it can hard the we want to keep the that simple last
so instead we trying to use a hamming window approach
the jokes
pretty good yeah
thank you
uh
i was wondering why do you choose a pulse position modulation as to a D conversion technique can have you
compared to this
like a at most that's
yeah exploit sparsity are quite a few of them and so on
yeah so in uh be five to look at a few uh are uh a two D time encoder techniques
but most of them
uh they this one a continuous time T S P proposed for prove process to with this and it kind
of uh
it has a higher power than the um
P M I D C and it works in the analog domain
and can it to based on the main leaves no advantage that's fifty
i and they uh even a finite rate of innovation
kind of a little bit unstable stable than our method i
and uh
uh
kind of also needs close to a nyquist rate sampling and that's
sub nyquist sampling
that's to a major difference that done none of the other matt that seem to a query well it's sub
make the something so we look at the
P P at C
is there some deeper explanation why these are their methods that's fail and and this one is very suitable for
the
or location in combination with sparse
signal process
uh i
a i haven't really talked much log that
but i think that uh
signal lip and then make chair maybe
oh
i the signal dependent sampling each of its planes
i
like this five
this but that might work
okay
that
i don't at
we may have for a time for one more question
hmmm
mouth full
one for two