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