0:00:13 thank you um so this talk will be about a a strain random demodulator this is work that i did with my adviser rubber colour bank and also with while he buys well um so first a little history of sampling um we start with the the well known result uh from chan and and nyquist that if you have a band one band limited signal then um you can um perfectly reconstruct that signal if you have enough samples um another result that was done later by try for forced stand in their early seventies they looked at taking sub nyquist samples of a signal and then doing parameter estimation to identify the parameters of a single tone in a large band with um and more recently we have the results of compressed sensing um that explores sparsity um as a prior on the input signals um and some examples are um chirp sampling example playing and the random demodulator a a little um brief note about compressed sensing since i'm sure most of your well aware of of it uh we have an under determined system of linear equations and uh sparse input vector which were taking um when you're measurements of if our measurement matrix five um it satisfies a near i a tree um that's captured in the restricted isometry property then we can um give very nice statements about um signal reconstruction of that sparse vector and so um i another line of research that started back um um with pro only around the time of the french revolution is um more parameter estimation um probably was trying to um to approximate a curve um of this form um given a set of samples and so if this is what you're trying to if you're trying to fit your curve uh this curve then you need at least two and point um to be able to do that and from this approach we can see that the the channel approach gives you a a band limited signal has um one over T degrees of freedom per unit time and so if you sample um at the right one over T then you will have a sufficient representation of that signal um and this idea was generalized by um but early um more recently um in the idea of the rate of innovation and so you if any signal a more general than band limited if it can be described by a a um finite number of degrees of freedom for unit time um which you calls the rate of innovation then that is the necessary sampling rate to be able to describe that signal so now we will concentrate on um the um on compressed sensing and in particular the random demodulator which um was presented by drop and and his call there's um the basic idea behind the random demodulator is that you have a signal that you're trying to sample you multiplied by a a random waveform and then you integrate that um this product over a long time that's longer than the nyquist rate of the signal then you take samples at the output of that and the greater which X as a low-pass filter um this continuous time process can be um presented as a discrete discrete time process described by um this matrix T which represents a multiplication by the random waveform and the matrix H which is this um and integration operator um so our measurement matrix acting on sparse um frequency vectors is given by the product H times T times have where F is the um inverse fourier transform matrix so um and so this this rate that you're taking samples that is much lower than the nyquist rate of the sim of the signal of your input signal but are you really slow or the nyquist so um the key idea here is that this random waveform from generator of the random the modulator requires a a random waveform that switches at the nyquist rate of the input signal and the E um and the reason we want to sample at a rate lower the nyquist is because it is hard to build um high fidelity high rate um and to digital converters because of the um to limit that the capacitors in the circuits place on the time it takes to um change states of that and what to digital converter and so the general rule is that a doubling of the sampling rate um corresponds to a one bit reduction in the resolution of the at C and so generating and since this random waveform is generated from the same capacitors it maybe just as hard to generate a a um fast lease switching um waveform form as it is to build a high speed analog to digital converter and so um uh along the same lines will take and the side back to the days of magnetic recording um and the problem here is that um engineers were trying to write lots of data on these magnetic disks and the fact that uh um your idealise square poles you um can't actually record in practice so what you're left with is this um smooth poles and these smooth pulses have trouble if you try to space them too close together and so that transitions what you're trying to detect in the waveform are given by these altered these peaks are alternating and sign and the ability of your read back mechanism to detect the transitions is compromised um when the peaks are shifted and produced or changed and amplitude and so the challenge for these engineers was how to write data um while keeping the um transitions um separate enough in time so that you limit your intersymbol interference so there's solution was run like limited codes these are codes written as in or as the I data and there parameter by um D in K where D is the minimum number of zeros between um ones these the zeros in your sequence indicate a no transition in your in or you are waveform of the ones represent trend transition um and here K is the um maximum number of zeros between and he want so the maximum time um between transitions and so the um D represents the or or so separation between um transitions and K A and timing recovery which was necessary for these uh magnetic this magnetic recording situation um and so what what we come up with is a rate loss and increase correlations um when we use the D K constraint sequences um but the advantages manages rate gain from from and increase transition time from spacing are data bits more finely than the transitions in the waveform um it's only a an example of an oral a code that was used and um B M just drives is the two seven code and here we see we have a it's a rate one have code but we have a a factor of three rate again um in or minimum transition spacing and so we see a fifty percent gain in real coding density and so we use this same idea and the random demodulator and replace the unconstrained waveform form of uh the random demodulator with the constraint are wave form and so we have a a waveform that switches the or that has a a larger with between transitions um meaning that we can measure are um the nyquist rate of our signal can be higher given a fixed minimum transition with in our waveform but what do these correlations to for to the properties of our round of the modulator um with this again is are measurement matrix five um given by the product H D F and each entry is a a um some of randomly sign um fourier components of the for a matrix and so if we want five the satisfy the R I P then um we want to be a a your i some tree um which is captured in this tape the right here where we're saying that or are um measurement matrix is newly an orthonormal system and so if we look at how good is on an average um measure um and average system given our constraint sequences then we see that we have in expectation we have the identity matrix plus this matrix to which is given right here and you can see it um completely determined by the correlation in our seek one and so you have um and so we've converted this problem in two a problem looking at um in on average anyway at the matrix to and how it performs and are note that this um this function at a right here which will come up later is the inner product between the columns of this matrix H so so um looking at the spectral norm of the matrix to we look at it the gram matrix and so each sure the gram matrix is given by this um which depends on this function F had which we call the windowed spectrum because it is the so it's the spectrum of a are random random sequence which is the um for a transform of the autocorrelation function um but it's multiplied by this um window function but um as W over are in create which W O W is the um size of your input vectors and R as a number of measurements and so is that gets large which is the situation we're looking at this window gets larger and this windowed spectrum can be approximated by the actual spectrum of of the random waveform taking the account the lack of the in B zero term so this minus one right here so our gram matrix becomes a a diagonal matrix with the square of the spectrum on the diagonal and so are um spectral norm of the matrix to is or the worst-case case spectral norm of our matrix delta to is determined by the um the part of our spectrum that is farthest away from one and so if we look at some examples of specific examples of random waveforms we see um for the rounded the module a which uses and um on constrained independent seek once we have a flat spectrum um um which gives us a a um are matrix delta to is um identically zero and so we see that are um the system proof but provides uniform illumination for all sparse input vectors and for the second example we looked at a a repetition coded sequence and so these repetition codes take a a a um and independent ra to a sequence and repeat each entry one time so that every pair of entries is completely dependent um and this is a plot of the spectrum of such as once and you can see that at high frequencies the spectrum rolls off to zero which means that are um spectrum norm of our delta matrix is um because to one and we we will um do not expect that are I P to be satisfied if we use these repetition coded sequences and and all um because of these high frequencies are not well illuminated but so uh third um random way from that we consider are these uh we call in general rll sequences and so they are generated from a markov chain that imposes are um can straight and are K constraint on the um transition with of the waveform uh the autocorrelation of such a a sequence is given by this um which depends on the transition probability matrix of are um markov chain and also the output symbols where um the vector B here is just a a collection of all the outputs of symbols for each state and a a is the vector B point wise multiplied by the um but the stationary distribution of R markov chain and because are um vector B is orthogonal to the all ones vector um we have that the are autocorrelation correlation um decays geometrically at a rate that depends on the second largest eigenvalue of the matrix P so we here in this part we um illustrate the um geometric tk K um of the of the oral sequences and here you can see this group of uh plots right here's for D equals one and these are pure for D equals to so you can see that the lower the value of D the faster the correlation is in the sequel and there's also a slight dependence on K but that's much less pronounced than the dependence on D so here on on this this plot gives an example spectrum for uh D equals one K cost twenty rll sequence and here you can see that the spectrum um rolls off at high frequency but it does not roll off to zero um so the spectral norm of are don't to matrix um will not go to one and we will um have some expectation that that all P is satisfied and for to that you notice here that um the region marked one here um the for a low pass signals um um the spectrum is very close to one and for the high pass signals mark of the two spectrum it's very close to zero and so if we restrict our input signals to a low-pass and high-pass respectively then we would expect to see different performance in reconstruction and so that's what we did these plots show um the probability of reconstruction versus sparsity um for low frequency and high frequency signals and um i should know the these values for band with are normalized so that the both the unconstrained and the constraint sequences have the same um transition with minimum transition with but um or minimum with between transitions in the waveform and so we see that are constrained random demodulator performs better than the random demodulator for low pass signals um but it performs worse for high pass signals so what this tells is is that are modulating sequence can be chosen um based on its spectrum to eliminate different regions of are the of the spectrum of our input signals and finally there's also even if we don't restrict ourselves to a high pass a low pass signals we still see a uh and advantage in the um band with of the signals that we and um acquire and reconstruct so here you see um these are plots for the random demodulator and are constrained random the modulator and if you allow a um a thirteen percent reduction in your allowed sparsity that you can still see a twenty percent gain in the band with your able to you and so there is a trade-off between the sparsity and the a with of your input signals and finally um we note that are so are theoretical results on the are I P um include the this maximum dependence distance in and a random waveform and the matrix to and we see a slight um inflation in are allowed um sampling rate and so the take aways or that um there's is a uh band with increase if you given a restriction on the minimum transition with your waveform um given fidelity constraints and the tradeoff between sparsity and band with and that you can also make your demodulator tunable so based on the spectrum you can um the late different input signals thank you but i'm we should have time for maybe to question maybe you one a half i i in you um simulations are was you mean that you uh you have that perfect uh a binary sequence that the that you you putting it the system so you you you've constrained it but it is still but but sequence could yeah they're still perfect also so didn't you motivation for doing this was the the these uh and the uh as i stand of the magnetic media argument is that these things would be but that's is yeah have have you looked to will the effect is that would be a how we see is that C uh in cat show in in the the fine make sure so we did a look at that but we had trouble finding a way to capture that in the discrete model that we're using so so is that is that gonna as a uh uh a a a a a big problem with using this is uh in in it's um that so we think that use so using these constraints sequences what um read the effect of that is in perfect imperfect as um that was our motivation for using these constraint sequences but i don't know um specifically what the effects of on putting this and the practise you have a time for for equal question um and it was that you shall can you elaborate a little bit how how you all that a discrete in in frequency see or yeah i um i i a okay uh oh then we will vol so the four well um yeah i okay you yeah okay thank you very much i