Samling and Distortion radeoffs for Bandlimited Periodic Signals Elaheh ohammadi and Farokh arvasti Advanced Communications Research Institute ACRI Deartment of Electrical Engineering Sharif University of echnology Email: el mohammadi@ee.sharif.edu marvasti@sharif.edu Abstract In this aer the otimal samling strategies uniform or nonuniform and distortion tradeoffs for stationary bandlimited eriodic signals are studied. We justify and use both the average and variance of distortion as the erformance criteria. o comute the otimal distortion one needs to find the otimal samling locations as well as the otimal re-samling filter. A comlete characterization of the otimal distortion for the rates lower than half the Landau rate is rovided. It is shown that nonuniform samling outerforms uniform samling. In addition this nonuniform samling is robust with resect to missing samles. Next for the rates higher than half the Landau rate we find bounds that are shown to be tight for some secial cases. An extension of the results for random discrete eriodic signals is discussed with simulation results indicating that the intuitions from the continuous domain carry over to the discrete domain. I. INRODUCION Consider the roblem of finding the best locations for samling a continuous signal nonuniform samling to minimize the reconstruction distortion. o find the otimal samling oints one should utilize any available rior information about the structure of the signal. Furthermore samling noise should be taken into account. Due to its imortance nonuniform samling and its stability analysis has been the subject of numerous works see [] for an overview of nonuniform samling. However there has been relatively less work for a fully Bayesian model of the signal and samling noise where one is interested in the statistical average distortion of signal reconstruction over a class of signals. In this aer we study this roblem for stationary eriodic bandlimited signals. he best locations of nonuniform samling via an adative samling method is considered in [] for a first order arkov source. he first order arkov assumtion allows the authors to comute the distortions in terms of the length of consecutive samling times. A tradeoff among samling rate information rate and distortion of uniform samling is derived in [3] for Gaussian sources corruted by noise. he authors in [4] look at choosing the samling oints for a stationary Gaussian signal with auto-correlation function Rτ = ρ τ. It is shown that uniform samling is otimal in this case. atthews in [5] considers a wide-sense stationary rocess and comutes the linear minimum mean square error SE estimate from sub-nyquist uniform samles. he authors in [6] comute the otimal re-and ost-filters for a signal corruted by noise when uniform samling is used. he authors in [7] derive a rate distortion function for lossy reconstruction of a multi-comonent signal when only a subset of the signal comonents are samled. hey only consider uniform and random samling strategies. oreover there has been some work in comressed sensing on the tradeoffs between samling rate and the reconstruction error. In [8] a samling rate-distortion region is found; it is shown that arbitrarily small but constant distortion is achievable with a constant measurement rate and a er-samle signal-to-noise ratio. he authors in [9] define a samling-rate distortion function for an i.i.d. source and derive lower bounds on the achievable erformance. In this aer we consider a continuous stationary eriodic signal. he signal is assumed to be bandlimited with at most N N + non-zero Fourier series coefficients from frequency N ω 0 to N ω 0 where ω 0 = π/ is the fundamental frequency. Since there are N N + free variables the signal can be uniquely reconstructed from = noiseless samles Landau rate. But as we consider taking noisy samles at time instances that we choose unique reconstruction is not feasible and distortion is inevitable. We also consider a re-samling filter in our model. See Fig. for a descrition of our roblem. he Fourier series coefficients and the samling noises are all random variables. herefore both the inut signal and its reconstruction from its samles are random signals and so is the reconstruction distortion. We are interested in minimizing the exected value and variance of this distortion by choosing the best samling oints and re-samling filter. We denote the minimum of the exected value and variance of the distortion by and resectively. A formal definition of and is given in Section II. A small guarantees a good average erformance over all instances of the random signal while a small guarantees that with high robability we are close to the romised average distortion for a given random signal see [0 Aendix A] for further justification. herefore our goal is to find the tradeoffs between the number of samles the re-samling filter the variance of noise σ and the otimal exected value and variance of distortion. Here we rovide tight results or bounds on the tradeoffs among various arameters such as distortion samling rate and samling noise. When we are below half the Landau rate N we find the otimal average distortion
and variance. Interestingly we show that the minimum of both and are obtained at the same samling locations and the same re-samling filter. Furthermore these two minima are obtained when we do not use a re-samling filter on the signal bandwidth and use a nonuniform samling rocedure in which the samles are arbitrary chosen from the set 0 /N /N 3/N N /N. Observe that this set contains uniform samling oints at half the Landau rate. hus for < N the otimal samles are nonuniform. It is worth to note that the samling locations only deend on the bandwidth of the signal; they are otimal for all values of the noise variance σ and N. oreover the samling oints are robust with resect to missing samles. Note that the otimal samling oints are any arbitrary oints from the set 0 /N /N 3/N N /N. hus if we samle at these ositions and we miss some samles i.e. getting < samles instead of samles the set of samle oints is still a subset of 0 /N /N 3/N N /N and hence otimal. herefore if we knew in advance that we get samles this knowledge would not have heled us. For above half the Landau rate we rovide some lower and uer bounds that are shown to be tight in some cases. When N < i.e. between half the Landau rate and the Landau rate we find otimal average distortion when N using a non-uniform set of samling locations. In addition when > uniform samling is shown to be otimal under certain constraints. Whenever we find and exlicitly the minima are achieved simultaneously at the same otimal samling oints and the re- samling filter. his aer is organized as follows: Section II formally defines the roblem. Section III resents the main results of the aer with the roofs given in [0]. Section IV rovides simulation results. II. PROBLE DEFINIION We consider a continuous stationary bandlimited eriodic signal defined as follows: St = l=n [A l coslω 0 t + B l sinlω 0 t] where ω 0 = π/ is the fundamental frequency. he summation is from l = N to N indicating that the signal is bandlimited. We assume that A l and B l for N l N are mutually indeendent Gaussian rv s distributed according to N 0 for > 0. hus the signal ower is N/ where N = N N +. he discrete version of the signal is S[n] = l=n [A l coslω 0 n + B l sinlω 0 n] where ω 0 = π/ for some integer. Our model is deicted in Figure. he signal St given in is assed through a re-samling filter Hω to roduce St. he signal St is samled at time instances t = t t t where t i [0. hese samles are then corruted by Zt an i.i.d. zero-mean Gaussian noise denoted as N 0 σ. hus our observations are Y i = St i + Z i i =. An estimator uses the noisy samles Y Y Y to reconstruct the original signal denoted by Ŝt. he incurred samling distortion given by SE criterion is equal to Ŝt St dt. his distortion is a random variable. Our goal is to comute the minima of the exected value and variance of this random variable i.e. to minimize D = E Ŝt St dt 3 V = Var Ŝt St dt. 4 We are free to choose Hω and the samling locations t i s i. herefore the otimal distortion is defined as follows: = = min H t t E min H t t Var Ŝt St dt Ŝt St dt where the minimization is taken over the samling locations and the re-samling filter. We assume that Hω is a real linear time invariant LI filter such that Hlω 0 for all N l N meaning that the frequency gain of the signal is at most one; in other words we assume that the filter is assive and hence cannot increase the signal energy in each frequency. In articular all ass filters Hω = satisfy this assumtion. he reason for introducing this assumtion is that we can always normalize the ower gain of the filter. III. AIN RESULS In this section first two general lower bounds on the average and variance of distortion are given and then the main results of the aer are stated. he roofs of the main results are given in [0 Aendices A and B]. Note that SNR = N/σ. A. wo General Lower Bounds on the Average and Variance of Distortion Lemma. For any noise variance σ > 0 and any choice of the re-samling filter the following lower bounds hold for all values of : + + + SNR + SNR. When we do not use a re-samling filter on the signal bandwidth i.e. Hlω 0 = for l = N N equality in
Fig.. Samling-Distortion odel. he signal St is assed through re-samling filter. he outut St is samled at times t t t. A noise Z is added to the signal at the time of samling. he noisy observations are then used to recover the signal. the above equations holds if and only if for any time instances t i and t j one of the following two equations holds: t i t j = m N for some integer m 5 t i t j = m + N + N for some integer m. 6 Lemma. For any σ > 0 and any choice of the re-samling filter the following lower bounds hold for all values of : N + SNR 7 +. SNR 8 Furthermore when we do not use a re-samling filter on the signal bandwidth equality in the above equation holds if the samling time instances satisfy t i= ejπk i = 0 for k N +. Note that for < Lemma gives better lower bounds on and whereas for > Lemma gives tighter ones. B. Samling Distortion radeoffs for N he samling-rate distortion tradeoffs are studied when the samling rate is lower than the Landau rate. heorem. For N the otimal average and variance of distortion for any given re-samling filter are given by = + 9 + SNR = + + SNR. 0 Both the minimal average distortion and its variance are obtained with the re-samling filter Hlω 0 = for l = N N and choosing distinct time instances arbitrarily from the following set of N samles 0 N N 3 N N N. he otimal interolation formula for this set of samling oints is given by Ŝt = N + σ i= l=n coslω 0 t t i Y i = N + σ i= cos ω 0 t t i Nω0 sin t t i sin ω0 t t i Y i where Y i is the noisy samle of the signal at t = t i. Remark. Consider 9. Observe that for fixed values of and σ the minimum distortion is linear in. But it is not linear in N as SNR in the denominator is N/σ excet when SNR goes to infinity. In the case of noiseless samles σ = 0 the SNR will be infinity and the minimal distortion will be /. o intuitively understand this equation observe that there are free variables and we can recover of them using the samles. herefore we will have free variables giving a total distortion of / as the ower of each sinusoidal function is /. oreover the maximum distortion is / which is obtained when σ = SNR = 0 or = 0. Observe that when SNR is large + σ = + SNR which is linear in both and σ. Observe that when N is increasing in both and σ. C. Samling-Distortion radeoffs for N < For the samling rates N < we have: heorem. For N < and any choice of resamling filter the otimal average distortion can be bounded as follows + + SNR in which + + Num + SNR + SNR + SNR + N Num Num = min fn N N + SNR 3 and fa b is equal to b if r = 0 fa b = b r + if r > b r if 0 < r b
where r is the remainder of dividing a by b. he lower bound given in is tight when N for any arbitrary. o achieve the otimal distortion no re-samling filter is necessary and the otimal time instances can be chosen from the set 0 N N N N N + N N + N + N N N + N + N D. Samling Distortion radeoffs for. Here the samling rates above the Landau rate are studied. heorem 3. For and any choice of the re-samling filter the following lower bounds on otimal average and variance of distortion hold: N + 4.SNR +. 5 SNR oreover these lower bounds are tight with uniform samling i.e. t i = i/ i = and without filtering on the signal bandwidth if for each k in the interval k we have k. In articular uniform samling is otimal when >. Also the reconstruction formula for the otimal set of samling oints is given by Ŝt = + coslω 0 t t i Y i σ i= l=n = N + σ cos ω 0 t t i 6 i= Nω0 sin t t i sin ω0 t t i Y i 7 where Y i is the noisy samle of the signal at t = t i. Remark. When goes to infinity the minimum distortion = O/ goes to zero regardless of the samling noise value. o see this observe that when is very large > the lower bounds given in 4 and 5 are tight. Remark 3. Unlike the case of N where otimal samling oints could be found without any need to know N we show in [0] that this is no longer the case for >. Besides the uniform samling strategy is not otimal in general. E. Discrete Signals Consider a real eriodic discrete signal of the form S[n] = l=n [A l coslω 0 n + B l sinlω 0 n] 8 where ω 0 = π for some integer and N N < / are natural numbers. Observe that is the eriod of the discrete signal and A l jb l is the l-th DF coefficient of the discrete signal S[n]. Suose that we have noisy samles at time instances t t t. We want to use these samles to reconstruct the discrete signal S[n]. If the reconstruction is Ŝ[n] = N l=n [Âl coslω 0 n + ˆB l sinlω 0 n] the distortion n= S[n] Ŝ[n] is roortional to l A l Âl + B l ˆB l. hus the formulation for minimizing the distortion is the same as that of the continuous signals. he only additional restriction is that t i s should be integers. Whenever the otimal samling oints in the continuous formulation turn out to be integer values they are the otimal oints in the discrete case. And when the otimal samling oints are not integers simulation results with exhaustive search show that their closest integer values are either otimal or nearly otimal. For examle suose that the signal eriod is = 5 and N N = 3 4. In this case N and the otimal samling oints in the continuous case are t t t 3 0 3.75 7.5.5 + τ for some τ [0 ]. he rounds of these oint for τ =. are otimal in the discrete case i.e. any choice of time instances from the set 5 8 is otimal. On the other hand when the signal eriod is = 5 and N N = 4 6. Here again N and t t t 3 t 4 Ϝ = 0.5 5 7.5 0.5+τ are the only continuous otimal oints. Choosing the time instances from their closest integers yields D = 8.0069. However the otimal oints are 8 9 resulting in = 8.0068. Note that the distance between the first two otimal oints is which cannot be achieved by choosing any arbitrary value of τ since the distance between any of the oints in Ϝ is.5. IV. SIULAION RESULS he maximum of the lower bounds given in Lemma and Lemma is deicted in Fig.. In this figure the distortion of the uniform samling without any re-samling filter on the signal bandwidth is also deicted. he erformance of the uniform samling is close to otimal for N its curve almost matches that of the lower bound in Lemma which is otimal for N. We observe that near the Landau rate increasing the number of samling oints does not decrease the distortion. Here the curve for uniform samling reaches its maximum at = 6 whereas the Landau rate is 8. he curve reaches the second lower bound at = 35. hus uniform samling is otimal for the rates above the Nyquist rate > = 34. Finally the exlicit uer bound on the erformance of the samling oints given in heorem is also deicted. As shown in the figure this uer bound can be below the distortion of the uniform samling. In Fig. 3 two random signals are generated using arameters N N σ = 7 4 0.. heir reconstruction with = 3 samles are also deicted emloying the otimal SE method and the iterative method of []. he samling oints are otimally chosen. he average reconstruction distortion SE over all random signals with the given arameters is D = 0.5093. In the to subfigure the
distortion of the otimal SE method is D = 0.000 and the distortion of the iterative method is D iterative = 0.0379. In the bottom subfigure the distortions of the otimal SE and iterative methods are D =.6038 and D iterative =.604 resectively. We see that the erformance of the iterative method is near otimal in the high SNR scenario. Fig. 4 uses the same arameters for generating the signal but the variance of the noise is 4 low SNR scenario. In this case the average distortion over all random signals is D =.9444. he distortions of the to subfigure are D = 0.44 for the otimal SE method and D iterative = 0.3079 for the iterative method. For subfigure b they are D =.8347 and D iterative =.9700 resectively. ACKNOWLEDGEN he authors would like to thank Amin Gohari and Salman Beigi for helful discussions. Fig.. he average distortion for N N σ = 9 9. REFERENCES [] F. arvasti Nonuniform Samling heory and Practice Kluwer Academic/Plenum Publishers New York 000. [] S. Feizi V. K. Goyal and. edard ime-stamless Adative Nonuniform Samling for Stochastic Signals IEEE rans. Signal Process. vol. 60 no. 0. 5440-5450 Oct. 0. [3] A. Kinis A. Goldsmith. Weissman Y. Eldar Distortion Rate Function of Sub-Nyquist Samled Gaussian Sources Corruted by Noise in Proc. Allerton Conf. on Comm. Control and Comuting 03. 90-908. [4] J. Wu and N. Sun Otimum Sensor Density in Distortion-tolerant Wireless Sensor Networks IEEE rans. Wireless Commun. vol. no. 6. 056-064 June 0. [5]. atthews On the Linear minimum-mean-squared-error Estimation of an Undersamled Wide-sense Stationary Random Process IEEE rans. Signal Process. vol. 48 no.. 7-75 Jan. 000. [6] D. Chan and R. Donaldson Otimum re- and ost-filtering of Samled Signals with Alication to Pulse odulation and Data Comression systems IEEE rans. on Commun. echnol. vol. 9. 4-57 Ar. 97. [7] V. P. Boda and P. Narayan Samling Rate Distortion in Proc. IEEE Int. Sym. on Inf. heory ISI 04. 3057-306. [8] G. Reeves and. Gastar he Samling Rate-Distortion radeoff for Sarsity Pattern Recovery in Comressed Sensing IEEE rans. Inf. heory vol. 58 no. 5. 3065-309 ay 0. [9] C. Guo and. E. Davies Samle Distortion for Comressed Imaging IEEE rans. Signal Process. vol. 6. 643-644 Dec. 03. [0] E. ohammadi and F. arvasti Samling and Distortion radeoffs for Bandlimited Periodic Signals Jan. 05. [Online]. Available: htt://arxiv.org/abs/405.3980. [] F. arvasti. Analoui and. Gamshadzahi Recovery of Signals from Nonuniform Samles Using Iterative ethods IEEE rans. Signal Process. vol. 39 no. 4. 87-878 Ar. 99. Fig. 3. wo random signals and their reconstructions in high SNR. Fig. 4. wo random signals and their reconstructions in low SNR.