Sensitivity of the Random Demodulation Framework to Filter Tolerances Pankiewicz, Pawel Jerzy; Arildsen, Thomas; Larsen, Torben

Transcription

1 Aalborg Universitet Sensitivity of the Random Demodulation Framework to Filter Toleranes Pankiewiz, Pawel Jerzy; Arildsen, Thomas; Larsen, Torben Published in: Proeedings of the European Signal Proessing Conferene Publiation date: 2011 Doument Version Aepted author manusript, peer reviewed version Link to publiation from Aalborg University Citation for published version (APA): Pankiewiz, P J, Arildsen, T, & Larsen, T (2011) Sensitivity of the Random Demodulation Framework to Filter Toleranes Proeedings of the European Signal Proessing Conferene, General rights Copyright and moral rights for the publiations made aessible in the publi portal are retained by the authors and/or other opyright owners and it is a ondition of aessing publiations that users reognise and abide by the legal requirements assoiated with these rights? Users may download and print one opy of any publiation from the publi portal for the purpose of private study or researh? You may not further distribute the material or use it for any profit-making ativity or ommerial gain? You may freely distribute the URL identifying the publiation in the publi portal? Take down poliy If you believe that this doument breahes opyright please ontat us at vbn@aubaaudk providing details, and we will remove aess to the work immediately and investigate your laim Downloaded from vbnaaudk on: deember 23, 2018

2 SENSITIVITY OF THE RANDOM DEMODULATION FRAMEWORK TO FILTER TOLERANCES Pawel J Pankiewiz, Thomas Arildsen, Torben Larsen Aalborg University Faulty of Engineering and Siene Department of Eletroni Systems DK-9220 Aalborg, Denmark {pjp, tha, tl}@esaaudk wwwsparsesamplingom ABSTRACT The aim of the present paper is to demonstrate the impat of low-pass filter non-idealities on ompressed sensing signal reonstrution in the random demodulator (RD) arhiteture The random demodulator is a ompressed sensing (CS) aquisition sheme apable of aquiring signals in ontinuous time One of the main advantages of the system is the possibility to use off-the-shelf omponents to implement this sub- Nyquist framework Low-pass filtering plays an important role in the RD analog aquisition proess, whih needs to be modeled arefully in the digital part of the ompressive sensing reonstrution Having a omplete model of the analog front-end, CS algorithms ondut almost perfet reonstrution taking far less samples than for traditional Nyquist-rate sampling This paper investigates reonstrution sensitivity to distortion in the impulse response of the low-pass filter aused by passive omponent value flutuations The authors simulate ommon CS reovery algorithms and show that the worst-ase performane degradation due to filter omponent toleranes an be substantial, whih requires speial attention when designing reonstrution algorithms for RD 1 INTRODUCTION For more than half a entury, traditional signal aquisition shemes have relied on Shannon-Nyquist sampling theory, whih ditates sampling at a rate that is higher than twie the highest frequeny of the sampled signal In many ases, this onstraint makes analog-to-digital onverters (ADCs) power hungry devies The reently emerged theory of CS [1, 2] states that this lower bound on the sampling frequeny an be signifiantly redued when the proessed signal is sparse or ompressible within a ertain basis CS theory exploits a priori knowledge of the signal struture, where signal sparsity is modeled by expressing the signal as the linear ombination of a few elements taken in a partiular ditionary (orthogonal or redundant) [3] Compressed sensing is still mainly a mathematial onept, and only few suessful attempts to sample ontinuoustime signals and hardware implementations have been reported [4, 5, 6] One of the sampling arhitetures that exploits the signal sparsity is alled random demodulator, the blok diagram of whih is shown on Fig 1 RD modulates the signal by multiplying it with a high-rate pseudo-random alternating sign sequene, smearing the signal frequeny The work of Pawel J Pankiewiz and Thomas Arildsen is finaned by The Danish Counil for Strategi Researh under grant number hipping sequene Analog filter seed ADC Figure 1: Random demodulator struture [4] ontent aross the entire spetrum A low-pass filter assures no aliasing is introdued by the following low-rate ADC apturing ompressed samples The main advantage of this design is that it is easy to implement using robust, low-power and ommonly available off-the-shelf omponents Thanks to its design simpliity, the RD hardware CS system is less exposed to the effets of non-ideal omponents ompared to other CS hardware realizations [7] Considering the fat that the random demodulator operates at relatively lower sampling rates, omponent non-idealities suh as lok jitter of an ADC or mixer distortion an be negleted One important aspet of the RD system is the measurement matrix defined within a reonstrution algorithm It represents the analog sampling proess where the pseudo-random sequene and impulse response of the filter need to be modeled preisely In reality, no physial devie an be modeled perfetly Due to filter omponent toleranes, the impulse response of the hardware low-pass filter will generally differ from the one modeled in the reonstrution algorithm In this paper, we study the sensitivity of ompressed sensing to this mismath between the ideal and the atual analog front-end represented by the measurement matrix For that purpose we demonstrate a MATLAB simulation framework and disuss results of our analysis 2 METHODOLOGY 21 The Random Demodulator When dealing with ompressed sensing as it was initially defined [1, 3], we an distinguish two main operations that an be arried out separately Namely, the data aquisition proess and the subsequent reovery of the signal of interest by means of a reonstrution algorithm In order to apply om- ym y2 y1

3 where P (n) represents Fourier series oeffiients of the hipping sequene p (t) The ontinuous-time multi-band signal is then multiplied with the p (t) sequene, whih is arried out by a mixer: y(t) = x(t) p (t), t [0,T ) (4) random demodulation ym y3 y2 y1 DSP reovery Figure 2: Proess of ompressed sensing aquisition pressed sensing to ontinuous signals, one needs an analog sampling front-end that will provide a signal representation that is ompatible with the CS framework The random demodulator is one of the analog aquisition shemes, whih has been suessfully employed in the ompressed sampling field [4, 8, 7] This data aquisition proess provides nonadaptive linear projetions of the analog input signal that are strutured to diretly onform to the CS approah: y = ΦΨα, (1) where: y R M 1 represents ompressed measurements, Φ R M N, (M N) desribes system measurement matrix, Ψ C N N is signal sparsifying basis and α C N 1 is a sparse vetor ontaining oeffiients, most of whih are zero Fig 2 represents the general model of the CS system with RD framework inluded 22 System design The analog input signal model in our analysis is assumed to be a multi-tone signal that is band-limited and sparse in the frequeny domain Assuming finite energy in the signal it an be modeled as: x(t) = N n=1 α n ψ n (t), t [0,T ), (2) where α n denotes Fourier series oeffiients, T represents proessed interval and [ ψ n (t) = exp j 2π ] T nt The random demodulator aquisition sheme onsists of 3 operations: A) demodulation; B) low-pass filtering; and C) sampling The operations performed by the analog front-end have to be modeled in the measurement matrix Φ Properly defined Φ enables the reonstrution algorithm to suessfully reover the sparse representation of the signal from the ompressed measurements Demodulation is arried out by multipliation by a randomly alternating zero-mean ±1 sequene, often alled hipping sequene, whih is produed by a random number generator The hipping sequene needs to be generated at minimum Nyquist frequeny of the input signal Continuous-time demodulation an be desribed by p (t) = + n= P (n)exp [ 2π T t ], t [0,T ), (3) This orresponds to frequeny smearing, whih is equal to onvolution within the frequeny domain [7] The original sparse signal ontains few tones, so it is suffiient to examine a small portion of the spetrum in order to extrapolate them For that reason, we perform lowpass filtering to prevent aliasing, and we sample with a low-rate ADC at rate T s This operation produes ompressed samples that are enoded with the representation of the original sparse signal [7] The disrete measurement vetor y an be haraterized as a linear transformation of the disrete oeffiient vetor α In CS it is expressed by the transform matrix A = ΦΨ The ompressed measurement vetor y an be modeled as follows [8]: + y[m] = x(τ) p (τ)h(mt τ)dτ t=mt = N n=1 + α n ψ n (τ) p (τ)h(mmt τ)dτ (5) In the ompressed sensing framework, (5) is approximated by the measurement (sensing) matrix Φ The measurement matrix, modeling the basi priniple of the random demodulator, ontains M N C pseudo-random ±1s per row, where N defines the signal length, M is the number of measurements, and C is a saling onstant depending on the impulse response representation 3 ANALOG FRONT-END AND MEASUREMENT MATRIX The measurement matrix represents a model of the operations undergone by the signal during aquisition It enables the reovery to find the sparsest solution to the system In addition to the hipping sequene, that needs to be refleted in the Φ matrix, a preise model of the filter needs to be inluded Beause the sampling system aquires samples in the time domain, the filtering operation is modeled through its impulse response h(t) In order to visualize how the matrix Φ is onstruted, one might onsider two matrix fators representing the hipping sequene and impulse response of the filter, P and H, respetively Φ = HP (6) The demodulation proess multiplies eah sequene value p,n with the orresponding signal input x n within speified period P = diag{p,1,, p,n } The struture of the impulse response matrix is a little bit more ompliated The size of the matrix depends on how densely the filter impulse response is disretized Eah row of the matrix orresponds to one proessed sample The offset for eah row orresponds to the deimation, whih is

4 diretly proportional to the sub-sampling fator h n h n 1 h n τ h n H = h 3 h 2 h h n h 3 h 2 h 1 The above system of linear dependenies is an approximation of the infinite dimensional analog arhiteture Given the above model in (6) and the aquired data, whih is desribed by the CS model in (1), we an try to solve the underdetermined system of equations and reonstrut the sparse vetor α orresponding to the sampled input signal, linearly transformed by the sparsifying matrix In order to obtain the sparsest solution, in priniple we would have to solve the ombinatorial searh implied by: min α α 0 subjet to y = ΦΨα Solving (7) is an NP-hard problem Fortunately, the problem an be relaxed to a onvex problem, based on l 1 -norm minimization, and the sparse vetor an be reovered min α α 1 subjet to y = ΦΨα This partiular approah is alled Basis Pursuit [9] and it is one of the onvex optimization methods used to reover signals within the CS framework Apart from onvex optimization approahes, there is a group of methods alled Greedy Pursuit where the sparse solution is omputed one step at a time by adding new signal omponents that yield least reovery approximation error In our work, both types of methods have been employed to investigate reovery sensitivity to measurement matrix deviations due to filter impulse response variations In the desription of the performed simulations, we speify the exat algorithms used 31 Modeling non-ideal effets Following the methodology of RD, briefly desribed in this paper, we need a disrete model of the analog front-end in order to perform ompressed sensing The system annot disard the information that is arried by the signal of interest Sine the aquisition proess takes plae in the time domain, we aim to design a disrete model of the analog sampler that losely approximates the time-domain system As mentioned, we represent the low-pass filtering operation by a disretized model of the filter impulse response It is worth noting that our modeling approah entails a trade-off between omputational omplexity in the signal reonstrution and density of the disretized filter impulse response We assume known filter arhiteture and parameters from whih the filter transfer funtion H(s) an be alulated, whih an be further represented in the disrete-time domain, where one an obtain an approximation of the impulse response In order to investigate CS reonstrution sensitivity to hanges in the filter response we have developed a rather simple, but for the purpose appropriate, MATLAB framework We have employed a 4th-order Butterworth low-pass (7) (8) L1 C1 L2 C2 Figure 3: 4th-order low-pass filter filter, with indutive passive network arhiteture depited in Fig 3 Assuming that our input impedane is represented by R (50 Ω) we an derive a transfer funtion 1, H(s) for this filter: where H(s) = 1 As 4 + Bs 3 +Cs 2 + Ds + 1, (9) A = C 1 C 2 L 1 L 2 B = C 1 C 2 L 2 R C = L 1 C 2 + L 2 C 2 + L 1 C 1 D = RC 2 + RC 1 Using bilinear transform (aka Tustin s method), one an transform the model to disrete-time (Z) domain, obtain impulse response and reate the H matrix representing the filter In reality, the passive omponent values, due to the manufaturing proess, deviate from their ideal tehnial speifiation In partiular, eah omponent is produed under a proess whih annot be ontrolled with suffiient preision The proess an be modeled with a Gaussian distribution where the mean µ is the target value Quality assessment of the omponents (indutors, apaitors, resistors) selets them aording to standard guidelines (eg IEC ) for hoosing exat produt dimensions within a given set of onstraints Depending on the design budget and omponent dimensions we end up with fabriated passive omponents with values different by up to, for example 1% or 10% from what we aimed for Hene, the H matrix designed aording to ideal values will be an inaurate representation of the filter devie When possible, one might try to alibrate the algorithm aording to performed measurements on the partiular devie, sine omponent values due to prodution toleranes are stati Fig 4 illustrates 4th-order Butterworth filter impulse response deviations from the expeted response Two worst-ase senarios are depited, where all the apaitors deviate by 5% and indutors by 10% from their expeted values Another problem with the elements in the hardware realization is that nominal values of the omponents might flutuate dynamially due to external fators like eg, ambient temperature variations Dynami omponent flutuations diretly introdue hange to the filter response, whih annot be alibrated real time We would like to point out that the problem of filter non-idealities, as well as RD hardware implementation issues is more omplex and some of the other possible problems have been disussed in [7, 10], where some ompensation shemes have been proposed The idea of alibrating the devie has been pointed out by [7] 1 Assuming high load impedane we have negleted it for the purpose of simpliity 2 IEC, Preferred number series for resistors and apaitors, Standard IEC 60063, International Eletrotehnial Commission, 1963

5 Amplitude 8 x ideal impulse response orner 1 impulse response 4 orner 2 impulse response Samples Figure 4: Impulse responses of the ideal omponent value low-pass filter and two worst ase omponent deviations This alibration ould help to derease the impat of passive element flutuations, but only in limited ases 32 Simulation sheme A MATLAB framework has been designed to simulate omponent deviations and examine the reonstrution error performane of different CS reonstrution algorithms Algorithm 1 briefly desribes the simulation methodology In our experiments, we use a multi-tone signal with five tones distributed within a 100Hz to 21kHz range, where 21 khz represents the maximal signal frequeny The input x is represented by N = 9000 samples; oversampling imitates the analog front-end behavior Additionally, the input signal is orrupted by additive white Gaussian noise (AWGN) and its signal-to-noise ratio (SNR) is set to 50dB The low-pass filter was synthesized with a 500 Hz ut-off frequeny ( f ut ), where the following values from (9) were alulated: A = 2053E 14, B = 8428E 11, C = 3459E 07 and D = 8318E 4 Sub-sampling operates at rate 1 khz (twie the f ut ) The analog front-end for eah input vetor x outputs a 1000-samples ompressed vetor y, also known as an observation vetor (M = 1000) The trivial initial onditions have been hosen to assure almost perfet signal reonstrution, with high probability [7] Additionally the low frequenies that were hosen for the input signal were ditated by omputational limitations of our simulation framework The RD measurement matrix, whih is our entral objet of interest, is generated from Tustin approximation of an analog filter input response and pre-generated hipping sequene The horizontal size of the measurement matrix N orresponds to one proessed period [0, T ] of an oversampled input signal This dense measurement system representation enables aurate impulse response representation Modeling the low-pass filter behavior with emphasis on omponent deviations enables the following benhmarking strategy for reovery sensitivity: the reonstrution system is generated aording to the ideal filter speifiation, while the atual analog sampling simulation uses a filter with non-ideal omponent values As one may guess from the system representation in (9) or Fig 4, reonstrution without appropriate knowledge of the atual deviations in omponent values will inrease the reovery error In order to investigate the performane degradation of the CS reonstrution, three ommon reovery methods have been utilized: Basis Pursuit, Basis Pursuit De-Nosing and Orthogonal Mathing Pursuit [9, 11] Orthogonal Math- Algorithm 1 RD aquisition and reonstrution Input: test signal x (multitone, frequeny sparse) Generate hipping sequene P, reate P matrix Delare num of measurements M, (M < N) F s Set ut-off frequeny F ut = F s 2 anti-aliasing Synthesize LP filter and obtain H R M N h(n) Create sparsity basis DFT matrix Ψ N N for omponent K 1 to K nth do Change nominal value aording to tolerane Tol Perform random demodulation on x Reonstrut sparse vetor ˆα, using BP, BPDN, OMP Reover input x: ˆx = Ψ ˆα Obtain reovery error vetor: x ˆx Calulate quality of the reonstrution SNR SNR = 20 log 10 ( x 2 x ˆx ) 2 Compare performanes of RD with deviated filter values SNR tol to the ideal ase SNR ideal end for ing Pursuit (OMP) is the anonial greedy algorithm for the sparse approximation In order to solve an underdetermined system of equations using the mentioned algorithms, the following MATLAB toolboxes have been utilized: CVX 3, SPGL1 4 and Sparsify 5 4 RESULTS We perform two types of experiments First, we simulate the nominal values ase and the 16 worst-ase senarios, ie the worst-ase ombinations of deviating values of omponents C 1, C 2, L 1, and L 2 from (9) We onsider two apaitors and two indutors in the low-pass filter (Fig 3), deviating from expeted values by ±5% and ±10%, respetively Performing reonstrution with the hanged values, we were able to estimate reovery performane boundaries for worstase senarios Fig 5 illustrates signal reovery performane (SNR tol values ) of all three algorithms solving underdetermined system of equations low-pass filtered with worst-ase omponent toleranes Eah separate orner has been labeled as (1,16) The ideal figure of merit in the plot indiates reovery performane in ase of the ideal math between measurement matrix and atual filter omponent values Ideal ase reonstrution returns an almost perfetly reovered signal ( 489 db) for onvex optimization algorithms BP, BPDN and 46 db using the greedy approah (OMP) Sine the amount of noise added to the input is very small, it is not a surprise that the onvex methods perform alike during benhmark (maximum differene 06 db) OMP appeared to reat slightly different in eah test ase but the reonstrution degradation for all of the methods is substantial Results of the worst-ase simulations indiate a drop in SNR down to 40 db Although the extreme ases do not our with high probability, when designing RD signal aquisition systems for ommerial purposes one annot neglet this effet Sine we are fousing on pratial issues related to the implementation, we have also performed Monte Carlo 3 CVX: 4 SGPL1: 5 SPARSIFY:wwwpersonalsotonauk/tb1m08/sparsify/sparsifyhtml

6 50 [db] SNR 1 Basis Pursuit (CVX) Orthogonal Mathing Pursuit Basis Pursuit De-Noising ideal [ orners ] Figure 5: Performane of 16 worst-ase ombinations under BP, BPDN and OMP reovery methods number of runs µ = 2437 db BP µ = 1964 db OMP µ= 2450 db SNR [db] SNR [db] SNR [db] BPDN Figure 6: Monte Carlo simulation results, 500 runs (3σ trunated Gaussian distribution) with multitone input signal with AWGN at 50dB SNR Histograms represent number of simulation runs to SNR of the reonstrution simulations with 500 runs, using the same initial onditions for the input signal Component values for eah examination iteration were randomly hosen from a trunated Gaussian distribution The trunation fator has been set aording to the same omponent tolerane values as for the worst-ase senario tests (5%, 10%) Fig 6 shows the simulation results The observed average reonstrution quality drop for BP and BPDN is 24 db, and 29 db in the ase of OMP 5 CONCLUSIONS In this paper we have presented an investigation of the ommon ompressed sensing reovery methods, utilized with the random demodulator setup The researh elaborates on systems suffering from omponent deviations of a filter loated in the analog front-end Speial attention has been paid to the representation mismath in the measurement matrix between the ideal and atual analog front-end model The problem analysis emphasizes the implementation aspets When we deal with a real system, sensitivity of the reonstrution framework plays an important role Ongoing system flutuations due to temperature or power supply hanges, as well as stati omponent deviations are diffiult to avoid A simple, but rather representative filter model was hosen to be tested under random demodulator setup, with ommon ompressed sensing reovery tehniques applied Worst-ase senario simulations reveal substantial reovery degradation Reonstrution error inrease of up to 40 db has been observed Through Monte Carlo experiments we extend the sope of our methodology towards physial realizations, indiating mean reovery error inrease of 24 db for onvex optimization methods and 29 db for the greedy approah Performed simulations learly show that ommon CS reovery shemes are highly sensitive to the low-pass filter omponent toleranes This problem deserves speial attention when ompressed sensing is used to sample moderate frequeny analog signals using the RD framework Moreover, utilizing low ost off-the-shelf parts without making additional improvements to the reovery sheme will result in poor system performane Future work of this researh will aim at improving the reonstrution robustness to the measurement matrix inauray in modeling the analog front-end The ompressed sensing method of sampling has great potential, but in order to make it eonomially reasonable for the industry to adopt, improved reovery methods need to be developed REFERENCES [1] E J Candès, Compressive sampling, in Proeedings of the International Congress of Mathematiians, vol 3, pp , 2006 [2] D Donoho, Compressed sensing, IEEE Transations on Information Theory, vol 52, no 4, pp , 2006 [3] E J Candès and M Wakin, An introdution to ompressive sampling, IEEE Signal Proess Mag, vol 25, no 2, pp 21 30, 2008 [4] S Kirolos, J Laska, M Wakin, M Duarte, D Baron, T Ragheb, Y Massoud, and R Baraniuk, Analog-toinformation onversion via random demodulation, in IEEE Dallas/CAS Workshop on Design, Appliations, Integration and Software, 2006, pp 71 74, 2006 [5] M Mishali, Y Eldar, O Dounaevsky, and E Shoshan, Xampling: Analog to digital at sub-nyquist rates, Ciruits, Devies & Systems, IET 2011, vol 5, no 1, pp 8 20, 2011 [6] T Ragheb, J Laska, H Nejati, S Kirolos, R Baraniuk, and Y Massoud, A prototype hardware for random demodulation based ompressive analog-to-digital onversion, in 51st Midwest Symposium on Ciruits and Systems, MWSCAS 2008, pp 37 40, 2008 [7] J Tropp, J Laska, M Duarte, J Romberg, and R Baraniuk, Beyond Nyquist: Effiient sampling of sparse bandlimited signals, IEEE Trans Inf Theory, 2010, vol 56, no 1, pp , 2010 [8] J Laska, S Kirolos, M Duarte, T Ragheb, R Baraniuk, and Y Massoud, Theory and implementation of an analog-toinformation onverter using random demodulation, in IEEE International Symposium on Ciruits and Systems, ISCAS 2007, pp , 2007 [9] S S Chen, D L Donoho, and M A Saunders, Atomi deomposition by basis pursuit, SIAM Review, vol 43, no 1, pp , 2001 [10] Y Chen, M Mishali, Y C Eldar, and A O Hero, Modulated wideband onverter with non-ideal lowpass filters, in IEEE International Conferene on Aoustis, Speeh and Signal Proessing, 2010 ICASSP, pp , 2010 [11] T Blumensath and M Davies, Gradient pursuits, IEEE Trans Signal Proess, vol 56, no 6, pp , 2008 The simulation framework desribed in this artile as well as relevant doumentation are available online at: