TO APPEAR IN THE IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 1. Non-Uniform Wavelet Sampling for RF Analog-to-Information Conversion

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1 TO APPEAR IN THE IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 1 Non-Uniform Wavele Sampling for RF Analog-o-Informaion Conversion Michaël Pelissier and Chrisoph Suder Absrac Feaure exracion, such as specral occupancy, inerferer energy and ype, or direcion-of-arrival, from wideband radio-frequency (RF) signals finds use in a growing number of applicaions as i enhances RF ransceivers wih cogniive abiliies and enables parameer uning of radiional RF chains. In power and cos limied applicaions, e.g., for sensor nodes in he Inerne of Things, wideband RF feaure exracion wih convenional, Nyquis-rae analog-o-digial converers is infeasible. However, he srucure of many RF feaures (such as signal sparsiy) enables he use of compressive sensing (CS) echniques ha acquire such signals a sub-nyquis raes. While such CS-based analog-o-informaion (A2I) converers have he poenial o enable low-cos and energy-efficien wideband RF sensing, hey suffer from a variey of real-world limiaions, such as noise folding, low sensiiviy, aliasing, and limied flexibiliy. This paper proposes a novel CS-based A2I archiecure called non-uniform wavele sampling (NUWS). Our soluion exracs a carefully-seleced subse of wavele coefficiens direcly in he RF domain, which miigaes he main issues of exising A2I converer archiecures. For muli-band RF signals, we propose a specialized varian called non-uniform wavele bandpass sampling (NUWBS), which furher improves sensiiviy and reduces hardware complexiy by leveraging he muli-band signal srucure. We use simulaions o demonsrae ha NUWBS approaches he heoreical performance limis of l 1-norm-based sparse signal recovery. We invesigae hardware-design aspecs and show ASIC measuremen resuls for he wavele generaion sage, which highligh he efficacy of NUWBS for a broad range of RF feaure exracion asks in cos- and power-limied applicaions. Index Terms Analog-o-informaion (A2I) conversion, cogniive radio, compressive sensing, Inerne of Things (IoT), radiofrequency (RF) signal acquisiion, waveles, specrum sensing. I. INTRODUCTION FOR nearly a cenury, he cornersone of digial signal processing has been he Shannon Nyquis Whiaker sampling heorem [1]. This resul saes ha signals of finie energy and bandwidh are perfecly represened by a se of uniformly-spaced samples a a rae higher han wice he maximal frequency. I is, however, well-known ha signals wih cerain srucure can be sampled well-below he Nyquis rae. For example, Landau esablished in 1967 ha muliband signals occupying N non-coniguous frequency bands of bandwidh B can be represened using an average sampling rae no lower han wice he sum of he bandwidhs (i.e., 2NB) [2]. In 26, Landau s concep has been generalized by Candès, Donoho, Romberg, and Tao in [3], [4] o sparse signals, i.e., M. Pelissier was a visiing researcher a he School of Elecrical and Compuer Engineering (ECE), Cornell Universiy, Ihaca, NY, from CEA Univ. Grenoble Alpes, CEA, LETI, F-38 Grenoble, France ( michael.pelissier@cea.fr). C. Suder is wih he School of ECE, Cornell Universiy, Ihaca, NY ( suder@cornell.edu). Websie: vip.ece.cornell.edu signals ha have only a few nonzero enries in a given ransform basis, e.g., he discree Fourier ransform (DFT). These resuls are known as compressive sensing (CS) and find poenial use in a broad range of sampling-criical applicaions [5]. In essence, CS fuses sampling and compression: insead of sampling signals a he Nyquis rae followed by convenional daa compression, CS acquires jus enough compressive measuremens ha guaranee he recovery of he signal of ineres. Signal recovery hen explois he concep of sparsiy, a srucure ha is presen in mos naural and man-made signals. CS has he poenial o acquire signals wih sampling raes wellbelow he Nyquis frequency, which may lead o significan reducions in he sampling coss and/or power consumpion, or enable an increase he bandwidh of signal acquisiion beyond he physical limis of analog-o-digial converers (ADCs) [6]. As a consequence, CS is commonly believed o be a panacea for wideband radio-frequency (RF) specrum awareness applicaions [7] [9]. A. Challenges of Wideband Specrum Awareness In RF communicaion, here is a growing need in providing radio ransceivers wih cogniive abiliies ha enable awareness and adapabiliy o he specrum environmen [9]. The main goals of suiable mehods are o capure a variey of RF parameers (or feaures) o dynamically allocae specral resources [1] and/or o une radiional RF-chain circuiry wih opimal parameer seings in real-ime, e.g., o cancel ou srong inerferers using a unable noch filer [11], [12]. The RF feaures o be acquired for hese asks are mainly relaed o wideband specrum sensing [8] and include he esimaion of frequency occupancy, signal energy, energy variaion, signalo-noise-raio, direcion-of-arrival, ec. [7], [13]. For mos wideband specrum sensing asks, one is ypically ineresed in acquiring large bandwidhs (e.g., several GHz) wih a high dynamic range (e.g., 8 db or more). However, achieving such specificaions wih a single analog-o-digial converer (ADC) is an elusive goal wih curren semiconducor echnology [6]. A pracicable soluion is o scan he enire bandwidh in sequenial manner. From a hardware perspecive, his approach relies on radiional RF receivers as pu forward by Armsrong in 1921 [14]. The idea is o mix he incoming RF signal wih a complex sinusoid (whose frequency can be uned) eiher o a lower (and fixed) frequency or direcly o baseband. The signal is hen sampled wih an ADC operaing a lower bandwidh. While such an approach enjoys widespread use mainly due o is excellen specral seleciviy, sensiiviy, and dynamic range he associaed hardware requiremens

2 2 TO APPEAR IN THE IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS RF signal A2I radiional RF fronend RF feaures DSP baseband daa 111 Fig. 1. Overview of a cogniive radio receiver: A radiional RF fron-end is enhanced wih an analog-o-informaion (A2I) converer ha exracs RF feaures direcly from he incoming analog RF signals. The A2I converer enables parameer uning o reduce design margins in he RF circuiry and assiss specrum awareness asks in he digial signal processing (DSP) sage. (for wideband unable oscillaors and highly-selecive filers) and sweeping ime may no mee real-world applicaion consrains [11]. This aspec is paricularly imporan for he Inerne of Things (IoT), in which devices mus adhere o sringen power and cos consrains, while operaing in a mulisandard environmen (e.g., conaining signals from 3GPP NB- IoT, IEEE g/15.4k/11.ah, SigFox, and LoRa). Hence, here is a pressing need for RF feaure exracion mehods ha minimize he power and sysem coss, while offering flexibiliy o a variey of environmens and sandards. B. Analog-o-Informaion (A2I) Conversion A promising soluion for such wideband specrum sensing applicaions is o use CS-based analog-o-informaion (A2I) converers ha leverage specrum sparsiy [8], [15] [17]. Indeed, one of he main advanages of CS is ha i enables he acquisiion of larger bandwidhs wih relaxed samplingrae requiremens, hus enabling less expensive, faser, and poenially more energy-efficien soluions. While a large number of CS-based A2I converers have been proposed for specrum sensing asks [8], [11], [15], [16], [18], he generallypoor noise sensiiviy of radiional CS mehods [19], [2] and he excessive complexiy of he recovery sage [16], [21] prevens heir sraighforward use in low-power, cos-sensiive, and laency-criical applicaions, which are ypical for he IoT. Forunaely, for a broad range of RF feaure exracion asks, recovery of he enire specrum or signal may no be necessary. In fac, as i has been noed in [22], only a small number of measuremens may be required if one is ineresed in cerain signal feaures and no he signal iself. This key observaion is crucial for a broad range of emerging energy or cos-consrained applicaions in he RF domain, such as sensors or acuaors for he IoT, wake-up radio, specrum sensing, and radar applicaions [7], [23] [25]. In mos of hese applicaions, he informaion of ineres has, informally speaking, a rae far below he physical bandwidh, which is a perfec fi for CS-based A2I converers ha have he poenial o acquire he relevan feaures direcly in he RF (or analog) domain a low cos and low power. Figure 1 illusraes a cogniive radio receiver ha is assised wih an A2I converer specifically designed for RF feaure exracion. The A2I converer bypasses convenional RF circuiry and exracs a small se of feaures direcly from he incoming RF signals in he analog domain. The acquired feaures can hen be used by he RF fron-end and/or a subsequen digial signal processing (DSP) sage. Such an A2I-assised RF fron-end enables one o opimally reconfigure he key parameers of a radiional RF chain according o he specral environmen. This capabiliy can also be used o assis radiional RF ransceivers by providing means o eliminae over-design margins hrough adapaion o he specrum environmen via radio link-qualiy esimaion and inerferer localizaion [9], which is relevan in power- and cos-limied IoT applicaions. C. Conribuions This paper proposes a novel CS-based A2I converer archiecure for cogniive RF receivers. Our approach, referred o as non-uniform wavele sampling (NUWS), combines wavele preprocessing wih non-uniform sampling in order o alleviae he main issues of exising A2I converer soluions, such as signal noise, aliasing, and sringen clocking consrains, which enables a broad range of RF feaure exracion asks. For RF muli-band signals, we propose a specialized varian called nonuniform wavele bandpass sampling (NUWBS), which combines radiional bandpass sampling wih NUWS. This soluion builds upon (i) wavele sample acquisiion using highly over-complee and hardware-friendly Gabor frames or Morle waveles and (ii) a suiable measuremen selecion sraegy ha idenifies he relevan waveles required for RF feaure exracion. We use sysem simulaions o demonsrae he efficacy of NUWBS and show ha i approaches he heoreical phase ransiion of l 1 -norm-based sparse signal recovery in ypical muli-band RF applicaions. We invesigae hardware-implemenaion aspecs and validae he effecive inerference rejecion capabiliy of NUWBS. We conclude by showing ASIC measuremen resuls for he wavele generaion sage in order o highligh he pracical feasibiliy of he wavele generaor, which is a he hear of he proposed A2I converer archiecure. D. Paper Ouline The res of he paper is organized as follows. Secion II provides an inroducion o CS and discusses exising A2I converer archiecures for sub-nyquis RF signal acquisiion. Secion III presens our non-uniform wavele sampling (NUWS) mehod and he specialized varian for muli-band signals called nonuniform wavele bandpass sampling (NUWBS). Secion IV discusses opimal measuremen selecion sraegies and provides simulaion resuls. Secion V discusses hardware implemenaion aspecs of NUWBS. We conclude in Secion VI. E. Noaion Lowercase and uppercase boldface leers denoe column vecors and marices, respecively. For a marix A, we represen is ranspose and Hermiian ranspose by A T and A H, respecively. The M M ideniy marix is I M. The enry on he kh row and lh column of A is [A] k,l = A k,l and he lh column is [A] :,l = a l ; he kh enry of he vecor a is [a] k = a k. We wrie R Ω A = [A] Ω,: and R Ω a Ω = [a] Ω o resric he rows of a marix A and he enries of a vecor a o he index se Ω, respecively. Coninuous and discree-ime signals are denoed by x() and x[n], respecively.

3 M. PELISSIER AND C. STUDER 3 II. CS TECHNIQUES FOR RF SIGNAL ACQUISITION We sar by inroducing he basics of CS and hen review he mos prominen A2I converer archiecures for RF signal acquisiion, namely non-uniform sampling (NUS) [15], [16], [26] [29], variable rae sub-nyquis sampling [8], [3] [32], and random modulaion [33], [34], which includes he modulaed wideband converer and Xampling [11], [35] [37]. For each of hese archiecures, we briefly discuss he pros and cons from a RF specrum sensing and hardware design sandpoin. A. Compressive Sensing (CS) Basics Le x C N be a discree-ime, N-dimensional complexvalued signal vecor ha we wish o acquire. We assume ha he signal x has a so-called K-sparse represenaion s C N, i.e., he vecor s has K dominan non-zero enries in a known (uniary) ransform basis Ψ C N N wih x = Ψs and Ψ H Ψ = I N. In specrum sensing applicaions, one ypically assumes sparsiy in he DFT basis, i.e., Ψ = F H is he N- dimensional inverse DFT marix. CS acquires M compressive measuremens as y i = φ i, s + n i for i = 1, 2,..., M, where φ i C N are he measuremen vecors and n i models measuremen noise. The CS measuremen process can be wrien in compac marix-vecor form as follows: y = Φx + n = Θs + n. (1) Here, he vecor y C M conains all M compressive measuremens, he ih row of he sensing marix Φ M N corresponds o he measuremen vecor φ i, he M N effecive sensing marix Θ = ΦΨ models he join effec of CS and he sparsifying ransform, and n C M models mesuremen noise. The main goal of CS is o acquire far fewer measuremens han he ambien dimension N, i.e., we are ineresed in he case M N; his implies ha he marix Θ maps K-sparse signals of dimension N o a small number of measuremens M. Given a sufficien number of measuremens, ypically scaling as M K log(n), ha saisfy cerain incoherence properies beween he measuremen marix Φ and he sparsifying ransform Ψ, one can use sparse signal recovery algorihms ha generae robus esimaes of he sparse represenaion s and hence, enable he recovery of he signal x = Ψs from he measuremens in y; see [5], [38] for more deails on CS. B. A2I Converer Archiecures While sparse signal recovery is ypically carried ou in sofware [39] or wih dedicaed digial circuiry [16], [21], he CSbased A2I conversion process modeled by (1) is implemened direcly in he analog domain. The nex paragraphs summarize he mos prominen A2I converer archiecures ha perform CS measuremen acquisiion wih mixed-signal circuiry. 1) Non-Uniform Sampling: Non-uniform sampling (NUS) is one of he simples insances of CS. In principle, he NUS sraegy samples he incoming signal a irregularly spaced ime inervals by aking a random subse of he samples of a convenional Nyquis ADC [15], [16]. For his scheme, he sensing marix Φ is given by he M N resricion operaor R Ω = [I N ] Ω,: ha conains of a subse Ω he rows of he S&H non-uniform clock low-rae ADC Fig. 2. High-level archiecure of non-uniform sampling (NUS). A sampleand-hold (S&H) sage acquires a random subse of Nyquis-rae samples of a wideband signal x() and convers each sample x[n] o he digial domain. ideniy marix I N, where Ω = M is he cardinaliy of he sampling se. The effecive sensing marix Θ = R Ω I N Ψ in (1) conains he M rows of he sparsifying basis Ψ indexed by Ω. More specifically, NUS can be modeled as y = R Ω I N x + n = Θ NUS s + n (2) wih Θ NUS = R Ω I N F H, where we assume DFT sparsiy. As shown in [38], randomly-subsampled Fourier marices enable faihful signal recovery from M K log 4 (N) compressive measuremens. Hence, NUS no only enables sampling raes close o he Landau rae [2], bu is also concepually simple. A high-level archiecure of NUS, as depiced in Figure 2, consiss of a sample-and-hold (S&H) sage and an ADC supporing he shores sampling period used by he NUS [15], [16]. The main challenge of NUS is in he acquisiion of a wideband analog inpu signal. While he average sampling rae can be decreased significanly, he ADC sill needs o acquire samples from wideband signals wih frequencies poenially reaching up he maximal inpu signal frequency. This key observaion has wo consequences: Firs, NUS requires a sampling clock operaing a he ime resoluion of he order of he Nyquis rae, which is ypically power expensive. Second, NUS is sensiive o iming jier: informally speaking, if he inpu signal changes rapidly, a small error in he sampling ime can resul in a large error in he acquired sample. 2) Variable Rae Sub-Nyquis Sampling: Variable-rae sub- Nyquis sampling builds upon he fundamenals of bandpass sampling [4]. In principle, his A2I conversion sraegy undersamples he inpu signal wih muliple branches (i.e., a bank of parallel bandpass sampling sages) wih sampling raes ha differ from one branch o he oher. There exis wo main insances of his concep, namely muli-rae sampling (MRS) ha uses a fixed se of sampling frequencies for each branch [8], [31], [32] and he Nyquis-folding receiver (NYFR) ha modulaes he sampling frequencies [41]. Boh approaches rely on he fac ha he signal of ineres is aliased a a paricular frequency when undersampled a a given rae on a given branch, bu he same signal may experience aliasing a a differen frequency when sampled a a differen rae on anoher branch. Empirical resuls show ha his approach enables signal recovery for a sufficienly large number of branches [3]. From a hardware perspecive, MRS is relaively simple as i avoids any randomness during he sampling sage and each branch performs convenional bandpass sampling. Neverheless, MRS faces he same issues of radiional bandpass sampling [4]: i suffers from noise folding, i.e., wideband noise in

4 4 TO APPEAR IN THE IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS he signal of ineres is folded (or aliased) ino he compressive measuremens, which resuls in reduced sensiiviy [19], [2]. 3) Random Modulaion: Random modulaion (RM) is used by a broad range of A2I converers. Exising archiecures firs muliply he analog inpu signal by a pseudo-random sequence, inegrae he produc over a finie ime window, and sample he inegraion resul. The random-modulaion preinegraor (RMPI) [5], [34] and is single branch counerpar, he random demodulaor (RD) [18], [33], are he mos basic insances of his idea. However, modulaing he signal wih a (pseudo-)random sequence is only suiable for very specific signal classes, such as signals ha are well-represened by a union of sub-spaces [37]. In addiion, he (pseudo-)random sequence generaor mus sill run a Nyquis rae. The main advanage of he modulaed wide-band converer (MWC) is o reduce he bandwidh of he S&H o run a sub-nyquis raes [35], [37]. Indeed, he MWC avoids a fas sampling sage and, insead, requires a high-speed mixing sage which is ypically more wideband. A recen soluion ha avoids some of he drawbacks of RM is he quadraure A2I converer (QAIC) [17]. This mehod relies on convenional down-conversion before RM, hus focusing on a small RF band raher han he enire bandwidh. C. Limiaions of Exising A2I Converers While numerous A2I converer archiecures have been proposed in he lieraure, heir limied pracical success is a resul of many facors. From a heoreical perspecive, one is generally ineresed in acquisiion schemes ha minimize he number of measuremens while sill enabling faihful recovery of a broad range of signal classes. From a hardware perspecive, he key goals are o minimize he bandwidh requiremens, he number of branches, and he power consumpion, while being unable o he applicaion a hand. Finally, suiable A2I converers should exhibi high sensiiviy and be robus o hardware impairmens and imperfecions. We now summarize he key limiaions of exising A2I converer archiecures as discussed in Secion II-B wih hese desirables in mind. Mos of he discussed A2I converers rely on random mixing or sampling. Such archiecures eiher require large memories o sore he random sequences or necessiae efficien means for generaing pseudo-random sequences [42]. In addiion, such unsrucured sampling schemes preven he use of fas linear ransforms (such as he fas Fourier ransform) in he recovery algorihm, which resuls in excessively high signal processing complexiy and power consumpion [16], [21]. From a hardware perspecive, large pars of he analog circuiry of many A2I converers mus sill suppor bandwidhs up o he Nyquis rae, even if he average sampling rae is significanly reduced. For example, NUS [15] and MRS [32] require S&H circuiry and ADCs designed for he full Nyquis bandwidh. Similarly, he RD and RMPI require sequence generaors ha run a he Nyquis rae. Anoher drawback of many A2I converers, especially for MRS or he MWC [11], [35] [37], is ha hey require a large number of branches, which resuls in large silicon area and poenially high power consumpion. A more fundamenal issue of mos CS-based A2I converer soluions for wideband RF sensing applicaions is noise wavele ransform S&H non-uniform clock low-rae ADC Fig. 3. High-level archiecure of non-uniform wavele sampling (NUWS): Concepually, NUWS firs performs a coninuous wavele ransform Wx() of he inpu signal x(), followed by NUS as shown in Figure 2 o obain wavele samples x[n]. A pracical hardware archiecure is discussed in Secion V. folding [19], [2], which prevens heir use for applicaions requiring high sensiiviy, such as aciviy deecion of low- SNR signals. In addiion, mos A2I converers lack versailiy or adapabiliy o he applicaion a hand, i.e., mos sysem parameers are fixed a design ime and signal acquisiion is nonadapive (one canno selec he nex-bes sample based on he hisory of acquired samples). However, adapive CS schemes have he poenial o significanly reduce he acquisiion ime or he complexiy of signal recovery [43]. III. NON-UNIFORM WAVELET (BANDPASS) SAMPLING We now propose a novel CS-based A2I converer ha miigaes some of he drawbacks of exising A2I converer soluions. Our approach is referred o as non-uniform wavele sampling (NUWS) and essenially acquires wavele coefficiens direcly in he analog domain. We firs inroduce he principle of NUWS and hen adap he mehod o muli-band signals, resuling in non-uniform wavele bandpass sampling (NUWBS). We hen highligh he advanages of NUWS and NUWBS compared o exising A2I converers for RF feaure exracion. A. NUWS: Non-Uniform Wavele Sampling The operaing principle of NUWS is illusraed in Figure 3. In conras o NUS (cf. Figure 2), NUWS firs ransforms he incoming analog signal x() ino a wavele frame Wx() (see Secion IV-A for he basics on waveles) and hen performs NUS o acquire a small se of so-called wavele samples x[n]. As illusraed in Figure 4(a), NUS is equivalen o muliplying he inpu signal x() wih a Dirac comb followed by he acquisiion of a subse of samples (indicaed by black arrows). In conras, as shown in Figure 4(b), NUWS muliplies he inpu signal x() wih waveles, inegraes over he suppor of each wavele, and samples he resuling wavele coefficiens. From a high-level perspecive, NUWS has he following advanages over NUS. Firs, he coninuous wavele ransform W reduces he bandwidh of he inpu signal x(), which relaxes he bandwidh of he S&H circui and he ADC (see Secion IV for he deails). Second, NUWS enables full conrol over a number of parameers, such as he sample ime insans, wavele bandwidh, and cener frequency. In conras, NUS has only one degree-of-freedom: he sample ime insans. In discree ime, he sensing marix Φ for NUWS can be described by aking a small se Ω of rows of a (possibly overcomplee) wavele frame W H C W N, where W H conains a specific wavele on each row and W M corresponds o he oal number of waveles. Hence, he sensing marix of

5 Higher Bandwih M. PELISSIER AND C. STUDER 5 (a) (b) T NYQ (c) Ts T NYQ Higher scale T NYQ Fig. 4. Illusraion of he sampling paerns of NUS, NUWS, and NUWBS. NUS muliplies he incoming signals wih a puncured Dirac comb; NUWS muliplies he incoming signals wih a series of carefully-seleced waveles; NUWBS uses a wavele comb ha is sub-sampled in ime and wih waveles of variable cenral frequency in order o filer he sub-bands of ineres. NUWS is Φ = R Ω W H, where M = Ω is he number of wavele samples. We can describe he NUWS process as y = R Ω W H x + n = Θ NUWS s + n (3) wih he effecive sensing marix Θ NUWS = R Ω W H F H where we, once again, assumed sparsiy in he DFT domain. 1 The necessary deails on waveles are provided in Secion IV-A. By comparing (3) wih (2), we see ha NUS subsamples he inverse DFT marix, whereas NUWS subsamples he (possibly overcomplee) marix (FW) H, which is he Hermiian of he Fourier ransform of he enire wavele frame. We can wrie he acquisiion of he frequency-sparse signal s as y = R Ω (FW) H s + n, (4) which implies ha each wavele sample is equivalen o an inner produc of he Fourier ransform of he wavele, i.e., ŵ i = Fw i, wih he sparse represenaion s as y i = ŵ i, s + n i, i Ω. As we will discuss in deail in Secion IV-A, he considered waveles essenially correspond o bandpass signals wih a given cener frequency, bandwidh, and phase (given by he sample ime insan). Thus, each wavele sample corresponds o poinwise muliplicaion of he sparse signal specrum wih he bandpass filer equivalen o he Fourier ransform of he wavele. Figure 5 illusraes his propery and shows he absolue value of he marix (FW) H for he complex-valued Morle wavele [44] wih six scales. Evidenly, each wavele capures a differen porion of he specrum wih a differen phase (phase differences are no visible) and bandwidh. We noe ha for he Morle wavele, he bandwidh and cener frequency of each wavele depends on he scale. B. NUWBS: Non-Uniform Wavele Bandpass Sampling Non-uniform wavele bandpass sampling (NUWBS) is a special insance of NUWS opimized for muli-band RF signals. 1 Depending on he applicaion, oher sparsiy bases Ψ han he inverse DFT marix F H can be used; an invesigaion of oher bases is ongoing work. Frequency Fig. 5. Absolue value of he produc beween he Hermiian of he complexvalued marix and inverse DFT marix [W H F H ] k,l. We see ha each scale focuses on a differen frequency band, whereas he bandwidh wihin each scale is fixed and he phase changes for differen waveles. The capabiliy of handling such signals is of paricular ineres for non-coniguous carrier aggregaion, a promising echnology o enhance IoT hroughpu needs [45]. Figure 6(a) illusraes a ypical muli-band scenario in which RF signals occupy muliple non-coniguous frequency bands ha may be sparsely populaed; in addiion, here may be inerferers ouside he sub-bands of ineres. A sandard way o acquire muli-band signals is o use a filerbank wih one dedicaed filer and RF receiver per sub-band. Besides requiring high complexiy and power, and suffering from lack of flexibiliy, such designs are ypically unable o exploi signal sparsiy wihin he sub-bands. Tradiional bandpass sampling [4] or NUS [26] [29] for muli-band signals would resul in several issues. Firs and foremos, noise and inerferers ouside he sub-bands of ineres will ineviably fold (or alias) ino he measuremens a phenomenon known as noise folding [19], [2], [4]. Furhermore, for NUS, he a-priori informaion on he occupied sub-bands is generally no exploied during he acquisiion process. In sark conras o hese mehods, NUWBS explois he muli-band srucure and sparsiy wihin each sub-band, while being resilien o inerferers or noise ouside of he bands of ineres. As illusraed in Figure 4(c), NUWBS muliplies he incoming signals wih a wavele comb on a regular sampling grid, sub-sampled in ime wih respec o he Nyquis rae. Unlike NUWS, here are no overlaps beween waveles, which prevens he need for a large number of branches ypically one branch per sub-band is sufficien. Furhermore, he cener frequencies of he waveles can be focused on he sub-bands of ineres, which renders his approach resilien o ou-of-band noise and inerferers, effecively reducing noise folding and aliasing wihou he need for a filer bank. Finally, as shown in Secion IV-C, NUWBS is able o leverage CS and achieves near-opimal sampling raes, i.e., close o he Landau rae. The operaing principle of NUWBS is illusraed in Figure 6. Every NUWBS measuremen acs like a filer, which removes ou-of-band noise and inerference (see Figure 6(b)). Then, as

6 6 TO APPEAR IN THE IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS (a) (b) Δf f c1 f c2 f c1 Wavele Band-Pass f c2 Non Uniform Sampling (c) noise f max= f nyq /2 Fig. 6. Illusraion of a muli-band RF signal (a) consising of wo sparselypopulaed sub-bands and an inerferer (red). NUWBS firs performs wavele bandpass sampling o exrac boh sub-bands (b); hen, NUS is used o minimize he number of wavele samples, effecively reducing he sampling rae (c). shown in Figure 6(c), by aking a subse of wavele samples (i.e., wavele bandpass sampling), NUWBS reduces he average sampling rae. Tradiional recovery mehods for CS can hen be used o recover he muli-band signals of ineres. From a mahemaical viewpoin, NUWBS can be modeled as in (3) wih he differences ha he subse of samples Ω is adaped o he sub-bands of ineres and he wavele samples are on a regular sub-sampled grid wih non-overlapping waveles. C. Advanages of NUWS and NUWBS Waveles find broad applicabiliy in wireless communicaion sysems, including source coding, modulaion, inerference miigaion, and signal de-noising [46], [47]. Neverheless, CS-based mehods ha rely on wavele sampling are raher unexplored, especially when dealing wih RF signals. A noable excepion is he paper [48], in which a muli-channel acquisiion scheme based on Gabor frames is proposed ha explois he sparsiy in he ime-frequency domain. In conras o NUBS/NUWBS, his approach relies on a parallel se of Gabor sampling branches, where each Gabor wavele has a fixed bandwidh and he sampling rae is reduced using he MWC. We nex summarize he benefis of wavele sampling and he advanages of NUWS/NUWBS o RF applicaions. 1) Tunabiliy and Robus Feaure Acquisiion: Waveles offer a broad range of parameers including ime insan, cener frequency, and bandwidh (see Secion IV-A for he deails). This flexibiliy can be exploied o adap each measuremen o he signal or feaure class a hand or o improve robusness o ou-of-band noise and inerferers, or aliasing. For NUWBS, we ake advanage of his propery by focusing each wavele sample on he occupied sub-bands, which yields improved sensiiviy by miigaing noise folding and inerference. 2) Adapive Feaure Exracion: The ree srucure of waveles across scales [49] is a well-exploied propery in daa compression [5]. In RF applicaions, one can exploi his propery o develop adapive feaure exracion schemes ha firs idenify RF aciviy on a coarse scale (e.g., in a wide frequency band) and hen, adapively zoom in o sub-bands ha exhibi aciviy for a more deailed analysis. This approach avoids radiional frequency scanning and has he poenial o enable faser RF feaure exracion han non-adapive schemes. 3) Srucured Sampling: A broad range of CS-based A2I converer soluions focuses on randomized or unsrucured sampling mehods. Such mehods ypically require large sorage (for he sampling marices) and high complexiy during signal recovery. In conras, srucured sensing approaches are known o avoid hese drawbacks [51]. Waveles exhibi a high degree of srucure and heir paramerizaion requires low sorage. Furhermore, recovery algorihms ha rely on fas (inverse) wavele ransforms are compuaionally efficien [52]. 4) Relaxed Hardware Consrains: From a hardware perspecive, random sequences or clock generaion circuiry ha operaes a Nyquis raes can in conras o NUS and RD be avoided due o he sub-nyquis operaion of NUWS and NUWBS. Hence, he associaed clock synhesis and clockree managemen can be relaxed [53]. In addiion, by subsampling he wavele coefficiens, we can furher reduce he ADC sampling raes. Due o he signal correlaion wih he wavele prior o sampling, he bandwidh requiremens of he S&H circui and he ADC are relaxed as well. In addiion, NUWBS prevens overlapping waveles, which enables he use of a small number of parallel sampling branches. This propery reduces he circui area and power consumpion. As we will show in Secion V, widely-unable waveles can be generaed efficienly in analog hardware. IV. WAVELET DESIGN AND VALIDATION OF NUWBS This secion summarizes he basics of waveles and hen, discusses wavele selecion and design for NUWS/NUWBS. We finally validae NUWBS for muli-band RF sensing. A. Wavele Prerequisies For he sake of simpliciy, we will use boh coninuous-ime and discree-ime signal represenaions and ofen swich in beween wihou making he discreizaion sep explici. 1) Wavele Basics: A wavele is a coninuous waveform ha is effecively limied in ime, has an average value of zero, and bounded L 2 -norm (ofen normalized o one). Waveles for signal processing were inroduced by Morle [44] who showed ha coninuous-ime funcions x() in L 2 can be represened by a so-called wavele ψ s,δ () ha is obained by scaling s R + and shifing δ R a so-called moher wavele ψ(). The scaling and shifing operaions can be made formal as follows: ψ s,δ () = 1 ( ) δ ψ, s R +, δ R. (5) s s The so-called wavele coefficien Wx s,δ of a signal x() for a given wavele ψ s,δ () a scale s and wih ime shif δ, is defined as he following inner produc [54], [55]: Wx s,δ = x, ψ s,δ = R x() 1 s ψ ( δ s ) d. (6) In words, each wavele coefficien Wx s,δ compares he signal x() o a shifed and scaled version ψ s,δ () of he

7 M. PELISSIER AND C. STUDER 7 Fig. 7. Real par of he produc beween he Hermiian of he Gabor and inverse DFT marix A = R{W H F H } a a given cener frequency and for various ime shifs. Signals far away from he cener frequency are aenuaed, which effecively miigaes ou-of-band noise, inerference, and aliasing. moher wavele ψ(). By comparing he signal o waveles for various scales and ime shifs, we arrive a he coninuous wavele ransform (CWT) Wx s,δ. The CWT represens onedimensional signals in a highly-redundan manner, i.e., by wo coninuous parameers (s, δ). All possible scale-ime aoms can be colleced in an (overcomplee) frame given by D = { ψ s,δ () δ R, s R +}. In pracice, one is ofen ineresed in selecing a suiable subse of scales and shifs ha enable an accurae (or exac) represenaion of original signal s() of ineres. In wha follows, we are paricularly ineresed in waveles ha can be generaed efficienly in hardware; such waveles are discussed nex. 2) Gabor Frame: The Gabor ransform is a well-known analysis ool o represen a signal simulaneously in ime and frequency, similarly o he shor-ime Fourier ransform (STFT). The se of Gabor funcions (ofen called Gabor frame) is, sricly speaking, no a wavele basis he formalism, however, is very similar [56], [57]. Gabor frames consis of funcions (or aoms) ψ f c ν,δ k () = p( δ k )e j2πf c ν, (7) which are paramerized by he cener frequencies fν c and ime shifs δ k of a windowing funcion p(), where ν = 1, 2,... and k = 1, 2,... are he indices of discree frequency and ime shifs, respecively. In pracice, one ofen uses a Gaussian windowing funcion p() ha is characerized by he widh (or duraion) parameer τ. Based on [56], he ime and frequency represenaion of he uni l 2 -norm Gabor aoms wih a Gaussian window are defined as follows: ψ f c ν,δ k () = τπ 1 4 ( e j2πf c ν ( δk) e δk τ ) 2 (8) Ψ f c ν,δ k (f) = (τ 2π) 1 2 e j2πδ k f e (πτ(f f c ν ))2. (9) There exiss a rade-off when choosing he widh parameer τ: a large widh increases he frequency resoluion while lowering he ime resoluion, and vice versa. As i can be seen from (9), he Fourier represenaions of Gabor aoms decay exponenially fas, which is he reason for heir excellen frequency-rejecion properies, i.e., signals sufficienly far away of he cener frequency f c ν are srongly aenuaed. This filering effec of Gabor aoms is illusraed in Figure 7, which shows he real par of he marix (FW) H for one paricular cener frequency f c ν and various ime shifs δ k. Clearly, signals ha are sufficienly far apar from he cener frequency f c ν will be filered. 3) Complex-Valued Morle Wavele (C-Morle): In conras o he Gabor frame, he complex-valued Morle (C-Morle) wavele uses windowing funcions whose widh parameer is linked o he cenral frequency (cf. Figure 5) [54], [55]. Recall from (5) ha higher scales correspond o he mos sreched Fig. 8. Frequency domain ampliude of C-Morle waveles for 6 scales as shown in Figure 5; he bandwidh of he waveles increases wih he cenral frequency, which is in conras o Gabor aoms ha have consan bandwidh. waveles (in ime) and hence, waveles measure long ime inervals for feaures conaining low-frequency informaion and shorer inervals for high-frequency informaion. In fac, he widh of a C-Morle is linked o he cenral frequency so ha here is a consan number of oscillaions per effecive wavele duraion. More formally, he C-Morle shows a consan qualiy facor Q across scales. As a resul, he C-Morle waveles coincide wih (8) and saisfy he addiional consrain accross scales ha he cenral frequency f c ν and he wavele bandwidh BW p saisfy he following condiion: Q = f c ν/bw p = f c ντ ν πα 2. Here, he parameer α is.33 for a 1 db referred bandwidh. Figure 8 shows he specrum ampliude for six scales of he C-Morle wavele family for a given qualiy facor clearly he wavele bandwidh is linked o he scale. B. Wavele Selecion for he Design of NUWBS We are, in principle, free in choosing he widh, frequency, and ime insan of each wavele. In pracice, however, we are ineresed in waveles ha can be generaed efficienly in hardware, enable he use of a small number of branches, and exrac he RF feaures of ineres. We now ouline how o selec suiable wavele parameers for NUWBS. 1) Parameer Selecion: As deailed in Secion III-B, NUWBS firs performs a projecion of he inpu signal on a selec se of waveles (or aoms) and hen, subsamples he wavele coefficiens. Since Gabor frames conain a highly redundan se of aoms, i may a firs seem couner-inuiive o use an overcomplee frame expansion W H C W N wih W N of he signal x as our ulimae goal is o reduce he number of measuremens. I is hus criical o selec a suiable subse Ω of aoms ha enables robus signal recovery or feaure exracion wih a minimum number of measuremens M = Ω N W. As we will see, he high redundancy urns ou o be beneficial as i allows us o selec a poenially beer subse of measuremen, e.g., compared o NUS ha can only selec a subse of rows of he Fourier marix. If we are ineresed in opimizing our se Ω of wavele samples for sparse signal recovery, which is he original

8 8 TO APPEAR IN THE IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS moivaion of CS, hen we can minimize he so-called muual coherence [5] beween he sub-sampled sensing marix R Ω W H and he sparsifying basis Ψ = F H, defined as µ m (R Ω W H, F H ) = max i,k [R ΩW] i, [F H ] k. (1) The muual coherence is relaed o he minimum number of measuremens M ha are required o guaranee recovery of K-sparse signals [5], [58]. Hence, we wish o find an opimal se Ω of cardinaliy M ha minimizes (1); unforunaely, his is a combinaorial opimizaion problem. We herefore resor o a qualiaive analysis and heurisics o idenify a suiable se of waveles ha enables he recovery of sparse signals. According o he closed-form expression in (8), he Gabor aoms are characerized by he widh parameer τ. Our goal is o find he opimal widh parameer ˆτ, depending on he inpu signal (e.g., is bandwidh). Inuiively, he widh parameer τ should be linked o he bandwidh BW RF of he RF signal. In fac, he effecive wavele widh should be designed so ha each wavele measuremen y i, i = 1, 2,..., M, collecs enough informaion over he bandwidh of ineres or, in oher words, he pulse specrum should be as fla as possible over he bandwidh of ineres. We can make his inuiion more formal by considering he so-called local muual coherence [3], [59] µ m (W H s, F H Σ ) = max i,k [W s] i, [F H Σ ] k (11) beween he wavele sampling marix Ws H a a paricular scale s and he sparsifying basis limied o he subse of frequencies Σ of ineres (e.g., limied o he poenially acive or occupied sub-bands). From he Gabor frame definiion in (8), we can compue a closed form expression of he muual coherence beween a Gabor frame having a fixed widh parameer τ and he Fourier basis. Assuming ha he aom s cenral frequency fν c is cenered o he band of ineres and is a muliple value of he frequency resoluion f = f Nyq /N, we can compue he local muual coherence defined in (1) as follows: µ m (W H s, F H Σ ) = (τ 2π) 1/2 Figure 9 shows he evoluion of he muual coherence as well as he heoreical lower bound (he purple horizonal line) given by 1/ N [42], [6]. The curves in his figure are obained by seing he dimension o N = 256 and compuing inner producs beween he rows of Gabor frame and he rows of he inverse discree Fourier resriced o he band of ineres F H Σ. The (local) muual coherence is hen compued according o (1) (and Eq. 11). The individual poins on he curves are obained by uning he wavele bandwidh BW p divided by he occupied RF bandwidh BW RF. Noe ha he shorer he aom (or wavele) duraion τ, he wider is bandwidh BW p is. As a resul, he energy of he sensing vecor is spread in he frequency domain and hence, capures informaion of all frequencies wihin he sub-band of ineres. The limi τ corresponds o he Dirac comb (he bandwidh ends o infiniy) for which he muual coherence is known o reach he Welch lower bound [61]. The limi τ corresponds o he case in which he sensing vecors are localized in he frequency domain, i.e., he measuremens are maximally coheren wih he Fourier Wavele Comb Sampling Dirac Comb Sampling Fig. 9. Muual coherence µ m beween he Gabor frame W H and he inverse DFT marix F H as a funcion of he wavele bandwidh (BW p) relaive o he RF signal bandwidh BW p/bw RF. basis. Hence, for wavele sampling, we can deermine he widh parameer τ o mach he signal of ineres. In pracice, one can rade-off filering performance (o miigae noise folding and aliasing) versus measuremen incoherence (o reduce he number of required CS measuremens). 2) Gabor Time-Shif Selecion: Besides selecing he opimal widh parameer τ of he Gabor aoms, we have o idenify suiable frequencies f c ν and ime shifs δ k. Consider, for example, he muli-band signal shown on he lef side in Figure 1, where we assume ha we know he coarse locaions of he poenially acive sub-bands (e.g., deermined by a given sandard), bu no he locaions of he non-zero frequencies wihin each subband (e.g., he frequency slos used for ransmission). For simpliciy, he axes have been normalized so ha he y-axis sands for he frequency index ν and he x-axis sands for he ime shif index k. We define he following parameers: he sub-sampling raio γ = f Nyq /BW RF is he raio beween he Nyquis frequency f Nyq and he bandwidh of each sub-band BW RF ; he aggregae bandwidh BW ag is he oal bandwidh of all occupied sub-bands, i.e., in our example BW ag = 2BW RF. Equivalenly, he aggregae bandwidh can be expressed by he cardinaliy of he occupied frequency indices Σ so ha BW ag = f Σ, where f is he bandwidh per frequency bin. Our proposed Gabor frequency and ime shif selecion sraegy relies on wo principles. Firs, in order o acquire informaion in a given sub-band, we consider a fixed cenral frequency cenered in he sub-band of ineres. Second, in he ime domain we perform bandpass sampling wih he goal of mixing he signal of ineres o (or near o) baseband. This means ha insead of sampling a all of he available ime shifs defined by he Nyquis rae (shown by he verical black dashed lines in Figure 1), we only acquire a subse defined by he sub-sampling facor γ (he red circles in Figure 1), effecively performing wavele bandpass sampling. As a resul wo adjacen Gabor aoms will no overlap in ime since, by consrucion, he sampling rae is inversely proporional o he pulse duraion. In he case of wo sub-bands he aggregae sampling rae is

9 Subband n 1 Subband n 2 Δf S /2 S /2 M. PELISSIER AND C. STUDER 9 Nyquis grid g f c Frequency index... N wavele aom wavele sub-sampling grid f c 1 g/2 g Time index Fig. 1. Time-frequency grid of he Gabor aoms o be acquired via NUWBS; our approach makes use of a-priori knowledge of he occupied frequency bands; Aom selecion relies on consan frequency and bandpass sampling in ime for 2 sub-bands; he used parameers are BW ag = 2 BW RF = 2 16 f, γ = 16, and N = 256 samples. se o 2f Nyq /γ equivalen o he aggregae bandwidh equal o 2BW RF. We noe ha in addiion o bandpass sampling, we can perform NUS on he acquired wavele samples o furher reduce he sampling raes. As shown nex, his is ypically feasible in he case where we know a-priori ha he sub-bands are sparsely populaed. C. Performance Validaion of NUWBS We now demonsrae he efficacy of NUWBS for specral aciviy deecion in a muli-band RF applicaion. In paricular, we simulae an empirical phase ransiion [39], [62] ha characerizes he probabiliy of correc suppor recovery, i.e., he rae of correcly recovering he rue acive frequency bins from NUWBS measuremens. As a reference, we also include he heoreical phase ransiion of l 1 -norm based sparse signal recovery for a Gaussian measuremen ensemble [62]. We use N = 256 frequency bins and wo acive sub-bands wih a oal number of Σ = 32 poenially acive frequency bins. The signals wihin hese bins are assumed o be K Σ sparse. The NUWBS measuremens are seleced as discussed in Secion IV-B and illusraed in Figure 1, i.e., we form he M N marix Θ NUWBS = R Ω W H F H by fixing he frequency f c ν a he cener of each sub-band and use a subsampling raio per branch of γ = 2N/ Σ = 16. We generae measuremen-sparsiy pairs (M, K), and for each pair, we generae a synheic K-sparse signal wihin he wo allowed sub-bands; he K non-zero coefficiens are complex-valued numbers of uni ampliude and random phases. For suppor recovery, we use orhogonal maching pursui [6], resriced o he sub-bands of ineres, i.e., we assume ha he sub-band suppor Σ is known a-priori bu no he acive coefficiens wihin hese sub-bands. We perform suppor se recovery for 1 Mone Carlo rials and repor he average success rae. Figure 11 shows he empirical phase ransiion, where whie areas indicae zero errors for suppor se recovery. The x-axis... N Fig. 11. Empirical phase ransiion graph of NUWBS for muli-band signal acquisiion compared o he heoreical l 1 -norm phase ransiion for a Gaussian measuremen ensemble (shown wih he dashed purple line). NUWBS exhibis similar performance as he heoreical phase ransiion, which demonsraes ha NUWBS enables near-opimal sample raes. shows he normalized compression raio, i.e., he number of measuremens compared o he oal sub-band widh M/ Σ ; he y-axis shows he normalized sparsiy level, i.e., he fracion of non-zeros compared o he oal sub-band widh K/ Σ. We see ha NUWBS exhibis a similar success-rae profile as prediced by he heoreical phase ransiion (i.e., recovery will fail above and succeed below he dashed purple line), which is valid in he asympoic limi for l 1 -norm based sparse-signal recovery from Gaussian measuremens. This key observaion implies ha NUWBS in combinaion wih he aom selecion sraegy discussed in Secion IV-B exhibis near-opimal sample complexiy in muli-band scenarios. We emphasize ha even for he relaively small dimensionaliy of he simulaed sysem (i.e., N = 256), NUWBS is already in saisfacory agreemen wih he heoreical performance limis for sparse signal recovery. V. IMPLEMENTATION ASPECTS OF NON-UNIFORM WAVELET (BANDPASS) SAMPLING This secion discusses hardware implemenaion aspecs o highligh he pracical feasibiliy of NUWS/NUWBS and heir advanages over exising A2I converer soluions. A. Archiecure Consideraions of NUWBS Figure 12 shows he criical archiecure deails for NUWBS ha uses Gabor frames or C-Morle waveles. The coninuousime inpu signal x() is firs muliplied (or mixed) wih a wavele comb p c (). The resuling signal is hen inegraed over a period T s (for each wavele) and subsampled a a rae f s. The rae f Nyq of he inpu signal x = [x 1,..., x N ] T reduces o a uniform sub-sampling rae f s of he measuremens y = [y 1,..., y M ] T such ha NT Nyq = MT s. For uniform subsampling a rae f s (i.e., we do no perform NUS of he wavele samples), he compression raio N/M is proporional o he sub-sampling raio κ = f Nyq /f s. If we randomly selec a subse of samples of he sample sream (in addiion o uniform subsampling), hen we can furher lower he (average) sampling

10 1 TO APPEAR IN THE IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS [x 1....x N ] T Samples@fs... y[1] y[2] y[m] T acq =N.T NYQ x() p c () [Ts] fs T acq =M.T s Ts Ts /f c T NYQ Fig. 12. Generic serial NUWBS archiecure o acquire Gabor frame or wavele samples. NUWBS firs muliplies he inpu signal x() wih a wavele comb p c() a rae 1/T s and inegraes he resul. One hen akes a random se of wavele samples and quanizes hem using an ADC. A wavele is defined by is cenral frequency f c and widh parameer τ (he effecive pulse duraion). rae, effecively implemening CS. The sampling diversiy of NUWBS comes from he wavele parameers seings, namely he widh parameer τ and he cenral frequency f c. While he archiecure depiced in Figure 12 is purely serial, one can deploy muliple parallel branches o (i) furher increase he diversiy of he CS acquisiion sage, (ii) reduce he ADC rae by inerleaved processing, or (iii) sense muliple subbands. In addiion, a muli-branch archiecure can simplify he circuiry for each branch by fixing he cener frequency, pulse widh, delay, or pulse rae per branch. For such an archiecure, each branch performs wavele bandpass sampling a a given scale wih fixed bandwidh and cener frequency. B. Idealisic CWT Bandpass Sampling versus Realisic Serial Wavele Bandpass Sampling This secion discusses he commonaliies and differences beween idealisic CWT bandpass sampling and he serial wavele bandpass sampling archiecure shown in Figure 12. 1) Analysis of CWT Bandpass Sampling: The wavele coefficien Wx s,δ of he signal x() a a scale s and ime shif δ is defined in (6). Assume ha he ime-shif parameer δ is coninuous so ha CWT is a coninuous funcion in δ. Then, he scalar produc in (12) can be rewrien using he convoluion operaor as follows [54]: ADC Wx s (δ) = (x ψ s )(δ). (12) Here, ψs (u) = 1 s ψ ( u s ). We can now compue he Fourier ransform F in he ime-shif parameer δ o obain F{Wx s (δ)} = X(f) Ψ s (f), (13) where Ψ s (f) is he Fourier ransform of he wavele ψs (u) given by Ψ s (f) = F { ψs (u) } = sψ ( sf) (14) and Ψ(f) is he Fourier ransform of he moher wavele ψ(). From (13), we see ha he CWT Wx s (δ) is equivalen o filering he inpu signal X(f) wih he ransfer funcion H CWT (f) = Ψ s (f). We can now analyze he resul of bandpass sampling applied o he funcion Wx s (δ). To his end, we assume a sampling rae f s well-below he Nyquis bandwidh of he inpu signal x() and below he bandwidh of he moher wavele, i.e., f s BW p f Nyq. We have he following discree-ime oupu signal y[ = nt s ] = n=+ n= Wx s (nt s )δ( nt s ) sampled a f s = 1/T s. The oupu signal y[ = nt s ] corresponds o he bandpass sampled version of signal x() afer filering i wih he ransfer funcion H CWT (f) = Ψ s (f). In conras o classical bandpass sampling, he iniial CWT exracs a paricular frequency band defined by he cener frequency f c and he bandwidh BW p of he waveles. In words, CWT bandpass sampling is an effecive combinaion of filering and mixing via bandpass sampling. I is imporan o realize ha his scheme requires access o he coninuous-ime CWT of he signal prior o sub-sampling. In pracice, however, we do no have access o he CWT insead, we have o make use of he wavele sampling archiecure shown in Figure 12. Performing a CWT in hardware is infeasible and would require an excessively large number of branches, i.e., a dedicaed branch per ime shif δ or convoluion resul every T Nyq second as he CWT aoms have infinie suppor. In conras, he archiecure proposed in Figure 12 performs a convoluion of he inpu signal wih he aom ψ s (δ) every T s second (insead of T Nyq ) in a serial manner. While boh approaches are similar, here are imporan differences in he filering capabiliies. To his end, we invesigae he ou-of-suppor Σ (ou-of-band inerference) rejecion performance for serial wavele bandpass sampling ha can be implemened (cf. Figure 12) and he idealisic CWT bandpass sampling approach. 2) Analysis of Serial Wavele Bandpass Sampling: Consider he case in which boh he wavele cener frequency and bandwidh remains consan for he enire wavele comb. This is he case of he Gabor frame projecion repored on a single branch in Figure 1. We will use Figure 13, which illusraes he specrum represenaion, o assis our discussion. The inpu signal x() in Figure 13(a) is firs muliplied (mixed) wih a wavele comb p c () shown in Figure 13(b). The mixing resul z() can be expressed as follows: n=+ z() = x()p c () = x() p( nt s ), n= where p() is he considered wavele. The Fourier ransform of he signal z() shown in Figure 13(c) is given by k=+ Z(f) = f s X(f) P (f) δ(f kf s ), (15) k= which reveals ha he specrum of he mixed signal Z(f) is he convoluion beween he Fourier ransform of he inpu signal X(f) (cf. Figure 13(a)) and a Dirac comb weighed by he Fourier ransform of he wavele P (f). According o (9) he Gaussian envelope of Gabor aoms or C-Morle waveles show

11 M. PELISSIER AND C. STUDER 11 (a) - X(f) (b) - P c (f) (c) - Z(f) (d) - Y(f) (e) - Y d (f) f u LPF f i... -f s /2 P(f)= sy(sf) f NYQ /2... f s f NYQ /2 f s f s f s /2 f NYQ /2 f NYQ /2 Fig. 13. Illusraion of he signal (useful f u and inerferer f i ) specrum evoluion along he serial wavele bandpass sampling; The wavele sampling rae is f s wih consan wavele parameers seings (τ, f c ). The inpu signal specrum X(f) is convolved wih weighed Dirac comb, filered by a sinc low-pass filer, and decimaed a rae f s. exponenially fas decay, which implies ha he infinie sum can effecively be reduced o a small number of Dirac dela funcions shown in Figure 13(b). Furhermore, we see ha NUWBS effecively reduces noise folding by pre-filering he specrum wih he pulse P (f) prior o band-pass sampling; his is conras o convenional band-pass sampling in which noise from he enire Nyquis bandwidh folds ino each sample [4]. In order o mach his approach wih he bandpass CWT approach discussed in Secion V-B1, we can see ha p() corresponds o he wavele aom a he scale s and ime shif δ = wih Fourier ransform ( )} 1 s P (f) = F{ Ψ = sψ(sf). (16) s By comparing (14) wih (16), we see ha one is he complex conjugae of he oher. In he archiecure shown in Figure 12, he mixing produc z() is low-pass filered. A ypical filer ha can be implemened corresponds o an inegraion over a recangular window of duraion T s. The frequency-domain represenaion of his inegraor corresponds o he cardinal sine (sinc) funcion. Hence, he Fourier ransform Y (f) of he filered and mixed signal Z(f) shown in Figure 13(d) is Y (f) = Z(f)T s sinc(t s f), (17) where we define sinc(u) = sin(πu)/(πu). In he archiecure shown in Figure 12, he signal y() is finally decimaed by a facor κ such ha κ = f Nyq /f s, i.e., he enire Nyquis band is folded ino he frequency range [ f s /2, f s /2]. Hence, he sample sream of he decimaed signal is y d [nt s ] = y[nκ ] wih = 1/f Nyq and he Fourier ransform of he discree signal y d [nt s ] shown in Figure 13(e) is given by Y d [e 2jπf ] = 1 κ κ 1 r= Y (e 2j f r κ ). (18) According o his equaion, he serial wavele bandpass sampling mehod collapses all he sinc-filered and Gaussian weighed convoluion producs ino he band [ f s /2, f s /2]. As illusraed in Figure 13(e), because of he sub-sampling process, he oupu frequency locaion is folded o {f i /f s }f s wih {f i /f s } he fracional par beween he inerference frequency f i and he wavele repeiion rae, equal in our case o he oupu sampling frequency f s. The equivalen filering effec is given by H WBS (f) = = κ/2 1 k= κ/2 κ/2 1 k= κ/2 sinc(t s (f kf s ))P (kf s ) sinc(t s (f kf s ))e (πτkfs)2. (19) The expression in (19) highlighs he ou-of-band rejecion capabiliies of he proposed (realisic) serial wavele bandpass sampling approach in comparison wih he (idealisic) CWT bandpass mehod compued in (13). We emphasize ha he major differences beween he serial wavele bandpass sampling approach and he CWT baseband sampling comes from he fac ha he equivalen filer ransfer funcion differs from a mixure of sinc-shaped for he former (see Eq. 19) o a Gaussian shape (wih infinie suppor) for he laer (see Eq. 14). 3) Simulaion Resuls: We now validae he serial wavele bandpass sampling scheme and more specifically evaluae he ou-of-band rejecion capabiliies. As illusraed in Figure 13(a), he inpu signal x() is complex-valued and builds upon a useful signal locaed a f u wihin he band of ineres (we assume f s is a sub-muliple of f c ) and an ou-of-band inerference signal a f i locaed f i apar from our signal of ineres. The signal x() is sub-sampled by a uniform wavele comb a rae f s = 1/(4τ). We consider a sampling rae of f s = 1 GHz. The wavele parameers, such as widh parameer τ and cenral frequency f c, remain consan over he frame while he ime shif is adjusing o he sampling posiion. As shown in (15) and illusraed in Figure 13(b), serial wavele bandpass sampling is, in he frequency domain, equivalen o a Dirac comb whose ampliude is weighed by he wavele (or pulse) envelope P (f). Since f s < BW p (because 1/f s = T s = 4τ) emporal overlapping among waveles is avoided, several Dirac funcions are included wihin he pulse envelope cenered on carrier frequency f c. As illusraed in Figure 13(c), each convoluion down-convers he useful signal o he origin and ou-of-band inerferences ino baseband. Then, he inegraion over a ime period T s low-pass filers he signals ha are close o baseband (see Figure 13(d)). Figure 14 summarizes he ou-of-band rejecion characerisics of he serial wavele bandpass sampling approach H WBS (f) and provides a comparison wih he idealisic CWT H CWT (f). This analysis quanifies he ou-of-band alias rejecion capabiliy of NUWBS. Our analyical expressions from (19) and (14) are shown wih coninuous lines; simulaion resuls are indicaed wih plus (+) markers, respecively, in blue for wavele bandpass sampling and green for he idealisic CWT mehod. Evidenly, our simulaions coincide wih he heoreical resuls in (19) and (14). We also observe ha he filer characerisics of serial wavele bandpass sampling is inbeween he ideal equivalen Gaussian filer and he sandard

12 12 TO APPEAR IN THE IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS Vcc Frequency conrol signal L R C OUTp RF Wavele OUTn Digial Base Band Wavele shaper gm PRBS@T s Ibias gm Fig. 14. Comparison beween ou-of-band inerference rejecion beween he CWT and wavele bandpass sampling in he case of wavele sampling rae equal o four imes he wavele bandwidh (i.e., T s = 4τ = 1 ns). sinc filer associaed o he T s recangular windows inegraion. As a resul, we achieve a rejecion of 5 dbc, which remains o be lower han he idealisic CWT rejecion bu (i) wih more han 23 db improvemen wih respec o he sandard sinc filer and (ii) can, as shown nex, be implemened in hardware. C. Wavele Generaor Circui The key missing piece of he proposed NUWS and NUWBS approach is he unable wavele generaor circui. For RF applicaions, wavele generaion in he ime domain can be realized by leveraging exensive prior work in he field of ulra-wideband (UWB) impulse echnology [63]. For insance, in our previous work [64], we have demonsraed a circui for low-power pulse generaion a 8 GHz wih variable pulse repeiion rae. Here, we sugges o adap he design in [64], for unable and wideband wavele generaion. Figure 15 shows a corresponding circui diagram. The core of he oscillaor relies on a cross-coupled NMOS pair loaded by an RLC resonaor highlighed, which is commonly used for volage conrol oscillaor (VCO) circuis. Waveles are generaed across he LC ank a RF frequency as soon as he bias curren I bias is applied o he cross-coupled pair. Figure 16 shows a ypical chronogram of he proposed wavele generaion circui. The bias curren duraion is adjused by a digial base-band pulse shaper o enable variable bandwidh. A clock signal running a rae f s is combined wih a pseudo-random bi-sequence (PRBS) running a he low sub-sampling rae f s in order o swich he biasing source on and off. As a resul a non-uniform pulse paern is generaed ailored o he NUWBS soluion. Finally, a variable volage applied o he varacor C in Figure 15 enables us o une he cener frequency o he RF sub-band of ineres. In order o validae he wavele generaor circuiry for RF applicaions up o 8 GHz, physical measuremens have been performed on an ASIC fabricaed in a 13 nm CMOS echnology. Figure 17 shows he power specral densiy (PSD) of he wavele depending on he bandwidh or cenral frequency. Measuremens are provided a Mp/s, Mp/s, and Fig. 15. Circui schemaic of a wavele pulse generaor wih variable bandwidh and cenral frequency capabiliies. The circui acs as a Volage Conrol Oscillaor (VCO) swiched on according o a sub-nyquis PRBS sequence. Clock 1 Frequency V1 conrol signal V 1 PRBS Digial Base Band wavele shaper RF Wavele T s T NYQ Fig. 16. Signal chronogram involved in he conrol of he circui schemaic shown in Figure 15: The frequency conrol signal, he digial base band pulse shaper, and he clock and PRBS signals are running a sub-nyquis raes Mp/s wih ampliude up o 16 mv for 5 Ω impedance. The Tekronix TDS6124C high speed scope is se o 5 Ω impedance o avoid any reflecions wih lab equipmen ha could aler he wavele waveform. A high iming resoluion mode wih digial inerpolaion beween he 25 ps real samples is seleced o provide a 5 ps iming resoluion. The 1 db wavele bandwidh is unable from 3 MHz o 1 GHz and he cenral frequency range from 7.3 GHz o 8.5 GHz. In addiion o being flexible, he wavele generaion is power efficien, i.e., only requires 6 pj/pulse, and remains swiched off in beween wo successive wavele generaion phases (i.e., his is duycycled soluion). Our ASIC measuremens resuls demonsrae a feasible wavele funcion generaor wih a broad range of uning capabiliies in erms of cenral frequency, bandwidh, and repeiion rae operaing in he range of RF frequencies. These resuls pave he way for a complee NUWS/NUWBS inegraion including signal mixing and sampling sage. VI. CONCLUSIONS We have proposed a novel analog-o-informaion (A2I) conversion mehod for compressive-sensing (CS)-based RF

13 M. PELISSIER AND C. STUDER Mp/s Finally, a deailed exploraion of oher applicaions ha may benefi of NUWS/NUWBS and are in need of low power and low cos feaure exracion is lef for fuure work. 56 Mp/s ACKNOWLEDGMENTS The work of M. Pelissier was suppored by he Enhanced Euroalens fellowships program & Carno Insiu. The work of C. Suder was suppored by Xilinx, Inc. and by he US Naional Science Foundaion under grans CCF , ECCS- 1486, and CAREER CCF The auhors hank O. Casañeda for his help wih he manuscrip preparaion. 112 Mp/s Fig. 17. ASIC measuremen resuls: Illusraion of variable wavele rae generaion (28/56/112 Mp/s); Zoom in on a single wavele specrum illusraing he wavele cenral frequency and bandwidh uning capabiliies. feaure exracion. Our approach, referred o as non-uniform wavele sampling (NUWS), combines wavele preprocessing wih non-uniform sampling (NUS), which miigaes he main issues of exising analog-o-informaion (A2I) archiecures, such as ou-of-band noise, inerference, aliasing, and flexibiliy. In addiion, NUWS avoids circuiry ha mus adhere o Nyquis rae bandwidhs. From an RF feaure exracion sandpoin, NUWS can be adaped o he signals of ineres by uning heir duraion, cener frequency, and ime insan per acquired wavele sample. For muliband RF signals, we have developed a specialized varian of NUWS called non-uniform wavele bandpass sampling (NUWBS). For his mehod, we have discussed a wavele selecion sraegy ha enables adapaion o he a- priori knowledge of he sub-bands of ineres. Using simulaion resuls, we have shown ha NUWBS achieves near-opimal sample complexiy already for relaively small dimensions, i.e., NUWBS approaches he heoreical phase ransiion of l 1 - norm-based sparse signal recovery wih Gaussian measuremen ensembles. We have furhermore analyzed he rejecion rae of NUWBS agains ou-of-band inerferers. To demonsrae he pracical feasibiliy of our A2I feaure exracor, we have proposed a suiable wavele generaion circui ha enables he generaion of unable wavele pulses in he GHz regime. The proposed NUWS and NUWBS mehods are promising sraegies for A2I converer archiecures ha overcome he radiional limiaions of exising soluions in power and cos limied applicaions. Our soluions find poenial broad use in a variey of RF receivers argeing specrum awareness or assising convenional RF chains wih uning parameers. Boh of hese advanages render our soluions useful for he Inerne of Things, for which power and cos efficiency and RF feaure exracion are of umos imporance. There are many avenues for fuure work. The design of a complee NUWS/NUWBS-based RF feaure exracor ASIC is ongoing work. A heoreical analysis of he recovery properies for NUWS/ NUWBS is a challenging open research problem. REFERENCES [1] C. E. Shannon, Communicaion in he presence of noise, Proc. IRE, vol. 37, no. 1, pp. 1 21, Jan [2] H. J. Landau, Sampling, daa ransmission, and he Nyquis rae, Proc. IEEE, vol. 55, no. 1, pp , Oc [3] E. J. Candès, J. Romberg, and T. Tao, Robus uncerainy principles: exac signal reconsrucion from highly incomplee frequency informaion, IEEE Trans. Inf. Theory, vol. 52, no. 2, pp , Feb. 26. [4] D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, vol. 52, no. 4, pp , Apr. 26. [5] E. J. Candès and M. B. Wakin, An inroducion o compressive sampling, IEEE Signal Process. Mag., vol. 25, no. 2, pp. 21 3, Mar. 28. [6] B. Murmann, ADC performance survey , Jul [Online]. Available: hp://web.sanford.edu/~murmann/adcsurvey.hml [7] S. K. Sharma, E. Lagunas, S. 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