Reconstruction of Frequency Hopping Signals From Multi-Coset Samples

Transcription

1 1 Reconstruction o Frequency Hopping Signals From Multi-Coset Samples Chia Wei Lim and Michael B. Wakin arxiv: v1 [cs.it] 22 Mar 216 Abstract Multi-Coset (MC) sampling is a well established practically easible scheme or sampling multiband analog signals below the Nyquist rate. MC sampling has gained renewed interest in the Compressive Sensing (CS) community due partly to the act that in the requency domain MC sampling bears a strong resemblance to other sub-nyquist CS acquisition protocols. In this paper we consider MC sampling o analog requency hopping signals which can be viewed as multiband signals with changing band positions. This nonstationarity motivates our consideration o a segment-based reconstruction ramework in which the sample stream is broken into short segments or reconstruction. In contrast previous works ocusing on the reconstruction o multiband signals have used a segmentless reconstruction ramework such as the modiied MUSIC algorithm. We outline the challenges associated with segmentbased recovery o requency hopping signals rom MC samples and we explain how these challenges can be addressed using conventional CS recovery techniques. We also demonstrate the utility o the Discrete Prolate Spheroidal Sequences (DPSS s) as an eicient dictionary or reducing the computational complexity o segment-based reconstruction. I. INTRODUCTION Frequency hopping (FH) signals which arise in spreadspectrum multiple access (SSMA) communication systems have long been used in military radios and more recently appear in the Bluetooth protocol as a orm o communication oering excellent anti-jam capability. An FH signal rapidly changes its transmission requency over a bandwidth that is much larger than its original bandwidth. Over time the FH signal can be viewed as a concatenation o multiple short duration bursts (hops) each having a requency that is pseudorandomly chosen rom a preselected set o possible hopping requencies [1] [2]. Typically accurate synchronization is required amongst cooperative FH receivers (having a priori knowledge o hopping requencies) in a network. However in the absence o such knowledge (subsequently reerred to as the blind setting) constant monitoring over the entire transmission spectrum requires costly wideband receivers with large instantaneous bandwidths. This imposes a severe burden on the analogto-digital converters (ADCs) used in state-o-the-art digital receivers. With conventional uniorm sampling and reconstruction the Nyquist sampling theorem [3] dictates a minimum sampling rate o twice the transmission bandwidth which can be on the order o tens o gigahertz. CWL is with DSO National Laboratories o Singapore; lchiawei@dso.org.sg. MBW is with the Department o Electrical Engineering and Computer Science Colorado School o Mines; mwakin@mines.edu. This work was partially supported by DSO National Laboratories o Singapore and by NSF grant CCF In spite o the high Nyquist limit the act that each transmitted hop is narrowband suggests the possibility o sub-nyquist rate sampling even in the blind setting. As we discuss several previous works in the literature have considered sub-nyquist rate sampling o blind multiband signals in particular via Multi-Coset (MC) sampling but ew have considered the blind FH signal model. A multiband signal is one whose spectrum is concentrated on a small number o narrow bands and the preix blind indicates a lack o knowledge o the center requencies and widths o the signal bands. The act that the transmission requency o the blind FH signal changes over time limits the easibility o previously proposed blind multiband signal recovery techniques most o which do not require deliberate segmenting o MC sample streams into inite-length segments or signal recovery. In the sequel we reer to such a recovery ramework as segment-less and discuss a variant o the classical MUSIC algorithm [4] or this ramework in Section II-C. Due to the non-stationary nature o blind FH signals (changing transmission requencies) there is a need to consider an intentional segmenting o MC sample streams into segments or signal recovery. In the sequel we reer to such a recovery ramework as segment-based. In this paper we assume there exists a ront-end MC sampler which is explained in Section II-B producing sub- Nyquist rate periodic nonuniorm samples o a sum o multiple analog FH signals. We highlight a salient aspect o the MC sampler which acilitates segment-based recovery and we advocate the need or such a recovery ramework or the blind analog FH signal model. As we discuss segment-based recovery which essentially is a inite dimensional ramework is an appropriate setup in which Compressive Sensing (CS) signal recovery approaches can be used to recover blind analog FH signals rom MC samples. Speciically we show that the resulting linear system can be recast as the classical CS multiple measurement vector (MMV) problem and that CS MMV solvers can used or signal recovery. Further we propose the use o a Discrete Prolate Spheroidal Sequence (DPSS) vector-based dictionary which reduces the scale o the signal recovery problem thereby improving solver latency. A. Blind Analog Frequency Hopping (FH) Signal Model In this section we introduce the blind analog requency hopping (FH) signal model. We denote this class o signals as H(N B T ) where N is the number o FH radios B is the maximum bandwidth o each narrowband requency hop in Hz and T is the minimum hop repetition interval (HRI) in seconds. Formally signals x(t) belonging to H(N B T ) can

2 2 be written as where x(t) = N n i 1 i=1 k= g ik (t kt i τ i )e j(2π ikt+θ ik ) (1) g ik (t) = r i (t)m ik (t) n i is the number o hops or the ith radio r i (t) is a timelimited and essentially bandlimited window characteristic o the ith radio m ik (t) is the baseband modulated signal containing the inormation symbols transmitted by the ith radio s kth hop T i is the ith radio s HRI τ i is the delay oset o the ith radio used to model the asynchronous transmission nature 1 o the radios and satisies τ i T i ik is the carrier requency o the ith radio s kth hop and θ ik is the carrier phase o the ith radio s kth hop. As is evident rom (1) signals belonging to H(N B T ) are parameterized by numerous variables characteristic o each FH radio. In particular our reerence to H(N B T ) as a blind FH signal model relects the act that we assume prior knowledge o N B and T only where B sup{ : G ik () > ɛ} (2) G ik () denotes the continuous time Fourier transorm (CTFT) o g ik (t) ɛ denotes a small positive constant (since g ik (t) is an essentially bandlimited window) and T min i T i. The hopping requencies are assumed to satisy ik [ min max ] which corresponds to a continuum o possible hopping requencies. 2 This condition allows the signal model to include hopping requencies that are commonly reerred to as o-grid and in general need not be an integer multiple o some undamental requency. The on-grid assumption along with an assumed knowledge o are commonly adopted or the sake o simpliying blind FH signal models in the literature. A plot o the spectrogram o a signal belonging to H( ) is shown in Fig. 1. B. Previous Works Among the ample literature available with regards to sub- Nyquist rate MC sampling and reconstruction o multiband signals only literature relevant to this paper is mentioned in the sequel. The interested reader is reerred to [5] [17] or more inormation on related works. While the Nyquist sampling theorem holds or any bandlimited signal Landau [5] showed that the required average (both uniorm and nonuniorm) sampling rate or multiband signals can be reduced to the Landau rate which is the Lebesgue measure o the signal s spectral support. Early works on nonuniorm sampling and reconstruction o bandlimited 1 In the synchronous transmission case all radios have the same T i and they transmit at the same time such that τ i = τ i. 2 We assume that min and max are known as with any other typical signal acquisition setup. Fig. 1: Spectrogram o a signal belonging to H( ). In this plot all the emitters have an identical HRI o 1 ms corresponding to a hop rate o 1 hops/s. signals [6] [7] and multiband signals [9] required a priori knowledge o the signal s underlying spectral support. Feng and Bresler [8] [1] addressed the nonuniorm sampling and reconstruction o multiband signals in the blind setting using sub-nyquist rate MC samples. Reconstruction was achieved by irst using a modiied MUSIC algorithm to obtain the signal s unknown spectral support and subsequently using standard least squares to recover the multiband signal. It was shown that such a sampling scheme can approach the Landau rate asymptotically. Venkataramani and Bresler [11] derived explicit multiband signal reconstruction ormulas using sub-nyquist rate MC samples and derived error bounds on the peak error and the energy o the aliasing error (due to multiband signal modeling mismatch) or the sampling/reconstruction system. Further analysis o the system perormance in the presence o additive noise was also provided. Using the previously computed bounds as perormance measures it was subsequently shown [12] that optimizations such as optimal sampling pattern design and optimal base sampling rate can urther improve reconstruction perormance. Bresler [13] discussed the connections between the previous works on blind spectrum sensing and those o CS in particular or the multiband signal model. It was argued that blind spectrum sensing using the MC sampler provides eicient sub- Nyquist rate sampling with reconstruction costs linear in the amount o data and with robustness to noise. Mishali and Eldar [14] reexamined the sampling o multiband signals using sub-nyquist MC samples. It was shown that no sampling scheme can have a worst case perormance better than the MC sampler or the multiband signal model. In contrast to previous blind spectrum sensing works the data covariance matrix was irst used to obtain the subspace containing the multiband signal and subsequently a CS-type solver was used to recover the support. Kochman and Wornell [15] investigated the impact o a inite sensing interval on the recovery o the multiband signals underlying spectral sparsity structure (rom sub-nyquist MC samples) using inite sharp transition band windows and

3 3 characterized the associated redundancy. Davenport and Wakin [16] addressed the sub-nyquist rate sampling and reconstruction o analog multiband signals using a modulated DPSS based dictionary or eiciently representing the sampled multiband signals. Instead o the MC sampling protocol that work considered taking random linear projections o Nyquist rate samples o the multiband signal. Liu et al. [18] proposed a CS ramework or the interception o FH signals using conventional CS random projections without knowledge o the hopping sequence o the underlying FH signal. In contrast to classical CS applications where one seeks to reconstruct the FH signal the goal o [18] was to perorm classiication in the compressive domain. Subsequently in [19] a compressive detection strategy or FH signals was proposed with theoretical detection and alse alarm rates derived rom previous scanning spectrum analyzer results. Then in [2] the previously proposed compressive detection strategy was extended to multiple FH signals. We note that the FH on-grid signal model was used in all three papers. A comparison o all pertinent works against this paper can be ound in Table I. In this table RP denotes Random Projections MB denotes the blind multiband signal model FH- On denotes the FH on-grid signal model FH-O denotes the FH o-grid signal model Cov denotes the use o the data covariance matrix or support recovery Cov+CS denotes the use o the data covariance matrix and a CS-type solver or support recovery and CS denotes the use o a CS-type solver or support recovery. We note that no preerred support recovery technique was indicated or used in [15] although the three recovery techniques were mentioned as possible ways to recover the signal support. TABLE I: Comparison o pertinent previous works against this paper in terms o sampling scheme signal model and support recovery approach. Sampling Scheme Signal Model Support Recovery MC RP MB FH-On FH-O Cov Cov+CS CS [8] [1] [12] [14] [15] [16] [18] [2] This paper C. Need or a Segment-Based Reconstruction Framework In a typical idealized analysis o an MC sampling system signal acquisition and reconstruction take place over an ininite-length sub-nyquist sample stream. In practical inite systems o course such an ininite stream o samples cannot be acquired or processed. Over suitably long segments o samples in time one might hope to achieve signal reconstruction perormance that is comparable to what could be achieved using an ininite segment size; as mentioned previously [15] studies the implications o a inite segment size on multiband signal reconstruction rom MC samples. 3 In some applications 3 We note that the dierences between our work and that o [15] include the segment-based recovery ramework the H(N B T ) signal model and the application. one could simply increase the segment length (capturing samples over a longer time duration) to achieve the desired quality o reconstruction. In particular when considering a multiband signal the spectral support o the signal (and thus the signal s Landau rate) remains ixed over any time duration. Unortunately or FH signals observed over multiple hops the spectral support grows approximately linearly with the time duration over which the signal is observed. Thereore successul FH signal reconstruction at a minimum average sampling rate requires a processing segment size during which one expects the spectral support to have minimal growth. In particular it is necessary to intentionally partition the sample stream into segments that have duration commensurate with the minimum HRI T. To deal with these short-duration sample streams we develop a signal reconstruction ramework that is explicitly segment-based. D. Paper Organization In Section II we irst review the MC sampling protocol and the segment-less blind recovery o multiband signals rom sub-nyquist MC samples. We then outline various CS signal recovery concepts and discuss useul properties o DPSS vectors. In Section III we begin with a discussion o the MC Discrete-Time Equivalent Linear Measurement System (MC- DTLMS). We then show how this ramework acilitates segment-based blind analog FH signal recovery. Moving on we explain the utility o DPSS vectors as an eicient dictionary or representing sampled FH signals. Accordingly the use o the DPSS dictionary leads to a reduction in the scale o the signal recovery problem. In Section IV we use simulations to evaluate the signal reconstruction perormance rom sub-nyquist rate MC samples using our segment-based recovery ramework. We conclude in Section V with open questions. II. PRELIMINARIES In this section ater establishing some deinitions and notation we briely review the undamental principles o MC sampling and the application o MC sampling in blind multiband signal reconstruction. The interested reader is reerred to [8] [1] [13] [15] [21] or more details. We also discuss certain CS concepts which will be helpul or subsequent developments in this paper. A. Deinitions and Notation The CTFT o a continuous time signal x(t) is denoted by X() where X() = x(t)e j2πt dt and the discrete time Fourier transorm (DTFT) o a discrete time signal x(k) sampled at an interval o T s seconds/sample is denoted by X(e j2πts ) where X(e j2πts ) = x(k)e j2πtsk. k=

4 4 x(t) c 1 T c c i T c c q T c t = klt c y 1 (k) t = klt c y i (k) t = klt c y q (k) (a) A C(T c L q C) MC sampling system (b) An MC sampling pattern with T c = 1 L = 6 q = 3 and C = {1 2 5}. Fig. 2: The MC sampling scheme. t 1 LTc LT c LT c LT c LT c Y 1 () Y 2 () (a) Signal spectrum. = A X() 1 LTc X () X 1 () X 2 () X 3 () (b) The requency domain linear system o (4) where Y i() is the ith entry o y() with q = 2 L = 4 and X 1() =. Fig. 3: Spectral slicing in an MC system. where X() denotes the CTFT o the input signal x(t). Rearranging (3) in matrix-vector orm gives the ollowing: As a unction o X(e j2πts ) is periodic with period 1/T s. Matrices shall be denoted by bold uppercase letters and vectors by bold lowercase letters. For a given matrix A C m n A denotes the Hermitian transpose o A. The space spanned by the columns o A is reerred to as the range space o A and denoted by range(a). where y() = Ax() [ 1/(2LT c ) 1/(2LT c )] (4) e j2πc1tc Y 1 (e j2πltc ) y() = LT c. C q e j2πcqtc Y q (e j2πltc ) B. Multi-Coset (MC) Sampling MC sampling is a periodic multi-channel nonuniorm sampling protocol. We denote an MC sampling system by C(T c L q C) where T c is a base sampling interval that is less than or equal to the input signal s uniorm Nyquist sampling interval T nyq L > is an integer q is the number o MC channels and the set C contains q distinct integers c i such that c i L 1. In the ith channel (also reerred to as the ith active coset) the input signal x(t) is irst oset by c i T c seconds in time and then sampled uniormly at the interval o LT c seconds. On average the MC sampler acquires nonuniorm samples o the input signal at the rate o q LT c samples/second and such samples are reerred to as subq Nyquist when LT c < 1 T nyq. Figures 2a and 2b show the MC sampling system and an example o a MC sampling pattern respectively. The MC sample stream at the output o the ith coset is given by y i (k) = x((kl + c i )T c ) and has the corresponding DTFT Y i (e j2πltc ) = 1 L 1 ( X + l ) e j2πcitc(+ l ) LTc LT c LT c l= (3) [ 1/(2LT c ) 1/(2LT c )] the matrix A C q L contains entries with and A il = e j2πcil/l (5) x() = [ X () X L 1 () ] T C L ( X l () = X + l LT c 1() F = ) 1() [ 1/(2LTc)1/(2LT c)] { 1 : F : / F denotes the the indicator unction o the set F. In words (4) expresses the spectrum o the channel outputs y() (observed) as linear measurements o spectral slices o the input signal x() (unknown) with each spectral slice having a width o 1/(LT c ) Hz. In particular these spectral slices X l () correspond to spectral slices o the input signal shited to the baseband interval [ 1/(2LT c ) 1/(2LT c )]. Notwithstanding the matrix-vector orm o (4) this equation holds or all values o [ 1/(2LT c ) 1/(2LT c )] and thereore describes an ininite number o linear systems. Figures 3a and 3b show the input signal spectrum and the corresponding conceptual spectral slicing o the input signal at a requency resolution o 1/LT c respectively.

5 5 In the ollowing sections we review the necessary conditions or (4) to have a unique solution and in the context o blind spectrum sensing or the modiied MUSIC algorithm proposed previously [8] [1] to solve (4) without having to discretize. C. Blind Multiband Signal Reconstruction From Sub-Nyquist Rate MC Samples In the context o the blind multiband signal model where only a ew spectral bands are occupied one can expect x() to contain a similar number o nonzero slices. Suppose the index set I M contains the p indices o the nonzero slices o x(). Then (4) can be equivalently reormulated as y() = A IM x IM () (6) where A IM C q p is the submatrix ormed by taking the columns o A corresponding to the indices in I M and x IM () C p is the reduced vector containing only the nonzero slices o x(). A well-known necessary condition or a unique solution to (6) or any x IM () requires that A IM have ull column rank or all possible index sets I M. In turn this condition requires spark(a) = p + 1 where spark(a) is the minimum number o linearly dependent columns o A. For the matrix A with entries as given in (5) the interested reader is reerred to [13] or a detailed discussion on actors aecting A which in turn aect signal reconstruction quality. Assuming the necessary condition or a unique solution to (4) is satisied we now discuss previously proposed blind multiband signal reconstruction techniques in the segment-less reconstruction ramework. When the index set I M is known solving (4) can be reduced to solving (6) and its solution is simply given as the ollowing: x IM () = A I M y() (7) where A I M is the pseudoinverse o A. Let us remind the reader that solving (7) is equivalent to solving or the nonzero spectral slices o the input signal at spectral resolution 1/(LT c ) Hz. In practice multiband signal reconstruction is perormed in the time domain rather than the requency domain largely due to the convenience o time domain processing [8] [1] as discussed in the sequel. Due to the delays (c i T c s) introduced in each channel o the MC sampler and since the sampling rate o each channel is decimated by a actor o L time domain reconstruction begins with interpolation o the MC sample outputs to a sampling rate o 1/T c and then involves correcting or the delays. This can be easily achieved by computing ( ) k ci z i (k) = y i. (8) L Subsequently the interpolated (and delay corrected) MC time samples (z i (k) s) are linearly combined using A I M to obtain the time domain sequences o the input signal corresponding to the baseband spectral slices X l (). Finally each o these time domain sequences are requency shited to their original corresponding spectral slice location and summed to obtain a reconstructed sampled version x(k) o the input signal x(t). In short time domain signal reconstruction recovers x(k) by computing x(k) = 1 LT c q I M i=1 l=1 ( j2πlk β li z i (k) exp L ) where β li correspond to the entries o A I M. When the index set I M is unknown an additional step is required to irst obtain I M beore the above steps can be applied. This additional step is reerred to as the modiied MUSIC algorithm in [8] [1] [13]. First consider the covariance matrix o y() R y = y()y() d [ 1/(2LT [ c)1/(2lt c)] ] = A x()x() d A [ 1/(2LT c)1/(2lt c)] = AR x A (9) where R x denotes the covariance matrix o x(). Using (6) (9) can be reexpressed as R y = A IM R xim A I M (1) where R xim is the covariance matrix o x IM. In terms o its eigen-decomposition R y can be expressed as R y = UΛU where U contains the eigenvectors o R y and Λ is a diagonal matrix containing the eigenvalues o R y. I rank(r xim ) = I M = p (i.e. ull rank) then rank(r y ) = p since rank(a IM ) = p as necessitated by the unique solution condition. (The interested reader is reerred to [13] [21] or more details on this argument.) As such R y will have p nonzero eigenvalues and correspondingly R y = U x Λ x U x + U nλ n U n = U xλ x U x (11) where U x contains the eigenvectors corresponding to the p nonzero eigenvalues Λ x is a diagonal matrix containing the nonzero eigenvalues U n contains the eigenvectors corresponding to the zero eigenvalues o R y and Λ n is a matrix containing zero entries o appropriate dimensions. Eqn. (11) implies that range(u x ) = range(r y ) which in turn implies that range(u x ) = range(a IM ) due to (1). On the other hand range(u n ) is orthogonal to range(u x ) and hence is also orthogonal to range(a IM ). Thereore one can obtain I M by projecting all columns o A onto U n (by computing Un A) and constructing I M using indices corresponding to the columns that give the p smallest projections in terms o the l 2 -norm. We note that in order to estimate U n a necessary and suicient condition is that q p+1 so that there exists at least one eigenvector to represent the space spanned by U n. Correspondingly this condition also translates to a deterministic guarantee or perect multiband signal recovery when R x is ull rank. In the worst case scenario when rank(r x ) = 1 this guarantee becomes (the less avorable) q 2p. When R x is ull rank the deterministic guarantee results in a minimum average sampling rate o (p + 1)/(LT c ). Letting Ω L denote the spectral occupancy at resolution L we have Ω L = p/l. In this setting the modiied MUSIC algorithm is said to approach the Landau rate which equals

6 6 Ω/T c where Ω is the spectral occupancy asymptotically or almost all multiband signals since Ω L Ω as L. Ater I M has been identiied or a multiband signal one only needs to solve (7) or subsequent MC samples since the spectral support does not change. D. Finite-Dimensional CS Reconstruction Framework The emerging theory o CS has enabled the eicient acquisition o sparse or compressible signals via low-rate random projections. In its classical orm the CS ramework deals with a linear system o under-determined equations y = Ax (12) where the measurements y C q sensing matrix A C q L unknown signal vector x C L and q L. Under certain conditions it may be possible to recover x rom y (even though q L) or example i x L where x denotes the number o nonzero entries o x. A signal x is said to be p- sparse i x = p and compressible i it has p signiicant entries (with the rest o the entries being small). The matrix A satisies the Restricted Isometry Property (RIP) o order p with isometry constant δ p ( 1) i (1 δ p ) x 2 Ax 2 (1 + δ p ) x 2 holds or all p-sparse vectors x. It was shown previously [22] that exact recovery o x rom y can be achieved by solving a convex optimization problem (speciically l 1 minimization) provided A satisies the RIP o order 2p with δ 2p small. Further it was shown in [23] that one can obtain such an A (with entries as deined in (5)) by randomly selecting rows o the DFT matrix with high probability i q = O(p log 4 (L)). In addition numerous greedy iterative recovery algorithms such as OMP [24] and CoSaMP [25] have also been proposed. We note that the cited recovery algorithms have been shown to be robust to noise and return an approximate solution when x is compressible. The linear system o (12) is also reerred to the single measurement vector (SMV) problem in the CS literature. As an extension to the SMV linear system o (12) multiple measurement vector (MMV) linear systems have also been considered [26] [29]. In particular an MMV linear system o under-determined equations has the ollowing orm: Y = AX (13) where Y C q r A C q L and X C L r with an assumed structure in X. The assumed structure is typically in the orm o joint sparsity across the columns o X such that the index set containing the locations o the nonzero entries or every column o X is identical. For the MMV linear system numerous recovery algorithms such as S-OMP [26] [27] Mixed Norm [28] and the previously cited modiied MUSIC algorithm [8] [1] have been proposed. It has been shown in numerous works [1] [13] [27] [29] that guaranteeing a unique solution to the MMV linear system o (13) requires X spark(a) + rank(y ) 1 2 where X denotes the number o nonzero rows o X. While it has been suggested that the ability to recover the true support o X improves as the number r o measurement vectors increases (the number o columns o Y ) (13) reveals the important dependence o the recovery perormance on rank(y ). Intuitively one cannot expect to increase the ability to recover the true support o X when the additional columns o Y do not providing more inormation about the true support o X which is the case when an increase in r does not commensurate with an increase in rank(y ). As we discuss in the sequel this bound on the maximum number o nonzero rows o X can be used to establish a bound on the blind recovery o FH signals when the CS MMV ramework is used or recovery. E. Discrete Prolate Spheroidal Sequences (DPSSs) In a series o seminal papers Slepian et al. [3] [31] and Landau et al. [32] [33] derived and examined properties o the Prolate Spheroidal Wave Functions (PSWFs) which have important implications in time-requency analysis. Slepian later examined discrete versions o the PSWFs named Discrete Prolate Spheroidal Sequences (DPSSs) in [34]. In short PSWFs and correspondingly DPSSs have maximal energy concentration given a time interval (in which they are highly concentrated) and requency interval (in which they are strictly bandlimited). The PSWFs and DPSSs are deined to be the eigenunctions and eigenvectors respectively o a two step procedure which irst time-limits the unction (sequence) and then band-limits it. The DPSSs are parameterized by an integer N D Z + and W D where W D 1 2. Speciically the DPSSs are a collection o N D discrete time sequences that are strictly bandlimited to the digital requency W D s where s is the sampling requency and highly concentrated in the time index range { 1... N D 1}. Let B WD denote an operator that takes as input a discrete time signal bandlimits its DTFT to the requencies W D s and then returns the resulting discrete time signal. Next let T ND denote an operator that takes as input an ininite length discrete time signal and returns an ininite length discrete time signal with all entries outside the index range { 1... N D 1} set to zero. Formally DPSSs are deined as ollows. Deinition II.1. [34] Given N D and W D the Discrete Prolate Spheroidal Sequences (DPSSs) are a collection o N D real-valued discrete-time sequences s () s (1)... s (N D 1) that along with the corresponding scalar eigenvalues satisy 1 > λ () > λ (1) > > λ (N D 1) > B WD (T ND (s (l) )) = λ (l) s (l) or all l { 1... N D 1}. The DPSSs are normalized so that T ND (s (l) ) 2 = 1 or all l { 1... N D 1}.

7 7 The DPSSs are orthogonal both on { 1... N D 1} and on Z. In this work we are primarily interested in truncated time-limited DPSSs reerred to as DPSS vectors and denoted by D(N D W D ). Deinition II.2. [16] Given N D and W D the DPSS vectors s () s (1)... s (N D 1) R N D are deined by restricting the time-limited DPSSs to the index range n = 1... N D 1: s (l) [n] := T ND (s (l) )[n] = s (l) [n] (14) or all l n { 1... N D 1}. It ollows that D(N D W D ) orm an orthonormal basis or C N D (or or R N D ). A unique characteristic o DPSSs is that the irst 2N D W D eigenvalues cluster close to 1 while the remaining eigenvalues cluster close to. Formally this characteristic is captured in the ollowing lemmas. Lemma II.1. (Eigenvalues that cluster near one [34]). Suppose that W D is ixed and let ɛ ( 1) be ixed. Then there exist constants C 1 C 2 (where C 2 may depend on W D and ɛ) and an integer N (which may also depend on W D and ɛ) such that λ (l) 1 C 1 e C2N D or all l 2N D W D (1 ɛ) and all N D N. Lemma II.2. (Eigenvalues that cluster near zero [34]). Suppose that W D is ixed and let ɛ ( 1 2W D 1) be ixed. Then there exist constants C 3 C 4 (where C 4 may depend on W D and ɛ) and an integer N 1 (which may also depend on W D and ɛ) such that λ (l) C 3 e C4N D or all l 2N D W D (1 + ɛ) and all N D N 1. Alternatively suppose that W D is ixed and let α > be ixed. Then there exist constants C 5 C 6 and an integer N 2 (where N 2 may depend on W D and α) such that λ (l) C 5 e C6N D or all l 2N D W D + α log(n D ) and all N D N 2. As discussed urther in Section III-C we will be interested in using DPSS vectors to build a dictionary or representing sampled signal vectors. Lemmas II.1 and II.2 imply that only the irst 2N D W D DPSS vectors are required to capture the energy o length-n D signal vectors that arise rom time-limiting discrete-time signals that are bandlimited to requencies W D s. The ollowing theorem rom [16] gives an approximation guarantee or the representation o discrete-time approximately bandlimited truncated timelimited signals using slightly more than the irst 2N D W D DPSS vectors. Theorem II.1. [16] Let x(k) = T ND (x(k)) be a time-limited sequence and suppose that x(k) is approximately bandlimited to the requency range [ W D s W D s ] such that or some δ B WD x 2 2 (1 δ) x 2 2. Let x C N D denote the vector ormed by restricting x(k) to the indices k = 1... N D 1. Set k D = 2N D W D (1 + ɛ) and let Then or N D N 1 Q = [s () s (1) s (k D 1) ]. x P Q x 2 2 (δ + N D C 3 e C4N D ) x 2 2 where P Q x denotes the orthogonal projection o x onto the space spanned by the columns o Q and C 3 C 4 and N 1 are as speciied in Lemma II.2. III. BLIND ANALOG FH SIGNAL RECONSTRUCTION FROM MC SAMPLES We are now ready to answer the central question o this paper: How can one reconstruct signals belonging to H(N B T ) rom sub-nyquist rate MC samples? Unortunately it would be diicult to apply (4) directly in reconstructing an FH signal. In particular in order to obtain the term on the let-hand side o this equation y() it may be necessary to observe the channel outputs (y i (k) s) over a long time duration so that an accurate estimate o Y i (e j2πltc ) can be ormed. As we have explained while such long sensing intervals may be easible or multiband signals they may be ineasible or FH signals where the spectral support changes (and thus eectively increases) over time. Fortuitously as we will show this diiculty can be avoided by converting the ininite linear system o (4) into an equivalent orm in the discrete-time domain. We reer to this equivalent orm as the MC Discrete- Time Equivalent Linear Measurement System (MC-DTLMS). A. MC Discrete-Time Equivalent Linear Measurement System (MC-DTLMS) Our goal now is to highlight a salient aspect o the versatility o the MC sampling scheme which has not been emphasized in previous literature ocusing on multiband signal reconstruction. The MC-DTLMS is summarized in the ollowing lemma. Lemma III.1. The ininite linear system o (4) is equivalent to the system o equations where z(k) = Ax bb (k) k (15) z(k) = [ z 1 (k) z q (k) ] T z i (k) is as given in (8) A is as given in (5) where x bb (k) = [ x bb (k) x L 1bb (k) ] T X lbb (e j2πtc ) { X l () F otherwise (16) is the DTFT o the lth sample stream x lbb (k) and F = [ 1/(2LT c ) 1/(2LT c )]. Proo: See Appendix A. This straightorward but important lemma relates the interpolated oset corrected MC samples z i (k) to the sample

8 8 z 1 (k) k z 2 (k) k = A x bb (k) k x 1bb (k) k FH1 FH2 FH3 x 2bb (k) k x 3bb (k) Fig. 4: An illustration o Proposition III.1: the discrete-time equivalent linear system o (15) corresponding to Figure 3b. streams o baseband signals x lbb (k) having DTFTs identical to the spectral slices X l () at a sampling rate o 1/T c. As our primary interest is the reconstruction o Nyquist rate samples o the input signal x(t) the use o Lemma III.1 in the sequel is appropriate since T c T nyq. While previous works on multiband signal reconstruction rom sub-nyquist MC samples have ocused on the linear system o (4) Lemma III.1 reveals a salient aspect o the MC sampler which deserves attention in this setting as summarized in the ollowing proposition. Proposition III.1. The index set containing the indices o the nonzero slices o x() in the requency domain linear system o (4) is identical to the index set containing the indices o the nonzero sample streams o x bb (k) in the discrete-time domain linear system o (15). Essentially Proposition III.1 states that there exists a direct correspondence between the slices o x() and the sample streams o x bb (k) even though the ormer corresponds to a vector ormulation in the requency domain while the latter corresponds to that in the time domain. Accordingly an allzero spectral slice o x() at index l corresponds to an all-zero sample stream in x bb (k) at the same index l. Figure 4 shows an illustration o Proposition III.1; compare with Figure 3b. The dierence between (4) and (15) is not particularly signiicant when sampling and reconstructing a multiband signal. In particular in the case o a multiband signal model one would expect that among the L spectral slices that comprise x() ew are nonzero. Similarly among the L ininite sequences that comprise x bb (k) ew o the sample streams x lbb (k) are nonzero. Reconstruction could be perormed in either domain. The dierence between (4) and (15) is signiicant when sampling and reconstructing an FH signal. In particular in the case o an FH signal model where samples are collected over a long time duration many slices o x() will be nonzero as we have explained previously. Consequently many o the sample steams x lbb (k) will also be nonzero. Speciically what this means is that or most values o l there will exist some k Fig. 5: A toy example o the type o structure in X bb C 1 2 or N = 3 (denoted by FH1 FH2 and FH3) with identical HRIs B 1/(LT c ) segment size r < round(t/t c ) as well as the locations o their corresponding nonzero entries in X bb. k such that x lbb (k). However the MC-DTLMS in (15) reveals a time-varying structure to sample steams x lbb (k) that is not apparent rom (4). In particular or any time index k the number o indices l or which x lbb (k) should be commensurate with the number o FH radios N. This suggests that or any k it may be possible to solve the linear system o (15) by exploiting the sparsity among the sample streams. Solving this system repeatedly or every single value o k may be expensive; thereore we propose to solve this system over time segments o duration commensurate with the minimum HRI T. All o this diers rom previous works on multiband signal reconstruction rom MC samples as discussed previously. We discuss pertinent aspects o solving (15) in the ollowing section. B. An MC Segment-Based Recovery Framework Formally we set up our MC segment-based signal reconstruction problem as ollows: Z = AX bb (17) where Z C q r with entry [Z] ij = z i (k j ) in its ith row and jth column A is as given in (5) and X bb C L r with entry [X bb ] ij = x i 1bb (k j ) in its ith row and jth column. The linear system o (17) is a restriction o (15) to a inite dimension (or segment size) r since the sample streams o x bb (k) correspond to ininite sequences. Accordingly solving (17) repeatedly over consecutive segments o size r recovers consecutive segments o baseband sequences X bb. In order to reconstruct sampled sequences o the input signal x(t) at rate 1/T c it is necessary to requency shit the baseband sequences to their corresponding spectral slice location and then sum the requency-shited sequences. We note that when the segment size r = 1 (17) turns into the classical SMV problem in CS. When r > 1 we have an MMV problem and the structure o X bb deserves special mention with regards to the H(N B T ) signal model. Fig. 5 shows a toy example o the type o structure that exists in X bb when B 1/(LT c ) and r < round(t/t c ); that is the segment size has a smaller duration than the HRI o the requency hoppers. We note that setting L such that B 1/(LT c ) is a commonly used spectral width coniguration in previous [8] [11] [13] [14] multiband signal reconstruction literature. We will make this assumption throughout the rest

9 9 o this paper. We are primarily interested in the setting where r < round(t/t c ) since the spectral support size o the H(N B T ) signal is minimal in the sense that each hopper may change requency at most once. Due to the deinition o B ( 1/(LT c )) in (2) we note that every hop transmitted results in at most 2 nonzero rows o entries over the columns (duration) or which the hop is present. There exist instances when there is only 1 nonzero row o entries i the range o active requencies corresponding to that particular hop alls within the boundaries o the spectral slice X l () or the lth row. Accordingly or segments when the N hoppers do not switch requencies one can expect at most 2N nonzero rows in X bb. On the other hand or segments when the N hoppers do switch requencies one can expect at most 4N nonzero rows 4 in X bb. The ollowing corollary provides the necessary condition or the existence o a unique solution to (17) or a given H(N B T ) signal. Corollary III.1. (Uniqueness Condition or MC Segmentbased Recovery Framework) Let x(t) H(N B T ). A unique solution exists to (17) i q > 8N rank(z) (18) or each segment o size r where 1 r < round(t/t c ) provided spark(a) = q + 1. Proo: See Appendix B. Corollary III.1 provides a worst case guarantee or obtaining a unique solution to the linear system o (17) and hence can be interpreted as a minimal sampling rate requirement. We note that while the 8N term o Corollary III.1 seems excessive it is due to a worst case consideration. On average i T = rnt c where n Z + with n 1 one can expect most o the segments to contain hopping signals where the hoppers do not switch requencies thereby reducing the 8N term to 4N or such segments. Let us remind the reader that when r round(t/t c ) the 8N term in (18) increases due to the growing spectral support o the H(N B T ) signal which in turn increases the minimum average MC sampling rate required. On the other hand one can possibly reduce the minimum average MC sampling rate (by reducing q) using a Z with a larger rank which is oten achievable by increasing the segment size r. Due to the opposing eects on (18) when increasing r the appropriate choice o r to use translates to a design trade-o that one must consider or a given H(N B T ) signal. We shall discuss this design aspect o MC segmentbased signal recovery urther in Section IV. C. An Eicient Dictionary or the MC Segment-based Recovery Framework We now discuss the utility o DPSS vectors as an eicient dictionary or reducing the scale o the recovery problem. Recall that the inite linear system o (17) requires solving or the r columns o unknowns in X bb given that Z C q r and A C q L. As we discuss in Section IV when solving (17) the resulting solver latency can be high when r is 4 A row is declared nonzero it contains at least one nonzero entry. large. However note that the each row o Z corresponds to a sequence that is oversampled by a actor o L (due to the interpolation pre-processing) and thus bandlimited to only a bandwidth o 1/(LT c ). In light o this observation the DPSS vectors can be used to reduce the dimension o the recovery problem. A unique characteristic o the DPSS vectors is their ability to provide eicient and high quality approximations o oversampled discrete-time bandlimited signals. In particular each row o Z (o length r) can be approximated with k D 2N D W D expansion coeicients by computing inner products with a collection o k D DPSS vectors conigured by setting N D = r and W D = 1/(2L). In short the use o such a DPSS vector-based dictionary provides a way or reducing the linear system o (17) into a smaller set o equations (which is aster to solve) without sacriicing the rank o the measurement matrix. Deine Z = ZQ where Z C q k D and Q R r k D is constructed as [ ] Q s () s (k D 1) with s (l) as deined in (14). Accordingly post-multiplying both sides o the linear system o (17) gives Z = A X bb (19) where X bb = X bb Q. Thus as an alternative to solving (17) one may solve the smaller system (19) instead and then inally compute X bb = X bb Q T since the normalized columns o Q are orthogonal to each other. We note that the linear o system o (19) is an approximation o the linear system o (17). Thereore depending on the application i exact FH signal reconstruction is not necessary one can potentially trade o FH signal reconstruction error or signiicant improvements in solver runtime latency by solving the linear system o (19) instead o (17). In particular the use o (17) results in a reduction actor o L (or k D = 2N D W D ) in terms o the number o unknown X bb columns as compared to that o X bb. Further deine the approximation error E a C L r associated with the approximation o the linear system o (17) with that o (19) as E a X bb X bb QQ T. (2) The ollowing lemma bounds this error. Lemma III.2. The Frobenius norm o the approximation error E a C L r as deined in (2) satisies E a F L(δ + rc 3 e C4r ) X bb F where constants δ C 3 and C 4 are as deined in Theorem II.1 provided the assumption in Theorem II.1 is satisied and k D = 2N D W D (1 + ɛ). Proo: See Appendix C.

10 1 We note that the assumption in Theorem II.1 is a mild condition on our H(N B T ) signal model that is mostly satisied in practice. As discussed previously in Theorem II.1 in practice one needs to use a value o k D that is slightly larger than 2N D W D or excellent approximation quality. We numerically demonstrate the dependence o the DPSS dictionary approximation quality on k D in Section IV. IV. NUMERICAL EXPERIMENTS We begin this section by discussing the H(N B T ) signal used in our numerical experiments. For all FH signals the ollowing time-limited and essentially bandlimited window was used: 2 T H t i t <.5T H 1 i.5 t <.95T H r(t) = 2 T H (T H t) i.95 t < T H otherwise where T H =.95HRI is the duration o one hop. In words or each hop g ik (t) ramps up m ik (t) or 5% o the hop duration maintains m ik (t) or 9% o the hop duration and ramps down m ik (t) or 5% o the hop duration. All FH hops were generated with a baseband waveorm having 4PSK modulation type with pulses shaped using the raised cosine ilter and conigured with an excess bandwidth o.3. The symbol rate o m ik (t) was adjusted such that its resulting bandwidth was 25kHz. An H(N 25 T ) signal was then generated by summing N individual FH signals together. We note that all requency hoppers were generated with identical HRIs equal to T and this knowledge o T was used to determine the appropriate duration o the segments used in the segmentbased recovery ramework. As we show (e.g. in Fig. 7) one can expect improved signal recovery perormance when the collection o FH signals contain other requency hoppers operating at a larger HRI. An MC C( q C) sampler was used to sample the FH signal at sub-nyquist rates which correspond to q < L using a randomly chosen sampling pattern C. We note that T c = corresponds to a sampling rate o 2.5MHz or the inal reconstructed H(N 25 T ) signal. We choose the SPGL1 MMV solver [35] as a representative CS MMV solver in our numerical experiments. The interested reader is reerred to [17] or a comparison o the perormance o various recovery algorithms proposed previously or blind multiband signal reconstruction. In Section III-B it was argued that choosing an appropriate value o the segment size r or a given H(N B T ) signal amounts to a design trade-o. For various values o r Fig. 6a shows the normalized mean square error (NMSE) deined as NMSE ˆx x 2 2 x 2 2 where ˆx denotes an estimate o the H(N B T ) signal and x denotes an observation interval o the H(N B T ) signal. We note that while a H(N B T ) signal with a duration o 1ms was used recovery o X bb was perormed over segments with varying r. Overall an estimate o X bb with duration 1ms (a) A plot o NMSE versus segment size r or a H( ) signal or q=2 3 5 with corresponding eective compression levels L/q= and 2. (b) A plot o mean support size versus segment size r or the same H( ) signal. Fig. 6: A plot o NMSE versus segment size r or dierent q s corresponding to a H( ) and a plot o its corresponding mean support size versus segment size r. was constructed by concatenating consecutive estimates o X bb obtained rom their corresponding segments. Fig. 6a shows a plot o NMSE versus r or various q values (which scale proportionally to the average sampling rates o the MC sampler) using the MC segment-based recovery ramework o (17) and Fig. 6b shows its corresponding mean support size versus segment size r. The mean support size o the H( ) signal is obtained by averaging the support size o the signal over all segments o size r or a signal duration o 1ms. In particular the underlying FH signal had an HRI o.2ms (5 hops/s) which is equivalent to a segment size o 5 (or T c = s). We note that the plots in Fig. 6a indicate that in the regime when r round(t/t c ) (i.e. the segment size is small relative to the HRI o the FH signal) increasing r decreases the overall NMSE o the recovered signal. This is due to the eect o increasing rank(z) which improves the MMV solver perormance due to the increased ability o inding the true support o X bb. On the other hand increasing r too much results in an increasing NMSE due to the increasing number o nonzero rows o X bb as discussed previously. Due to these opposing eects the plots in Fig. 6a suggest an optimum r round(t/(2t c )) which can be explained

11 11 (a) A plot o NMSE versus q or an FH signal corresponding to H(1 25 T ). (a) A plot o NMSE versus q or a collection o FH signals corresponding to H( ). (b) A plot o NMSE versus q or a collection o FH signals corresponding to H(7 25 T ). Fig. 7: A comparison o NMSE versus q or collections o FH signals corresponding to dierent N s using the MMV-Time domain linear system o (17) denoted by MMVT and using the modiied MUSIC algorithm denoted by MMUSIC. (b) A plot o NMSE versus q or a collection o FH signals corresponding to H( ). Fig. 8: A comparison o NMSE versus q or collections o signals corresponding to dierent T s using the MMV-Time domain linear system o (17) denoted by MMVT and that o (19) using the DPSS dictionary with k D = 2N D W D and 4N D W D. intuitively. In particular one would expect an optimum value o r round(t/t c ) as this is the smallest segment size or which the astest requency hoppers do not switch requency. However due to the asynchronous nature o the requency hoppers most requency hoppers will change requency once during an arbitrary segment o size r round(t/t c ). Setting r round(t/(2t c )) guarantees that on average some o the hoppers do not change requency during any given segment. This is corroborated with the plot in Fig. 6b showing the moderate increase (much less than doubling) in the mean size o the signal s support as the segment size increases rom 25 to 5. We next examine the impact o the parameters N T as well as the number o channels (q=2 3 5 and 8) on the overall H(N B T ) signal reconstruction quality in terms o NMSE. In these and subsequent numerical experiments we have used r = round(t/(2t c )). Fig. 7 shows a comparison o the NMSE versus q or collections o FH signals corresponding to N = 1 7 with HRIs corresponding to 5 hops/s and 1 hops/s using the MMV- Time domain linear system o (17) denoted by MMVT (blue and red curves) and the modiied MUSIC algorithm denoted by MMUSIC (black and green curves). We note that in general an increase in q leads to a decrease in NMSE and hence an improvement in signal reconstruction quality. As the HRIs decrease (i.e. increasing hopping rates) the increase in NMSE (or a ixed q) can be attributed to the increase in number o hops thereby increasing the number o nonzero rows o X bb. We note that with each switching o requency additional rows in X bb become nonzero due to the transient signal amplitude changes involved as the FH signal ramps up or down. When N = 7 the increase in NMSE (due to switching) is less obvious since NMSE is largely dominated by the eect o a large N. In addition NMSE versus q plots (the black and green curves) corresponding to FH signal reconstruction using the modiied MUSIC algorithm denoted by MMUSIC show a consistent poorer FH signal reconstruction quality. We note that in this setup the MMUSIC algorithm was used to estimate the support o each segment o the collection o FH signals. Let us remind the reader that or the H(N B T ) signal model support recovery using the MMUSIC algorithm is not expected to work well as it consistently underestimates the size o the support o the collection o FH signals. As such subsequent FH signal reconstruction on an underestimated support size accounts or the signiicantly higher NMSE.