Xampling: Signal Acquisition and Processing in Union of Subspaces

XAMPLING: SIGNAL ACQUISITION AND PROCESSING IN UNION OF SUBSPACES

1

Xampling: Signal Acquisition and Processing in Union of Subspaces

arXiv:0911.0519v2 [cs.IT] 25 Sep 2010

Moshe Mishali, Student Member, IEEE, Yonina C. Eldar, Senior Member, IEEE and Asaf Elron

Abstract—We introduce Xampling, a unified framework for signal acquisition and processing of signals in a union of subspaces. The main functions of this framework are two. An analog projection narrows down the input bandwidth prior to sampling with commercial devices. A nonlinear algorithm detects the input subspace prior to conventional signal processing. We build the framework bottom-up, from the application layer to the abstract setting, in three steps. First, we study Xampling for lowrate signal acquisition by conducting a thorough comparative study between two sub-Nyquist acquisition strategies, the random demodulator and the modulated wideband converter (MWC), in terms of model robustness and hardware and software complexities. Second, we develop an algorithm that enables convenient signal processing at sub-Nyquist rates from samples obtained by the MWC. Third, with the intuition we gain from the previous stages, we describe the line of reasoning underlying the proposed Xampling framework. In addition, we show that a variety of sampling approaches for union classes fit nicely into our framework, supporting the generality of Xampling. Index Terms—Analog to digital conversion, baseband processing, compressed sensing, digital signal processing, modulated wideband converter, sub-Nyquist, Xampling.

I. I NTRODUCTION IGNAL processing methods have changed substantially over the last several decades. The number of operations that are shifted from analog to digital is constantly increasing, leaving amplifications and fine tunings to the traditional frontend. Sampling theory, the gate to the digital world, is the key enabling this revolution. Traditional sampling theorems assume that the input lies in a predefined subspace [1]. The most prevalent example is bandlimited sampling, according to the theorem of Shannon-Nyquist [2], [3]. Recently, nonlinear union of subspaces (UoS) models have been receiving growing interest in the context of analog sampling [4]–[11]. The UoS setting captures uncertainty in the signal by allowing several possible subspace descriptions, with the exact signal subspace unknown a-priori. In contrast to classic subspace sampling, the theory of sampling over UoS is yet to be completed. In particular, to date, there is no equivalent to the oblique projection operator

S

This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. The authors are with the Technion—Israel Institute of Technology, Haifa 32000, Israel (email: [email protected]; [email protected]; [email protected]). M. Mishali is supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities. Y. C. Eldar is currently on leave at Stanford, USA. Her work was supported in part by the Israel Science Foundation under Grant no. 1081/07 and by the European Commission in the framework of the FP7 Network of Excellence in Wireless COMmunications NEWCOM++ (contract no. 216715).

which reconstructs the signal when its exact subspace is known for almost all sampling functions [1], [12]–[18]. Interestingly, the lack of a complete theory did not withhold the development of numerous stylized applications [6]–[10], [14], [19]– [23], aiming at reducing the sampling rate below Nyquist by exploiting the UoS model. The acquisition and reconstruction methods of [6]–[10], [14], [19]–[23] are substantially different from each other, raising the question of whether the apparent distinct approaches can be derived from a common framework. The first and main contribution of this paper is a unified and pragmatic framework for acquisition and processing of UoS signal classes, referred to as Xampling. We build the Xampling framework bottom-up by leveraging insights and pragmatic considerations resulting from the applications layer into a unified generic architecture for the abstract setting. In the next section, we present the Xampling architecture which consists of two main functions: lowrate analog to digital conversion (X-ADC), in which the input is projected onto a low-dimensional subspace prior to sampling with commercial devices, and lowrate digital signal processing (X-DSP), in which the input subspace is detected prior to signal processing in the digital domain. In both case the X prefix hints at the rate reduction. To simplify the presentation, we briefly introduce the Xampling architecture in the next section, and establish it more generally only later on, in Section VI. In between, we build Xampling via the development of two standalone contributions, which provide initial motivation and practical insights into the proposed framework. Our first step in building the Xampling framework is to study signal acquisition in a representative UoS setting. We do that by a thorough comparison between the random demodulator (RD) [22] and modulated wideband converter (MWC) [7] systems. These methods apply compressed sensing (CS) ideas to reduce the sampling rate of spectrally-sparse signals below the Nyquist rate. Our contribution is a detailed technical comparison, which reveals prominent advantages for the MWC in terms of model robustness and efficient hardware realization. In addition, a numerical analysis on comparable inputs shows that the RD imposes severe computational loads, orders of magnitude higher than in the MWC approach. These aspects are not evident from the original publications [7], [22]. In fact, to the naked eye, the strategies may seem similar, at least visually. We draw several conclusions in an attempt to nail the source for the significant differences we encounter. As a third contribution, we study signal processing at subNyquist rates, which is challenging since conventional DSP methods assume their input data stream is given at the Nyquist rate. We develop a digital algorithm, named Back-DSP, that provides the MWC with a smooth interface to existing DSP

2


software. Our algorithm consists of several lowrate processing steps, which together detect the exact signal subspace, thereby gaining backward compatibility to conventional processing methods. The alternative approach of [24], which suggests the development of processing methods that are specialized to the union setting, is discussed and compared to. As a nice feature, we show that once the Back-DSP algorithm is applied, the input can be reconstructed more efficiently than the original method of [7]. Numerical simulations demonstrate backward compatibility in typical noisy wideband scenarios. With the insights collected, we reach the final contribution – establishing the Xampling framework. We first show that a wide range of UoS applications [6]–[10], [14], [19]–[23] fit elegantly into the proposed sampling structure. In addition, we explain the line of reasoning behind Xampling, using supporting examples from the previous sections. The consequence is that our Xampling framework is sufficiently general to capture a variety of UoS applications and at the same time specific enough to hint on potential improvements and warn from design caveats. The paper is organized as follows. Section II describe the UoS model formally. The Xampling framework is introduced in Section III. A comparison between the RD and MWC architectures is presented in Section IV. Following, in Section V, we develop and simulate the Back-DSP algorithm. We establish the Xampling framework in Section VI. II. U NION OF S UBSPACES We begin by describing the setup of signals within a UoS and delineate example sampling problems that can be derived from this model. Let x(t) be an input signal in the Hilbert space H = L2 (R). The signal x(t) is assumed to lie in a UoS of H, namely within a parameterized family of subspaces 4 [ x(t) ∈ U = Aλ , (1) λ∈Λ

where Λ is a list of indices, and each individual subspace Aλ ∈ H. The key property of the UoS model is that the input x(t) resides within Aλ∗ for some λ∗ ∈ Λ, but a-priori, the exact subspace index λ∗ is unknown. The union (1) over all possible signal locations forms U, which is a nonlinear set, since the sum (or a linear combination) of x1 (t), x2 (t) ∈ U does not lie in U, in general. Typically, U is a true subset of the linear affine space ( ) X Σ = x(t) = αλ xλ (t) : αλ ∈ R, xλ (t) ∈ Aλ . (2) λ∈Λ

In essence, the UoS model permits uncertainty in the exact signal subspace, opening the door to interesting sampling problems. We describe two example union models, which emphasize the difference between the nonlinear set U and the linear space Σ. A first application of (1) is sampling of multiband signals. A multiband input x(t) has sparse spectra, such that its continuous-time Fourier transform (CTFT) X(f ) is supported on N frequency intervals, or bands, such that the individual widths do not exceed B Hz. The top of Fig. 5 illustrates a

typical multiband spectra. When the band positions are known and fixed, the signal model is linear, since the CTFT of any combination of two inputs is supported on the same frequency bands. This scenario is typical in communication, when a receiver intercepts several radio-frequency (RF) transmissions, each modulated on a different high carrier frequency fi . Knowing the band positions, or the carriers fi , allows the receiver to demodulate a transmission of interest to baseband, that is to shift the contents from the relevant RF band to the origin. Subsequent sampling and processing are carried out at a low rate. When the carrier frequencies fi are unknown, the set of all possible multiband signals that occupy up to N B Hz of the spectrum results in a UoS model, where each Aλ corresponds to a specific combination of band positions. In this scenario, the transmissions can lie anywhere below fmax . At first sight, it may seem that sampling at the Nyquist rate fNYQ = 2fmax ,

(3)

is necessary, since every frequency interval below fmax appears in the support of some x(t) ∈ U. On the other hand, since each specific x(t) ∈ U fills only a portion of the Nyquist range (only N B Hz), one would intuitively expect to be able to reduce the sampling rate below fNYQ . Standard demodulation cannot be used since fi are unknown, which makes this sampling problem challenging. In Section IV-B, we describe the MWC architecture [7] which acquires and processes signals from the multiband union at a low rate, proportional to N B. Another interesting UoS example are signals with finite rate of innovation (FRI), originally introduced by Vetterli et al. in [23], [25]. An FRI signal has finitely many degrees of freedom within a time interval T . For example, the amplitudes d` in x(t) =

L X `=1

d` g(t − t` ),

t ∈ [0, T ],

(4)

contribute L degrees of freedom on each interval T . If the time delays t` are fixed, then (4) defines a linear space of inputs. A bank of L matched filters with responses g(t − t` ) can be used to determine d` and accordingly the input x(t). For unknown delays, the union over all possible t` ∈ [0, T ] results in an FRI model with 2L unknowns, t` , d` , 1 ≤ ` ≤ L. An FRI union such as (4) is encountered, for example, when a channel with multipath fading generates echoes of a transmitted pulse g(t) in various unknown delays and attenuations [8], or in ultrasonic imaging [10]. As before, the UoS model introduces a challenging sampling problem, namely how to sample x(t) at a rate that is proportional to the rate of innovations 2L/T , rather than to Nyquist rate of x(t), which is dictated by the potentially large bandwidth of g(t). In Section VI, we discuss variants of FRI union models and sampling solutions. Other UoS applications are also described in that section. All these methods share the same rationale – the sampling rate is reduced by exploiting the fact that the input belongs to a single subspace Aλ∗ , even though the exact subspace index λ∗ is unknown. While sharing the same rationale, the actual sampling strategies are quite different from one another. The Xampling framework which we propose next attempts to unify the sampling of UoS signal classes.

MISHALI, ELDAR AND ELRON

3

Reduce analog bandwidth prior to sampling

x(t)

Union projection P :U →S

ADC device

Linear, Analog

Commercial

Reduce digital complexity Gain backward compatability

y[n]

Detection x(t) ∈ Aλ∗

Subspace DSP

Subspace reconstruction

Nonlinear

Lowrate, Standard

Lowrate, Standard

X-ADC

x ˆ(t)

X-DSP

Fig. 1: Xampling – A pragmatic framework for signal acquisition and processing in union of subspaces.

III. X AMPLING In this section, we introduce Xampling – our proposed framework for acquisition and digital processing of UoS signal models. A. Unified Goals We define the sampling problem for the union set (1) as the design of a system which provides: 1) ADC: an acquisition operator which converts the analog input x(t) ∈ U to a sequence y[n] of measurements, 2) DSP: a toolbox of processing algorithms, which uses y[n] to perform classic tasks, e.g., estimation, detection, data retrieval etc., and 3) DAC: a method for reconstructing x(t) from the samples y[n]. In addition, we require that the above goals are accomplished with ADC + DSP + DAC → minimum use of resources. (5) The last requirement is what distinguishes Xampling from traditional subspace sampling. In (5) we mean that the hardware and software complexities involved in the ADC, DSP and DAC tasks should be proportional to the dimensions of the individual subspaces Aλ rather than to the large dimensions of the affine space Σ. This includes minimizing the number of devices in the acquisition stage, reducing the sampling rate, decreasing the processing speed and memory requirements, and reconstructing x(t) with a minimal number of operations. We emphasize that without aiming at minimal use of resources, one could conceptually treat U by sampling the space Σ instead, for which traditional signal acquisition and processing methods apply. However, this technically-correct approach often leads to practically-infeasible solutions with a tremendous waste of expensive hardware and software resources. For example, in the multiband union, Σ is the fmax bandlimited space, for which no rate reduction is possible. Similarly, in the FRI union, Σ has the high bandwidth of g(t), and again no rate reduction is possible. B. Architecture The Xampling framework we propose for union sampling has the high-level architecture presented in Fig. 1. The first two

blocks, termed X-ADC, perform the conversion to digital of x(t). A linear operator P maps, or projects, the input set U to a low dimensional bandlimited space S. This operation involves analog processing of x(t). Then, a commercial ADC device takes pointwise samples of the projected signal P {x(t)}, resulting in the sequence of samples y[n]. The role of P in Xampling is to narrow down the analog bandwidth, so that lowrate ADC devices can be used subsequently. As we demonstrate in the example applications of multiband and FRI sampling, the challenge is to project the entire family of subspaces (1), simultaneously, to a low dimensional subspace S, without prior knowledge on the exact subspace Aλ∗ that the signal belongs to. The design of P needs to wisely exploit the union structure, in order not to lose vital signal information while reducing the bandwidth. In the digital domain, Xampling consists of three computational blocks. A nonlinear step detects the signal subspace Aλ∗ from the lowrate samples. Once the index λ∗ is determined, we gain backward compatibility, meaning standard DSP methods apply and commercial DAC devices can be used for signal reconstruction. The combination of nonlinear detection and standard DSP is referred to as X-DSP. As we shall demonstrate, besides backward compatibility, the nonlinear detection decreases computational loads, since the subsequent DSP and DAC stages need to treat only the single subspace Aλ∗ , complying with (5). The important point is that the detection stage can be performed efficiently from the lowrate samples. Xampling is a generic template architecture. It does not specify the exact acquisition operator P or nonlinear detection method to be used. These are application-dependant functions. Our goal in introducing Xampling is to propose a high-level system architecture and a basic set of guidelines, which we support in the next sections. These guidelines include: 1) an analog pre-processing unit, prior to signal acquisition, so as to reduce the input bandwidth, 2) a linear projection P (rather than nonlinear analog processing), 3) commercial ADC devices for actual acquisition, 4) subspace detection in software, and 5) standard DSP and DAC methods. Later on in this paper, we support the validity of Xampling in two steps. In the next two sections we build initial motivation for Fig. 1, via two independent contributions in the

4


lowpass

pointwise sampler

Analog

Digital cutoff b

maxmial rate r samples/sec

Fig. 2: The front-end of a practical ADC has an inherent bandwidth limitation, which is modeled as a lowpass filter preceding the pointwise acquisition [7].

context of sub-Nyquist sampling of spectrally-sparse signals. We intentionally begin with the application perspective, in order to gain intuitions and practical insights from the subNyquist settings. Later on, in Section VI we consider other UoS applications that fit the Xampling framework. Overall, the practical perspective shows that the architecture of Fig. 1 is sufficiently general to capture a variety of UoS applications, and at the same time the scheme is specific enough to hint on design caveats and potential improvements. In Section VI, we further support Xampling for the generic setting (1) using the intuition built. As we show the framework essentially follow from two basic assumptions: (A1) DSP is the main purpose of signal acquisition, and (A2) The ADC device has limited bandwidth. The DSP assumption (A1) highlights the ultimate use, in our opinion, of any sampling system – substituting old fashion analog processing by modern software algorithms. DSP is perhaps the most profound reason for signal acquisition. Hardware development can rarely compete with the convenience and flexibilities that software environments provide. In many applications, therefore, DSP is what essentially motivates the ADC. C. Modeling Practical ADC Devices In order to motivate assumption (A2), we zoom into the drawing of the ADC device of Fig. 1. In the signal processing community, an ADC is often modeled as an ideal pointwise sampler that takes snapshots of x(t) at a constant rate of r samples/second. The sampling rate r is the main parameter that is highlighted in the datasheets of popular ADC devices; see online catalogues [26], [27] for many examples. For most analysis purposes, the first-order model of pointwise acquisition approximates the true ADC operation sufficiently well. Another property of practical devices, also listed in datasheets, is about to play a major role in the UoS settings – the analog bandwidth power b. The parameter b measures the −3 dB point in the frequency response of the ADC device, which stems from the responses of all the circuitries comprising the internal front-end. Consequently, inputs with frequencies up to b Hz can be reliably converted. Any information beyond b is attenuated and distorted. Fig. 2 depicts an ADC model in which the pointwise sampler is preceded by a lowpass filter with cutoff b, in order to take into account the bandwidth limitation [7]. In Xampling, the input signal x(t) belongs to a union set U which typically has high bandwidth, e.g., multiband signals whose spectrum

reaches up to fmax or FRI signals with wideband pulse g(t). This explains the necessity of an analog projection P to reduce the bandwidth prior to the actual ADC. The next stage can then employ commercial devices with low analog bandwidth b. Assumption (A2) of Xampling basically says that we expect the conversion device to behave according to the model of Fig. 2. The X-ADC can be realized on a circuit board, chip design, optical system and so on. In all these platforms, the front-end has certain bandwidth limitations which can be modeled as a lowpass filter with cutoff b that precedes a pointwise sampler. Note that perfect reconstruction of the input is not listed as a design goal or a base assumption of Xampling. Truly, proving the ability to recover x(t) exactly, even if in an idealized noiseless setting, is important since this is the theoretical certificate that the sampling strategy is a oneto-one mapping with no information loss. Nonetheless, in practical applications, reconstructing x(t) is often of less interest. Retrieving the information it carries is usually the prime goal. For example, in modern communication the goal is to transfer a stream of digital bits over a noisy medium. The transmitter encodes digital bits to analog symbols, modulates them on a carrier frequency, and transmits an RF signal x(t). The receiver intercepts a noisy version of x(t) and aims at retrieving the message bits with minimal error. Decoding the message bits usually represents the logical end of the physical communication path, and the starting point of higher logical layers. Continuous reconstruction appears sometimes in the physical layer, e.g., when using relay stations to re-transmit x(t) after local improvements. We point out that lowrate processing, X-DSP, has practical advantages even when high ADC rate is acceptable, since it reduces the computational loads in the processing units. The compounded usage of both X-ADC and X-DSP is for mainstream applications, where reducing the rate of both signal acquisition and processing is of interest. IV. X-ADC: S UB -N YQUIST S IGNAL ACQUISITION We now consider a representative union model for the study of lowrate signal acquisition in Xampling. We choose the class of spectrally-sparse signals with unknown spectral support, so that the energy of x(t) concentrates on a small, yet unknown, portion of its Nyquist range. We utilize Xampling to study the differences between two acquisition methods – the RD [22] and MWC [7] systems. To the naked eye, the approaches may seem similar: both mix the input with certain waveforms prior to sampling. We conduct a detailed technical comparison between the RD and MWC, revealing significant differences in terms of model robustness, hardware complexity and computational loads. Following the comparison, we attempt to reason the differences by the role CS ideas play in each approach. These insights serve as guidelines in building the signal acquisition part of the Xampling framework. A. Random Demodulator The RD approach views spectrally-sparse signals as consisting of a discrete set of tones on a uniform grid, which are sampled according to the system that is depicted in Fig. 3 [22].


Z

f (t)

5

t= t

n R

y[n]

1 t− R

pc (t) Seed

“CS block” Solve (9)

s

Spectrum of x(t) B

fp

Signal sythesis fˆ(t) (6)

Multitone model

0

∆

Pseudorandom ±1 generator at rate W

(Q−1)∆ 2

0

¯lfp

lfp

ci0 ˜l

ci˜l

cil

Fig. 3: Block diagram of the random demodulator [22].

fmax ci0 ¯l

ci¯l

Q∆ 2

f

˜lfp

+

Signal model. A multitone signal f (t) consists of a sparse combination of integral frequencies: X f (t) = aω ej2πωt , (6) ω∈Ω

where Ω is a finite set of K out of an even number Q of possible harmonics Ω ⊂ {0, ±∆, ±2∆, · · · , ±(0.5Q − 1)∆, 0.5Q∆} .

(7)

The model parameters are the tone spacing ∆, number of active tones K and grid length Q. The Nyquist rate is Q∆. Sampling. The input signal f (t) is mixed by a pseudorandom chirping sequence pc (t) which alternates at a rate of W . The mixed output is then integrated and dumped at a constant rate R, resulting in the sequence y[n], 1 ≤ n ≤ R. The development in [22] uses a normalized grid and the following parameter setup W ∈ Z. (8) R It was proven in [22] that if W/R is an integer and under the setting of (8), the vector of samples y = [y[1], . . . , y[R]]T can be written as ∆ = 1,

W = Q,

y = Φx,

R ∈ Z such that

x = Fs,

ksk0 ≤ K.

(9)

The matrix Φ has dimensions R×W , effectively capturing the mechanism of integration over W/R Nyquist intervals, where the polarity of the input is flipped on each interval according to the chirping function pc (t). See Fig. 7(a) in the sequel for the exact entries of Φ. The W -squared discrete Fourier transform (DFT) matrix F accounts for the sparsity in the frequency domain. The vector s has Q entries sω which are up to a constant scaling from the corresponding tones amplitude aω . Since the signal has only K active tones, ksk0 ≤ K, where the `0 -norm counts the number of nonzero entries. Reconstruction. The unknown in (9) is s. Observe that y = ΦFs does not determine s by itself, since ΦF is underdetermined, i.e., has less rows than columns, R < W . An underdetermined system has a nontrivial null space and infinitely many solutions in general. Among these solutions, (9) requires the one with ksk0 ≤ K. This type of problem has received extensive treatment in the CS literature, where Φ is referred to as the sensing matrix and F is termed the sparsifying basis. Under mild conditions on ΦF, (9) has a unique sparse solution s [20], [21], [28]. Whilst finding a sparse solution is NP-hard in general, several polynomialtime CS techniques are known to coincide with the true s under certain conditions on ΦF. Example techniques include

ci0 l +

Spectrum of yi [n]

Spectrum of yi0 [n]

Fig. 5: Spectrum slices of x(t) are overlayed in the spectrum of the output sequences yi [n]. Since 1/T ≥ B, eq. (12), a single band occupies at most 2 adjacent spectrum slices. In the example, channels i and i0 realize different linear combinations of the spectrum slices centered around lfp , ¯ lfp , ˜ lfp . For simplicity, the aliasing of the negative frequencies is not drawn.

`1 minimization, a.k.a., basis pursuit [29], and greedy-type algorithms; cf. [30]. In Fig. 3, the CS block refers to solving (9) with a polynomial-time CS algorithm. Correct recovery requires a sampling rate on the order of [22] R ≈ 1.7K log(W/K + 1).

(10)

Once the sparse s is found, the amplitudes aω are determined from sω by constant scaling, and the output fˆ(t) is synthesized according to (6). B. Modulated Wideband Converter Signal model. The MWC system samples multiband signals x(t) by treating the analog formulation directly without a finite discretization as in (6). A multiband signal x(t) has sparse spectra in the CTFT, so that X(f ) is supported on N frequency bands, with individual widths not exceeding B Hz. The bands positions are anywhere below fmax . Fig. 5 illustrates a typical multiband spectra. Sampling. The MWC samples multiband signals according to the scheme of Fig. 4, which consists of an analog frontend with m channels. In the ith channel, the input signal x(t) is multiplied by a periodic waveform pi (t) with period T , lowpass filtered by h(t) with cutoff 1/2T , and then sampled at rate fs = 1/T . The mixing operation scrambles the spectrum of x(t), such that a portion of the energy of all bands appears in baseband. Specifically, since pi (t) is periodic, it has a Fourier expansion pi (t) =

∞ X

2π

cil ej T lt .

(11)

l=−∞

In the frequency domain, mixing by pi (t) is tantamount to convolution between X(f ) and the CTFT of pi (t). The latter is a weighted Dirac-comb, with Dirac locations on f = lfp and weights cil , where fp = 1/T . Thus, conceptually, the spectrum

6


p1 (t) y1 [n]

h(t)

Continuous to finite (CTF) block Frame construction eq. (16)-(17)

T -periodic pi (t) Lowpass x(t)

t = nT

V

Detect active slices λ “CS block” Solve MMV (18)

Compute pseudo-inverse eq. (20)

yi [n] 1/Ts

y[n] Foreach l ∈ λ :

pm (t) ym [n]

h(t) RF front-end

Slices recovery eq. (21) real time

e−j2πlt/T

z[n] h(t)

zl (t)

X

x ˆ(t)

Lowrate ADC (Low analog bandwidth)

Fig. 4: Block diagram of the modulated wideband converter. The input pass through m parallel branches, where it is mixed with a set of periodic functions pi (t), lowpass filtered and sampled at a low rate. In the digital domain, the CTF block detects the active spectrum slices from a frame of consecutive vector samples y[n]. The active sequences zl [n], l ∈ λ are recovered at realtime and x ˆ(t) is reconstructed accordingly.

is divided into slices of width 1/T , and a weighted-sum of these slices is shifted to the origin [7]. The lowpass filter h(t) transfers only the narrow band frequencies up to fs /2 from that mixture to the output sequence yi [n]. The aliased output is illustrated in Fig. 5. Whilst aliasing is often considered as an undesired effect, here it is deliberately utilized to shift various frequency regions to baseband, simultaneously. The basic parameter setting is [7] 1 ≥ B. (12) T An advanced configuration enables to collapse the number of branches m by a factor of q at the expense of increasing the sampling rate of each channel by the same factor, so that fs = qfp but the overall sampling rate mfs is unchanged. In principle, the advanced configuration allows to collapse the MWC system to a single sampling branch using q = m. The choice of periodic functions pi (t) is flexible: The highest Dirac frequency needs to exceed fmax , so that spectrum slices from the entire Nyquist-range of x(t) are aliased to the origin. Mathematically, this requires nonzero coefficients cil for all −L ≤ l ≤ L, such that L satisfies L/T ≥ fmax . In principle, any periodic function with high-speed transitions within the period T can satisfy this requirement. One possible choice for pi (t) is a sign-alternating function, with M = 2L + 1 sign intervals within the period T [7]. Imperfect sign alternations are allowed, as long as periodicity is maintained [31]. The waveforms pi (t) need low mutual correlation in order to capture different mixtures of the spectrum. Popular binary patterns, e.g., the Gold or Kasami sequences, are especially suitable for the MWC [32]. Reconstruction. Denote by zl [n] the sequence that would have been obtained if the signal was mixed by a pure sinusoid ej2πlt/T and lowpass filtered. In other words, zl [n] are the samples of the lth spectrum slice. Since the system is linear, modulating by pi (t) and lowpass filtering is equivalent to the m ≥ 4N,

f p = fs =

weighted sum yi [n] =

L X

cil zl [n].

(13)

l=−L

Note that for l > L, zl [n] = 0, since it exceeds fmax . In matrix form, the vector of samples y[n] = [y1 [n], . . . , ym [n]]T obtained at time instant t = nT satisfies the underdetermined system y[n] = Cz[n], kz[n]k0 ≤ 2N, (14) with C an m × M matrix whose entries are cil , and z[n] = [z−L [n], . . . , z0 [n], . . . , zL [n]]T . The 2N -sparsity of z[n] stems from (12), which ensures that the signal energy occupies at most 2N slices of the spectrum [7]. Fig. 5 demonstrates this property. In principle, we can solve for the sparsest solution z[n] of (14), for every n, and then reconstruct x(t) by properly re-positioning the slices on the spectrum. A more efficient approach exploits the fact that z[n] are jointly sparse over time, so that the index set λ = {l : zl [n] 6= 0} ,

(15)

is independent of the time instant n. Therefore, λ can be estimated from several consecutive samples, which increases the robustness of the estimate. In Xampling terminology, λ determines the input subspace x(t) ∈ Aλ , where the possible subspaces are defined according to their spectral support at resolution 1/T . For brevity, we omit the superscript of λ∗ in (15) and in the sequel. The continuous to finite (CTF) block, which is depicted in Fig. 4, aims at a robust detection of λ. It builds a frame (or a basis) from the measurements using X Frame construct y[n] −−−−−−−−−→ Q = y[n]yH [n] (16) n

Frame decompose

−−−−−−−−−−−→ Q = VVH ,

(17)


7

Once λ is found, (14) reduces to y[n] = Cλ z[n], with Cλ being the appropriate column subset of C. The last CTF step computes the pseudo-inverse

−50 0.32

0.34

0.36

0.38

0.4

−50 0.32

Time (sec)

0.36

0.38

0.4

Time (sec)

(a)

(b)

Perfect sync. (∆=1)

5 ppm deviation (∆=1.000005)

5

5

4

4

3

2

1

0 −500

0.34

3

2

1

0

500

0 −500

Frequency (Hz)

0

500

Frequency (Hz)

(c)

(d)

Fig. 6: Effects of non-integral tones on the output of the random demodulator. The top (bottom) panels plot the recovered signal in the time (frequency) domain.

(20)

The reconstruction from that point on is carried out in real time; one matrix-vector multiplication per incoming vector of samples y[n] recovers zλ [n] = C†λ y[n],

Amplitude

(18)

The notation kUk0 counts the number of nonidenticallyzero rows. In CS, (18) is termed a multiple measurement vectors (MMV) system. Polynomial-time MMV solvers were developed in [6], [30], [33]–[38]. The crux of the CTF is that the indices of the nonidenticallyzero rows of U, that solves the finite underdetermined system (18), coincide with the index set λ that is associated with the continuous signal x(t). This property is proved in [7]. The CS-block in Fig. 4 refers to recovery with a polynomial-time MMV solver. The required sampling rate is on the order of [7] mfs ≈ 4N B log(M/2N + 1). (19)

−1 H C†λ = (CH Cλ . λ Cλ )

0

Amplitude (×104 )

kUk0 ≤ 2N.

5 ppm deviation (∆=1.000005) 50

0

Amplitude (×104 )

V = CU,

Perfect sync. (∆=1) 50

Amplitude

where (·)H denotes hermitian conjugate. Typically, Q is constructed from 2N time snapshots y[n], and the (optional) decomposition allows removal of the noise space [7]. Then, the CTF solves the following underdetermined system, which is independent of n:

(21)

with zλ [n] denoting the entries of z[n] indicated by λ. Standard DAC techniques reconstruct x ˆ(t) via lowpass interpolation of zl [n], l ∈ λ and modulation to the proper positions on the spectrum. At first sight, the RD and MWC technologies seem similar, at least in their sampling stages, which involve mixing followed by either integration in Fig. 3 or lowpass filtering in Fig. 4. The difference is in the details which we study below. We focus the comparison on model sensitivity, hardware complexity and computational loads. The study of these differences will provide guidelines for the design of the X-ADC block of the Xampling framework. C. Comparison – Model Sensitivity The multitone model is sensitive to the exact choice of tones grid. To see this, we repeat the developments of [22] for an unnormalized multitone model, so that ∆ is a free parameter, and consider W, R which are not necessarily integers. The measurements still obey the underdetermined system (9) as before, where now W ∈ Z, (22) W = Q∆, R = NR ∆, R and NR is the number of samples taken by the RD. We refer to the technical report [39] for the exact derivation of (22). The equalities in (22) imply that the rates W, R need to be perfectly synchronized with the tones spacing ∆. If (22) does not hold, either due to hardware imperfections so that the

rates W, R deviate from their nominal values, or due to model mismatch so that the actual spacing ∆ is different than what was assumed, then the reconstruction error grows large. The following toy-example demonstrates this sensitivity. Let W = 1000, R = 100 Hz, with ∆ = 1 Hz. Construct f (t) by drawing K = 30 locations uniformly at random on the tones grid and normally-distributed amplitudes aω . In our simulation, basis pursuit gave exact recovery fˆ(t) = f (t) for ∆ = 1. For 5 part-per-million (ppm) deviation in ∆ the squared-error reached ∆ = 1 + 5 · 10−6

→

kf (t) − fˆ(t)k2 = 37%. kf (t)k2

(23)

Fig. 6 plots f (t) and fˆ(t) in time and frequency, revealing many spurious tones due to the model mismatch. The equality W = Q in the normalized setup (8) hints at the required synchronization, though the dependency on the tones spacing is implicit since ∆ = 1. With ∆ 6= 1, this issue appears explicitly. Since the publication of the technical report [39], this problem was studied in [40] and [41], where it is referred to as nonintegral harmonics or sensitivity to basis mismatch, respectively. In comparison, the MWC setup (12) requires no synchronization with the multiband parameters; only inequalities are used in (12). The bands position are not restricted to any specific displacement with respect to the spectrum slices. Furthermore, a single band can split between slices, as depicted in Fig. 5. D. Comparison – Hardware Complexity To study the difference in the seemingly-similar sampling stages, we compare the complexities of the analog projection and continuous reconstruction. In addition, we quantify the sampling rate that is required in practice.

8


Nyquist-equivalent of MWC Nyquist-equivalent of RD

z1 [n]

h(t) x[1] t= f (t)

Z

t

Apply sensing matrix Φ

n W

1···1 1 · · · 1.

1 t− W

x[Q]

H

W R

..

1···1

±1

..

y[1]

exp(j2πlt/T ) x(t)

. ±1

y[NR ]

t = nT h(t)

zl [n]

Apply sensing matrix C = [cil ]

D

h(t)

Nyquist-rate (not sparse)

y[1]

y[m]

zM [n] Nyquist-rate (2N -sparse)

(a)

(b)

Fig. 7: The Nyquist-equivalents of the RD (a) and MWC (b) sample the input at its Nyquist rate and apply the sensing matrix digitally.

Figure 7 shall assists us in this discussion. The figure depicts the Nyquist-equivalent of each method, which is the system that samples the input at its Nyquist rate and then computes the relevant sub-Nyquist samples by applying the sensing matrix digitally. The RD-equivalent integrates and dumps the input at rate W , and then applies Φ on Q serial measurements, x = [x[1], · · · , x[Q]]T . To coincide with the sub-Nyquist samples of Fig. 3, Φ = HD is used, where D is diagonal with ±1 entries, according to the values pc (t) takes on t = n/W , and H sums over W/R entries [22]; see the figure. The MWCequivalent has M channels, with the lth channel demodulating the relevant spectrum slice to the origin and sampling at rate 1/T , which results in zl [n]. The sensing matrix C is applied on z[n]. While sampling according to the equivalent systems of Fig. 7 is a clear waste of resources, it enables us to view the internal workings of each strategy. Note that the reconstruction algorithms remain the same; it does not matter whether the samples were actually obtained at a sub-Nyquist rate, according to Figs. 3 or 4, or if they were computed after sampling according to the boxes in Fig. 7. Analog projection. The complexity of an analog design stems from the accuracy that the hardware needs to satisfy. In the RD approach time-domain properties of the hardware dictate the necessary accuracy. For example, the impulseresponse of the integrator needs to be a square waveform with a width of 1/R seconds, so that H has exactly W/R consecutive 1’s in each row. For a diagonal D, the sign alternations of pc (t) need to be sharply aligned on 1/W time intervals. If either of these properties is nonideal, then the sensing matrix becomes nonlinear and signal dependent. Precisely, (9) becomes [22] y = H(x)D(x)x.

(24)

A noninteger ratio W/R affects both H and D [22]. Since f (t) is unknown, x, H(x) and D(x) are also unknown. Consequently, the reconstruction becomes ill-posed. It is suggested in [22] to train the system on example signals, so as to approximate a linear system. Note that if (22) is not satisfied, the DFT expansion also becomes nonlinear and signal-dependent x = F(∆)s. The form factor of the RD is

the time-domain accuracy that can be achieved in practice. The MWC requires periodicity of the waveforms pi (t) and lowpass response for h(t), which are both frequency-domain properties. The sensing matrix C is constant as long as pi (t) are periodic, regardless of the time-domain appearance of these waveforms. Nonideal time-domain properties have therefore no effect on the MWC. The lowpass filter h(t) can be designed using standard RF techniques. Ripples and non-smooth transitions in the frequency response are compensated digitally [42]. The consequence is that stability in the frequency domain dictates the form factor of the MWC. RF design techniques excel in the wideband range. For example, 2 GHz periodic functions were demonstrated in a circuit prototype of the MWC [31]. More broadly, circuit publications report the design of high-speed sequence generators up to 23 and even 80 GHz speeds [43], [44], where stable frequency properties are verified experimentally. Accurate time-domain appearance is not considered a design factor in [43], [44], and is in fact not maintained in practice as shown in [31], [43], [44]. Sampling rate. Minimizing the sampling rate is the most immediate implication of the Xampling requirement (5) for minimal use of resources. In theory, both the RD and MWC approach the minimal rate for their model. The log factors in (10) and (19) stem from the use of polynomial-time CS algorithms, instead of searching for the sparse solution bruteforce. In practice, these factors are typically small. The RD system, however, requires in addition an integer ratio W/R; see (8) and (22). In general, a substantial rate increase may be needed to meet this requirement. The MWC does not limit the rate granularity. Continuous reconstruction. The RD synthesizes fˆ(t) using (6). Realizing (6) in hardware can be excessive, since it requires K oscillators, one per each active tone. Computing (6) digitally needs a processing rate of W , and then a DAC device at the same rate. Thus, the synthesis complexity scales with the Nyquist rate. The MWC reconstructs x ˆ(t) using commercial DAC devices, running at the low rate fs = 1/T . It needs 2N branches. In Section V, we improve this result by a factor of 2, for multiband inputs that carry narrowband communications. To compare the complexities, we note that,


9

TABLE I: Model and Hardware Comparison

relevant pseudo-inverse: (21) for the MWC or sΩ = (ΦF)†Ω y for the RD, where Ω indicates the active tones, cf. (7). RD (multitone) MWC (multiband) As before, the size of Φ results in long delay and huge Model parameters K, Q, ∆ N, B, fmax memory length for collecting the samples. The number of System parameters R, W, NR m, 1/T scalar multiplications (Mult.-ops.) for applying the pseudoSetup (8) (12) inverse reveals again orders of magnitude differences. We Sensitive, eq. (22), Fig. 6 Robust expressed the Mult.-ops. per block of samples, and in addition Form factor time-domain appearance frequency-domain stability scaled them to operations per clock cycle of a 100 MHz DSP Requirements accurate 1/R integration periodic pi (t) processor. sharp alternations pc (t) We conclude the table with our estimation of the technology ADC topology integrate-and-dump commercial barrier of each approach. Computational loads and memory Rate (theory) approach minimal (10) approach minimal (19) requirements in the digital domain are the bottleneck of the (practice) gap due to (8) RD approach. Therefore the size of CS problems that can DAC 1 device at rate W N devices at rate fs be solved with available processors limits the recovery. We estimate that W ≈ 1 MHz may be already quite demanding wideband continuous inputs require prohibitively large K, W using convex solvers, whereas W ≈ 10 MHz is probably the 1 to be adequately represented on a discrete grid of tones. barrier using greedy methods . The MWC is limited by the In contrast, despite of the infinitely many frequencies that technology for generating the periodic waveforms pi (t), which comprise a multiband input, N is typically small. Table I depends on the specific choice of waveform. The estimated barrier of 23 GHz refers to the potential according to [43], summarizes the model and hardware comparison. [44]. Our barrier estimates are roughly consistent with the hardware publications of these systems: [46], [47] report E. Comparison – Computational Loads the implementation of (single, parallel) RD for Nyquist-rate To compare between the computational loads of the RD W = 800 kHz. An MWC prototype demonstrates faithful and MWC, we consider an example. Following a suggestion reconstruction of fNYQ = 2 GHz wideband inputs [31]. in [22], we treat multiband signals in the RD framework by discretizing the continuous frequency axis to a grid of Q = F. Discussion: Compressed Sensing of Analog Signals fNYQ tones, out of which only K = N B are active. Table II The influential works by Donoho [20] and Candès et al. [21] compares between the RD and MWC for an input with 10 coined the CS terminology, in which the goal is to reduce GHz Nyquist rate and 300 MHz spectral occupancy. For the the sampling rate below Nyquist. These pioneering works RD we consider two discretization configurations, ∆ = 1 Hz established CS via a study of underdetermined systems, where and ∆ = 100 Hz. The table reveals high computational loads the sensing matrix abstractly replaces the role of the sampling that stem from the discretization step. We also included the operator, and the ambient dimensions allegedly stand for the sampling rate and DAC speeds to complement the previous high Nyquist rate. In practice, however, underdetermined syssection. The notation in the table is self-explanatory, though tems let alone do not hint at the actual sub-Nyquist sampling we would like to emphasize a few aspects. of analog signals. One cannot apply a sensing matrix on a set The sensing matrix Φ = HD of the RD has dimensions of Nyquist rate samples, as performed in Fig. 7, since that Φ : R × W ∝ K × Q (huge).

(25)

The dimension scales with the Nyquist rate; already for Q = 1 MHz Nyquist-rate input, there are 1 million unknowns in (9). The sensing matrix C of the MWC has dimensions fNYQ (small). (26) B For the comparable spectral occupancy we consider, Φ has dimensions that are 6 to 8 orders of magnitude higher, in both the row and column dimensions, than the MWC sensing matrix C. The size of the sensing matrix is a prominent factor since it affects many digital complexities: the delay and memory length that are associated with collecting the measurements, the number of multiplications when applying the sensing matrix on a vector and the storage requirement of the matrix. See the table for a numerical comparison of these factors. We also compare the reconstruction complexity, in the more simple scenario that the support is fixed. In this setting, the recovery is merely a matrix-vector multiplication with the C: m×M ∝ N ×

would contradict the whole idea of reducing the sampling rate to begin with. The previous sections demonstrate how two extensions of CS to continuous signals can be significantly different in many practical aspects. In an attempt to reason the encountered differences, we would like to draw several operative conclusions from our study: 1. sparsify the input prior in the analog domain prior to sensing, 2. avoid finite discretization of analog signals, 3. match the union projection to the technology that generates the input signals, and 4. decouple CS algorithms from actual signal reconstruction. In our opinion, the first point, analog sparsifying, is a central pilar in extending CS to analog signals. It is tightly related to finite discretization as we explain below. Refer to the Nyquist-equivalent systems in Fig. 7. The MWC translates x(t) into a high-rate sparse vector, z[n] with 1 A bank of RD channels was studied in [45]. The parallel system duplicates the analog issues and its computational complexity is not improved by much.

10


TABLE II: Discretization Impact on Computational Loads

Model

Sampling setup

Underdetermined system Preparation Collect samples Delay Complexity Matrix dimensions Apply matrix] Storage] Realtime (fixed support) Memory length Delay Mult.-ops. (per window) (100 MHz cycle) Reconstruction Technology barrier (estimated) §

Discretization spacing K tones out of Q tones alternation speed W

RD ∆ = 1 Hz 300 · 106 10 · 109 10 GHz

MWC ∆ = 100 Hz 3 · 106 10 · 107 10 GHz

rate R, eq. (10), theory 2.9 GHz eq. (8), practice 5 GHz (9): y = HDFs, ksk0 ≤ K Num. of samples NR NR /R

5 · 109 1 sec

2.9 GHz 5 GHz

5 · 107 10msec

Φ = HDF = NR × Q 5 · 109 × 1010 O(W log W ) O(W ) sΩ = (ΦF)†Ω y

5 · 107 × 108

NR 5 · 109 NR /R 1 sec KNR 1.5 · 1018 KNR /((NR /R) · 100M) 1.5 · 1010 1 DAC at rate W = 10 GHz CS algorithms (∼10 MHz)

5 · 107 10msec 1.5 · 1014 1.5 · 106

with q = 1; in practice, hardware size is collapsed with q > 1 [31].

at most 2N nonzeros, and then applies the sensing matrix C to reduce the rate. In contrast, the RD generates the Nyquist-rate vector x prior to sensing, but x is not sparse. Consequently, in order to apply CS algorithms in the digital domain, the RD needs to incorporate a basis under which x is sparse. The DFT matrix F is chosen for that purpose, at the expense of requiring the input to have finite parametrization to begin with; cf. (6). In the MWC approach, since z[n] was made sparse by the hardware, a discrete sparsifying basis is not needed and undesired effects of finite discretization are avoided. Generalizing this point, we suggest that when realizing a sensing approach in hardware, the system needs to have an equivalent representation that consists of two stages: one which sparsifies the input, namely translates x(t) to a sparse vector, and another which applies the sensing matrix. Once the hardware accounts for the analog sparsity, we avoid the delicate match between discrete and continuous transforms, which requires finite discretization of the input and leads to sensitivity issues. This explains how the first two conclusions above connect together, hinting that analog sparsifying hardware is a fundamental guideline for extending CS to analog signals. Moving on, based on the hardware comparison, we suggest that the union projection P in Xampling applications should use the same technology that is used to construct the input. Spectrally-sparse signals, as the name hints, are described by frequency-domain properties. Thus, an RF-based projection can be used to capture a wideband range of this type of signals. Accurate time-domain properties cannot be met for high RF rates. The underlying idea is that when using the same technology there are no essential limitations on the input range; the potential input range of the MWC scales with any

N bands width B m channels§ M Fourier coefficients fs per channel total rate (18): V = CU, kUk0 2N snapshots of y[n] 2N/fs

6 50 MHz 35 195 51 MHz 1.8 GHz ≤ 2N

12 · 35 = 420 235nsec

C=m×M 35 × 195 O(mM ) O(mM ) (21): zλ [n] = C†λ y[n] 1 snapshot of y[n] 35 1/fs 19.5nsec 2N m 420 2N mfs /100M 214 N = 6 DACs at individual rates fs = 51 MHz Waveform generator (∼23 GHz)

] for the RD, taking into account the structure HDF.

advance in RF technology, which also defines the possible range of multiband transmissions de-facto. Finally, we point out the role CS has in the RD and MWC schemes. In the former, CS is used to directly reconstruct the coefficient vector s, and thus the complexity scales with the Nyquist rate of the input. The latter decouples CS from the actual reconstruction. CS is applied on (18) for the sole purpose of detecting the index set λ; the values of the sparse solution U are not used throughout. This implies that CS should take the role of subspace detection, as is outlined in the Xampling architecture of Fig. 1. Sparse reconstruction is an NP-hard problem. Even if approximated by polynomial-time methods, the runtime can be quite considerable. Isolating CS to one instance of subspace detection, e.g., CTF, therefore avoids repeated executions of the CS block. G. Discussion: CS Universality The previous discussion on sparsifying basis sheds the notion of CS universality in new light. Indeed, the idea that the hardware incorporates knowledge on the sparsity basis may seem contradicting the popular wisdom of this concept. Discrete CS is often introduced as sensing y = Φx and sparsity basis x = Ψs, where universality refers to the attractive property of sensing with Φ without knowledge of Ψ, so that Ψ enters only in the reconstruction algorithm. Consequently, one can get the (wrong) impression that the sparsity basis has effectively no impact on the physical sampler. This impression is further emphasized with the default choice of the identity basis Ψ = I in many CS publications, which is justified by no loss of generality, since Ψ is conceptually absorbed into the sensing matrix Φ. We propose to revisit these assumptions when it comes to realizing the sensing in hardware.


To set things up, we slightly furnish the meaning of universality: once the input is sparsified, there is freedom in choosing the sensing matrix. For example, for multiband signals, the sparsifying part generates z[n] from x(t). The MWC uses the coefficients matrix C for sensing. In our previous work [19], which preceded the MWC [7], we used periodic nonuniform sampling (PNS) to sense the same z[n], where instead of cil the combination coefficients depended on the time shifts of the PNS sequences. The CTF reconstruction holds similarly in both cases. In principle, any other sensing matrix can be used. Nonetheless, since, in practice, sampling according to the Nyquist-equivalent system is wasteful, the trick is choose a sensing matrix which can be combined with the sparsifying part, so that the hardware does not go actually thru Nyquistrate sampling of the input. The MWC achieves this goal via periodic mixing, while PNS uses time-delays to obtain a similar effect. This is the meaning of universality we propose in Xampling. In the sequel, we describe the method of [14], which also combines a sparsifying stage with a sensing matrix. Note that the MWC has an advantage over PNS, since the Achilles heel of the latter is the pointwise acquisition of the wideband input, which necessitates an ADC with bandwidth b ≈ fNYQ [7]. Thus, the theoretical rate reduction that PNS promises does not translate to significant hardware savings in practice. We used the examples of RD and MWC to study the aspects of signal acquisition and continuous reconstruction. This brings us to the contribution of the next section, which addresses the DSP challenge in sub-Nyquist systems. V. X-DSP: S UB -N YQUIST S IGNAL P ROCESSING Popular DSP algorithms assume an input stream at the Nyquist rate. A fundamental reason for processing at the Nyquist rate is the clear relation between the spectrum of x(t) and that of x(nT ), so that digital operations can be easily substituted for their continuous counterparts. Digital filtering is an example where this relation is successfully exploited. Since the power spectral densities of continuous and discrete random processes are associated in a similar manner, estimation and detection of parameters of analog signals can be performed by DSP. When sampling below Nyquist, this key relation no longer holds in general. In this section, we study how to perform DSP in sub-Nyquist systems. We develop an algorithm that provides backwardcompatible DSP for the MWC. Towards the end of this section, we compare our algorithm with the approach of [24].

11

the width of a spectrum slice is equal to the (maximal) width of an individual band. A multiband signal can be described in a quadrature representation as [48]: N/2

x(t) =

X

Ii (t) cos(2πfi t) + Qi (t) sin(2πfi t),

(27)

i=1

where Ii (t), Qi (t) are real-valued narrowband signals, and fi are a relatively high carrier frequencies. Classic communication methods obey (27), including analog amplitude, phase- and frequency-modulation (AM/PM/FM). Modern digital communication transmit bits using techniques, such as frequency- and phase-shift keying (FSK/PSK), which also conform with (27). In these communication techniques, the message of interest is encoded in Ii (t), Qi (t), which is therefore referred to as the information signals. The carrier fi itself does not contain signal information. Baseband processing means the ability to obtain samples of Ii (t), Qi (t), using computational complexity that is proportional to N B. In particular, it implies that interpolation to the Nyquist rate fNYQ is forbidden. The MWC reconstructs the contents of the active spectrum slices, zl [n], l ∈ λ. Whilst zl [n] determines x(t), the information signals Ii (t), Qi (t) are not organized in zl [n] as standard DSP algorithms expect to receive. For example: in Fig. 5, the energy of the ith band splits between two consecutive sequences zl−1 [n], zl [n]. A single slice may, in general, contain several information bands. Moreover, even when zl [n] contains a single band, conventional software does not accommodate the lack of a nominal value for the carrier fi . The fact that fi is somewhere within a slice width, e.g., a range of 1/T = 51 MHz in the example of Table II, does not help, since standard software packages can tolerate only slight offsets in the nominal fi ; those that presumably occur due to slight frequency shifts between the transmitter and receiver oscillators. The algorithm we develop below addresses all these aspects. It outputs an accurate estimate of fi and samples of the pair Ii (t), Qi (t), per each band 1 ≤ i ≤ N/2. For the development, we need to assume that I(t), Q(t) are random with zero cross-correlation, E[I(t1 )Q(t2 )] = 0 for all t1 , t2 . In practice, it means that I(t), Q(t) carry uncorrelated information messages. This holds for AM, by definition, and for many digital communication techniques, when using a preceding source coding stage [48]. The algorithm does not assume any specific modulation technique; the only essential assumption is the quadrature form (27) and zero cross-correlation between I(t), Q(t). We refer to the proposed algorithm as Back-DSP.

A. Baseband Processing We begin by defining the backward-compatibility we would like to gain for multiband signals. When the carrier frequencies fi of a multiband input are known, the receiver demodulates a bandpass subspace around a carrier frequency fi of interest to the origin, producing a baseband signal si (t), which is sampled at a low rate to a sequence si [n]. DSP takes place from that point. In the union settings, our goal is to reach the same sequences si [n], despite the lack of information on the carriers fi . For simplicity, in this section 1/T = B is assumed, so that

B. Algorithm Description The Back-DSP algorithm consists of three steps, which are depicted in Fig. 8: 1. Refinement of the coarse support estimate λ to the actual band edges [ai , bi ]. Here, we rely on two additional model parameters: the minimal width of a single band Bmin and the smallest spacing between bands ∆min . These quantities are often known in communication,

12


Algorithm 1: Backward-compatible DSP (Back-DSP) Operation Step 1: Band edges estimation [ai , bi ] 1.1 for each l ∈ λ(

Simulation

z−l [n]

I2,0.5B {z±l [n]} l 6= 0 xl [n] = z0 [n] l=0 1.2 for each l ∈ λ, l 6= 0 Estimate PSD of xl [n] Threshold → fine support estimate 1.3 unite too adjacent intervals, ≤ ∆min (combat noise) prune too narrow intervals, ≤ Bmin (false alarms) retain only N/2 “powerful” intervals (model assumption)

interpft

1.

2. z[n]

Fine support detection Band isolation

3. Digital balanced quadricorrelator

[ai , bi ] Ii [n] si [n]

fi

Standard DSP ˜ i [n] Qi [n] packages Q

B 2

0

−B

B

< Bmin < ∆min B

B

0

All-pass π

0

same slice band split 0

π

π

0 si (t)

High-pass

firpmord, firpm Ap = 10−6 , As = 10−2

0

xl [n]

content out-of-band

Band-pass

π 2

π xl+1 [n]

see text

cos(ωi t)

lowpass sin(ωi t)

+

lowpass

ˆ(t) P x

per band analog reconstruction

Fig. 8: Digital algorithm for baseband processing with the MWC.

though uncertainty in the values Bmin , ∆min has little effect on the performance, as described later on; 2. Generating si [n] per band 1 ≤ i ≤ N/2. This step processes zl [n] and incorporates the edges [ai , bi ]; and 3. Estimating fi using a digital version of the balanced quadricorrelator (BQ) [48]. The information signals Ii (t), Qi (t) are obtained upon completion at no additional cost. Algorithm 1 outlines the operations that are carried out in each step of Back-DSP. We specify the MATLAB commands (in verbatim style) that are used in our implementation. Visual illustrations are also presented in Algorithm 1. A software package of the Back-DSP algorithm is available online in [49]. Step 1. For convenience, the complex-valued zl [n] are converted to real-valued counterparts xl [n], taking into account the conjugate-symmetry of x(t). The sequence xl [n] is obtained by re-positioning zl [n], z−l [n] on both sides of the origin. Mathematically, xl [n] = I2,0.5B {z±l [n]}, where 4

0

− B2

0

Digital → Analog

I˜i [n]

B 2

Threshold

pwelch eq. (30)

Step 3: Carrier estimate fi , i = 1, . . . , N/2: 3.1 Upsample ↑ 3 and frequency-shift 3.2 Apply BQ, Fig. 9

λ

0

xl [n]

PSD estimate

Filter out-of-band contents → si [n]

Digital domain

− B2

zl [n]

Low-pass

Step 2: Isolate sequence si [n] per band, i = 1, . . . , N/2: 2.1 Stitch bands ( energy xl [n] s˜i [n] = I4,0.5B {z±l [n]} + I4,B {z±(l+1) [n]} 2.2

Illustration

Ir,F {z±l [n]}=(zl [n] ↑ r)e−j2πF n + (z−l [n] ↑ r)ej2πF n , (28) and ↑ r denotes rate increase by a factor of r, with the appropriate post-filtering. By abuse of notation, here and in

the sequel the same index n is used before and after the rate conversion, where the context resolves the ambiguity. The case l = 0 ∈ S, has x0 [n] = z0 [n]. To detect the band edges [ai , bi ], we estimate the power spectral density (PSD) of xl [n]. We used the Welch PSD estimation method [50], with a windows overlapping ratio of 50%. The window size is set with accordance to the frequency resolution 2B fres = min(Bmin , ∆min ), Wsize ≥ . (29) fres In (29), the shortest possible window is used. The PSD (l) estimation produces Pxx [k] for 1 ≤ k ≤ K ≈ Wsize /2. A logarithmic threshold log10 (Threshold) =

K 1 X (l) log10 Pxx [k], K

(30)

k=1

(l)

translates Pxx [k] to a binary decision on the energy concentration. To mitigate undesired noise effects; support regions that are closer than ∆min are united, and isolated regions with widths smaller than Bmin are pruned. Our final estimate of the band edges [ai , bi ] comprises the N/2 most powerful bands, according to the PSD values. Step 2. The purpose of this step is to obtain a sequence si [n] for each 1 ≤ i ≤ N/2, such that si [n] contains the entire contribution of exactly one band. Using the edges [ai , bi ] we identify the cases of band split, namely when the energy of si (t) resides in adjacent spectrum slices xl [n], xl+1 [n] for some 0 ≤ l ∈ λ; see Fig. 5 for example. In such cases, merging occurs via s˜i [n] = I4,0.5B {z±l [n]} + I4,B {z±(l+1) [n]},

(31)

otherwise, s˜i [n] = xl [n] for a frequency band [ai , bi ] that lie in a single spectrum slice. As a result, s˜i [n] contains the entire


LPF s(t)

13

vI (t)

d dt

cos(ω0 t)

+

sin(ω0 t) LPF

vQ (t)

vd (t)

d dt

Fig. 9: The analog balanced-quadricorrelator.

energy of the ith band, possibly with additional contributions due to other information bands. We use the estimated edges [ai , bi ] to filter s˜i [n] from the out of band contents. The illustrations in Algorithm 1 describes the type of filter that is used, which can be either low-, high-, band- or all-pass, depending on the locations of the other bands. The allowed ripples in the pass- and stop-bands are Ap , As , respectively. The filter order is often small, since the actual spacing between the bands relaxes the cutoff constraints. Step 3. The final step estimates the carriers fi . Rough estimates can be readily computed at the median frequencies (ai +bi )/2, though we observed that this estimate is inaccurate in noisy settings. To improve the estimate, we use the BQ whose circuit appears in Fig. 9 [48]. For brevity, the band index i is omitted. The BQ is an analog circuit that estimates the carrier frequency of a quadrature input s(t) that consists of a single pair of information signals. It is initialized with an angular frequency ω0 = 2πf0 and outputs vd (t) whose expected value is proportional to offset from the true carrier fc E[vd (t)] = −KG (fc − f0 )(E[I 2 (t)] + E[Q2 (t)]).

(32)

In practice, time averaging replaces the expectations. The constant KG in (32) captures the analog gains along the way of the mixers, filters and differentiators. In our algorithm, we implement a digital version of the BQ. We used FIR lowpass filters and approximated the continuous derivatives by the finite difference – a filter with the discrete impulse response [1, −1]. Note that a wide family of filters can substitute the true differentiators [48]. A fundamental requirement of the BQ, either in analog or digital, is that the first mixing yields non-overlapping copies of s(t) at ω0 ± ωc . To ensure this property, each si [n] is interpolated by a factor of three, and the positive and negative frequencies are re-positioned in angular positions [π/3, 2π/3], [−2π/3, −π/3], respectively. For example, when no merging occurs in Step 2.1, this computation boils down to I6,1.5B {z±l [n]} with the relevant l. The digital BQ is applied iteratively on the outcome, where ω0 = π/2 initializes the procedure, and the update rule at the end of each iteration is P vd [n] ω0new = ω0old + G P n . (33) 2 |s n i [n]| The loop gain G = 5 · 106 . The procedure monitors ω0 ∈ [π/3, 2π/3] and terminates upon convergence or if a predefined number of iterations is reached. Properties. Upon completion, the (samples of the) desired information signals Ii (t), Qi (t) of the ith band are instantly

available – the last BQ iteration computed them for the nodes vI (t), vQ (t) of Fig. 9. The rate of Ii [n], Qi [n] is either 6B or 12B, depending on the rate of si [n]. The recovered carrier fi and the detected band edges [ai , bi ] allow to reduce the rate of Ii [n], Qi [n] to minimum, i.e., 2(bi − ai ). Besides the information signals I(t), Q(t), the algorithm outputs additional useful information per band: the edges [ai , bi ], the isolated sequence si [n] and the carrier estimate fî . The latter is computed from the angular frequency ω0 that the BQ converged to as ω0 − π/3 , (34) fî = B l + c π/3 where c = 1 when merging was not required, and c = 2 otherwise. The carrier-frequency-offset (CFO) fî − fi is not expected to be zero, but rather to fall below the allowed tolerance of commercial standards, as if nominal fi value was specified. We report the actual CFO values in the next subsection. For applications in which the exact Bmin , ∆max are unknown, approximate values can be set. The uncertainty with respect to the true values may yield many possible support regions in steps (1.1)-(1.2). Nonetheless, the effect on the overall performance is minor, since only the N/2 powerful regions are selected in step 1.3. The exact band locations have only a negligible effect on the filter design in step 2.2. Furthermore, the BQ in step 3 is insensitive to inaccuracies in [ai , bi ]. Therefore, approximate values for Bmin , ∆max are sufficient in practice. We used Bmin = ∆max = B/8 in our simulations. As a nice feature, using the proposed algorithm, the original continuous reconstruction of Fig. 4 is replaced by a more efficient method that is depicted in Fig. 8. In [7], x(t) is reconstructed from zl [n] by interpolation and properly positioning the spectrum slices. Since the scenario of band splitting is fairly common, at most 2N spectrum slices may be active, hence the number of DAC and modulation branches. The present approach requires only N mixers and filters. In addition, note that once the information signals Ii (t), Qi (t) are obtained, error correction DSP algorithms can be employed to improve the overall robustness to noise. C. Simulations To evaluate the accuracy of the estimate fî , we simulated an example multiband model with N = 6, B = 50 MHz. Quadrature phase-shift P3 keying (QPSK) modulation was used to generate x(t) = i=1 xi (t) via s ! 2Ei X xi (t) = Ii [n]p(t − nTsym ) cos(2πfi t) (35) Tsym n ! X + Qi [n]p(t − nTsym ) sin(2πfi t) + n(t), n

where Ei = {1, 2, 3}, 1/Tsym = 30 MHz, p(t) = rcosine(t/Tsym ) are the symbol energy, rate and raisedcosine pulse shape with 30% rolloff. The carriers fi ∈ [0, 5] GHz, the bit streams Ii [n] = ±1, Qi [n] = ±1, and the additive

14


1

0.3 0.25

Percentage of simulations

Percentage of simultions per bin

0.35

99 % in 150 kHz

0.2 0.15 0.1 0.05 0 −1000

−500

0

500

1000

0.8 0.6

|CFO| within 0.4

200 kHz 100 kHz 70 kHz 30 kHz

0.2 0

0

5

10

15

CFO (kHz)

SNR [dB]

(a)

(b)

20

25

30

Fig. 10: The distribution of CFO for fixed SNR=10 dB (a). The curves (b) represent the percentage of simulations in which the CFO magnitude is within the specified range.

white Gaussian noise n(t) were all drawn independently at random. An MWC with the basic configuration (12) was used with m = 30 channels and sign alternating waveforms pi (t), M = 195 alternations points per period T . We assume the spectrum slices zl [n] were obtained successfully by the preceding stages of Fig. 4. For each one of 40 test signals, we executed the Back-DSP algorithm and measured the CFO fî − fi for i = 1, 2, 3. Fig. 10 reports the distribution of the CFOs encountered in our simulations for various signal-to-noise ratios (SNRs). Evidently, in most cases our algorithm approaches the true carriers as close as 150 kHz. For reference, the 40ppm CFO specifications of IEEE 802.11 standards tolerates this error for transmissions located around 3.75 GHz [51]. To verify data retrieval using the Back-DSP algorithm, we generated a single binary phase-shift keying (BPSK) transmission, such that the band energy splits between two adjacent spectrum slices. We executed the algorithm and used a Costasloop receiver [52] to extract the bits that were encoded in the BPSK transmission. We measured the bit error rate (BER), that is the number of erroneous bits at the output, in a Monte Carlo simulation. For each trial out of 2500, we redraw a carrier position that gives band split, and simulated 6000 bits passing through the analog sampler and the digital algorithms. We repeated the procedure for input SNRs of 3,5,7 and 9 dB. In total, about 15 million bits were simulated. Estimated BERs for 3 dB and 5 dB SNR, respectively, are better than 0.77·10−6 and 0.71 · 10−6 . No erroneous bits were detected for SNR of 7 and 9 dB. Lab experiments in [31] report correct continuous reconstruction of a mixture of AM and FM signals, whose energy overlays at baseband. D. Related Work The Back-DSP algorithm provides the MWC with a smooth interface to existing DSP packages. An interesting alternative which does not target at backward-compatibility is proposed in [24]. The approach, termed compressive signal processing (CSP), is studied for an underdetermined system y = Φx, where x stands for a set of Nyquist-rate samples. The CSP idea is to avoid recovery-then-processing of x (at the Nyquist rate), and instead to develop new processing methods that can be performed directly on y.

The approach relies on quantities that are invariant under the sensing operator Φ. For example, Euclidean distances are approximately preserved under an underdetermined mapping, if Φ satisfies certain embedding conditions [24]. In practice, however, CSP may result in high computational loads, since the stable embedding condition effectively requires Φ to behave as a near isometry on Nyquist-rate sparse vectors. For example, as Table II shows, processing of y requires computations on vector lengths of NR = 5·109 or NR = 5·107 entries, depending on the discretization spacing that is used to represent f (t), ∆ = 1 or ∆ = 100, respectively. Thus, while avoiding the reconstruction of x which has even higher dimensions, CSP does not lead to substantial savings in this case. More inherently, CSP calls for the development of a new toolbox of processing methods. The examples in [24] are limited to certain linear-type processing tasks, whose true values are approximated in the compressed domain. In addition, the embedding conditions implies that CSP performance depends on the exact values of the sensing matrix. Any hardware inaccuracy alters Φ and in turn propagates errors to the CSP algorithms. In contrast, Xampling proposes first to detect the signal subspace and then perform conventional subspace DSP. In essence, Xampling suggests that the DSP does not need to be aware of the source of its input. For example, for multiband transmissions, the data can either arrive from a demodulator that knows the carriers fi or from the MWC which does not incorporate such knowledge. In both cases, the information signals Ii (t), Qi (t) are given the same treatment. VI. E STABLISHING THE X AMPLING F RAMEWORK To establish Xampling, we now return to the general UoS model (1), in which x(t) ∈ L2 (R) belongs to some subspace Aλ∗ , but a-priori the exact subspace index λ∗ is unknown. We describe a variety of applications that can be derived from (1) and show that they fit elegantly to the architecture of Fig. 1. To further support Xampling, we present the line of reasoning that underlies the X-ADC and X-DSP blocks of Xampling, bearing in mind the insights collected from the previous sections. A. Xampling and UoS Applications Table III list a variety of UoS applications, of which we already discussed the MWC [7], PNS [19] and RD [22]. For each method, we specify the signal model and designate the cardinalities of the union family size Λ and the individual subspaces Aλ . Notice the common structure that is evident from the table – analog pre-processing and digital subspace detection. FRI sampling. The first works, by Vetterli et al. [23], [25], study periodic FRI signals, so that x(t) obeys (4) and in addition x(t) = x(t + T ) for all t. Sampling of FRI signals with finite duration, namely x(t) = 0 outside the interval 0 ≤ t ≤ T , is studied in [53]. In both scenarios, the analog bandwidth is reduced by lowpassing x(t) prior to sampling. These results are generalized in [10] who introduce a family of hardware filters, termed Sum-of-Sincs (SoS), for FRI acquisition. Certain SoS filters have finite impulse response which


15

TABLE III: Applications of union of subspaces Signal model multiband, Fig. 5

Application MWC [7] PNS [19]

multiband, Fig. 5 P f (t) = aω ej2πωt

RD [22]

Cardinality union subspaces finite ∞

Analog projection periodic mixing + lowpass

Subspace detection CTF

finite

∞

time shifts

CTF

finite

finite

sign flipping +

CS

ω

ω ∈ discrete grid Ω (4): x(t) =

FRI

L P `=1

integrate-dump

d` g(t − t` )

x(t) = x(t + T ) 0≤t≤T either of the above

∞ ∞ ∞

finite finite finite

lowpass lowpass SoS filtering

annihilating filter [23], [25] moments factoring [53] annihilating filter

Sequences of innovation [8], [9]

see (36)

∞

∞

lowpass, or periodic mixing + integrate-dump

MUSIC [54] or ESPRIT [55]

Sparse-SIS [14]

see (37)

finite

∞

filter-bank, Fig. 11

CTF

Structured UoS [6]

abstract

finite

finite

projection onto Riesz bases

CS

multiband

finite

∞

jittered undersampling (nonlinear)

n/a

periodic [23], [25] finite [53] periodic/finite [9], [10]

NYFR [56]

can simplify the hardware realization. SoS filters also stabilize the subspace detection and reduce computation loads [10]. Ultrasonic imaging was discussed as a possible application. A recent work [9] further extends FRI to signals of the form (4), where neither periodicity or finite duration is assumed. In [9], the delays t` ∈ R and their count L is effectively unlimited. The sampling rate in [9], [10] achieves the critical rate of innovations. Sequences of innovation. In [8], [9], the signal has the form L X X d` [n]g(t − nT − t` ). (36) x(t) = `=1 n

As mentioned, the setting [9] allows arbitrary, not necessarily periodic, time delays t` . Since the delays are continuous parameters, both the union size Λ and the individual subspaces are of infinite dimensions. Time-delay estimation in multi-path transmissions is a possible application, in which a receiver observes the channel output for T seconds and exploits the known shape of a short probing pulse g(t) to identify the L propagation paths of the medium. In contrast to traditional time-delay estimation methods which sample and process the intercepted signal at the Nyquist-rate of g(t), cf. [57], the method of [8] requires as low as 2L measurements per observation interval T for channel identification and signal reconstruction, regardless of the bandwidth of g(t). Sparse shift-invariant subspace (sparse-SIS). The model assumes that the signal is generated by shifts of k out of m Riesz generators a` (t) [14] x(t) =

m X X `=1 n

d` [n]a` (t − nT ),

(37)

where a-priori the indices ` of the chosen generators are unknown. This extends the spirit of classic sampling, where fully-SIS models are considered, namely (37) with all the m generators being active [12], [13], [15]. The filter-bank

depicted in Fig. 11 is used for signal acquisition. Closedform expressions for the filters s∗i (−t) are developed in [14] based on a given set of m filters h∗i (−t) that would work with k = m. Interestingly, the way s∗i (−t) are constructed in [14, Th. 2, eq. (41)] consists of a sparsifying term, which conceptually translates x(t) to the sparse dl [n] using the hi (t) filters, and then applying a sensing matrix to reduce the rate. This complies with the conclusions we listed earlier in Section IV-F. A follow-up publication [58] proves that the sampling filters need Nyquist-rate bandwidth. Mixed-signal design is shown beneficial in terms of reducing the front-end bandwidth [58]. Xampling has matured from these understandings and the general scheme of [14]. P

δ(t − nT )

d1 [n] x(t)

s∗1 (−t)

y1 [n]

a1 (t)

t = nT

x ˆ(t) CTF s∗p (−t)

yp [n] t = nT

P

δ(t − nT )

dm [n]

am (t)

Fig. 11: Sampling and reconstruction of sparse union of SIS [14].

Two methods that do not strictly fit the Xampling paradigm deserve attention. The publication [6] starts from an abstract model: a cloud of points x, structured in a UoS, within an arbitrary Hilbert space H. A closed-form expression for the analog projection P and a generic reconstruction algorithm are developed, provided that the individual subspaces Aλ obey some underlying structure [6]. The approach engages a promising direction towards a general UoS theory. Despite the somewhat limiting setup of a finite union and finite subspaces, the strategy hints at an important conclusion. To enable signal reconstruction in UoS settings, even in an abstract model,

16


Baseline assumption (A2)

Baseline assumption (A1)

The practical ADC model of Fig. 2 holds

Goal:

DSP is the prime goal of sampling

Pragmatic consideration:

Minimal use of resources (6)

Existing ADC technology is mature and optimized

+

+

+

Must reduce ambient dimensions

Choose commerical ADC

+

Fact:

Minimal use of resources (6)

Lowrate DSP

Standard DSP relies on spectral relations x(t) ↔ y[n]

+

Pragmatic consideration:

+

Fact:

Backward compatability

The inverse of union projection is nonlinear

Analog unit separated from the ADC Pragmatic consideration:

+

Goal:

Linear design is efficient (see text)

Linear projection

Fig. 12: Flowchart reasoning the X-ADC structure.

Detect signal subspace prior to DSP Pragmatic consideration:

+

Digital detection is efficient (see text)

Digital detection

Fig. 13: Flowchart reasoning the X-DSP structure.

some structure may be needed. This is consistent with the orthogonality of the spectrum slices in the MWC, the known pulse shape g(t) in FRI models and known Riesz generators a` (t) in the sparse-SIS model. Finally, we point out the Nyquist-folding receiver (NYFR) of [56] which suggests an interesting alternative route towards sub-Nyquist sampling. This method introduces a deliberate jitter in an undersampling grid, which results in induced phase modulations at baseband such that the modulation magnitudes depend on the unknown carrier positions. This strategy is exceptional as it relies on a nonlinear acquisition effect, which departs from the linear P that is proposed in Xampling and utilized in all previous applications. In principle, to enable recovery, one would need to infer the magnitudes of the phase modulations. We do not elaborate on [56] since a reconstruction algorithm was not reported yet, thought it is clear that it would be systematically different than subspace detection that is based on a linear projection P . B. Line of reasoning Figures 12 and 13 present flowcharts of the various considerations leading to the proposed Xampling framework. Since the ADC industry is mature and devices are well optimized with wide market availability, we suggest utilizing these devices for signal acquisition, while noting that the front-end is bandwidth limited as assumed in (A2). To comply with (5), the sampling rate must be reduced until it is proportional to the dimensions of the individual Aλ . This explains why P is needed. Since the bandwidth is not reduced when combining P with the sampler itself, as happens in PNS, P is realized as a separate unit. In Xampling, we prefer linear P for two reasons. First, in practice, a linear circuitry may be relatively less difficult to stabilize than a nonlinear circuit, since a linear device has usually small deviations around the working point. Second, from the theoretical perspective, analyzing a nonlinear sampling scheme can be involved [59]. For these reasons, we suggest in Xampling to realize linear

P as a preceding analog unit, separated from the ADC. Note that all the applications listed in Table III, except the NYFR, have linear P . The reason behind the X-DSP block is sketched in Fig. 13. We want lowrate DSP in order to decrease computational loads as (5) requires. The major consideration behind X-DSP is backward compatibility, which had an intuitive meaning in the context of multiband sampling – the same DSP algorithms that can be performed when the band positions are known, can also be used, effectively in the same manner, when the carrier frequencies are unknown. Extending the concept, suppose that there exists a toolbox of processing algorithms for the single subspace scenario, namely when sampling only Aλ∗ . DSP in the union refers to providing a comparable package of algorithms when the subspace index λ∗ is unknown. The literature provides a vast amount of DSP algorithms for various tasks, which were developed and debugged over the years for subspace scenarios. To utilize the existing rich variety of algorithms, we need to restore the clear relation between the spectrum of x(t) and that of y[n], which lies at the heart of conventional DSP methods, and which was lost when the front-end operator P reduced the input bandwidth. This is accomplished by the subspace detection stage of X-DSP. The discussion on CSP in Section V implies that without a preceding subspace detection, the processing options are quite limited, and may come at the expense of substantially high processing rates, against (5). Since in CSP the relation x(t) ↔ y[n] is nonlinear due to the nonlinearity of the union, the starting point for software development may be more complicated, compared to conventional DSP. Backwardcompatibility and convenient development are therefore the main reasons underlying X-DSP. Finally, note that Xampling suggests a digital detection step, rather than a hardware mechanism, that would work in parallel to signal acquisition to indicate the signal subspace.


17

Single subspace

Union of subspaces

Aλ2

A

Aλ3

x(t)

y[n]

θ

Sampling space

S

(a)

Aλ1

x(t)

S

In Xampling applications, the abstract choice of S materializes to the design of hardware elements: periodic functions pi (t) in the MWC, time delays in PNS, shaping of the sampling filter response in FRI methods, and a certain sensing matrix in sparse-SIS. In viewing the role S has in hardware, one can come up with intuitive selections, e.g., low crosscorrelation sequences for the MWC [32].

(b)

Fig. 14: Geometry of signal acquisition in (a) traditional subspace sampling, and in (b) union of subspaces modeling.

For example, in the multiband scenario, one could think of M power detectors which give binary indications for the presence of signal energy in each spectrum slice. Obviously the na¨ıve realization involves excessive hardware resources, against (5). In viewing the modern trend of shifting as much processing operations as possible to the DSP, we prefer digital detection. C. Union Projection Design To conclude the framework, we address the challenge in the preprocessing operator P of X-ADC. Since P is linear, sampling of P {x(t)} has the nice geometric interpretation that is depicted in Fig. 14. The choice of P is tantamount to choosing a subspace S to which x(t) is projected [1]. In classic sampling theorems, the signal is assumed to lie in a single subspace A, Fig. 14(a). The samples are inner products y[n] = hx(t), sn (t)i with a given set of functions sn (t), and the direct sum condition A⊥ ⊕ S = H,

(38)

is a prerequisite for (oblique) reconstruction [1]. Here S = span{sn (t)} and A⊥ is the complement of A in H = L2 (R). Roughly speaking, S should be chosen such that the angle θ it creates with the input subspace A is other than 90◦ . The recovery is more robust to noise as θ tends to zero. The trivial choice S = A results in θ = 0. The design of P in the union setting, or equivalently the choice of the sampling space S, is inherently more complicated, since there is no known oblique-like reconstruction operator. The existence of (stable) recovery requires constants 0 < α, β < ∞ such that [4], [5] αkx − ykH ≤kSx − Sykl2 ≤ βkx − ykH , ∀x, y ∈ Aλ1 + Aλ2 ,

(39)

λ1 , λ2 ∈ Λ.

We deduce from (39) that S needs to preserve the geometry of the union in the reduced dimensions, so that no two subspaces Aλ coincide in S after projection. The challenge in satisfying (39) is apparent in the example illustrated in Fig. 14(b); while the projection preserves the union geometry, the images of Aλ1 , Aλ2 are close to each other and can be perturbed in a noisy environment. In other words, in contrast to the single subspace scenario where choosing S requires minimization over the scalar θ, the UoS setting requires to simultaneously minimize the angle between S and each of the subspaces Aλ , and at the same time maximize the angle between each pair of images of Aλ .

VII. C ONCLUDING REMARKS Union of subspaces models appear nowadays at the research frontier in sampling theory. The ultimate goal is to build a complete sampling theory for UoS models of the general form (1) and then derive specific sampling solutions for applications of interest. Although several promising advances have already been made [4]–[6], [14], this esteemed goal is yet to be accomplished. In this paper, we proposed a unified framework for treating UoS signals, which we built bottom-up, by leveraging insights and pragmatic considerations from the applications layer to the high-level architecture of Fig. 1. The discussion on spectrallysparse signals and the applications listed in Table III show that Xampling is broad enough to capture a multitude of engineering solutions, under the same logical flow of operations. The core contributions which assisted in developing Xampling are two. First, we examined signal acquisition thru the extension of CS to continuous signals, as utilized in the RD and MWC approaches. Our technical comparison showed that the somewhat visual resemblance can be quite misleading. Major differences were found in three practical aspects: model sensitivity, hardware complexity and computational loads, where the MWC was found to outperform the RD in all three parameters. Numerical examples demonstrated the consequences of analog discretization; severe computational loads and model sensitivities. Based on this study, we have drawn several guidelines for Xampling applications, and in particular to the extension of CS to analog signals. The most important conclusion is to design the analog hardware so that it combines two goals: sparsifying the input and sensing. The development of the Back-DSP algorithm shed light on lowrate signal processing in union models. It also provided another viewpoint on the difference between the RD and MWC systems. While the MWC interfaces smoothly with standard DSP packages, the RD approach needs either reconstructing the input at the Nyquist rate or inventing a new toolbox of compressive processing methods. We foresee three roles for Xampling in the study of UoS theory and applications. First, it can hint on potential improvements and warn from design caveats in existing applications, as we have shown in Sections IV and V. Second, the template scheme of Fig. 1 can serve as a substrate for developing future sampling strategies for UoS models. Third, our hope is that these ideas will inspire future developments and eventually be extended to a complete generalized sampling theory for UoS models. The nomenclature Xampling was chosen to highlight the important aspects of our framework. Xampling is designed to treat UoS models. The X prefix symbolically represents

18


the intersection of two signal subspaces, as visualized in Fig. 14(b). It therefore distinguishes the union models from classic results in the sampling literature that were stated for single subspace models [1], though Xampling still hints that our framework is only a sub-field of generalized sampling theory [1], [60]–[62]. The fact that the prefix is integrated into the noun symbolizes the impact of the union model on the entire data path: both the signal acquisition and processing methods need to be modified appropriately to enjoy the advantages of the union structure. With regard to extending CS to analog signals, Xampling conveys a guideline for designers. Breaking-through the Nyquist barrier necessitates balancing between CS and sampling by combining traditional concepts from sampling theory together with recent developments from the CS literature. Xampling is literally pronounced as CSSampling (phonetically /k"sæmplIN/), so as to symbolize the necessity of this synergy in practice. Finally, it was recently suggested to us [63] that X can stand for “extreme sampling”, hinting at the very low rates. R EFERENCES [1] Y. C. Eldar and T. Michaeli, “Beyond bandlimited sampling,” IEEE Signal Process. Mag., vol. 26, no. 3, pp. 48–68, May 2009. [2] C. E. Shannon, “Communication in the presence of noise,” Proc. IRE, vol. 37, pp. 10–21, 1949. [3] H. Nyquist, “Certain Topics in Telegraph Transmission Theory,” Trans. AIEE, vol. 47, no. 2, pp. 617–644, Apr. 1928. [4] Y. M. Lu and M. N. Do, “A theory for sampling signals from a union of subspaces,” IEEE Trans. Signal Process., vol. 56, no. 6, pp. 2334–2345, Jun. 2008. [5] T. Blumensath and M. E. Davies, “Sampling theorems for signals from the union of finite-dimensional linear subspaces,” IEEE Trans. Inf. Theory, vol. 55, no. 4, pp. 1872–1882, April 2009. [6] Y. C. Eldar and M. Mishali, “Robust recovery of signals from a structured union of subspaces,” IEEE Trans. Inf. Theory, vol. 55, no. 11, pp. 5302–5316, Nov. 2009. [7] M. Mishali and Y. C. Eldar, “From theory to practice: Sub-Nyquist sampling of sparse wideband analog signals,” IEEE J. Sel. Topics Signal Process., vol. 4, no. 2, pp. 375–391, Apr. 2010. [8] K. Gedalyahu and Y. C. Eldar, “Time delay estimation from low rate samples: A union of subspaces approach,” IEEE Trans. Signal Process., vol. 58, no. 6, pp. 3017–3031, Jun. 2010. [9] K. Gedalyahu, R. Tur, and Y. C. Eldar, “Multichannel sampling of pulse streams at the rate of innovation,” [Online] arXiv.org 1004.5070, Apr. 2010. [10] R. Tur, Y. C. Eldar, and Z. Friedman, “Low rate sampling of pulse streams with application to ultrasound imaging,” [Online] arXiv.org 1003.2822, Mar. 2010. [11] W. U. Bajwa, K. Gedalyahu, and Y. C. Eldar, “Identification of underspread linear systems with application to super-resolution radar,” submitted to IEEE Trans. Signal Process.; [Online] arXiv.org 1008.0851, Aug. 2010. [12] C. de Boor, R. A. DeVore, and A. Ron, “The structure of finitely generated shift-invariant spaces in l2 (Rd ),” J. Funct. Anal, vol. 119, no. 1, pp. 37–78, 1994. [13] O. Christensen and Y. C. Eldar, “Generalized shift-invariant systems and frames for subspaces,” J. Fourier Anal. Appl., vol. 11, no. 3, pp. 299–313, 2005. [14] Y. C. Eldar, “Compressed sensing of analog signals in shift-invariant spaces,” IEEE Trans. Signal Process., vol. 57, no. 8, pp. 2986–2997, Aug. 2009. [15] ——, “Uncertainty relations for shift-invariant analog signals,” IEEE Trans. Inf. Theory, vol. 55, no. 12, pp. 5742 – 5757, Dec. 2009. [16] ——, “Sampling and reconstruction in arbitrary spaces and oblique dual frame vectors,” J. Fourier Analys. Appl., vol. 1, no. 9, pp. 77–96, Jan. 2003. [17] Y. C. Eldar and T. Werther, “General framework for consistent sampling in Hilbert spaces,” International Journal of Wavelets, Multiresolution, and Information Processing, vol. 3, no. 3, pp. 347–359, Sep. 2005.

[18] O. Christensen and Y. C. Eldar, “Oblique dual frames and shift-invariant spaces,” Applied and Computational Harmonic Analysis, vol. 17, no. 1, 2004. [19] M. Mishali and Y. C. Eldar, “Blind multi-band signal reconstruction: Compressed sensing for analog signals,” IEEE Trans. Signal Process., vol. 57, no. 3, pp. 993–1009, Mar. 2009. [20] D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1289–1306, April 2006. [21] E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006. [22] J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: Efficient sampling of sparse bandlimited signals,” IEEE Trans. Inf. Theory, vol. 56, no. 1, pp. 520–544, Jan. 2010. [23] M. Vetterli, P. Marziliano, and T. Blu, “Sampling signals with finite rate of innovation,” IEEE Trans. Signal Process., vol. 50, no. 6, pp. 1417–1428, 2002. [24] M. A. Davenport, P. T. Boufounos, M. B. Wakin, and R. G. Baraniuk, “Signal processing with compressive measurements,” IEEE J. Sel. Topics Signal Process., vol. 4, no. 2, pp. 445–460, 2010. [25] I. Maravic and M. Vetterli, “Sampling and reconstruction of signals with finite rate of innovation in the presence of noise,” IEEE Trans. Signal Process., vol. 53, no. 8, pp. 2788–2805, 2005. [26] “A/D Converters,” Analog Devices Corp., 2009, [Online]. Available: http://www.analog.com/en/analog-to-digital-converters/ad-converters/ products/index.html. [27] “Data converters,” Texas Instruments Corp., 2009, [Online]. Available: http://focus.ti.com/analog/docs/dataconvertershome.tsp. [28] D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via `1 minimization,” Proc. Natl. Acad. Sci., vol. 100, pp. 2197–2202, Mar. 2003. [29] S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Scientific Computing, vol. 20, no. 1, pp. 33– 61, 1999. [30] S. F. Cotter, B. D. Rao, K. Engan, and K. Kreutz-Delgado, “Sparse solutions to linear inverse problems with multiple measurement vectors,” IEEE Trans. Signal Process., vol. 53, no. 7, pp. 2477–2488, July 2005. [31] M. Mishali, Y. C. Eldar, O. Dounaevsky, and E. Shoshan, “Xampling: Analog to digital at sub-Nyquist rates,” to appear in Circuits, Devices & Systems, IET; [Online] arXiv.org 0912.2495. [32] M. Mishali and Y. C. Eldar, “Expected-RIP: Conditioning of the modulated wideband converter,” in Information Theory Workshop, 2009. ITW 2009. IEEE, Oct. 2009, pp. 343–347. [33] J. A. Tropp, “Algorithms for simultaneous sparse approximation. Part I: Greedy pursuit,” Signal Process. (Special Issue on Sparse Approximations in Signal and Image Processing), vol. 86, pp. 572–588, Apr. 2006. [34] ——, “Algorithms for simultaneous sparse approximation. Part II: Convex relaxation,” Signal Process. (Special Issue on Sparse Approximations in Signal and Image Processing), vol. 86, pp. 589–602, Apr. 2006. [35] J. Chen and X. Huo, “Theoretical results on sparse representations of multiple-measurement vectors,” IEEE Trans. Signal Process., vol. 54, no. 12, pp. 4634–4643, Dec. 2006. [36] M. Mishali and Y. C. Eldar, “Reduce and boost: Recovering arbitrary sets of jointly sparse vectors,” IEEE Trans. Signal Process., vol. 56, no. 10, pp. 4692–4702, Oct. 2008. [37] Y. C. Eldar and H. Rauhut, “Average case analysis of multichannel sparse recovery using convex relaxation,” IEEE Trans. Inf. Theory, vol. 56, no. 1, pp. 505–519, Jan. 2010. [38] M. E. Davies and Y. C. Eldar, “Rank awareness in joint sparse recovery,” submitted to IEEE Trans. Inf. Theory; [Online] arXiv.org 1004.4529, Apr. 2010. [39] M. Mishali, Y. C. Eldar, and A. Elron, “Xampling: Signal acquisition and processing in union of subspaces,” CCIT Report no. 747, EE Dept., Technion; [Online] arXiv.org 0911.0519v1, Oct. 2009. [40] M. F. Duarte and R. G. Baraniuk, “Spectral compressive sensing,” [Online]. Available: http://www.math.princeton.edu/∼mduarte/images/ SCS-TSP.pdf, 2010. [41] Y. Chi, A. Pezeshki, L. Scharf, and R. Calderbank, in ICASSP 2010, Mar. 2010, pp. 3930 –3933. [42] Y. Chen, M. Mishali, Y. C. Eldar, and A. O. Hero III, “Modulated wideband converter with non-ideal lowpass filters,” in ICASSP 2010, 2010, pp. 3630–3633. [43] E. Laskin and S. P. Voinigescu, “A 60 mW per Lane, 4 × 23-Gb/s 27 −1 PRBS Generator,” IEEE J. Solid-State Circuits, vol. 41, no. 10, pp. 2198–2208, Oct. 2006.


[44] T. O. Dickson, E. Laskin, I. Khalid, R. Beerkens, X. Jingqiong, B. Karajica, and S. P. Voinigescu, “An 80-Gb/s 231 − 1 pseudorandom binary sequence generator in SiGe BiCMOS technology,” IEEE J. Solid-State Circuits, vol. 40, no. 12, pp. 2735–2745, Dec. 2005. [45] Z. Yu, S. Hoyos, and B. M. Sadler, “Mixed-signal parallel compressed sensing and reception for cognitive radio,” in ICASSP’08, 2008, pp. 3861–3864. [46] T. Ragheb, J. N. Laska, H. Nejati, S. Kirolos, R. G. Baraniuk, and Y. Massoud, “A prototype hardware for random demodulation based compressive analog-to-digital conversion,” in Circuits and Systems, 2008. MWSCAS 2008. 51st Midwest Symposium on, 2008, pp. 37–40. [47] Z. Yu, X. Chen, S. Hoyos, B. M. Sadler, J. Gong, and C. Qian, “Mixedsignal parallel compressive spectrum sensing for cognitive radios,” International Journal of Digital Multimedia Broadcasting, 2010. [48] F. Gardner, “Properties of frequency difference detectors,” IEEE Trans. Commun., vol. 33, no. 2, pp. 131–138, Feb. 1985. [49] A. Elron, M. Mishali, and Y. C. Eldar, “Carrier frequency recovery from sub-Nyquist samples: Online documentation and simulations,” available: http://webee.technion.ac.il/Sites/People/YoninaEldar/Info/software/ FR/FR.htm. [50] P. Welch, “The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms,” IEEE Trans. on Audio and Electroacoustics, vol. 15, no. 2, pp. 70–73, 1967. [51] “Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specifications: High-speed physical layer in the 5 GHz band,” IEEE Std. 802.11a-1999. [52] E. Hagemann, “The Costas loop,” Jun. 2001, [Online]. Available: http: //archive.chipcenter.com/dsp/colarch.html. [53] P. L. Dragotti, M. Vetterli, and T. Blu, “Sampling moments and reconstructing signals of finite rate of innovation: Shannon meets Strang ndash;Fix,” IEEE Trans. Signal Process., vol. 55, no. 5, pp. 1741–1757, May 2007. [54] R. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propag., vol. 34, no. 3, pp. 276–280, Mar. 1986, first presented at RADC Spectrum Estimation Workshop, 1979, Griffiss AFB, NY. [55] R. Roy and T. Kailath, “Esprit-estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust., Speech, Signal Process., vol. 37, no. 7, pp. 984–995, Jul. 1989. [56] G. L. Fudge, R. E. Bland, M. A. Chivers, S. Ravindran, J. Haupt, and P. E. Pace, “A Nyquist folding analog-to-information receiver,” in Proc. 42nd Asilomar Conf. on Signals, Systems and Computers, Oct. 2008, pp. 541–545. [57] M. A. Pallas and G. Jourdain, “Active high resolution time delay estimation for large BT signals,” IEEE Trans. Signal Process., vol. 39, no. 4, pp. 781–788, 1991. [58] Y. C. Eldar and V. Pohl, “Recovering signals from lowpass data,” IEEE Trans. Signal Process., vol. 58, no. 5, pp. 2636–2646, May 2010. [59] T. G. Dvorkind, Y. C. Eldar, and E. Matusiak, “Nonlinear and non-ideal sampling: Theory and methods,” IEEE Trans. Signal Process., vol. 56, no. 12, pp. 5874–5890, 2008. [60] M. Unser, “Sampling – 50 years after Shannon,” Proceedings of the IEEE, vol. 88, no. 4, pp. 569–587, Apr. 2000. [61] P. P. Vaidyanathan, “Generalizations of the sampling theorem: Seven decades after Nyquist,” IEEE Trans. Circuit Syst. I, vol. 48, no. 9, pp. 1094–1109, Sep. 2001. [62] Y. C. Eldar and T. G. Dvorkind, “A minimum squared-error framework for generalized sampling,” IEEE Trans. Signal Processing, vol. 54, no. 6, pp. 2155–2167, Jun. 2006. [63] R. Nowak, personal communications, 2009.

19

Xampling: Signal Acquisition and Processing in Union of Subspaces

Xampling: Signal Acquisition and Processing in Union of Subspaces

Suggest Documents

Biomedical signal acquisition, processing and transmission using ...

Signal Acquisition and Processing in the Magnetic ... - Telfor Journal

Methods of Acquisition and Signal Processing for ... - ROMJIST

Methods of Acquisition and Signal Processing for ... - ROMJIST

EMG signal acquisition, processing and analysis for ... - Google Sites

Oblique projections in discrete signal subspaces of l2 and ... - CiteSeerX

Advances in Signal Processing

Multirate Signal Processing - Signal Processing for Communications

Multirate Signal Processing - Signal Processing for Communications

Signals and Signal Processing

Cyclic LTI Systems in Digital Signal Processing - Signal Processing ...

Signal Processing

Signal Processing

Multirate Digital Signal Processing - Signal and Image Processing ...

Signal Processing

signal processing

Signal-To-Noise-Ratio of Signal Acquisition In Global ... - CiteSeerX

Signal-To-Noise-Ratio of Signal Acquisition In Global ... - CiteSeerX

Language Processing and Acquisition in Languages

Individual Differences in Language Acquisition and Processing

Individual Differences in Language Acquisition and Processing

Signal Analysis and Signal Processing Recitation #7

(EMG) Signal Acquisition

Morphological processing in reading acquisition - University of ...