## Blind Multiband Signal Reconstruction: Compressed Sensing for Analog Signals

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### BibTeX

@MISC{Mishali_blindmultiband,

author = {Moshe Mishali and Yonina C. Eldar},

title = {Blind Multiband Signal Reconstruction: Compressed Sensing for Analog Signals },

year = {}

}

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### Abstract

We address the problem of reconstructing a multiband signal from its sub-Nyquist pointwise samples, when the band locations are unknown. Our approach assumes an existing multi-coset sampling. Prior recovery methods for this sampling strategy either require knowledge of band locations or impose strict limitations on the possible spectral supports. In this paper, only the number of bands and their widths are assumed without any other limitations on the support. We describe how to choose the parameters of the multi-coset sampling so that a unique multiband signal matches the given samples. To recover the signal, the continuous reconstruction is replaced by a single finitedimensional problem without the need for discretization. The resulting problem is studied within the framework of compressed sensing, and thus can be solved efficiently using known tractable algorithms from this emerging area. We also develop a theoretical lower bound on the average sampling rate required for blind signal reconstruction, which is twice the minimal rate of known-spectrum recovery. Our method ensures perfect reconstruction for a wide class of signals sampled at the minimal rate. Numerical experiments are presented demonstrating blind sampling and reconstruction with minimal sampling rate.

### Citations

1922 | Compressed sensing
- Donoho
(Show Context)
Citation Context ...results do not involve any probability model. In contrast, the common approach in compressive sensing assumes random sampling operators and typical results are valid with some probability less than 1 =-=[13]-=-,[19],[21],[22]. The paper is organized as follows. In Section II we formulate our reconstruction problem. The minimal density theorem for blind reconstruction is stated and proved in Section III. A b... |

1827 | Atomic decomposition by basis pursuit
- Chen, Donoho, et al.
- 1998
(Show Context)
Citation Context ...thms for MMV systems can be classified into two groups. The first group contains algorithms that seek the sparsest solution matrix , e.g., convex relaxations from the basis pursuit family [13], [30], =-=[31]-=- or matching pursuit [14], [29], [32] with a termination criterion based on the residual. The other group contains methods that approximate a sparse solution according to user specification, e.g., mat... |

1447 | Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information
- Candès, Romberg, et al.
(Show Context)
Citation Context ...ts do not involve any probability model. In contrast, the common approach in compressive sensing assumes random sampling operators and typical results are valid with some probability less than 1 [13],=-=[19]-=-,[21],[22]. The paper is organized as follows. In Section II we formulate our reconstruction problem. The minimal density theorem for blind reconstruction is stated and proved in Section III. A brief ... |

1166 | Matching pursuits with time-frequency dictionaries
- Mallat, Zhang
- 1993
(Show Context)
Citation Context ...oach. Existing algorithms for MMV systems can be classified into two groups. The first group contains algorithms that seek the sparsest solution matrix U0, e.g. Basis Pursuit [17] or Matching Pursuit =-=[18]-=- with a termination criterion based on the residual. The other contains methods that approximate a sparse solution according to user specification, e.g. Matching Pursuit with a predetermined number of... |

824 |
Stable signal recovery from incomplete and inaccurate measurements
- Candeś, Romberg, et al.
- 2006
(Show Context)
Citation Context ...procedure. This demonstrates that formulating the recovery problem within the CS framework, as implied by Proposition 3, is beneficial since other robust methods can be adapted, e.g., [15], [24], and =-=[33]-=-. D. Comparison With Discretization We now describe a discretization approach for blind recovery and compare it with SBR4 and SBR2. In the computer simulations our approach is also performed over a di... |

569 | Multiple Emitter Location and Signal Parameter Estimation - Schmidt - 1986 |

391 | X.: Uncertainty principles and ideal atomic decomposition - Donoho, Huo - 2001 |

236 | Algorithms for simultaneous sparse approximation
- Tropp
- 2006
(Show Context)
Citation Context ...MMV with a unique sparsest solution. It is however a well-known CS result that finding is solvable but NP-hard [13]. Several suboptimal efficient algorithms for finding are given in [13], [14], [20], =-=[29]-=-, and [30]. Some of them can indicate a success recovery of . We explain which class of algorithms has this property in Section IV-A. VI. SBR ALGORITHMS The theoretical results developed in the previo... |

193 |
Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics
- Kruskal
- 1977
(Show Context)
Citation Context ...sparse if the number of non-zero values in v is no greater than K. Using the ℓ0 pseudo-norm the sparsity of v is expressed as ‖v‖0 ≤ K. We use the following definition of the Kruskal-rank of a matrix =-=[14]-=-: Definition 1: The Kruskal-rank of A, denoted as σ(A), is the maximal number q such that every set of q columns of A is linearly independent. Observe that for every f ∈ F0 the system of (16) has less... |

154 | Sparse solutions to linear inverse problems with multiple measrement vectors
- Rao, Kreutz-Delgado
- 1998
(Show Context)
Citation Context ...requires finding a sparsest solution matrix which is an NP-hard problem [12]. Several sub-optimal efficient methods have been developed for this problem in the compressed sensing (CS) literature [15],=-=[16]-=-. In our algorithms, any of these techniques can be used. Numerical experiments on random constructions of multi-band signals show that both SBR4 and SBR2 maintain a satisfactory exact recovery rate w... |

130 | Basis Pursuit
- Chen
- 1995
(Show Context)
Citation Context ... try a different MMV approach. Existing algorithms for MMV systems can be classified into two groups. The first group contains algorithms that seek the sparsest solution matrix U0, e.g. Basis Pursuit =-=[17]-=- or Matching Pursuit [18] with a termination criterion based on the residual. The other contains methods that approximate a sparse solution according to user specification, e.g. Matching Pursuit with ... |

129 | Adaptive greedy approximations
- Davis, Mallat, et al.
- 1997
(Show Context)
Citation Context ...ry indication. Thus, if a signal cannot be recovered this indication prevents further processing of invalid data. The CTF block requires finding a sparsest solution matrix which is an NP-hard problem =-=[12]-=-. Several sub-optimal efficient methods have been developed for this problem in the compressed sensing (CS) literature [15],[16]. In our algorithms, any of these techniques can be used. Numerical expe... |

108 |
Necessary density conditions for sampling and interpolation of certain entire functions
- Landau
- 1967
(Show Context)
Citation Context ...ne is blindness, namely a design that does not assume knowledge of the band locations. Blindness is a desirable property as signals with different band locations are processed in the same way. Landau =-=[1]-=- developed a minimal sampling rate for an arbitrary sampling method that allows perfect reconstruction. For multi-band signals, the Landau rate is the sum of the band widths, which is below the corres... |

93 | The retrieval of harmonics from a covariance function, Geophys - Pisarenko |

77 | Theoretical results on sparse representations of multiple measurement vectors
- Chen, Huo
- 2006
(Show Context)
Citation Context ...lock requires finding a sparsest solution matrix which is an NP-hard problem [12]. Several sub-optimal efficient methods have been developed for this problem in the compressed sensing (CS) literature =-=[15]-=-,[16]. In our algorithms, any of these techniques can be used. Numerical experiments on random constructions of multi-band signals show that both SBR4 and SBR2 maintain a satisfactory exact recovery r... |

71 |
Random filters for compressive sampling and reconstruction
- Tropp, Wakin, et al.
- 2006
(Show Context)
Citation Context ...ssical problem addressed in the CS literature is the recovery of discrete and finite vectors. An adaptation of CS results to continuous signals was also considered in a set of conferences papers (see =-=[21]-=-,[22] and the references therein). However, these papers did not address the case of multi-band signals. In [22] an underlying discrete model was assumed so that the signal is a linear combination of ... |

66 |
A theory for sampling signals from a union of subspaces
- Lu, Do
- 2008
(Show Context)
Citation Context ...ect reconstruction with arbitrary sampling and reconstruction. As we show the lower bound is twice the Landau rate and no more than the Nyquist rate. This result is based on recent work of Lue and Do =-=[20]-=- on sampling signals from a union of subspaces. The heart of this paper is the development of a spectrum-blind reconstruction (SBR) scheme for multi-band signals. We assume a blind multi-coset samplin... |

64 | Reduce and boost: Recovering arbitrary sets of jointly sparse vectors
- Mishali, Eldar
- 2008
(Show Context)
Citation Context ...nd reconstruction and present the CTF block in 1 Since submission of the report [18] and the conference version [19], we have further expanded the analog CS framework in several follow-up papers; see =-=[20]-=-–[23]. Authorized licensed use limited to: Technion Israel School of Technology. Downloaded on June 7, 2009 at 06:02 from IEEE Xplore. Restrictions apply.MISHALI AND ELDAR: BLIND MULTIBAND SIGNAL REC... |

48 |
The theory of bandpass sampling
- Vaughan, Scott, et al.
- 1991
(Show Context)
Citation Context ...te is the sum of the band widths, which is below the corresponding Nyquist rate. Uniform sampling of a real bandpass signal with a total width of 2B Hertz on both sides of the spectrum was studied in =-=[2]-=-. It was shown that only special cases of bandpass signals can be perfectly reconstructed from their uniform samples at the minimal rate of 2B samples/sec. Kohlenberg [3] suggested periodic non-unifor... |

47 | An Uncertainty Principle for Cyclic Groups of Prime Order
- Tao
- 308
(Show Context)
Citation Context ...H 2009 Finding a universal pattern , namely one that results in a fully Kruskal-rank , is a combinatorial problem. Several specific constructions of sampling patterns are proved to be universal [11], =-=[26]-=-. In particular, choosing a prime integer renders every pattern universal [26]. In addition, a random pattern drawn uniformly among all combinations is universal with high probability; see [24] for de... |

46 |
Theory and Implementation of an Analog-To-Information Converter using Random Demodulation
- Laska, Kirolos, et al.
(Show Context)
Citation Context ...l problem addressed in the CS literature is the recovery of discrete and finite vectors. An adaptation of CS results to continuous signals was also considered in a set of conferences papers (see [21],=-=[22]-=- and the references therein). However, these papers did not address the case of multi-band signals. In [22] an underlying discrete model was assumed so that the signal is a linear combination of a fin... |

41 | Robust recovery of signals from a union of subspaces - Eldar, Mishali |

39 |
Spectrum-blind minimum-rate sampling and reconstruction of multiband signals
- Feng, Bresler
- 1996
(Show Context)
Citation Context ...-coset sampling renders the reconstruction applicable to a wide set of multi-band signals but not to all of them. Although spectrum-blind reconstruction was mentioned in two conference papers in 1996 =-=[6]-=-,[7], a full spectrumblind reconstruction scheme was not developed in these papers. It appears that spectrum-blind reconstruction has not been handled since then. We begin by developing a lower bound ... |

38 | On Unique Localization of Mul- tiple Sources by Passive Sensor Arrays - Wax, Ziskind |

35 | Minimum rate sampling and reconstruction of signals with arbitrary frequency support
- Herley, Wong
- 1999
(Show Context)
Citation Context ...at the proposed sampling rate approaches that dictated by Landau. As we prove in Section III, perfect blind reconstruction requires a higher sampling rate. In subsequent publications, Herley and Wong =-=[10]-=- suggested a half-blind system. Similar ideas were later suggested in [11]. 1053-587X/$25.00 © 2009 IEEE Authorized licensed use limited to: Technion Israel School of Technology. Downloaded on June 7,... |

34 |
Exact Interpolation of Bandlimited Functions
- Kohlenberg
- 1953
(Show Context)
Citation Context ...f the spectrum was studied in [2]. It was shown that only special cases of bandpass signals can be perfectly reconstructed from their uniform samples at the minimal rate of 2B samples/sec. Kohlenberg =-=[3]-=- suggested periodic non-uniform sampling with an average sampling rate of 2B. He also provided a reconstruction scheme that recovers any bandpass signal exactly. Lin and Vaidyanathan [4] extended his ... |

29 |
Maximal sparsity representation via ℓ1 minimization
- Donoho, Elad
- 2003
(Show Context)
Citation Context ...is used to provide a sufficient condition.Theorem 2: Suppose ¯x is a solution of y = Ax. If ‖¯x‖0 ≤ σ(A)/2 then ¯x is the unique sparsest solution of the system. Theorem 2 and its proof are given in =-=[11]-=-, [15] with a slightly different notation of Spark(A) instead of the Kruskal-rank of A. Note that the condition of the theorem is not necessary as there are examples that the sparsest solution ¯x of y... |

28 | Perfect reconstruction formulas and bounds on aliasing error in sub-Nyquist nonuniform sampling of multiband signals
- Venkataramani, Bresler
- 2000
(Show Context)
Citation Context ...rks lack the blindness property as the information about the band locations is used in the design of both the sampling and the reconstruction stages. Herley and Wong [5] and Venkataramani and Bresler =-=[8]-=- suggested a blind multi-coset sampling strategy that is called universal in [8]. The authors of [8] also developed a detailed reconstruction scheme for this sampling strategy, which is not blind as i... |

23 |
Periodically nonuniform sampling of bandpass signals
- Lin, Vaidyanathan
- 1998
(Show Context)
Citation Context ...c. Kohlenberg [3] suggested periodic non-uniform sampling with an average sampling rate of 2B. He also provided a reconstruction scheme that recovers any bandpass signal exactly. Lin and Vaidyanathan =-=[4]-=- extended his work to multi-band signals. Their method ensures perfect reconstruction from periodic non uniform sampling with an average sampling rate equal to the Landau rate. Both of these works lac... |

17 |
Maximal sparsity representation via `1 minimization
- Donoho, Elad
- 2003
(Show Context)
Citation Context ...to provide a sufficient condition for a unique solution in (20). Theorem 2: Suppose is a solution of .If then is the unique sparsest solution of the system. Theorem 2 and its proof are given in [13], =-=[27]-=- with a slightly different notation of instead of the Kruskal-rank of . Note that the condition of the theorem is not necessary as there are examples in which the sparsest solution of is unique while ... |

12 |
Further results on spectrum blind sampling of 2D signals
- Venkataramani, Bresler
(Show Context)
Citation Context ...etecting the spectral support, prior to sampling, is impossible or too expensive to implement. Reconstruction under partial knowledge of the support was addressed in a series of conference papers [5]–=-=[7]-=-. These works do not assume the exact support but require that the band locations obey a certain mathematical condition. Sampling is carried out by a multi-coset strategy, independent of the band loca... |

9 |
Universal minimum-rate sampling and spectrum-blind reconstruction for multiband signals
- Feng
- 1997
(Show Context)
Citation Context ...f techniques from this field, such as MUSIC [8]. An alternative method was proposed in [7] under the assumption that the support obeys a strict mathematical condition. Similar results were derived in =-=[9]-=-. This approach, however, suffers from two main drawbacks. First, as detailed in Section V-B, if the support violates the condition of [7], then the samples match many possible input signals, thus hin... |

9 | Compressed sensing of analog signals - Eldar |

4 | Distributed compressed sensing,” [Online]. Available: http://www.dsp.ece.rice.edu/cs/DCS112005.pdf - Baron, Wakin, et al. |

4 |
Uncertainty relations for analog signals
- Eldar
(Show Context)
Citation Context ...construction and present the CTF block in 1 Since submission of the report [18] and the conference version [19], we have further expanded the analog CS framework in several follow-up papers; see [20]–=-=[23]-=-. Authorized licensed use limited to: Technion Israel School of Technology. Downloaded on June 7, 2009 at 06:02 from IEEE Xplore. Restrictions apply.MISHALI AND ELDAR: BLIND MULTIBAND SIGNAL RECONSTR... |

3 | Eldar, “Spectrum-blind reconstruction of multiband signals
- Mishali, C
(Show Context)
Citation Context ...ling is presented in Section IV. We develop our main theoretical results on spectrum-blind reconstruction and present the CTF block in 1 Since submission of the report [18] and the conference version =-=[19]-=-, we have further expanded the analog CS framework in several follow-up papers; see [20]–[23]. Authorized licensed use limited to: Technion Israel School of Technology. Downloaded on June 7, 2009 at 0... |

2 |
Minimum rate sampling and reconstructtion of signals with arbitrary frequency support
- Herley, Wong
- 1999
(Show Context)
Citation Context ... the Landau rate. Both of these works lack the blindness property as the information about the band locations is used in the design of both the sampling and the reconstruction stages. Herley and Wong =-=[5]-=- and Venkataramani and Bresler [8] suggested a blind multi-coset sampling strategy that is called universal in [8]. The authors of [8] also developed a detailed reconstruction scheme for this sampling... |

1 |
Properties of universal sampling patterns for multi-band signals
- Mishali, Eldar, et al.
- 2007
(Show Context)
Citation Context ...niversal pattern C, namely one that results in a fully Kruskal-rank A, is a combinatorial process. Several specific constructions of sampling patterns that are proved to be universal are given in [8],=-=[10]-=-. In particular, choosing L to be prime renders every pattern universal [10]. To summarize, choosing a universal pattern allows recovery of any x(t) satisfying (20) when the band locations are known i... |