## Efficient sampling of sparse wideband analog signals (2008)

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Venue: | in Proc. Conv. IEEE in Israel (IEEEI), Eilat |

Citations: | 7 - 3 self |

### BibTeX

@INPROCEEDINGS{Mishali08efficientsampling,

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

title = {Efficient sampling of sparse wideband analog signals},

booktitle = {in Proc. Conv. IEEE in Israel (IEEEI), Eilat},

year = {2008}

}

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### Abstract

Periodic nonuniform sampling is a known method to sample spectrally sparse signals below the Nyquist rate. This strategy relies on the implicit assumption that the individual samplers are exposed to the entire frequency range. This assumption becomes impractical for wideband sparse signals. The current paper proposes an alternative sampling stage that does not require a full-band front end. Instead, signals are captured with an analog front end that consists of a bank of multipliers and lowpass filters whose cutoff is much lower than the Nyquist rate. The problem of recovering the original signal from the low-rate samples can be studied within the framework of compressive sampling. An appropriate parameter selection ensures that the samples uniquely determine the analog input. Moreover, the analog input can be stably reconstructed with digital algorithms. Numerical experiments support the theoretical analysis. Index Terms — Analog to digital conversion, compressive sampling, infinite measurement vectors (IMV), multiband sampling. 1.

### Citations

1864 | Compressed sensing - Donoho |

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(Show Context)
Citation Context ...e of signs will work, except with probability exponentially small in M [14]. In fact, recent work on compressive sampling shows that a random choice of signs ensures that signal acquisition is stable =-=[9]-=-. A matrix A is said to have the restricted isometry property (RIP) of order K, if there exists 0 ≤ δK < 1 such that (1 − δK)‖x‖ 2 ≤ ‖Ax‖ 2 ≤ (1 + δK)‖x‖ 2 (18) for every K-sparse vector x [9]. When A... |

331 |
The restricted isometry property and its implications for compressed
- Candès
- 2008
(Show Context)
Citation Context ... every K-sparse vector x [9]. When A = SF satisfies the RIP of order 4N, then the matrices AS and (AS)† are well conditioned for every possible frequency subset S ⊆ F0 with |S| ≤ 2N. It was proved in =-=[15]-=- that basis pursuit can recover the K-sparse solution x of y = Ax for an underdetermined A, if A has δ2K < √ 2 − 1. The mean squared error of the recovery in the presence of noise or model mismatch wa... |

320 | A simple proof of the restricted isometry property for random matrices,” Constructive Approximation
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(Show Context)
Citation Context ...rix whose entries are equally likely to be ±1/ √ m. The RIP of order K holds with high probability for the matrix A = SF when m ≥ CK log(M/K), where C is a positive constant independent of everything =-=[17]-=-. The log factor is necessary [18]. In practice, we empirically evaluate the stability of the system since the RIP cannot be verified computationally.11 IV. NUMERICAL EVALUATION To evaluate the empir... |

221 | Algorithms for simultaneous sparse approximation—part I: greedy pursuit
- Tropp, Gilbert, et al.
- 2006
(Show Context)
Citation Context ...m the sequences of samples, we apply the reduction from an IMV system to an MMV system, as described in Fig. 2. We solve the resulting MMV systems using simultaneous orthogonal matching pursuit [11], =-=[12]-=-. More precisely, we evaluate the performance on 100 noisy test signals of the form x(t) + w(t), where x is a multiband signal and w is a white Gaussian noise process. The multiband signals consist of... |

145 | Sparse solutions to linear inverse problems with multiple measrement vectors
- Rao, Kreutz-Delgado
- 1998
(Show Context)
Citation Context ...ls from the sequences of samples, we apply the reduction from an IMV system to an MMV system, as described in Fig. 2. We solve the resulting MMV systems using simultaneous orthogonal matching pursuit =-=[11]-=-, [12]. More precisely, we evaluate the performance on 100 noisy test signals of the form x(t) + w(t), where x is a multiband signal and w is a white Gaussian noise process. The multiband signals cons... |

73 | Theoretical results on sparse representations of multiple-measurement vectors
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- 2006
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Citation Context ...d where (AS)† = (AH S AS) −1AH S is the Moore–Penrose pseudoinverse. For unknown support S, (2) is still invertible if K = |S| is known, and every set of 2K columns from A is linearly independent [6]–=-=[8]-=-. In general, solving (2) for x(Λ) is NP-hard because it may require a combinatorial search. Nevertheless, recent advances in compressive sampling and sparse approximation delineate situations where p... |

62 | Reduce and boost : Recovering arbitrary sets of jointly sparse vectors
- Mishali, Eldar
- 2008
(Show Context)
Citation Context ...II reviews essential background material. In Section III, we describe the system design and a frequency-domain analysis that leads to an infinite measurement vectors (IMV) system. Applying ideas from =-=[6]-=-, we reduce the problem of locating the frequency bands to a finite-dimensional compressive sampling problem. We then derive an appropriate choice of parameters for the sampling system. Section IV pre... |

55 | Blind multiband signal reconstruction: Compressed sensing for analog signals
- Mishali, Eldar
(Show Context)
Citation Context ...ype, was analyzed in [2], which established that exact recovery is possible when the band locations are known. The blind case, in which the band locations are unknown, has been extensively studied in =-=[3]-=-. Unfortunately, the sampling front ends proposed in [1]–[3] are impractical for wideband applications because they require ADCs whose sampling rate is matched to the Nyquist rate of the input signal,... |

45 |
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- 1959
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Citation Context ... these shortcomings using a hybrid optic–electronic system at the expense of size and cost. In this paper, we analyze a practical sampling system inspired by the recent work on the random demodulator =-=[5]-=-. This system multiplies the input signal by a random square wave alternating at the Nyquist rate, then it performs lowpass filtering, and samples the signal at a lower rate. Our system consists of a ... |

40 | Robust recovery of signals from a union of subspaces
- Eldar, Mishali
- 2008
(Show Context)
Citation Context ...r of the recovery in the presence of noise or model mismatch was also shown to be bounded under the same condition. Similar conditions were shown to hold for other recovery algorithms. In particular, =-=[16]-=- proved a similar argument for a mixed ℓ2/ℓ1 program, for the MMV setting. Thus, if A = SF has the RIP of order 4N, then it implies the stability of the recovery using Fig. 2, when the mixed-norm prog... |

28 |
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(Show Context)
Citation Context ...0 In blind recovery, S is unknown, and the following CS result is used to ensure the uniqueness. A K-sparse vector x is the unique solution of y = Ax if every 2K columns of A are linearly independent =-=[7]-=-. Clearly, this condition translates to m ≥ 4N and the condition on S of the theorem. The parameter selection of Theorem 1 guarantees an average sampling rate m/T ≥ 4NB. Depending on whether fNYQ/B is... |

27 | Bresler: “Perfect reconstruction formulas and bounds on aliasing error in sub-nyquist nonuniform sampling of multiband signals
- Venkataramani, Y
- 2000
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Citation Context ...signals has shown that it is possible to reduce the sampling rate by acquiring samples from a periodic but nonuniform grid [1]. Multi-coset sampling, a specific strategy of this type, was analyzed in =-=[2]-=-, which established that exact recovery is possible when the band locations are known. The blind case, in which the band locations are unknown, has been extensively studied in [3]. Unfortunately, the ... |

21 |
Vaidyanathan: “Periodically nonuniform sampling of bandpass signals
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(Show Context)
Citation Context ...must exploit its structure in an intelligent way. Previous work on multiband signals has shown that it is possible to reduce the sampling rate by acquiring samples from a periodic but nonuniform grid =-=[1]-=-. Multi-coset sampling, a specific strategy of this type, was analyzed in [2], which established that exact recovery is possible when the band locations are known. The blind case, in which the band lo... |

18 | On random±1 matrices: singularity and determinant
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- 2006
(Show Context)
Citation Context ...m is computationally difficult because one must check the rank of every set of 4N columns from S. It is known that a random choice of signs will work, except with probability exponentially small in M =-=[14]-=-. In fact, recent work on compressive sampling shows that a random choice of signs ensures that signal acquisition is stable [9]. A matrix A is said to have the restricted isometry property (RIP) of o... |

14 | Sampling theorems for signals from the union of linear subspaces
- Blumensath, Davies
- 2007
(Show Context)
Citation Context ...ly to be ±1/ √ m. The RIP of order K holds with high probability for the matrix A = SF when m ≥ CK log(M/K), where C is a positive constant independent of everything [17]. The log factor is necessary =-=[18]-=-. In practice, we empirically evaluate the stability of the system since the RIP cannot be verified computationally.11 IV. NUMERICAL EVALUATION To evaluate the empirical performance of the proposed s... |

2 | Multirate synchronous sampling of sparse multiband signals,” Arxiv preprint arXiv:0806.0579 - Fleyer, Rosenthal, et al. - 2008 |