### Citations

344 | Sampling Signals with Finite Rate of Innovation
- Vetterli, Marziliano, et al.
- 2002
(Show Context)
Citation Context ...an[k](S k T φn)(t) , t ∈ R, (1) n=1 k∈Z with certain coefficient sequences an = {an[k]}k∈Z for every n =1,...,N. The number RI = N/T associated with ST (φ) is usually be called the rate of innovation =-=[9]-=-. To guarantee a unique and stable representation of every signal in ST (φ) in terms of the coefficient sequences an, the generators φ and the shift parameter T are typically chosen such that the sequ... |

184 |
Fractal functions and wavelet expansions based on several scaling functions
- GERONIMO, HARDIN
- 1994
(Show Context)
Citation Context ... where Ω:=2π/T. Moreover, the N × N matrix Gφ(ω) := 1 T Φ(ω)∗ Φ(ω) is called the Grammian associated with the sequence φ. Therewith, we can reformulate the condition for φ to be a Riesz basis. Namely =-=[2]-=-, φ is a Riesz basis with frame bounds Aφ,Bφ if and only if Aφ IN ≼ Gφ(ω) ≼ Bφ IN , a.e. ω ∈ [−Ω/2, Ω/2] . (4) Examples of SI spaces include multiband signals [10] and spline functions [11], [12]. Exp... |

158 | The structure of finitely generated shift-invariant spaces in L2(Rd
- Boor, DeVore, et al.
- 1994
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Citation Context ...the signal structure is available then it may be possible to interpolate it from the given data. Our focus here is on signals that lie in shift-invariant (SI) spaces, generated by multiple generators =-=[1]-=-–[3]. For these signals, we derive a necessary condition on the cutoff frequency of the low pass filter (LPF) and sufficient conditions on the generators such that x can be recovered from its lowpasse... |

151 |
Cardinal Spline Interpolation
- Schoenberg
- 1973
(Show Context)
Citation Context ...sis. Namely [2], φ is a Riesz basis with frame bounds Aφ,Bφ if and only if Aφ IN ≼ Gφ(ω) ≼ Bφ IN , a.e. ω ∈ [−Ω/2, Ω/2] . (4) Examples of SI spaces include multiband signals [10] and spline functions =-=[11]-=-, [12]. Expansions of the type (1) are also encountered in communication systems, when the analog signal is produced by pulse amplitude modulation. III. RECOVERY CONDITIONS The first question we addre... |

148 | From theory to practice: Sub-nyquist sampling of sparse wideband analog signals
- Mishali, Eldar
- 2010
(Show Context)
Citation Context ... be chosen almost arbitrarily. VI. APPLICATIONS AND EXAMPLES A special choice of periodic functions that are easy to implement in practice are binary sequences. These mixing functions were studied in =-=[13]-=- in the context of sparse multiband sampling. More specifically, pm, m = 1,...,M are chosen to attain the values ±1 over intervals of length T/P where P is a given integer. Formally, pm(t) =α (m) n , ... |

107 | Blind multiband signal reconstruction: Compressed sensing for analog signals
- Mishali, Eldar
- 2009
(Show Context)
Citation Context ...ion for φ to be a Riesz basis. Namely [2], φ is a Riesz basis with frame bounds Aφ,Bφ if and only if Aφ IN ≼ Gφ(ω) ≼ Bφ IN , a.e. ω ∈ [−Ω/2, Ω/2] . (4) Examples of SI spaces include multiband signals =-=[10]-=- and spline functions [11], [12]. Expansions of the type (1) are also encountered in communication systems, when the analog signal is produced by pulse amplitude modulation. III. RECOVERY CONDITIONS T... |

99 | Reduce and boost: Recovering arbitrary sets of jointly sparse vectors
- Mishali, Eldar
- 2008
(Show Context)
Citation Context ...rates Fig. 1. x(t) ✲ ✻ ✲ω − Ωc 2 Ωc 2 ✲ y(t) Lowpass filtering of x(t) with bandwidth Ωc =2π/Tc. lower than Nyquist. These ideas rely on the recently developed framework for analog compressed sensing =-=[4]-=-–[7]. In our setting, they translate to reducing the LPF bandwidth, or the number of modulators. Because of the limited space we will not give proofs of the presented results, but we refer to [8] for ... |

73 | Compressed sensing of analog signals in shift-invariant spaces
- Eldar
- 2009
(Show Context)
Citation Context ...ossible if the bandwidth of the LPF is increased. VII. CONNECTION WITH SPARSE ANALOG SIGNALS In this section we want to discuss the incorporation of sparsity into our approach. To this end, we follow =-=[5]-=- to describe sparsity of analog signals in SI spaces. Specifically, we assume that only K out of the N generators φn are active, so that at most K of the sequences {an[k]}k∈Z are nonzero. In [5], it w... |

40 | A Hadamard matrix of order 428
- Kharaghani, Tayfeh-Rezaie
(Show Context)
Citation Context ...xing sequences pm are chosen such that Q is invertible. This can be achieved by choosing Q as a Hadamard matrix of order N. It is known that Hadamard matrices exists at least for all orders up to 667 =-=[14]-=-. In the previous example, ΦK(ω) was an N × N invertible matrix for all ω ∈ [−Ω/2, Ω/2]. According to Prop. 2 a recovery of the signal x without preprocessing is therefore possible if the bandwidth Ωc... |

31 | Time-delay estimation from low-rate samples: A union of subspaces approach
- Gedalyahu, Eldar
(Show Context)
Citation Context ...s Fig. 1. x(t) ✲ ✻ ✲ω − Ωc 2 Ωc 2 ✲ y(t) Lowpass filtering of x(t) with bandwidth Ωc =2π/Tc. lower than Nyquist. These ideas rely on the recently developed framework for analog compressed sensing [4]–=-=[7]-=-. In our setting, they translate to reducing the LPF bandwidth, or the number of modulators. Because of the limited space we will not give proofs of the presented results, but we refer to [8] for deta... |

21 | Generalized Shift-Invariant Systems and Frames for Subspaces
- Christensen, Eldar
(Show Context)
Citation Context ...signal structure is available then it may be possible to interpolate it from the given data. Our focus here is on signals that lie in shift-invariant (SI) spaces, generated by multiple generators [1]–=-=[3]-=-. For these signals, we derive a necessary condition on the cutoff frequency of the low pass filter (LPF) and sufficient conditions on the generators such that x can be recovered from its lowpassed ve... |

11 | Beyond bandlimited sampling: Nonlinearities, smoothness and sparsity
- Eldar, Michaeli
- 2009
(Show Context)
Citation Context ...amely [2], φ is a Riesz basis with frame bounds Aφ,Bφ if and only if Aφ IN ≼ Gφ(ω) ≼ Bφ IN , a.e. ω ∈ [−Ω/2, Ω/2] . (4) Examples of SI spaces include multiband signals [10] and spline functions [11], =-=[12]-=-. Expansions of the type (1) are also encountered in communication systems, when the analog signal is produced by pulse amplitude modulation. III. RECOVERY CONDITIONS The first question we address is ... |

3 | Recovering signals from lowpass data
- Eldar, Pohl
- 2010
(Show Context)
Citation Context ...sing [4]–[7]. In our setting, they translate to reducing the LPF bandwidth, or the number of modulators. Because of the limited space we will not give proofs of the presented results, but we refer to =-=[8]-=- for detailed proofs and for some more concrete examples. II. PROBLEM FORMULATION As usual, CN , L2 , and ℓ2 denote the N-dimensional Euclidean space, the space of square integrable function on the re... |