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## Sub-Nyquist Radar via Doppler Focusing (2013)

Citations: | 8 - 3 self |

### Citations

1665 | Matching pursuits with time-frequency dictionaries
- Mallat, Zhang
- 1993
(Show Context)
Citation Context ...y nonzero index denotes a target with delay . Given a set of sampled Fourier coefficients, a variety of CS techniques can be employed for recovery [11], for instance orthogonal matching pursuit (OMP) =-=[25]-=-, iterative hard thresholding (IHT) [26], or minimization (see [27] and references within). Choosing the coefficients at random produces favorable conditions for CS, aiding recovery in the presence of... |

1095 |
Multiple emitter location and signal parameter estimation
- Schmidt
- 1986
(Show Context)
Citation Context ...pectral analysis methods which require sampling a consecutive subset of coefficients, such as the annihilating filter [20], matrix pencil [21], or ESPRIT [22]. An alternative approach is to use MUSIC =-=[23]-=-, which does not require consecutive coefficients. The lower bound on can be achieved only when the noise is negligible and computational complexity is not of concern. When there is substantial noise ... |

723 |
Esprit-estimation of signal parameters via rotational invariance techniques
- Roy, Kailath
- 1989
(Show Context)
Citation Context ... i.e., . These parameters can be found using spectral analysis methods which require sampling a consecutive subset of coefficients, such as the annihilating filter [20], matrix pencil [21], or ESPRIT =-=[22]-=-. An alternative approach is to use MUSIC [23], which does not require consecutive coefficients. The lower bound on can be achieved only when the noise is negligible and computational complexity is no... |

667 | On the use of windows for harmonic analysis with the Discrete Fourier Transform
- Harris
- 1978
(Show Context)
Citation Context .... 4 we see an example of how windowing can reduce the effect of out-of-focus targets compared with no windowing (constant ). For a comprehensive review of windowing function design considerations see =-=[35]-=-. When attempting to support a set of targets amplitudes comprising a large dynamic range, a situation fairly common in real scenarios, the focusing operation must be performed with aggressive windowi... |

604 |
Estimating the dimension of a model
- Schwartz
- 1978
(Show Context)
Citation Context ...tection is performed iteratively until all targets have been detected, if is known, or until an amplitude threshold is met, if the model order is unknown. The latter case has been studied extensively =-=[30]-=-–[32] as a problem of estimating the number of sinusoids in a noisy sequence. In our simulations, since we wish to eliminate model order errors which influence recovery performance, we assume is known... |

542 | Detection of signals by information theoretic criteria - Wax, Kailath - 1985 |

446 |
Introduction to Spectral Analysis
- Stoı̈ca, Moses
- 1997
(Show Context)
Citation Context ...o recover the unknown ’s and ’s [9], i.e., . These parameters can be found using spectral analysis methods which require sampling a consecutive subset of coefficients, such as the annihilating filter =-=[20]-=-, matrix pencil [21], or ESPRIT [22]. An alternative approach is to use MUSIC [23], which does not require consecutive coefficients. The lower bound on can be achieved only when the noise is negligibl... |

341 | Sampling signals with finite rate of innovation
- Vetterli, Marziliano, et al.
- 2002
(Show Context)
Citation Context ...reedom (DOF): a delay, Doppler frequency and amplitude for each of the targets. Signals which can be described with a fixed number of DOF per unit of time are known as Finite Rate of Innovation (FRI) =-=[9]-=- signals. The proposed recovery process estimates these DOF from low rate samples. The concept of FRI together with the Xampling methodology enables sub-Nyquist rates using practical hardware [1], [10... |

324 | Iterative hard thresholding for compressed sensing,”
- Blumensath, Davies
- 2009
(Show Context)
Citation Context ...lay . Given a set of sampled Fourier coefficients, a variety of CS techniques can be employed for recovery [11], for instance orthogonal matching pursuit (OMP) [25], iterative hard thresholding (IHT) =-=[26]-=-, or minimization (see [27] and references within). Choosing the coefficients at random produces favorable conditions for CS, aiding recovery in the presence of noise. When the indices in are selected... |

258 | On sparse reconstruction from Fourier and Gaussian measurements,”
- Rudelson, Vershynin
- 2008
(Show Context)
Citation Context ...of noise. When the indices in are selected uniformly at random, it can be shown that if , for some positive constant , then obeys the desired Restricted Isometry Property (RIP) with large probability =-=[28]-=-. By satisfying the condition for RIP we are able to recover , using a CS recovery algorithm. VI. DELAY-DOPPLER RECOVERY USING DOPPLER FOCUSING In Section IV we introduced the concept of Doppler focus... |

217 |
Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise
- Hua, Sarkar
- 1990
(Show Context)
Citation Context ...n ’s and ’s [9], i.e., . These parameters can be found using spectral analysis methods which require sampling a consecutive subset of coefficients, such as the annihilating filter [20], matrix pencil =-=[21]-=-, or ESPRIT [22]. An alternative approach is to use MUSIC [23], which does not require consecutive coefficients. The lower bound on can be achieved only when the noise is negligible and computational ... |

151 | High-resolution radar via compressed sensing
- Herman, Strohmer
(Show Context)
Citation Context ...oving targets, and low signal-to-noise ratio (SNR). Several past works employ CS algorithms to this type of problem, but do not address sample rate reduction and continue sampling at the Nyquist rate =-=[4]-=-, [5]. Other ideas combine radar and CS in order to reduce the receiver’s sampling rate, but in doing so impose constraints on the radar transmitter and do not treat noise [6], or do not handle noise ... |

114 | Compressed Sensing: Theory and Applications
- Eldar, Kutyniok
(Show Context)
Citation Context ...d Fourier coefficients, a variety of CS techniques can be employed for recovery [11], for instance orthogonal matching pursuit (OMP) [25], iterative hard thresholding (IHT) [26], or minimization (see =-=[27]-=- and references within). Choosing the coefficients at random produces favorable conditions for CS, aiding recovery in the presence of noise. When the indices in are selected uniformly at random, it ca... |

108 | Compressive Radar Imaging - Baraniuk, Steeghs - 2007 |

101 | Structured compressed sensing: from theory to applications
- Duarte, Eldar
- 2011
(Show Context)
Citation Context ...he resulting set of focused Fourier coefficients can be concatenated and written as ... . . . ... (39) where we used (22). Since is -sparse, the dictionary must have greater than for perfect recovery =-=[33]-=-. The spark of a block-diagonal matrix equals the maximal spark of any single block, so . Since has rows, its spark is upper bounded by , and thus , the minimal number of Fourier coefficients required... |

98 | Blind multiband signal reconstruction: Compressed sensing for analog signals
- Mishali, Eldar
- 2009
(Show Context)
Citation Context ... and Doppler estimation, performing them sequentially rather than in parallel. A common approach to performing two-stage recovery uses the multiple measurement vector (MMV) framework, as performed in =-=[39]-=- in the context of undersampling of sparse wideband signals. MMV recovery jointly processes (18) for by stacking the sampled Fourier coefficient vectors and sparse target delay vectors as and accordin... |

84 | Sensitivity to Basis Mismatch in Compressed Sensing
- Chi, Scharf, et al.
- 2011
(Show Context)
Citation Context ...grid, local interpolation around detected grid points can be used to reduce quantization errors. In our simulations in Section VIII we perform a parabolic fit around detected indices. The analysis in =-=[24]-=- can be used to quantify these off-the-grid errors. Choosing a set of indices , we define the corresponding vector of Fourier coefficients (14) We can then write (12) in vector form as (15) where is a... |

76 |
Fundamentals of Radar Signal Processing
- Richards
- 2005
(Show Context)
Citation Context ...points in the map. The Doppler processing stage can be viewed as MF in the pulse dimension to a constant radial velocity target. As such, it increases the SNR by compared to the SNR of a single pulse =-=[15]-=-, [16]. Since a MF is the linear time-invariant (LTI) system which maximizes SNR, it follows that a factor increase is optimal for pulses. In Section VIwe show that the SNR achieved with Doppler focus... |

67 | Xampling: Analog to digital at sub-Nyquist rates
- Mishali, Eldar, et al.
- 2011
(Show Context)
Citation Context ...ple rate reduction. We also briefly treat target dynamic range and clutter. However, a full analysis of these important issues is left to future work. The sub-Nyquist Xampling (“compressed sampling”) =-=[1]-=- method we use is an ADC which performs analog prefiltering of the signal before taking point-wise samples. These compressed samples (“Xamples”) contain the information needed to recover the desired s... |

57 |
Estimating Two-Dimensional Frequencies by Matrix Enhancement and Matrix Pencil
- Hua
- 1992
(Show Context)
Citation Context ...f pulses required for Doppler focusing is within order of magnitude of the lower bound. Finally, the result in Theorem 1 coincides with the minimal sampling rate for two dimensional spectral analysis =-=[34]-=-. E. Practical Considerations We now describe a few practical issues, starting with computational efficiency. If one wishes to perform Doppler focusing on a uniform grid of Doppler frequencies, i.e., ... |

42 | Multichannel sampling of pulse streams at the rate of innovation - Gedalyahu, Tur, et al. - 2011 |

38 | Xampling: Signal acquisition and processing in union of subspaces
- Mishali, Eldar, et al.
- 2009
(Show Context)
Citation Context ... [17]. Since Doppler focusing yields such a problem, we now review how Xampling can be used to solve (10) at a sub-Nyquist sampling rate. A. Xampling The concept of Xampling, introduced in [1], [18], =-=[19]-=-, describes analog-to-digital conversion which acquires samples at sub-Nyquist rates while preserving the ability to perfectly reconstruct the signal. Xampling can be interpreted as “compressed sampli... |

32 |
Reduction of Sidelobe and Speckle Artifacts in Microwave Imaging
- Tsao
- 1988
(Show Context)
Citation Context ...to reduce masking of weaker targets and to remove spurious targets created by processing sidelobes. A similar subtraction is performed in many iterative algorithms such as OMP or the Clean Process of =-=[29]-=-. Detection is performed iteratively until all targets have been detected, if is known, or until an amplitude threshold is met, if the model order is unknown. The latter case has been studied extensiv... |

30 | Innovation rate sampling of pulse streams with application to ultrasound imaging
- Tur, Eldar, et al.
- 2011
(Show Context)
Citation Context ...s on the sampling rate BAR-ILAN AND ELDAR: SUB-NYQUIST RADAR VIA DOPPLER FOCUSING 1799 needed for perfect reconstruction. Practical sampling methods achieving these bounds are explained in [2], [11], =-=[17]-=- in the context of ultrasound and radar, both without Doppler. Another work [7] investigates the delay-Doppler estimation problem, but recovers the delays and Doppler frequencies in a two-stage proces... |

23 | Micro-Doppler effect in radar: Phenomenon, model, and simulation study,”
- Chen, Li, et al.
- 2006
(Show Context)
Citation Context ...on of targets with Doppler frequencies far enough from the nominal frequency ( here) increases significantly with proper windowing. Also, focus zone changes for different ’s. micro-Doppler phenomenon =-=[36]-=-. This effect is the modulation of the target’s main Doppler frequency caused by motion of the structure of the target around its main trajectory. For example, missile wingtips will exhibit different ... |

20 | Eldar, “Identification of parametric underspread linear systems and super-resolution radar
- Bajwa, Gedalyahu, et al.
- 2011
(Show Context)
Citation Context .... Other ideas combine radar and CS in order to reduce the receiver’s sampling rate, but in doing so impose constraints on the radar transmitter and do not treat noise [6], or do not handle noise well =-=[7]-=-. The work in [7] first estimates target delays and then uses these recovered delays to estimate Doppler frequencies and amplitudes. We refer to this technique as “two-stage recovery” in subsequent se... |

17 |
Estimating the Number of Sinusoids in Additive White Noise
- Fuchs
- 1988
(Show Context)
Citation Context ...on is performed iteratively until all targets have been detected, if is known, or until an amplitude threshold is met, if the model order is unknown. The latter case has been studied extensively [30]–=-=[32]-=- as a problem of estimating the number of sinusoids in a noisy sequence. In our simulations, since we wish to eliminate model order errors which influence recovery performance, we assume is known. It ... |

13 | Analysis of micro-doppler signatures
- Chen, Li, et al.
(Show Context)
Citation Context ...the sparsity assumption remains valid. At the heart of our model is the assumption that the target scene is composed of a small number of targets with discrete Doppler frequencies. Recent works [36], =-=[37]-=- show that for cases of micro-Doppler, there are a small number of dominant frequencies in the continuous Doppler spectrum, caused by distinct vibration modes, rotation rates or resonant frequencies. ... |

12 | Sub-Nyquist sampling: bridging theory and practice - Mishali, Eldar - 2011 |

9 |
Signal Processing Fundamentals and Applications for Communications and Sensing Systems. Artech House
- Minkoff
- 2002
(Show Context)
Citation Context ...on-coherent integration is a common practice in radar and has been analyzed extensively. Several sources develop approximations of the SNR increase for multiple pulses using non-coherent integration: =-=[40]-=- estimates it at for , while [15], [16] estimates where , with decreasing towards 0.5 as increases. Regardless of the exact value of , Doppler focusing, which compensates for the exact phase differenc... |

8 | Adaptive Compressed Sensing Radar Oriented Toward Cognitive Detection in Dynamic Sparse Target Scene - Zhang, Zhu, et al. - 2012 |

7 | A sub-Nyquist radar prototype: Hardware and algorithms
- Baransky, Itzhak, et al.
- 2012
(Show Context)
Citation Context ...ise samples. These compressed samples (“Xamples”) contain the information needed to recover the desired signal parameters using compressed sensing (CS) algorithms. This work expands on the results of =-=[2]-=-, adding Doppler to the target model and proposing a new method to estimate it. The same sampling technique and hardware that were used in [2], [3] are also applicable here, while the digital processi... |

6 | Compressed Beamforming in Ultrasound Imaging
- Wagner, Eldar, et al.
- 2012
(Show Context)
Citation Context ...of targets separated in Doppler by more than create almost no interference with each other. The idea of Doppler focusing comes from a similar function used in the context of ultrasound beamforming in =-=[11]-=-, [12]. There, in a method named “Dynamic focusing”, the signal returned to a set of linearly aligned transceivers is focused in a manner similar to how we focus pulses, where the Doppler frequency is... |

5 | Micro-Doppler Effect in Radar - Chen, Li, et al. - 2006 |

4 |
Xampling: Compressed sensing for analog signals,” in Compressed Sensing: Theory and Applications
- Mishali, Eldar
(Show Context)
Citation Context ..., [9], [17]. Since Doppler focusing yields such a problem, we now review how Xampling can be used to solve (10) at a sub-Nyquist sampling rate. A. Xampling The concept of Xampling, introduced in [1], =-=[18]-=-, [19], describes analog-to-digital conversion which acquires samples at sub-Nyquist rates while preserving the ability to perfectly reconstruct the signal. Xampling can be interpreted as “compressed ... |

3 | Fourier domain beamforming: The path to compressed ultrasound imaging,” Trans
- Chernyakova, Eldar
(Show Context)
Citation Context ...gets separated in Doppler by more than create almost no interference with each other. The idea of Doppler focusing comes from a similar function used in the context of ultrasound beamforming in [11], =-=[12]-=-. There, in a method named “Dynamic focusing”, the signal returned to a set of linearly aligned transceivers is focused in a manner similar to how we focus pulses, where the Doppler frequency is repla... |

2 | High-Resolution Range-Doppler Imaging by Coherent Block-Sparse Estimation - Demissie - 2012 |

2 |
Compressive radar imaging,” presented at the
- Baraniuk, Steeghs
- 2007
(Show Context)
Citation Context ...ling at the Nyquist rate [4], [5]. Other ideas combine radar and CS in order to reduce the receiver’s sampling rate, but in doing so impose constraints on the radar transmitter and do not treat noise =-=[6]-=-, or do not handle noise well [7]. The work in [7] first estimates target delays and then uses these recovered delays to estimate Doppler frequencies and amplitudes. We refer to this technique as “two... |

1 |
Richards,Noncoherent Integration Gain, and Its Approximation, Noncoherent Integration
- A
- 2010
(Show Context)
Citation Context ... in the map. The Doppler processing stage can be viewed as MF in the pulse dimension to a constant radial velocity target. As such, it increases the SNR by compared to the SNR of a single pulse [15], =-=[16]-=-. Since a MF is the linear time-invariant (LTI) system which maximizes SNR, it follows that a factor increase is optimal for pulses. In Section VIwe show that the SNR achieved with Doppler focusing al... |

1 |
Sub-Nyquist radar,” presented at the 9th Int
- Bar-Ilan, Eldar
- 2013
(Show Context)
Citation Context ...ndwidth, which is a critical parameter for high-resolution radar. B. Two-Stage Recovery To overcome the problematic scaling of the simultaneous recovery dictionary, two-stage recovery techniques [7], =-=[38]-=- separate delay and Doppler estimation, performing them sequentially rather than in parallel. A common approach to performing two-stage recovery uses the multiple measurement vector (MMV) framework, a... |

1 | The Chinese Remainder Theorem and Multi-PRF - Clark - 1984 |

1 | Noncoherent Integration Gain, and its Approximation - Richards - 2010 |

1 | Sub-Nyquist Radar - Bar-Ilan, Eldar - 2013 |