## Compressive spectral estimation for nonstationary random processes (2009)

Venue: | IN PROC. IEEE ICASSP-2009 |

Citations: | 2 - 1 self |

### BibTeX

@INPROCEEDINGS{Jung09compressivespectral,

author = {Alexander Jung and Georg Tauböck and Franz Hlawatsch},

title = {Compressive spectral estimation for nonstationary random processes},

booktitle = {IN PROC. IEEE ICASSP-2009},

year = {2009},

pages = {3029--3032},

publisher = {}

}

### OpenURL

### Abstract

We propose a “compressive” estimator of the Wigner-Ville spectrum (WVS) for time-frequency sparse, underspread, nonstationary random processes. A novel WVS estimator involving the signal’s Gabor coefficients on an undersampled time-frequency grid is combined with a compressed sensing transformation in order to reduce the number of measurements required. The performance of the compressive WVS estimator is analyzed via a bound on the mean square error and through simulations. We also propose an efficient implementation using a special construction of the measurement matrix.

### Citations

1864 | Compressed sensing
- Donoho
(Show Context)
Citation Context ...reconstruction, basis pursuit 1. INTRODUCTION The recently introduced methodology of compressed sensing (CS) enables the efficient reconstruction of sparse signals from a small number of measurements =-=[1]-=-. Here, we apply CS to nonstationary spectral estimation from a single observed process realization. We first present a spectral estimator for underspread nonstationary processes that is derived from ... |

802 |
Stable signal recovery from incomplete and inaccurate measurements
- Candès, Romberg, et al.
(Show Context)
Citation Context ...∈ C N . If the compressed dimension M and the “measurement matrix” Φ are chosen as discussed in Section 3.2, we can recover c from z up to a small error by means of the convex program (basis pursuit) =-=[9]-=- ĉ � arg min ‖c′ ‖1 subject to Φc ′ = z . (9) c ′ ‖c′ From the estimated Gabor coefficients ĉ (k,l) x contained in ĉ, wefinally obtain a compressive WVS estimate by substituting the ĉ (k,l) x for the ... |

299 |
Foundations of Time-Frequency Analysis
- Gröchenig
- 2001
(Show Context)
Citation Context ...satisfied. We now define a WVS estimator c WX(t, f) by replacing the WVS samples WX(kT, lF) in (2) with estimates. Here, these estimates are chosen as the squared magnitudes of the Gabor coefficients =-=[7]-=- of an observed realization x(t) using T,F as TF grid constants, Z � x(t) g ∗ (t − kT) e −j2πlFt dt , c (k,l) x t with a suitably chosen normalized window g(t) (i.e., ‖g‖2 =1). The WVS estimator is th... |

80 | Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements
- Rudelson, Vershynin
- 2006
(Show Context)
Citation Context ...c − cS‖1 √ , (12) S where the constant C depends only on δ3S and δ4S. In particular, if c is almost S-sparse (i.e., c ≈ cS), then (12) shows that the reconstruction error ĉ−c is small. It is shown in =-=[10]-=- that if Φ ∈ C M×N is constructed by randomly selecting M rows from a unitary N × N matrix U and normalizing the columns, a sufficient condition for (11) to be true with overwhelming probability 1 is ... |

73 | Adaptive covariance estimation of locally stationary processes
- Mallat, Papanicolaou, et al.
(Show Context)
Citation Context ...estimators for underspread processes include TF smoothed versions of the Wigner distribution of an observed realization x(t) of X(t) [2, 4] and estimators using a local cosine basis expansion of x(t) =-=[5]-=-. However, these estimators do not perform a compression of the measurements. Our contributions and the organization of this paper can be summarized as follows. In Section 2, we present a WVS estimato... |

30 |
Matched Weyl-Heisenberg Expansions of Nonstationary Environments
- Kozek
- 1996
(Show Context)
Citation Context ... grid geometry defined by T,F should be matched to the correlation TF geometry of X(t) as characterized by the EAF support S = [−τmax,τmax] × [−νmax,νmax]. This is achieved by letting T/F = τmax/νmax =-=[8]-=-. Choice of g(t). To obtain small correlation of the c (k,l) X ,theTF geometry of the Gabor analysis window g(t) should be matched to theTFgridgeometrydefinedbyT,F. Thisisachievedbyletting Tg/Fg = T/F... |

17 | Analysis, optimization, and implementation of low-interference wireless multicarrier systems
- Matz, Schafhuber, et al.
- 2005
(Show Context)
Citation Context ...n U = 1 √ FK ⊗ FL , KL where, e.g., FK is the K × K IDFT matrix, i.e., [FK] k,l � (k−1)(l−1) j2π e K ,and⊗denotes the Kronecker product. This construction allows us to adapt an algorithm described in =-=[11]-=- to our context. Let x denote a discrete-time (sampled) segment of x(t), obtainedwithasufficientlyhighsamplingratefs, and let the lengthNg vector g denote the discrete-time version of the (finite-leng... |

16 | A Time-Frequency Calculus for Time-Varying Systems and Nonstationary Processes with Applications
- Matz
- 2000
(Show Context)
Citation Context ... by the WWTF project SPORTS (MA 07-004). 2. WVS ESTIMATION BASED ON GABOR ANALYSIS We assume that the nonstationary process X(t) is underspread, which means that it has small “TF correlation moments” =-=[3, 6]-=- m (φ) X � R R φ(τ,ν) |ĀX(τ,ν)| dτ dν τ Rν R , |ĀX(τ,ν)| dτ dν τ ν M (φ) X � R R τ ν φ2 (τ,ν) | ĀX(τ, ν)|2 dτ dν R R τ ν |ĀX(τ,ν)|2 . (1) dτ dν Here, ĀX(τ,ν) � R t r ` ´ τ τ −j2πνt X t+ ,t− e dt is th... |

15 |
Quadratic time-varying spectral estimation for underspread processes
- Kozek, Riedel
- 1994
(Show Context)
Citation Context ...e in the TF plane are approximately uncorrelated [3]. Existing WVS estimators for underspread processes include TF smoothed versions of the Wigner distribution of an observed realization x(t) of X(t) =-=[2, 4]-=- and estimators using a local cosine basis expansion of x(t) [5]. However, these estimators do not perform a compression of the measurements. Our contributions and the organization of this paper can b... |

10 |
The Wigner-Ville spectrum of nonstationary random signals
- Flandrin, Martin
- 1997
(Show Context)
Citation Context ...ocorrelation rX(t1,t2) � E{X(t1)X ∗ R (t2)} and finite mean energy ĒX � t rX(t, t) dt (integrals and sums are from −∞ to ∞ unless noted otherwise). We wish to estimate the Wigner-Ville spectrum (WVS) =-=[2]-=- Z “ WX(t, f) � rX t + τ τ τ ” ,t− e 2 2 −j2πfτ dτ . The WVS can be viewed as a “time-dependent power spectrum” if X(t) is an underspread process, i.e., if process components that are not too close in... |

9 | Nonstationary spectral analysis based on time-frequency operator symbols and underspread approximations
- Matz, Hlawatsch
- 2006
(Show Context)
Citation Context ...fτ dτ . The WVS can be viewed as a “time-dependent power spectrum” if X(t) is an underspread process, i.e., if process components that are not too close in the TF plane are approximately uncorrelated =-=[3]-=-. Existing WVS estimators for underspread processes include TF smoothed versions of the Wigner distribution of an observed realization x(t) of X(t) [2, 4] and estimators using a local cosine basis exp... |

9 |
Second-order time-frequency synthesis of nonstationary random processes
- Hlawatsch, Kozek
- 1995
(Show Context)
Citation Context ...etter effective TF sparsity (smaller σX(S)). Using the approximations A1 and A2, (17) simplifies to Δε ≤ 32 α(S)Ē 2 ˆ 2 X TF α(S)σX(S)+σX(S) ˜ . 5. SIMULATION RESULTS Using the TF synthesis method of =-=[12]-=-, we generated 1000 realizations of a discrete-time random process of length 256 whose WVS is shown in Fig. 1(a). From the TF sparsity profile and EAF plotted in Fig. 2, we conclude that the process i... |

8 | Time-frequency localization from sparsity constraints
- Borgnat, Flandrin
- 2008
(Show Context)
Citation Context ...ors is small up to about N/M =3but increases beyond that point. The NMSE decrease for N/M between 1 and 2 may be due to a regularization effect of the CS recovery stage (such an effect is reported in =-=[13]-=- in a different context). We did not plot the MSE bound (15)–(17) because it is much larger than the empirical MSE (this lack of tightness is mostly due to the notoriously loose [10] CS error bound us... |