By Yanwei Wang

ISBN-10: 1598290002

ISBN-13: 9781598290004

Spectral estimation is critical in lots of fields together with astronomy, meteorology, seismology, communications, economics, speech research, clinical imaging, radar, sonar, and underwater acoustics. so much current spectral estimation algorithms are devised for uniformly sampled complete-data sequences. besides the fact that, the spectral estimation for info sequences with lacking samples is usually vital in lots of functions starting from astronomical time sequence research to artificial aperture radar imaging with angular range. For spectral estimation within the missing-data case, the problem is find out how to expand the prevailing spectral estimation suggestions to house those missing-data samples. lately, nonparametric adaptive filtering established thoughts were built effectively for numerous missing-data difficulties. jointly, those algorithms offer a finished toolset for the missing-data challenge dependent completely at the nonparametric adaptive filter-bank techniques, that are powerful and exact, and will supply excessive answer and occasional sidelobes. during this booklet, we current those algorithms for either one-dimensional and two-dimensional spectral estimation difficulties.

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**Extra resources for Spectral Analysis of Signals: The Missing Data Case (Synthesis Lectures on Signal Processing)**

**Example text**

In other words, Gγ (ωk1 , ωk2 ) and Gµ (ωk1 , ωk2 ) contain the columns of G(ωk1 , ωk2 ) that correspond to the indices in G and U, respectively. By introducing the following matrices: ˜γ G TGγ (ω0 , ω0 ) .. 40) 24 SPECTRAL ANALYSIS OF SIGNALS and TGµ (ω0 , ω0 ) .. 38) can then be written as min µ where ˜ α ˜ µµ + G ˜ γγ − α ˜ G 2 , ˆ 0 , ω0 )a L1 ,L2 (ω0 , ω0 ) α(ω .. 42) . 42) is easily obtained as ˜ ˜ HG ˆ = G µ µ µ −1 ˜ ˜H α G µ ˜ − Gγ γ . 44) A step-by-step summary of 2-D GAPES is as follows: Step 0: Obtain an initial estimate of {α(ω1 , ω2 ), h(ω1 , ω2 )}.

2 Two-Dimensional GAPES Let G be the set of sample indices (n1 , n2 ) for which the data samples are available, and U be the set of sample indices (n1 , n2 ) for which the data samples are missing. The set of available samples {yn1 ,n2 : (n1 , n2 ) ∈ G} is denoted by the g × 1 vector γ, whereas the set of missing samples {yn1 ,n2 : (n1 , n2 ) ∈ U} is denoted by the (N1 N2 − g ) × 1 vector µ. The problem of interest is to estimate α(ω1 , ω2 ) given γ. Assume we consider a K 1 × K 2 -point DFT grid: (ωk1 , ωk2 ) = (2πk 1 / K 1 , 2π k 2 /K 2 ), for k1 = 0, .

3(c) shows the image obtained by the application of APES to the full data with a 2-D filter of size 24 × 36. Note that the two closely located vertical lines (corresponds to the loader bucket) are well resolved by APES because of its super resolution. To simulate the gapped data, we create artificial gaps in the phase history data matrix by removing the columns 10–17 and 30–37, as illustrated in Fig. 3(d). In Fig. 3(e) we show the result of applying WFFT to the data where the missing samples are set to zero.

### Spectral Analysis of Signals: The Missing Data Case (Synthesis Lectures on Signal Processing) by Yanwei Wang

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