The Algorithm Study Of Band-limited Extrapolation

Let f be a finite energy function defined over the real line(f∈L~2): Its Fourier transform (?) is zero outside [-σ,σ]:(?)(ω)=0,ω(?)[-σ,σ].Since f isanalytic,given a piece of f ,say g: [-T, T]â†'C:g(t) = f(t), t∈[-T, T] ,we can expectto recover f(t) ,when t (?) [-T, T] from g .Hence, the continuous extrapolationproblem is (under the conditions stated above) is findf(t),t(?)[-T,T],given f(t),t∈[-T,T].When the signal does not include the frequency which is higher than somefrequency, we call the signal frequency-limited signal or band-limited signal (that isband-limited function). Band-limited function extrapolation is extensively applied inspectrum estimate ,angle-limited image reconstruction,and many other areas.Papoulis and Gerchebrg presented an iterative algorithm in which the iterationconverges to the extrapolation of the band-limited signal if the initial data isnoise-free.In order to implement the Gerchberg-Papoulis algorithm for theextrapolation of one-dimensional band-limited signals, Sanz and Huang suggested aconjecture (proofed in [31]) for the discrete approximation of Gerchberg-Papoulis algorithm. There are also many other extrapolation algorithms.In this paper, two new algorithms are suggested to solve the problem of band-limited extrapolation, one is Landweber iterative algorithm, another is conjugate gradient method. Then, convergence of the two new schemes is demonstrated. It is also proofed that when choosing the relaxtion coefficients of Landweber iteration as 1 ,this method is equivalent to the Gerchberg-Papoulis algorithm for band-limited signal extrapolation. Finally a number of illustrative numerical examples are presented and discussed. When we compute numerical examples, we based on Sanz and Huang's theorem to implement our new algorithms. The results show that Landweber iterative method and conjugate gradient iterative scheme are both more efficient than Gerchberg-Papoulis algorithm.