A deterministic sparse FFT algorithm for vectors with small support

Abstract

In this paper we consider the special case where a signal x ∈ C N is known to vanish outside a support interval of length m < N . If the support length m of x or a good bound of it is a-priori known we derive a sublinear deterministic algorithm to compute x from its discrete Fourier transform x ∈ C N . In case of exact Fourier measurements we require only O(m log m) arithmetical operations. For noisy measurements, we propose a stable O(m log N ) algorithm.

Real sparse fast DCT for vectors with short support

“…In this paper we want to find a deterministic algorithm for reconstructing x ∈ R N with short support of length m from its discrete cosine transform of type II, x II , if an upper bound M ≥ m is known and using, unlike in [3], only real arithmetic. In order to do so we adapt techniques used in [3,[12][13][14] for the FFT reconstruction of vectors with short support to the real DCT setting. There exist several different factorizations of the orthogonal matrix C II n , but the following one, see Lemma 2.2 in [11], has proven to be particularly useful in our case.…”

Sparse fast DCT for vectors with one-block support

“…The vector y is computed using an iterative procedure. As in [15,16] we use the 2 j -length periodizations y (j) of y ∈ R 2 J for j ∈ {0, . .…”

“…Using the close relation between the DFT and the DCT-II shown in Lemma 2.1, this problem can be transformed into deriving a sparse FFT algorithm for reconstructing y = (x T , (J N x) T ) T , which has a reflected block support, from y. For this purpose we extend recent approaches in [15,16] for FFT reconstruction of vectors with one-block support to our setting. We assume that y satisfies (1) in order to avoid cancellation of nonzero entries in the iterative algorithm.…”

“…We assume that y satisfies (1) in order to avoid cancellation of nonzero entries in the iterative algorithm. Similarly as in [15,16] we employ a divide-and-conquer technique to efficiently compute y from y.…”

“…Lemma 2.2 in [15] shows how the Fourier transform of a vector u ∈ R 2 j+1 , j ≥ 0, changes if its entries are cyclically shifted by 2 j . Lemma 3.2 Let u ∈ R 2 j+1 , j ≥ 0, and let the shifted vector u 1 := u 1…”