Phase Retrieval for Wide Band Signals

Abstract

This study investigates the phase retrieval problem for wideband signals. We solve the following problem: given f ∈ L 2 (R) with Fourier transform in L 2 (R, e 2c|x| dx), we find all functions g ∈ L 2 (R) with Fourier transform in L 2 (R, e 2c|x| dx), such that |f (x)| = |g(x)| for all x ∈ R. To do so, we first translate the problem to functions in the Hardy spaces on the disc via a conformal bijection, and take advantage of the inner-outer factorization. We also consider the same problem with additional constraints involving some transforms of f and g, and determine if these constraints force uniqueness of the solution.2010 Mathematics Subject Classification. 30D05, 30H10, 42B10, 94A12.

On the effect of zero-flipping on the stability of the phase retrieval problem in the Paley-Wiener class

“…if and only if f belongs to the Hardy space on the strip H 2 τ (S λ ) (see [18] and references therein). Here, H 2 τ (S λ ) is the collection of all holomorphic functions on the strip…”

“…However we will show that F a f is wide-banded, that is, its Fourier transform statisfies a square-integrability condition with an exponential weight. It turns out that this problem was solved in our previous work in [18].…”

“…|(C a,b − C b,b )(x)| dx = 2 Im a R |τ βa f (x) − τ β b f (x)||(Rγ b * τ β b f )(x)| dx = 2 Im a R |τ βa f (x) − τ β b f (x)| +∞ 0 ibs f (x − β b + s) ds dx = 2 Im a R |τ βa f (x) − τ β b f (x)| e −ib(x−β b ) L x−β b e iby f (y) dy dx Im a R |τ βa f (x) − τ β b f (x)| e Im b (x−β b ) L x−β b e − Im by | f (y)| dy dx ≤ 2 Im b C(b) • ω 2 ( f ; β a − β b )||f || 2 .For the last term, the bounds from Lemmas 3.5 and 3.7 imply thatR |(D a,b − D b,b )(x)| dx = 4 Im b R |(Rγ b * τ β b f )(x)|| Im a (Rγ a * τ βa f )(x) − Im b (Rγ b * τ β b f )(x)| dx ≤ 4 Im b C(b)||f || 2 || Im a (Rγ a * τ βa f ) − Im b (Rγ b * τ β b f )|| Im b C(b)C(a, b)||f || 2 2 + 4 √ 2π Im b C(b) • ω 2 ( f ; β a − β b )||f || 2 .Combining these three estimates from above, we getR (B a,b − B b,b )(x) + (C a,b − C b,b ) (x) + (D a,b − D b,b ) (x) dx ≤ R | (B a,b − B b,b ) (x)| + | (C a,b − C b,b ) (x)| + | (D a,b − D b,b ) (x)| dx ≤ 2 √ 2π + (2 + 4 √ 2π) Im b C(b) ω 2 ( f ; β a − β b )||f || 2 + 2C(a, b) + 4 Im b C(b)C(a, b) ||f || 2Finally, from(18), we obtain inf|c|=1 ||F a f − cF b f || 2 2 ≤ 2 āb a b F a f , F b f − ||f || 2 b C(b) ω 2 ( f ; β a − β b )||f || 2 + 2C(a, b) + 4 Im b C(b)C(a, b) ||f || 2(a, b) = 2C(a, b) + 4 Im b C(b)C(a, b), (22) so that C 2 (a, b) −→ 0 as a −→ b, we obtain the theorem.…”