S. R. Curtis, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform sign information,”IEEE Trans. Acoustics Speech Signal Process. ASSP-33, 643–657 (1985).

[CrossRef]

P. L. Van Hove, M. H. Hayes, J. L. Lim, A. V. Oppenheim, “Signal reconstruction. from signed Fourier transform magnitude,”IEEE Trans. Acoustics ASSP-31, 1286–1293 (1983).

[CrossRef]

A. Levi, H. Stark, “Signal restoration from phase by projections onto convex sets,”J. Opt. Soc. Am. 73, 810–822 (1983).

[CrossRef]

D. C. Youla, H. Webb, “Image restoration by the method of projections onto convex sets. Part I,”IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).

[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of projections onto convex sets. Part II,”IEEE Trans. Med. Imaging TMI-1, 95–101 (1982).

[CrossRef]

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,”IEEE Trans. Acoustics Speech Signal Process. ASSP-30, 140–154 (1982).

[CrossRef]

See, for example, R. V. Churchill, Complex Variables and Applications (McGraw-Hill, New York, 1948), p. 194.

S. R. Curtis, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform sign information,”IEEE Trans. Acoustics Speech Signal Process. ASSP-33, 643–657 (1985).

[CrossRef]

P. L. Van Hove, M. H. Hayes, J. L. Lim, A. V. Oppenheim, “Signal reconstruction. from signed Fourier transform magnitude,”IEEE Trans. Acoustics ASSP-31, 1286–1293 (1983).

[CrossRef]

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,”IEEE Trans. Acoustics Speech Signal Process. ASSP-30, 140–154 (1982).

[CrossRef]

M. H. Hayes, “The unique reconstruction of multidimensional sequences from Fourier transform magnitude or phase,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, to be published), Chap. 6.

P. L. Van Hove, M. H. Hayes, J. L. Lim, A. V. Oppenheim, “Signal reconstruction. from signed Fourier transform magnitude,”IEEE Trans. Acoustics ASSP-31, 1286–1293 (1983).

[CrossRef]

S. R. Curtis, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform sign information,”IEEE Trans. Acoustics Speech Signal Process. ASSP-33, 643–657 (1985).

[CrossRef]

S. R. Curtis, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform sign information,”IEEE Trans. Acoustics Speech Signal Process. ASSP-33, 643–657 (1985).

[CrossRef]

P. L. Van Hove, M. H. Hayes, J. L. Lim, A. V. Oppenheim, “Signal reconstruction. from signed Fourier transform magnitude,”IEEE Trans. Acoustics ASSP-31, 1286–1293 (1983).

[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of projections onto convex sets. Part II,”IEEE Trans. Med. Imaging TMI-1, 95–101 (1982).

[CrossRef]

A. Levi, H. Stark, “Image restoration by the method of generalized projections with applications to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).

[CrossRef]

A. Levi, H. Stark, “Signal restoration from phase by projections onto convex sets,”J. Opt. Soc. Am. 73, 810–822 (1983).

[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of projections onto convex sets. Part II,”IEEE Trans. Med. Imaging TMI-1, 95–101 (1982).

[CrossRef]

P. L. Van Hove, M. H. Hayes, J. L. Lim, A. V. Oppenheim, “Signal reconstruction. from signed Fourier transform magnitude,”IEEE Trans. Acoustics ASSP-31, 1286–1293 (1983).

[CrossRef]

D. C. Youla, H. Webb, “Image restoration by the method of projections onto convex sets. Part I,”IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).

[CrossRef]

D. C. Youla, H. Webb, “Image restoration by the method of projections onto convex sets. Part I,”IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).

[CrossRef]

P. L. Van Hove, M. H. Hayes, J. L. Lim, A. V. Oppenheim, “Signal reconstruction. from signed Fourier transform magnitude,”IEEE Trans. Acoustics ASSP-31, 1286–1293 (1983).

[CrossRef]

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,”IEEE Trans. Acoustics Speech Signal Process. ASSP-30, 140–154 (1982).

[CrossRef]

S. R. Curtis, J. S. Lim, A. V. Oppenheim, “Signal reconstruction from Fourier transform sign information,”IEEE Trans. Acoustics Speech Signal Process. ASSP-33, 643–657 (1985).

[CrossRef]

D. C. Youla, H. Webb, “Image restoration by the method of projections onto convex sets. Part I,”IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).

[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of projections onto convex sets. Part II,”IEEE Trans. Med. Imaging TMI-1, 95–101 (1982).

[CrossRef]

We work in a Hilbert space ℋ with the norm of g ∈ ℋ denoted by ||g|| and the inner product of g ∈ ℋ denoted by (g, f). For more details refer to Refs. 2, 3, 6, and 7.

M. H. Hayes, “The unique reconstruction of multidimensional sequences from Fourier transform magnitude or phase,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, to be published), Chap. 6.

A set C⊆ ℋ is said to be closed if given a sequence {fn}, fn∈ C∀ n, limn→∞||fn− f|| = 0 implies that f ∈ C, i.e., C is closed if it contains all its limit functions.

Let C′ be the set of all limit points of C in ℋ; then the closure of C(denoted by C¯) is C¯ = C∪ C′. A set is closed if and only if C=C¯.

m(A) denotes the (Lebesgue) measure of the set A, where A is any set of real numbers. (The Lebesgue measure of an interval is its length. The measure of an isolated point is zero. The measure of the real line is infinite.)

L2is the space of all square-integrable functions over −∞ < t< ∞.

The theorem that states this result uses a more general definition of the signum function that permits different partitions of the complex plane. For the case under consideration in this paper (the standard signum function) the theorem requires, in addition, that the sequence vanish at the origin.

A (complex) function is entire if it is analytic everywhere. The FT of an integrable function with finite support is entire.

See, for example, R. V. Churchill, Complex Variables and Applications (McGraw-Hill, New York, 1948), p. 194.

An alternative and more concise proof is that every causal function may be written as a sum of an even function fe and an odd function fo. Now, we prove the statement by contradiction. Assume that Im F(ω) = 0. Since Im Fe(ω) = 0 (an even function) we conclude that Im Fo(ω) = 0. But Re Fo(ω) = 0 (an odd function). Thus fo= 0 and f= fe, which contradicts our assumptions on f(f causal and f≠ 0).

See, for example, Refs. 2–4 below, where further references may be found.