Phaseless Signal Recovery in Infinite Dimensional Spaces Using Structured Modulations

Abstract

This paper considers the recovery of continuous signals in infinite dimensional spaces from the magnitude of their frequency samples. It proposes a sampling scheme which involves a combination of oversampling and modulations with complex exponentials. Sufficient conditions are given such that almost every signal with compact support can be reconstructed up to a unimodular constant using only its magnitude samples in the frequency domain. Finally it is shown that an average sampling rate of four times the Nyquist rate is enough to reconstruct almost every time-limited signal.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. 1.

    The variable \(t\) stands here for the one or two dimensional spatial dimension.

References

  1. 1.

    Balan, R., Bodmann, B.G., Casazza, P.G., Edidin, D.: Painless reconstruction from magnitudes of frame coefficients. J. Fourier Anal. Appl. 15(4), 488–501 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Balan, R., Casazza, P.G., Edidin, D.: On signal reconstruction without phase. Appl. Comput. Harmon. Anal. 20(3), 345–356 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Bauschke, H.H., Combettes, P.L., Luke, D.R.: Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization. J. Opt. Soc. Am. A 19(7), 1334–1345 (2002)

    Article  MathSciNet  Google Scholar 

  4. 4.

    Boche, H., Pohl, V.: On the calculation of the Hilbert transform from interpolated data. IEEE Trans. Inform. Theory 54(5), 2358–2366 (2008)

    Article  MathSciNet  Google Scholar 

  5. 5.

    Bodmann, B.G., Hammen, N.: Stable phase retrieval with low-redundancy frames. Adv. Comput. Math. (2014). doi:10.1007/s10444-014-9359-y

  6. 6.

    Burge, R.E., Fiddy, M.A., Greenaway, A.H., Ross, G.: The phase problem. Proc. R. Soc. Lond. A 350(1661), 192–212 (1976)

    Article  MathSciNet  Google Scholar 

  7. 7.

    Candès, E.J., Eldar, Y.C., Strohmer, T., Voroninski, V.: Phase retrieval via matrix completion. SIAM J. Imaging Sci. 6(1), 199–225 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Duffin, R., Schaeffer, A.C.: Some properties of functions of exponential type. Bull. Am. Math. Soc. 44, 236–240 (1938)

    Article  MathSciNet  Google Scholar 

  9. 9.

    Falldorf, C., Agour, M., Kopylow, C.V., Bergmann, R.B.: Phase retrieval by means of spatial light modulator in the Fourier domain of an imaging system. Appl. Opt. 49(10), 1826–1830 (2010)

    Article  Google Scholar 

  10. 10.

    Fickus, M., Mixon, D.G., Nelson, A.A., Wang, Y.: Phase retrieval from very few measurements. Linear Algebr. Appl. 449, 475–499 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Fienup, J.R.: Phase retrieval algorithms: a comparison. Appl. Opt. 21(15), 2758–2769 (1982)

    Article  Google Scholar 

  12. 12.

    Fienup, J.R., Marron, J.C., Schulz, T.J., Seldin, J.H.: Hubble space telescope characterized by using phase-retrieval algorithms. Appl. Opt. 32(10), 1747–1767 (1993)

    Article  Google Scholar 

  13. 13.

    Finkelstein, J.: Pure-state informationally complete and “really” complete measurements. Phys. Rev. A 70, 052107 (2004)

    Article  MathSciNet  Google Scholar 

  14. 14.

    Goodman, J.W.: Introduction to Fourier Optics. McGraw-Hill Comp, New York (1996)

    Google Scholar 

  15. 15.

    Hayes, M.H., Lim, J.S., Oppenheim, A.V: Signal reconstruction from phase or magnitude. IEEE Trans. Acoust. Speech Signal Process. ASSP-28, 6, 672–680 (1980)

  16. 16.

    Hörmander, L.: Linear Partial Differential Operators. Springer, Berlin (1976)

    MATH  Google Scholar 

  17. 17.

    Jaming, P.: The phase retrieval problem for the radar ambiguity function and vice versa. IEEE International Radar Conference, May 2010

  18. 18.

    Katkovnik, V., Astola, J.: Phase retrieval via spatial light modulators phase modulation in 4f optical setup: numerical inverse imaging with sparse regularization for phase and amplitude. J. Opt. Soc. Am A 29(1), 105–116 (2012)

    Article  Google Scholar 

  19. 19.

    Levenshtein, V.: On designs in compact metric spaces and a universal bound on their size. Discret. Math. 192, 251–271 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. 20.

    Levin, B.Y.: Lectures on Entire Functions. American Mathematical Society, Providence, RI (1997)

    MATH  Google Scholar 

  21. 21.

    Levin, B.Y., Ostrovskii, I.V.: Small perturbations of the set of roots of sine-type functions. Izv. Akad. Nauk SSSR Ser. Mat 43(1), 87–110 (1979)

    MathSciNet  Google Scholar 

  22. 22.

    Lu, Y.M., Vetterli, M.: Sparse spectral factorization: unicity and reconstruction algorithms. Proceedings 36th International Conference on Acoustics, Speech, and Signal Processing (ICASSP), May 2011, pp. 5976–5979

  23. 23.

    Marchesini, S.: Phase retrieval and saddle-point optimization. J. Opt. Soc. Am. A 24(10), 3289–3296 (2007)

    Article  Google Scholar 

  24. 24.

    Miao, J., Ishikawa, T., Shen, Q., Earnest, T.: Extending X-ray crystallography to allow the imaging of noncrystalline materials, cells, and single protein complexes. Annu. Rev. Phys. Chem. 59, 387–410 (2008)

    Article  Google Scholar 

  25. 25.

    Millane, R.P.: Phase retrieval in crystallography and optics. J. Opt. Soc. Am. A 7(3), 394–411 (1990)

    Article  Google Scholar 

  26. 26.

    Mönich, U.J., Boche, H.: Non-equidistant sampling for bounded bandlimited signals. Signal Process. 90(7), 2212–2218 (2010)

    Article  MATH  Google Scholar 

  27. 27.

    Ortega-Cerdà, J., Seip, K.: Multipliers for entire functions and an interpolation problem of Beurling. J. Funct. Anal. 162(2), 400–415 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  28. 28.

    Pohl, V., Yang, F., Boche, H.: Phase retrieval from low-rate samples. Sampl. Theory Signal Image Process. (2014). arXiv:1311.7045

  29. 29.

    Pohl, V., Yapar, C., Boche, H., Yang, F.: A phase retrieval method for signals in modulation-invariant spaces. Proceedings of 39th International Conference on Acoustics, Speech, and Signal Processing (ICASSP), May 2014

  30. 30.

    Rabiner, L., Juang, B.-H.: Fundamentals of Speech Recognition. Prentice Hall Inc, Englewood Cliffs (1993)

    Google Scholar 

  31. 31.

    Ross, G., Fiddy, M.A., Nieto-Vesperinas, M., Wheeler, M.W.L.: The phase problem in scattering phenomena: the zeros of entire functions and their significance. Proc. R. Soc. Lond. A 360(1700), 25–45 (1978)

    Article  MathSciNet  Google Scholar 

  32. 32.

    Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, Boston (1991)

    MATH  Google Scholar 

  33. 33.

    Thakur, G.: Reconstruction of bandlimited functions from unsigned samples. J. Fourier Anal. Appl. 17(4), 720–732 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  34. 34.

    Vladimirov, V.S.: Equations of Mathematical Physics. Marcel Dekker Inc, New York (1971)

    Google Scholar 

  35. 35.

    Xiao, X., Shen, Q.: Wave propagation and phase retrieval in Fresnel diffraction by a distorted-object approach. Phys. Rev. B 72, 033103 (2005)

    Article  Google Scholar 

  36. 36.

    Yang, F., Pohl, V., Boche, H.: Phase retrieval via structured modulations in Paley–Wiener spaces. Proceedings of 10th International Conference on Sampling Theory and Applications (SampTA), July 2013.

  37. 37.

    Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, Cambridge (2001)

    MATH  Google Scholar 

  38. 38.

    Zauner, G.: Quantum designs: foundations of a noncommutative design theory. Int. J. Quantum Inform. 9(1), 445–507 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  39. 39.

    Zhang, F., Pedrini, G., Osten, W.: Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation. Phys. Rev. A 75, 043805 (2007)

    Article  Google Scholar 

Download references

Acknowledgments

This work was partly supported by the German Research Foundation (DFG) under Grant PO 1347/2-1 and BO 1734/22-1.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Volker Pohl.

Additional information

Communicated by Thomas Strohmer.

Appendix: An Auxiliary Lemma

Appendix: An Auxiliary Lemma

Lemma 7.1

Let \(S\) be a sine-type function of type \(\sigma \) and let \(\Lambda = \{\lambda _n\}_{n\in \mathbb {Z} }\) be its zero set. If \(\lambda _{n_{0}} \in \Lambda \) is an arbitrary zero of \(S\) then

$$\begin{aligned} \sum _{n\ne n_{0}} \frac{1}{|\lambda _n - \lambda _{n_{0}}|^{2}} \le C < \infty \end{aligned}$$

with a constant \(C\) which depends only on \(S\) but not on \(n_{0}\).

Proof

Since \(S\) is a sine-type function there are constants \(A,B,H\) such that inequalities (6) hold. Furthermore there exists a constant \(\delta > 0\) such that \(|\lambda _{m} - \lambda _{n}| \ge \delta \) for all \(m\ne n\) and a constant \(\alpha >0\) such that \(|S'(\lambda _{n})| \ge \alpha \) for all \(\lambda _{n} \in \Lambda \) (see, e.g., [20, 37]). In particular, \(S\) is an entire function of exponential type \(\sigma \). Then the Phragmén–Lindelöf principle and (6) imply that \(S\) is bounded on every line parallel to \( \mathbb {R} \). Therefore there exists a constant \(M\) such that

$$\begin{aligned} |S(\xi + \mathrm {i} \eta )| \le M\, \mathrm {e} ^{\sigma |\eta |}\, \quad \text {for all}\ \xi , \eta \in \mathbb {R} . \end{aligned}$$
(32)

Set \(\widetilde{S}(z) := S(z + \lambda _{n_{0}})\) and write \(\lambda _{n_{0}} = \xi _{0} + \mathrm {i} \eta _{0}\). It follows from (6) that \(|\eta _{0}| \le H\) and (32) gives

$$\begin{aligned} \big |\widetilde{S}(\xi + \mathrm {i} \eta )\big | \le M\, \mathrm {e} ^{\sigma |\eta + \eta _{0}|} \le M\, \mathrm {e} ^{\sigma H}\, \mathrm {e} ^{\sigma |\eta |} \quad \text {for all}\ \xi , \eta \in \mathbb {R} . \end{aligned}$$

Consequently \(|\widetilde{S}(z)| \le \widetilde{M}\, \mathrm {e} ^{\sigma |z|}\) for all \(z\in \mathbb {C} \) and with the constant \(\widetilde{M} = M \mathrm {e} ^{\sigma H}\) which depends only on \(S\) but not on \(n_{0}\). The zeros of \(\widetilde{S}\) are \(\widetilde{\lambda }_{n} = \lambda _{n} - \lambda _{n_0}\), and we assume that they are ordered increasingly by their absolute values, i.e. such that \(0 = |\widetilde{\lambda }_{0}| < \delta \le |\widetilde{\lambda }_1| \le |\widetilde{\lambda }_{2}| \le \cdots \). Then we define \(Q(z) := \widetilde{S}(z)/z\). This is again an entire function of exponential type \(\sigma \) which satisfies

$$\begin{aligned}&|Q(z)| \le P\, \mathrm {e} ^{\sigma |z|}\quad \text {for all}\ z\in \mathbb {C} \quad \text {and with}\ P = \widetilde{M}\, \mathrm {e} ^{\sigma } = M\, \mathrm {e} ^{\sigma (H+1)},\nonumber \\&\big | Q(0) \big | = \big | \widetilde{S}'(0) \big | = \big | \widetilde{S}'(\widetilde{\lambda }_{0}) \big | \ge \alpha , \end{aligned}$$
(33)

and the zero set of \(Q\) is obviously \(\{\widetilde{\lambda }_n\}^{\infty }_{n=1}\). If \(n(r)\) denotes the number of zeros of \(Q\) for which \(|\widetilde{\lambda }_n| \le r\), then Jensen’s formula [20] and (33) imply

$$\begin{aligned} N(r) := \int ^{r}_{0} \frac{n(\tau )}{\tau }\, \mathrm {d} \tau = \frac{1}{2\pi } \int ^{\pi }_{-\pi } \ln |Q(r \mathrm {e} ^{ \mathrm {i} \theta })|\, \mathrm {d} \theta - \ln |Q(0)| \le \ln (P/\alpha ) + \sigma \, r. \end{aligned}$$

Since \(n(r)\) is non-decreasing, we have \(N(e r) \ge \int ^{e r}_{r} n(\tau )\, \tau ^{-1}\, \mathrm {d} \tau \ge n(r)\), where \(e = \mathrm {e} ^{1}\). This yields the upper bound \(n(r) \le \ln (P/\alpha ) + e\, \sigma \, r\). If we take \(r = |\widetilde{\lambda }_{n}|\) then \(n(r) = n\) and one gets \(n = n(r) \le \ln (P/\alpha ) + e\, \sigma \, |\widetilde{\lambda }_n|\). Now we have

$$\begin{aligned} \sum _{n\ne n_{0}} \big |\lambda _n - \lambda _{n_{0}}\big |^{-2} = \sum ^{\infty }_{n=1} \big |\widetilde{\lambda }_{n}\big |^{-2} = \sum ^{N-1}_{n=1} \big |\widetilde{\lambda }_{n}\big |^{-2} + \sum ^{\infty }_{n=N} \big |\widetilde{\lambda }_{n}\big |^{-2} \end{aligned}$$

where \(N \in \mathbb {N} \) was chosen as the smallest integer such that \(N \ge \ln (P/\alpha ) + 1\). Next we use for the first sum on the right hand side that \(|\widetilde{\lambda }_{n}| \ge \delta \) for all \(n \ge 1\). In the second sum we apply the bound \(|\widetilde{\lambda }_{n}| \ge [ n-\ln (P/\alpha ) ]/[e\, \sigma ]\) from above. This gives

$$\begin{aligned} \sum _{n\ne n_{0}} \frac{1}{\big |\lambda _n - \lambda _{n_{0}}\big |^{2}}&\le \frac{N-1}{\delta ^{2}} + \sum ^{\infty }_{n=N} \frac{ \mathrm {e} ^{2} \sigma ^{2}}{[n - \ln (P/\alpha )]^{2}}\\&\le \frac{\ln (P/\alpha ) + 1}{\delta ^{2}} + \mathrm {e} ^{2} \sigma ^{2} \sum ^{\infty }_{m=1} \frac{1}{m^2} = \frac{\ln (e P / \alpha )}{\delta ^{2}} + \frac{ \mathrm {e} ^{2} \pi ^{2}}{6}\, \sigma ^{2} =: C \end{aligned}$$

where the constant \(C<\infty \) is independent of \(n_0\) and depends only on \(S\). \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pohl, V., Yang, F. & Boche, H. Phaseless Signal Recovery in Infinite Dimensional Spaces Using Structured Modulations. J Fourier Anal Appl 20, 1212–1233 (2014). https://doi.org/10.1007/s00041-014-9352-3

Download citation

Keywords

  • Bernstein spaces
  • Interpolation
  • Phase retrieval
  • Sampling
  • Signal reconstruction

Mathematics Subject Classification

  • Primary 30D10
  • 94A20
  • Secondary 42C15
  • 94A12