Signal reconstruction from noisy-phase and -magnitude data

Alexander M. Taratorin and Samuel Sideman

Alexander M. Taratorin^{1,}^{2} and Samuel Sideman^{1}

^{1}When this research was performed, the authors were with the Heart System Research Center, The Julius Silver Institute, Department of Biomedical Engineering, Technion-Israel Institute of Technology, Haifa 32000,
Israel.

^{2}A. M. Taratorin is now with Guzik Technical Enterprises, 4620 Fortran Drive, San Jose, California 95134.

Signal reconstruction based on knowing either the magnitude or the phase of the Fourier transform of the signal is important in numerous applications, and the problem of signal reconstruction from noisy-phase and noisy-magnitude data is addressed. The proposed procedure relates to the deviations of the available magnitude and phase estimates from their exact values in the reconstruction algorithm by use of spectral prototype constraint sets. The properties of these new constraint sets for the magnitude and the phase of the Fourier transform are analyzed, and the corresponding projection operators are constructed. Simulation results indicate improvement of the performance of reconstructions from noisy-phase and -magnitude values based on these sets.

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Values of distance given by Eq. (8).
Standard deviations are averaged over 100 noise realizations. EM, standard deviation between the signal reconstructed from noisy magnitude and the original signal; EM2, standard deviation between the signal reconstructed by the M2 constraint set and the original signal.

Table 2

Performance of the Iterative Algorithm [Eq. (18)] Utilizing the M1 Constraint Set for Various Values of ∊

Values of distance given by Eq. (7).
Standard deviations are averaged over 100 noise realizations.
EM, standard deviation between the signal reconstructed from noisy magnitude and the original signal; EM1, standard deviation between the signal reconstructed by the MI constraint set and the original signal; AEM1, standard deviation between the signal reconstructed by the frequency-adaptive M1 constraint set and the original signal.

Table 3

Performance of the Iterative Algorithm [Eq. (17)] Utilizing the P1 Constraint Set for Various Values of σ

Values of distance given by Eq. (12).
Standard deviations are averaged over 100 noise realizations. EP, standard deviation between the signal reconstructed from the noisy phase and the original signal; EP1, standard deviation between the signal reconstructed by the P1 constraint set and the original signal.

Table 4

Performance of the Iterative Algorithm [Eq. (17)] Utilizing the P2 Constraint Set for Various Values of σ

Values of distance given by Eq. (13).
Standard deviations are averaged over 100 noise realizations. EP, standard deviation between the signal reconstructed from the noisy phase and the original signal; EP2, standard deviation between the signal reconstructed by the P2 constraint set and the original signal.

Tables (4)

Table 1

Performance of the Iterative Algorithm [Eq. (18)] Utilizing the M2 Constraint Set for Various Values of ∊

Values of distance given by Eq. (8).
Standard deviations are averaged over 100 noise realizations. EM, standard deviation between the signal reconstructed from noisy magnitude and the original signal; EM2, standard deviation between the signal reconstructed by the M2 constraint set and the original signal.

Table 2

Performance of the Iterative Algorithm [Eq. (18)] Utilizing the M1 Constraint Set for Various Values of ∊

Values of distance given by Eq. (7).
Standard deviations are averaged over 100 noise realizations.
EM, standard deviation between the signal reconstructed from noisy magnitude and the original signal; EM1, standard deviation between the signal reconstructed by the MI constraint set and the original signal; AEM1, standard deviation between the signal reconstructed by the frequency-adaptive M1 constraint set and the original signal.

Table 3

Performance of the Iterative Algorithm [Eq. (17)] Utilizing the P1 Constraint Set for Various Values of σ

Values of distance given by Eq. (12).
Standard deviations are averaged over 100 noise realizations. EP, standard deviation between the signal reconstructed from the noisy phase and the original signal; EP1, standard deviation between the signal reconstructed by the P1 constraint set and the original signal.

Table 4

Performance of the Iterative Algorithm [Eq. (17)] Utilizing the P2 Constraint Set for Various Values of σ

Values of distance given by Eq. (13).
Standard deviations are averaged over 100 noise realizations. EP, standard deviation between the signal reconstructed from the noisy phase and the original signal; EP2, standard deviation between the signal reconstructed by the P2 constraint set and the original signal.