Optimized sampling and processing of structured signals
Final Report Abstract
In this project, optimized sampling, reconstruction, and processing techniques for structured signals were analyzed, and a novel signal model for forensic DNA signals was developed. First, fundamental properties of different signal classes were studied. Novel results were obtained for the Bernstein space B∞π (bounded bandlimited signals) and the space BMO (signals of bounded mean oscillation), which naturally appears when studying the Hilbert transform for the space B∞π. The space B∞π is of particular importance in all applications where the peak value of a signal has to be controlled. For example, in wireless communication systems high peak-to-average power ratios (PAPRs) can overload power amplifiers and lead to undesirable out-of-band radiation. By determining the asymptotic growth behavior of the Hilbert transform for B∞π, the peak value problem of the Hilbert transform was solved for this space. As a further result of these findings, a new sampling theorem for bandlimited BMO functions was derived. Then, the behavior of system approximation processes for stable linear time-invariant (LTI) systems and signals in the Paley–Wiener space PW1π was analyzed. It was shown that a stable system approximation is not possible for pointwise sampling, because there exist signals and systems such that the approximation process diverges. This remains true even with oversampling, and shows a fundamental limit for the digital implementation of analog systems. In the investigation it turned out, that the use of more general measurement functionals can eliminate this divergence behavior. Additionally, it was demonstrated that the approximation process can be made more bandwidth efficient, by using a two channel approach, which allows to reduce the bandwidth of the approximation process to the input signal bandwidth. Further, the effects of thresholding and quantization on the system approximation process were studied for the space PW1π. The threshold operator leads to completely new phenomena in the approximation of stable LTI systems. It was proved that the set of signals which lead to a divergent system approximation is a residual set. Hence, for “almost all” signals in PW1π, thresholding leads to an instable system approximation process. Finally, forensic DNA signals were studied, and a novel signal model for the automatic processing and interpretation of DNA profiles was developed. DNA signals possess interesting structural properties. However, the automated analysis of forensic DNA signals is challenging, because a forensic DNA sample is usually a mixture, where more than one individual contributed. Additionally, noise and other distortions are an integral component of DNA signals, which is significant because crime labs are moving toward the analysis of samples with very low DNA mass, resulting in signals with very low signal-to-noise ratio (SNR). To develop algorithms for an automated DNA interpretation, a detailed understanding of the signal properties is necessary. By analyzing data from 643 single person samples, various properties of DNA signals were studied. Distribution classes that succinctly summarize peak-height distributions of baseline noise and allelic peaks through a small number of parameters were successfully identified. Further, a fully quantitative signal model for forensic DNA profiles that incorporates the variability in the allelic peak heights, stutter, and baseline noise was developed.
Publications
- New Perspectives on Approximation and Sampling Theory - Festschrift in honor of Paul Butzer’s 85th birthday, chapter “Signal and System Approximation from General Measurements”, pages 115–148. Applied and Numerical Harmonic Analysis. Birkhauser (Springer-Verlag), 2014
Holger Boche and Ullrich J. Mönich
- “A signal model for forensic DNA mixtures.” In 2014 Asilomar Conference on Signals, Systems and Computers, November 2014
Ullrich J. Mönich, Catherine Grgicak, Viveck Cadambe, Jason Yonglin Wu, Genevieve Wellner, Ken Duffy, and Muriel Médard
(See online at https://doi.org/10.1109/ACSSC.2014.7094478) - “A two channel approach for system approximation with general measurement functionals.” In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP ’15), 2015, 3881–3885
Ullrich J. Mönich and Holger Boche
(See online at https://doi.org/10.1109/ICASSP.2015.7178698) - “Probabilistic Characterisation of Baseline Noise in STR Profiles.” Forensic Science International: Genetics, 19:107–122, November 2015
Ullrich J. Mönich, Ken Duffy, Muriel Médard, Viveck Cadambe, Lauren Alfonse, and Catherine Grgicak
(See online at https://doi.org/10.1016/j.fsigen.2015.07.001) - “Strong Divergence for System Approximations.” Problems of Information Transmission, 51(3):240–266, July 2015
Holger Boche and Ullrich J. Mönich
(See online at https://doi.org/10.1134/S0032946015030047)
