Stepwise Iterative Fourier Transform: The SIFT

A program, designed specifically to study the respective effects of some common data problems on results obtained through stepwise iterative Fourier transformation of synthetic data with known waveform composition, was outlined. Included in this group were the problems of gaps in the data, different time-series lengths, periodic but nonsinusoidal waveforms, and noisy (low signal-to-noise) data. Results on sinusoidal data were also compared with results obtained on narrow band noise with similar characteristics. The findings showed that the analytic procedure under study can reliably reduce data in the nature of (1) sinusoids in noise, (2) asymmetric but periodic waves in noise, and (3) sinusoids in noise with substantial gaps in the data. The program was also able to analyze narrow-band noise well, but with increased interpretational problems. The procedure was shown to be a powerful technique for analysis of periodicities, in comparison with classical spectrum analysis techniques. However, informed use of the stepwise procedure nevertheless requires some background of knowledge concerning characteristics of the biological processes under study

A universal two-way approach for estimating unknown frequencies for unknown number of sinusoids in a signal based on eigenspace analysis of Hankel matrix

YesWe develop a novel approach to estimate the n unknown constituent frequencies of a noiseless signal that comprises of unknown number, n, of sinusoids of unknown phases and unknown amplitudes. The new two way approach uses two constraints to accurately estimate the unknown frequencies of the sinusoidal components in a signal. The new approach serves as a verification test for the estimated unknown frequencies through the estimated count of the unknown number of frequencies. The Hankel matrix, of the time domain samples of the signal, is used as a basis for further analysis in the Pisarenko harmonic decomposition. The new constraints, the Existence Factor (EF) and the Component Factor (CF), have been introduced in the methodology based on the relationships between the components of the sinusoidal signal and the eigenspace of the Hankel matrix. The performance of the developed approach has been tested to correctly estimate any number of frequencies within a signal with or without a fixed unknown bias. The method has also been tested to accurately estimate the very closely spaced low frequencies.Innovate U

FRIDA: FRI-Based DOA Estimation for Arbitrary Array Layouts

In this paper we present FRIDA---an algorithm for estimating directions of arrival of multiple wideband sound sources. FRIDA combines multi-band information coherently and achieves state-of-the-art resolution at extremely low signal-to-noise ratios. It works for arbitrary array layouts, but unlike the various steered response power and subspace methods, it does not require a grid search. FRIDA leverages recent advances in sampling signals with a finite rate of innovation. It is based on the insight that for any array layout, the entries of the spatial covariance matrix can be linearly transformed into a uniformly sampled sum of sinusoids.Comment: Submitted to ICASSP201

High-resolution sinusoidal analysis for resolving harmonic collisions in music audio signal processing

Many music signals can largely be considered an additive combination of multiple sources, such as musical instruments or voice. If the musical sources are pitched instruments, the spectra they produce are predominantly harmonic, and are thus well suited to an additive sinusoidal model. However, due to resolution limits inherent in time-frequency analyses, when the harmonics of multiple sources occupy equivalent time-frequency regions, their individual properties are additively combined in the time-frequency representation of the mixed signal. Any such time-frequency point in a mixture where multiple harmonics overlap produces a single observation from which the contributions owed to each of the individual harmonics cannot be trivially deduced. These overlaps are referred to as overlapping partials or harmonic collisions. If one wishes to infer some information about individual sources in music mixtures, the information carried in regions where collided harmonics exist becomes unreliable due to interference from other sources. This interference has ramifications in a variety of music signal processing applications such as multiple fundamental frequency estimation, source separation, and instrumentation identification. This thesis addresses harmonic collisions in music signal processing applications. As a solution to the harmonic collision problem, a class of signal subspace-based high-resolution sinusoidal parameter estimators is explored. Specifically, the direct matrix pencil method, or equivalently, the Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) method, is used with the goal of producing estimates of the salient parameters of individual harmonics that occupy equivalent time-frequency regions. This estimation method is adapted here to be applicable to time-varying signals such as musical audio. While high-resolution methods have been previously explored in the context of music signal processing, previous work has not addressed whether or not such methods truly produce high-resolution sinusoidal parameter estimates in real-world music audio signals. Therefore, this thesis answers the question of whether high-resolution sinusoidal parameter estimators are really high-resolution for real music signals. This work directly explores the capabilities of this form of sinusoidal parameter estimation to resolve collided harmonics. The capabilities of this analysis method are also explored in the context of music signal processing applications. Potential benefits of high-resolution sinusoidal analysis are examined in experiments involving multiple fundamental frequency estimation and audio source separation. This work shows that there are indeed benefits to high-resolution sinusoidal analysis in music signal processing applications, especially when compared to methods that produce sinusoidal parameter estimates based on more traditional time-frequency representations. The benefits of this form of sinusoidal analysis are made most evident in multiple fundamental frequency estimation applications, where substantial performance gains are seen. High-resolution analysis in the context of computational auditory scene analysis-based source separation shows similar performance to existing comparable methods

Multipath Parameter Estimation from OFDM Signals in Mobile Channels

We study multipath parameter estimation from orthogonal frequency division multiplex signals transmitted over doubly dispersive mobile radio channels. We are interested in cases where the transmission is long enough to suffer time selectivity, but short enough such that the time variation can be accurately modeled as depending only on per-tap linear phase variations due to Doppler effects. We therefore concentrate on the estimation of the complex gain, delay and Doppler offset of each tap of the multipath channel impulse response. We show that the frequency domain channel coefficients for an entire packet can be expressed as the superimposition of two-dimensional complex sinusoids. The maximum likelihood estimate requires solution of a multidimensional non-linear least squares problem, which is computationally infeasible in practice. We therefore propose a low complexity suboptimal solution based on iterative successive and parallel cancellation. First, initial delay/Doppler estimates are obtained via successive cancellation. These estimates are then refined using an iterative parallel cancellation procedure. We demonstrate via Monte Carlo simulations that the root mean squared error statistics of our estimator are very close to the Cramer-Rao lower bound of a single two-dimensional sinusoid in Gaussian noise.Comment: Submitted to IEEE Transactions on Wireless Communications (26 pages, 9 figures and 3 tables

Asymptotic Performance for Delayed Exponential Process

International audienceThe damped and delayed sinusoidal (DDS) model can be defined as the sum of sinusoids whose waveforms can be damped and delayed. This model is suitable for compactly modeling short time events. This property is closely related to its ability to reduce the time-support of each sinusoidal component. In this correspondence, we derive exact and approximate asymptotic Cramér–Rao bounds (CRBs) for the DDS model. This analysis shows that this model has better, or at least similar, theoretical performance than the well-known exponentially damped sinusoidal (EDS) model. In particular, the performance in the DDS case is significantly improved compared to that of the EDS for closely spaced sinusoids thanks to the nonzero time delays. Consequently, we can exploit the advantageous properties of the DDS model and, in the same time, we can keep high theoretical model parameter estimation accuracy