We consider the problem of hypothesis testing for detection of a signal in Gaussian noise. We assume that the vector of measurements is unobserved, and that our observations consist of phaseless inner products with a set of known measurement vectors. This is typical of the phase retrieval problem, where the goal is to recover the vector of measurements. We provide a simple estimator for the test statistic that does not necessitate a phaseless recovery method to reconstruct the measurements. Our analysis shows that for random measurement vectors, we can reconstruct the test statistic for any signal from a sufficient number of observations, quadratic in the signal length, using a simple least-squares approach. The primary advantage of this method its simplicity and computational efficiency, which comes at the expense of requiring many more measurements. We show that for Fourier measurements vectors, our approach works only when the signal is also a Fourier vector.

In this paper, a new method for multiple fundamental frequency estimation for speech and music signals is proposed. Applications of audio and speech processing include many well-reviewed algorithms for estimating the fundamental frequency of monophonic speech and music signals. In the case of polyphonic signals, it is more difficult to successfully estimate each of the fundamental frequencies, as reflected by the dearth of existing methods addressing this problem. In this paper, a new method based on the combination of the maximum likelihood and maximum a posteriori probability criteria is derived for fundamental frequencies tracking where each one of the fundamental frequencies is modeled by a first-order Markov process. The dominant signal is modeled as a harmonic source with unknown deterministic amplitudes, while the remaining signals, including other harmonic signals, are modeled as Gaussian interference sources with an unknown covariance matrix. After estimation of the dominant source, it is removed from the signal by projection of the signal into the null subspace spanned by the estimated signal. This procedure is iterated for all the harmonic sources in the data. The algorithm is tested with speech, music, and synthetic signals where in each case, two harmonic sources of the same kind were mixed. The performance of the proposed algorithm is evaluated and compared to an existing reference method in terms of gross-error-rate as a function of signal-to-interference ratio.