Asymptotic behavior of the minimum mean squared error threshold for noisy wavelet coefficients of piecewise smooth signals

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Asymptotic Behavior of the Minimum Mean Squared Error Threshold for Noisy Wavelet Coefficients of Piecewise Smooth Signals

This paper investigates the minimum risk threshold for wavelet coefficients with additive, homoscedastic, Gaussian noise, and for a soft-thresholding scheme. We start from N samples from a signal on a continuous time axis. For piecewise smooth signals, and for N ! 1, this threshold behaves as C p 2 log N , where is the noise standard deviation. The paper contains an original proof for this asymptotic behavior as well as an intuitive explanation. This behavior is necessary to prove the asymptotic optimality of a generalized cross validation procedure in estimating the minimum risk threshold

Noise reduction method for the heart sound records from digital stethoscope

In recent years, digital instruments have been widely used in the medical area with the rapid development of digital technology. The digital stethoscope, which converts the acoustic sound waves in to electrical signals and then amplifies them, is gradually replacing the conventional acoustic stethoscope with the advantage of additional usage such as restoring, replaying and processing the signals for optimal listening. As the sounds are transmitted in to electrical form, they can be recorded for further signal processing. One of the major problems with recording heart sounds is noise corruption. Although there are many solutions available to noise reduction problems, it was found that most of them are based on the assumption that the noise is an additive white noise [1]. More research is required to find different de-noising techniques based on the specific noise present. Therefore, this study is motivated to answer the research question: ‘How might the noise be reduced from the heart sound records collected from digital stethoscope with suitable noise reduction method’. This research question is divided into three sub-questions, including the identification of the noise spectrum, the design of noise reduction method and the assessment of the method. In the identification stage, five main kinds of noise were chosen and their characteristics and spectrums were discussed. Compared with different kinds of adaptive filters, the suitable noise reduction filter for this study was confirmed. To assess the effect of the method, 68 pieces of sound resources were collected for the experiment. These sounds were selected based on the noise they contain. A special noise reduction method was developed for the noise. This method was tested and assessed with those sound samples by two factors: the noise level and the noise kind. The results of the experiment showed the effect of the noise reduction method for each kind of noise. The outcomes indicated that this method was suitable for heart sound noise reduction. The findings of this study, including the analysis of noise level and noise kind, indicated and concluded that the chosen method for heart sound noise reduction performed well. This is perhaps the first attempt to understand and assess the noise reduction method with classified heart sound signals which are collected from the real healthcare environment. This noise reduction method may provide a de-noising solution for the specific noise present in heart sound