Hann function

Abstract: A noniterative method for the reconstruction of the short-time fourier transform (STFT) phase from the magnitude is presented. The method is based on the direct relationship between the partial derivatives of the phase and the logarithm of the magnitude of the un-sampled STFT with respect to the Gaussian window. Although the theory holds in the continuous setting only, the experiments show that the algorithm performs well even in the discretized setting (discrete Gabor transform) with low redundancy using the sampled Gaussian window, the truncated Gaussian window and even other compactly supported windows such as the Hann window. Due to the noniterative nature, the algorithm is very fast and it is suitable for long audio signals. Moreover, solutions of iterative phase reconstruction algorithms can be improved considerably by initializing them with the phase estimate provided by the present algorithm. We present an extensive comparison with the state-of-the-art algorithms in a reproducible manner.

Abstract: Summary In seeking to estimate the frequencies, decay rates, amplitudes and phases of the free oscillations of the Earth using observed seismic spectra, data windows are required to reduce the effects of mode—mode interference. Formal expressions for the variances and covariances of these estimated parameters are derived here, for the case of a single isolated mode and an arbitrary data window. Explicit results are given for a Hann and Blackman—Harris minimum four-term window as well as for a boxcar. The optimum or minimum variance record length for measuring frequencies and decay rates using a Hann window is 1.1 Q-cycles, and that for determining amplitudes and phases is 0.5 Q-cycles.

Abstract: Due to its simplicity and accuracy, quadratic peak interpolation in a zero-padded fast Fourier transform (FFT) has been widely used for sinusoidal parameter estimation in audio applications. We propose, as its natural extension, a method to estimate the first order amplitude and frequency modulation rates of time-varying sinusoidal components, as well as to correct biases in conventional amplitude, frequency and phase estimates. We derive exact formulas for Gaussian windows and obtain approximate formulas for often-used windows by introducing a simple window adaptation scheme. Experimental results show the average estimation biases of the AM and FM rates with a 30 ms Hann window are below 1% for typical AM/FM rates in speech.

Abstract: A non-iterative method for the construction of the Short-Time Fourier Transform (STFT) phase from the magnitude is presented. The method is based on the direct relationship between the partial derivatives of the phase and the logarithm of the magnitude of the un-sampled STFT with respect to the Gaussian window. Although the theory holds in the continuous setting only, the experiments show that the algorithm performs well even in the discretized setting (Discrete Gabor transform) with low redundancy using the sampled Gaussian window, the truncated Gaussian window and even other compactly supported windows like the Hann window. Due to the non-iterative nature, the algorithm is very fast and it is suitable for long audio signals. Moreover, solutions of iterative phase reconstruction algorithms can be improved considerably by initializing them with the phase estimate provided by the present algorithm. We present an extensive comparison with the state-of-the-art algorithms in a reproducible manner.