ディジタル信号処理による自然楽器音の分析

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In the past decade, the of development of digital processing technique of signals with computers has yielded a remarkable progress in the acoustic aspects of musical instrument tones and several papers of this analysis have been published. As a part systematic investigations of the analysis and synthesis of musical instrument tones, is reported here an application of the Discrete Fourier Transform to musical instrument tones with special regard to the temporal fluctuation of physical parameters of the tones. New mathematical procedure of error estimation is proposed, and an analysis of physical features with actual example is presented. Generally, the following three assumptions are postulated for the application of Discrete Fourier Transform to musical instrument tones : (1) The spectrum of a musical instrument tone is of harmonic constitution. (2) In the duration of a musical note, the fundamental frequency is held constant. (3) The results of the DFT analysis in a transient state represent averaged values for amplitude and frequency in the sampled interval. First, on these assumptions, a duration of a musical instrument tone was divided into sequential waves of short-period frames (Fig. 2). Next, a time series of spectrum of each frame was constructed, with the application of DFT to each frame. The number of samples in a frame set at 2^γ(γ : integer), since the FFT algorithm is based on 2, and the sampling frequency was set to make both the wave form and data window pitch synchronous. Since, however, the actual pitch of a musical instrument tone is regarded as fluctuating instantaneously, it is suspected that the pitch might become asynchronous with the window and errors might be brought into the results of the DFT analysis. Then, a new estimation procedure of errors is proposed, utilizing the properties of a convolution of the spectral window of the data window and the Fourier transform of singnals. The spectrum of a musical instrument sound wave from the DFT is represent by G'(mω)=Σ^^^m___<m=-m> G(mω_1)・W(nm-mω_1), where G(mω_1) is the Fourier transform of the signal and W(nm-mω_1) is an arbitrary spectrum window. Here G'(mω) is the theoretical spectrum including errors due to a slight variation of fundamental angular frequency ω_1 of the signal deviated from the angular frequency ω, which is pitchsynchronous with the data window. The errors were estimated by calculating the ratio of this spectral value to a modelized, true spectral value. If a Hamming window is used as the data window, the spectral error up to the 8th harmonic in the vicinity of the attack transient portions is less than 3 dB owing to the large fluctuation in pitch of actual musical instrument tone. However, in the steady state the error is of the order of 0. 1 to 0. 3 dB, showing that sufficient accuracy is achieved by this analytical method on the three assumption stated above. Although this analytical procedure is applied to various natural musical instrument tones, only the results of violin A_4 are presented here. In Fig. 8, it is shown that the attack transient form the violin tone is exponential with an attack time of 100 msec, and that even in the steady state, marked modulations in frequency (vibrato) and amplitude are observed.
社団法人日本音響学会の論文
1977-05-01

ディジタル信号処理による自然楽器音の分析

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