Instantaneous frequency and short term Fourier transforms: Application to piano sounds

Abstract

For more than 30 years and until nowadays, development of a system reproducing the functioning of human hearing has remained an aim difficult to reach. Recent methods for identification of the fundamental frequency of musical sounds obtain good results using information about the temporal evolution of the amplitude and frequency of individual sound partials. Piano sounds and polyphonic sounds, which can have several partials that are closely spaced in the frequency domain, have not been extensively tested by these procedures. In this paper, the Instantaneous Frequency (IF), as defined by the Hilbert transform, is used to obtain the frequency variations of piano sounds partials. The result implies that, for these sounds, the IF may contain modulations resulting in the separation of an apparent single sinusoid signal into two or more sinusoidal components at various times in the analysis process, which makes it impossible to use the temporal evolution of the frequency of partials for the procedure of note identification. The separation phenomenon also appears when the short term Fourier transform is used and can induce the detection of short-lived parasitic spectral peaks that must be taken into account by any note identification procedure based on the use of spectral information.

Transcription of piano recordings

“…In this paper, we focus on a transcription system for piano music [7][8][9][10][11]. The transcription system proposed is based on a heuristic approach, which has been found to achieve very good performance using simple signal processing techniques.…”

The song of the singing rod

“…An analysis (not presented here), employing the socalled instantaneous frequency method, [19][20][21][22][23][24] of the beat frequency reveals that it slightly increases over time as the rod vibration decays. One can audibly hear the increase in the beat frequency as well (increase of 1-2 Hz).…”

Sympathetic String Modes in the Concert Harp

“…The presence of several vibration modes having close frequencies is a common characteristic for numerous free stringed instruments. Various methods, applied to musical instruments, such as techniques based on the Hilbert transform [4,9] or High Resolution methods [4,10,11], allow the identification of spectral components.…”