Time–frequency analysis for music signals

Time–frequency analysis for music signals is one of the applications of time–frequency analysis. Musical sound can be more complicated than human vocal sound, occupying a wider band of frequency. Music signals are time-varying signals; while the classic Fourier transform is not sufficient to analyze them, time–frequency analysis is an efficient tool for such use. Time–frequency analysis is extended from the classic Fourier approach. Short-time Fourier transform (STFT), Gabor transform (GT) and Wigner distribution function (WDF) are famous time–frequency methods, useful for analyzing music signals such as notes played on a piano, a flute or a guitar.

Music is a type of sound that has some stable frequencies in a time period. Music can be produced by several methods. For example, the sound of a piano is produced by striking strings, and the sound of a violin is produced by bowing. All musical sounds have their fundamental frequency and overtones. Fundamental frequency is the lowest frequency in harmonic series. In a periodic signal, the fundamental frequency is the inverse of the period length. Overtones are integer multiples of the fundamental frequency.

In musical theory, pitch represents the perceived fundamental frequency of a sound. However the actual fundamental frequency may differ from the perceived fundamental frequency because of overtones.

Short-time Fourier transform is a basic type of time–frequency analysis. If there is a continuous signal x(t), we can compute the short-time Fourier transform by

where w(t) is a window function. When the w(t) is a rectangular function, the transform is called Rec-STFT. When the w(t) is a Gaussian function, the transform is called Gabor transform.

However, normally the musical signal we have is not a continuous signal. It is sampled in a sampling frequency. Therefore, we can’t use the formula to compute the Rec-short-time Fourier transform. We change the original form to

Let ${\displaystyle t=n\,\Delta t}$, ${\displaystyle f=m\,\Delta f}$, ${\displaystyle \tau =p\,\Delta t}$ and ${\displaystyle B=Q\,\Delta t}$. There are some constraints of discrete short-time Fourier transform:

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