Time–frequency analysis for music signals

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

${\displaystyle \mathbf {STFT} \left\{x(t)\right\}\equiv X(t,f)=\int _{-\infty }^{\infty }x(\tau )w(t-\tau )e^{-j2\pi f\tau }\,d\tau }$

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

${\displaystyle X(n\,\Delta t,m\,\Delta f)=\sum _{p=n-Q}^{n+Q}x(p\,\Delta t)e^{-j2\pi pm\,\Delta t\,\Delta f}\,\Delta t}$

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:

• ${\displaystyle \Delta t\,\Delta f={\frac {1}{N}},}$ where N is an integer.
• ${\displaystyle N\geq 2Q+1}$
• ${\displaystyle \Delta <{\frac {1}{2f_{\max }}}}$, where ${\displaystyle f_{\max }}$ is the highest frequency in the signal.

The Wigner distribution function ${\displaystyle W_{x}(t,f)}$ is:

${\displaystyle \mathbf {W} _{x}(t,f)=\int _{-\infty }^{\infty }x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi \tau \,f}\,d\tau ,}$

where x(t) is the signal, and x*(t) is the conjugate of the signal.