STFT例子

1.正弦信号

A single sinusoid of amplitude A and frequency \(\omega_{0}=2\pi f_{0}/F_{s}\)

\[\begin{array}{ll}x_{0}[t] &=Acos(\omega_{0}t)\\ &=\frac{A}{2}e^{i\omega_{0}t}+\frac{A}{2}e^{-i\omega_{0}t} \end{array}\]

The STFT of \(x_{0}[t]\) is given by

\[\begin{array}{ll}X_{0}[t] &=\frac{A}{2}e^{i\omega_{0}lL}\sum_{n=0}^{N-1}\omega[n]e^{-i(\omega_{k}-\omega_{0})n}+\frac{A}{2}e^{-i\omega_{0}lL}\sum_{n=0}^{N-1}\omega[n]e^{-i(\omega_{k}+\omega_{0})n}\\ &=\frac{A}{2}e^{i\omega_{0}lL}W(\omega_{k}-\omega_{0})+\frac{A}{2}e^{-i\omega_{0}lL}W(\omega_{k}+\omega_{0}) \end{array}\]

For the case \(\omega_{0}=0.5\), \(N=128\), and \(K=512\)

2.啁啾信号

Consider the chirp signal

\[x_{1}[t]=Acos(\omega_{0}t+\alpha_{0}t^{2}) \]

From a filter bank perspective, the chirp moves across the subbands as time progresses.

The progression of the chirp across the subbands is illustrated in Fig.12.6, which depicts the (non-subsampled) STFT filter bank subband signals for several subbands.

3.正弦合成信号

Consider a sum of weighted sinusoids

\[x_{2}[t]=\sum_{q}A_{q}cos(\omega_{q}t+\phi_{q}) \]

\[X_{2}[t]=\sum_{q}\frac{A_{q}}{2}e^{i(\omega_{q}lL+\phi_{q})}W(\omega_{k}-\omega_{q})+\sum_{q}\frac{A_{q}}{2}e^{-i(\omega_{q}lL+\phi_{q})}W(\omega_{k}+\omega_{q}) \]

4.语音

It is clear that the STFT magnitude changes from frame to frame, but some of the spectral peaks persist; there can be interpreted as persistent sinusoids in the speech signal.