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.
