In mathematics and Fourier analysis, a rectangular mask short-time Fourier transform (rec-STFT) is a simplified form of the short-time Fourier transform which is used to analyze how a signal's frequency content changes over time. In rec-STFT, a rectangular window (a simple on/off time-limiting function) is used to isolate short time segments of the signal. Other types of the STFT may require more computation time ( refers to the amount of time it takes a computer or algorithm to perform a calculation or complete a task) than the rec-STFT. The rectangular mask function can be defined for some bound (B) over time (t) as
w ( t ) = { 1 ; | t | ≤ B 0 ; | t | > B {\displaystyle w(t)={\begin{cases}\ 1;&|t|\leq B\\\ 0;&|t|>B\end{cases}}}
We can change B for different tradeoffs between desired time resolution and frequency resolution. Rec-STFT
X ( t , f ) = ∫ t − B t + B x ( τ ) e − j 2 π f τ d τ {\displaystyle X(t,f)=\int _{t-B}^{t+B}x(\tau )e^{-j2\pi f\tau }\,d\tau }
Inverse form
x ( t ) = ∫ − ∞ ∞ X ( t 1 , f ) e j 2 π f t d f where t − B < t 1 < t + B {\displaystyle x(t)=\int _{-\infty }^{\infty }X(t_{1},f)e^{j2\pi ft}\,df{\text{ where }}t-B<t_{1}<t+B}
Property Rec-STFT has similar properties with Fourier transform
Integration (a)
∫ − ∞ ∞ X ( t , f ) d f = ∫ t − B t + B x ( τ ) ∫ − ∞ ∞ e − j 2 π f τ d f d τ = ∫ t − B t + B x ( τ ) δ ( τ ) d τ = { x ( 0 ) ; | t | < B 0 ; otherwise {\displaystyle \int _{-\infty }^{\infty }X(t,f)\,df=\int _{t-B}^{t+B}x(\tau )\int _{-\infty }^{\infty }e^{-j2\pi f\tau }\,df\,d\tau =\int _{t-B}^{t+B}x(\tau )\delta (\tau )\,d\tau ={\begin{cases}\ x(0);&|t|<B\\\ 0;&{\text{otherwise}}\end{cases}}}
(b)
∫ − ∞ ∞ X ( t , f ) e − j 2 π f v d f = { x ( v ) ; v − B < t < v + B 0 ; otherwise {\displaystyle \int _{-\infty }^{\infty }X(t,f)e^{-j2\pi fv}\,df={\begin{cases}\ x(v);&v-B<t<v+B\\\ 0;&{\text{otherwise}}\end{cases}}}
Shifting property (shift along x-axis)
… excerpt ends here. Continue reading the full article.
