Rectangular mask short-time Fourier transform

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)

∫ t − B t + B x ( τ + τ 0 ) e − j 2 π f τ d τ = X ( t + τ 0 , f ) e j 2 π f τ 0 {\displaystyle \int _{t-B}^{t+B}x(\tau +\tau _{0})e^{-j2\pi f\tau }\,d\tau =X(t+\tau _{0},f)e^{j2\pi f\tau _{0}}}

• Modulation property (shift along y-axis)

∫ t − B t + B [ x ( τ ) e j 2 π f 0 τ ] d τ = X ( t , f − f 0 ) {\displaystyle \int _{t-B}^{t+B}[x(\tau )e^{j2\pi f_{0}\tau }]d\tau =X(t,f-f_{0})}

• special input

1. When x ( t ) = δ ( t ) , X ( t , f ) = { 1 ; | t | < B 0 ; otherwise {\displaystyle x(t)=\delta (t),X(t,f)={\begin{cases}\ 1;&|t|<B\\\ 0;&{\text{otherwise}}\end{cases}}}
2. When x ( t ) = 1 , X ( t , f ) = 2 B sinc ⁡ ( 2 B f ) e j 2 π f t {\displaystyle x(t)=1,X(t,f)=2B\operatorname {sinc} (2Bf)e^{j2\pi ft}}

• Linearity property

If h ( t ) = α x ( t ) + β y ( t ) {\displaystyle h(t)=\alpha x(t)+\beta y(t)\,},H ( t , f ) , X ( t , f ) , {\displaystyle H(t,f),X(t,f),}and Y ( t , f ) {\displaystyle Y(t,f)\,}are their rec-STFTs, then

H ( t , f ) = α X ( t , f ) + β Y ( t , f ) . {\displaystyle H(t,f)=\alpha X(t,f)+\beta Y(t,f).}

• Power integration property

∫ − ∞ ∞ | X ( t , f ) | 2 d f = ∫ t − B t + B | x ( τ ) | 2 d τ {\displaystyle \int _{-\infty }^{\infty }|X(t,f)|^{2}\,df=\int _{t-B}^{t+B}|x(\tau )|^{2}\,d\tau } ∫ − ∞ ∞ ∫ − ∞ ∞ | X ( t , f ) | 2 d f d t = 2 B ∫ − ∞ ∞ | x ( τ ) | 2 d τ {\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }|X(t,f)|^{2}\,df\,dt=2B\int _{-\infty }^{\infty }|x(\tau )|^{2}\,d\tau }

• Energy sum property (Parseval's theorem)

∫ − ∞ ∞ X ( t , f ) Y ∗ ( t , f ) d f = ∫ t − B t + B x ( τ ) y ∗ ( τ ) d τ {\displaystyle \int _{-\infty }^{\infty }X(t,f)Y^{*}(t,f)\,df=\int _{t-B}^{t+B}x(\tau )y^{*}(\tau )\,d\tau } ∫ − ∞ ∞ ∫ − ∞ ∞ X ( t , f ) Y ∗ ( t , f ) d f d t = 2 B ∫ − ∞ ∞ x ( τ ) y ∗ ( τ ) d τ {\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }X(t,f)Y^{*}(t,f)\,df\,dt=2B\int _{-\infty }^{\infty }x(\tau )y^{*}(\tau )\,d\tau }

Example of tradeoff with different B

From the image, when B is smaller, the time resolution is better. Otherwise, when B is larger, the frequency resolution is better.

Advantage and disadvantage

Compared with the Fourier transform:

• Advantage: The instantaneous frequency can be observed.
• Disadvantage: Higher complexity of computation.

Compared with other types of time-frequency analysis:

• Advantage: Least computation time for digital implementation.
• Disadvantage: Quality is worse than other types of time-frequency analysis. The jump discontinuity of the edges of the rectangular mask results in Gibbs ringing artifacts in the frequency domain, which can be alleviated with smoother windows.

See also

• Uncertainty principle