Rectangular mask short-time Fourier transform

Rec-STFT has similar properties with Fourier transform

• Integration

(a)

${\displaystyle \int _{-\infty }^{\infty }X(t,f)\,df=\int _{t-B}^{t+B}x(\tau )\int _{-\infty }^{\infty }e^{-j2\pi f\tau }\,df\,d\tau =\int _{t-B}^{t+B}x(\tau )\delta (\tau )\,d\tau ={\begin{cases}\ x(0);&|t|<B\\\ 0;&{\text{otherwise}}\end{cases}}}$

(b)

${\displaystyle \int _{-\infty }^{\infty }X(t,f)e^{-j2\pi fv}\,df={\begin{cases}\ x(v);&v-B<t<v+B\\\ 0;&{\text{otherwise}}\end{cases}}}$

• Shifting property (shift along x-axis)

${\displaystyle \int _{t-B}^{t+B}x(\tau +\tau _{0})e^{-j2\pi f\tau }\,d\tau =X(t+\tau _{0},f)e^{j2\pi f\tau _{0}}}$

• Modulation property (shift along y-axis)

${\displaystyle \int _{t-B}^{t+B}[x(\tau )e^{j2\pi f_{0}\tau }]d\tau =X(t,f-f_{0})}$

• special input

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

• Linearity property

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

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

• Power integration property

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

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

• Energy sum property (Parseval's theorem)

${\displaystyle \int _{-\infty }^{\infty }X(t,f)Y^{*}(t,f)\,df=\int _{t-B}^{t+B}x(\tau )y^{*}(\tau )\,d\tau }$

${\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }X(t,f)Y^{*}(t,f)\,df\,dt=2B\int _{-\infty }^{\infty }x(\tau )y^{*}(\tau )\,d\tau }$

Spectrograms produced from applying a rec-STFT on a function consisting of 3 consecutive cosine waves. (top spectrogram uses smaller B of 0.5, middle uses B of 1, and bottom uses larger B of 2.)

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

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.

• Uncertainty principle

1. Jian-Jiun Ding (2014) Time-frequency analysis and wavelet transform