Rectangular mask short-time Fourier transform

Rec-STFT has similar properties with Fourier transform

• Integration

(a)

: $\backslash int\_\{-\backslash infty\}^\backslash infty\; X(t,\; f)\backslash ,\; df\; =\; \backslash int\_\{t-B\}^\{t+B\}\; x(\backslash tau)\backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{-j\; 2\; \backslash pi\; f\; \backslash tau\}\backslash ,\; df\; \backslash ,\; d\backslash tau\; =\; \backslash int\_\{t-B\}^\{t+B\}\; x(\backslash tau)\backslash delta(\backslash tau)\; \backslash ,\; d\backslash tau=\backslash begin\{cases\}$

\ x(0); & |t|< B \\

\ 0; & \text{otherwise}

\end{cases}

(b)

: $\backslash int\_\{-\backslash infty\}^\backslash infty\; X(t,\; f)e^\{-j\; 2\; \backslash pi\; f\; v\}\; \backslash ,df\; =\backslash begin\{cases\}$

\ x(v); & v-B

\ 0; & \text{otherwise}

\end{cases}

• Shifting property (shift along x-axis)

:: $\backslash int\_\{t-B\}^\{t+B\}\; x(\backslash tau+\backslash tau\_0)\; e^\{-j\; 2\; \backslash pi\; f\; \backslash tau\}\backslash ,\; d\backslash tau\; =\; X(t+\backslash tau\_0,f)e^\{j\; 2\; \backslash pi\; f\; \backslash tau\_0\}$

• Modulation property (shift along y-axis)

:$\backslash int\_\{t-B\}^\{t+B\}\; [x(\backslash tau)\; e^\{j\; 2\; \backslash pi\; f\_0\; \backslash tau\}]\; d\backslash tau\; =\; X(t,f-f\_0)$

• special input

1. When $x(t)=\backslash delta(t),\; X(t,f)=\backslash begin\{cases\}$

\ 1; & |t|< B \\

\ 0; & \text{otherwise}

\end{cases}

1. When $x(t)=1,X(t,f)=2B\backslash operatorname\{sinc\}(2Bf)e^\{j\; 2\; \backslash pi\; f\; t\}$

• Linearity property

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

: $H(t,f)=\backslash alpha\; X(t,f)+\backslash beta\; Y(t,f)\; .$

• Power integration property

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

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

• Energy sum property (Parseval's theorem)

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

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

File:DifferentB.JPGs 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. [http://djj.ee.ntu.edu.tw/TFW.htm Jian-Jiun Ding (2014) Time-frequency analysis and wavelet transform]

Category:Fourier analysis

Category:Time–frequency analysis

Category:Transforms