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I am currently reading a paper about speech enhancement. The paper uses Spectral Subtraction Method. On that section, The paper stated that ω is the "discrete angular frequency index of the frames"

Question: What is the discrete angular frequency index? Is it part of Short Time Fourier Transform?

Assume that the noisy speech $\boldsymbol{y[n]}$ can be expressed as $\boldsymbol{y[n] = s[n] + d[n]}$, where $\boldsymbol{s[n]}$ is the clean speech and $\boldsymbol{d[n]}$ is the additive noise. As the enhancement is carried out according to the frame, the above model can be expressed as $$y(n,k) = s(n,k) + d(n,k),\quad n=0,1,2,\ldots,(N-1);\quad k=1,2,\ldots,N\tag{1}$$ Here $n$ is the discrete time index, $k$ is the frame number and $N$ is the length of the frame. $$y(\omega,k) = S(\omega,k) + D(\omega,k)\tag{2}$$ Here $\omega$ is the discrete angular frequency index of the frames.

The paper is:

Navneet Upadhyay and Rahul Kumar Jaiswal: "Single Channel Speech Enhancement: Using Wiener Filtering with Recursive Noise Estimation", Procedia Computer Science, Volume 84, 2016, Pages 22-30, doi:10.1016/j.procs.2016.04.061.

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$\omega$ is angular frequency in radians, $\omega =2\pi fT$, where $f$ is frequency (for example in Hz) and $T$ is sampling period (for example in seconds). This is revealed by Eq. 3:

$$Y(\omega, k) = \textstyle\sum_{n=-\infty}^\infty y(n)w(k-n)e^{-j\omega n}\tag{3},$$

which represents discrete-time Fourier transform (DTFT) of windowed signal $y(n)w(k-n)$.

It's not common to call $\omega$ "discrete angular frequency index", which gives just 3 google hits.

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  • $\begingroup$ I see. But I never heard of DTFT function exist in python especially Librosa, do you have any reference on how I compute in Python? Moreover, the paper also mentions that they use STFT in Fig.2 is STFT is the same with STFT? $\endgroup$ – Andreas Chandra Apr 26 at 9:53
  • $\begingroup$ @AndreasChandra sorry, I only wanted to answer the specific question. $\endgroup$ – Olli Niemitalo Apr 26 at 10:21

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