# What approximations are involved in STFT of a convolution?

I have two equations in time domain: $$x_k(t)=\sum_{i=1}^N s_i(t)\star h_{ki}(t)+n^a_k(t)$$ and $$x_k(t)=\sum_{i=1}^N s_i(t)\star h_{ki}(t)\star n^c_k(t)$$ which explain the additive and convolutive noise models for recorded sounds in microphones when $N$ sources are available.

Using the short-time Fourier transform (STFT) I will have:

$$X_k(m,f)=\sum_{i=1}^{N} S_i(m,f)\cdot H_{ki}(f)+N^a_k(m,f)$$ and
$$X_k(m,f)=\sum_{i=1}^{N} S_i(m,f)\cdot H_{ki}(f)\cdot N^c_k(m,f)$$

I assumed that the mixing system, $H(f)$ is time-invariant. What are the other approximations? The only thing I can think of is that impulse response, $h(t)$ is shortened to the window length in the STFT process.

Regards, Ati

• You can include the window function centered around $m$ explicitly in your equation for $X_k(m,f)$. Then there are no approximations. – Atul Ingle Jan 31 '17 at 15:00
• Thanks a lot. However, it seems that there are some approximation as I am asked about them :-) – Ati Jan 31 '17 at 15:12