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I am doing my project on speaker recognition. One of my friends is doing his project on speech recognition. Both of us need to calculate the MFCC for feature extraction.

Is the MFCC going to be the same one for both of us? Is there any difference between the algorithm of MFCC for speech and that for speaker recognition?

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The features are the same. You might encounter differences in some of the implementation details (window sizes, number of mel filters, number of extracted coefficients) reported in research papers, but these are not significant.

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Both speech recognition and speaker recognition require some set of features to distinguish one speaker (or speech section) from another.

Suppose you have two speakers $S_1$ and $S_2$ and two words $w_1$ and $w_2$. Let's call $u_{ij}$ the utterance of speaker $i$ of word $j$. Then the MFCC's of the speech are $M(u_{ij})$.

One way to pose the speaker recognition problem is to determine which speaker said a new utterance, $u_{\rm new}$. Assume that the utterance is word $J$. To do this, we just select the speaker such that $$ I = \arg\min_{\forall i} \lVert u_{iJ} - u_{\rm new}\rVert $$

How this works will be determined by the inter-speaker variation (i.e. how much the MFCCs of different speakers saying the same word differ).

One way to pose the speech recognition problem is to select: $$ J = \arg\min_{\forall j} \lVert \bar{u}_{j} - u_{\rm new} \rVert $$ where $\bar{u}_{j}$ is the average MFCC of all speakers saying word $j$.

How this works will be determined by the inter-word variation (i.e. how much the MFCCs of different words being said by different speakers differ).

So, yes, the MFCCs for both applications are (or can be) the same.

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