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I am not from electrical or electronic background so my knowledge will be lacking.

MFCC is represented by 39 values for each window frame. 12 values are the mel filter-bank and we get 13th value by taking DCT[ Is this right ]? So rest are the delta and double delta and their energy.

Below is the equation for calculating mel frequency cepstrum:

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It appears to me that it gives a single value for a window frame. I understand that S[m] is the log energies for each M filter. Does c[n] refers to coefficients of n the frame?

Isn't the equation 6.145 summing the log energies over M filters?. If there are 13 mel filters(M=13) then equation 6.145 appears to be the sum of 13 log energies which gives 1 value. Isn't this logic right.

I need to understand how 13 values are found from equation 6.145

Naveen Gabriel
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    Does this answer help? – jojeck Jul 22 '20 at 21:28
  • not entirely. I have edited my question. Can you check ? – Naveen Gabriel Jul 23 '20 at 08:05
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    All 13 coefficients are from DCT. I can try to bake a full answer later. In your equation, $S[m]$ is the log-energy in $m$'th mel filter bank. Let's say in one frame of audio you have 13 log-energies. You get vector $S$ with 13 values. Now you fit $\cos$ functions. You can either fit 13 $\cos$ functions to get 100% accurate representation, giving you 13 MFCC's (DCT coefficients). Or you can take first 6 coefficients, ignoring the 6 highest. This will give you the overall shape of the spectrum without fine details (represented by higher coefficients). – jojeck Jul 23 '20 at 08:07
  • I have updated the equation but i guess i got a bit idea about it. So c[n] are the different coefficients for a window frame? where n ranges from 0 to 12 ? – Naveen Gabriel Jul 23 '20 at 08:25
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    See that you get $c[n]$ where $n$ goes up to $M-1$. $n$ is also inside the cosine function. As you take higher coefficients ($c[6]$, etc.) the cosine function has higher frequency. – jojeck Jul 23 '20 at 08:26
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    Yes, precisely! – jojeck Jul 23 '20 at 08:26
  • Yes. I missed that. Thanks for being patient. – Naveen Gabriel Jul 23 '20 at 08:27

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