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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

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    $\begingroup$ Does this answer help? $\endgroup$
    – jojeck ♦
    Commented Jul 22, 2020 at 21:28
  • $\begingroup$ not entirely. I have edited my question. Can you check ? $\endgroup$
    – Naveen Gabriel
    Commented Jul 23, 2020 at 8:05
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    $\begingroup$ 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). $\endgroup$
    – jojeck ♦
    Commented Jul 23, 2020 at 8:07
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    $\begingroup$ 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. $\endgroup$
    – jojeck ♦
    Commented Jul 23, 2020 at 8:26
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    $\begingroup$ Yes, precisely! $\endgroup$
    – jojeck ♦
    Commented Jul 23, 2020 at 8:26

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