# Going from an MFCC coefficient to Hz range?

I have not worked with MFCCs before, but I am faced with the observation that the 5th MFCC of a signal is actually of interest for me in my speech research. In order to understand why that one is important for me, I need to make sure that I understand what it represents.

If I know the number of Mel frequency spaced filters applied to the signal, and the sample rate, can I assume that the 5th coefficient is related to the 5th mel spaced filtered regions?

If not, how do I learn what in the frequency domain is represented by the 5th MFCC?

In my understanding you need a little more information. You need to know the minimum and maximum frequency and number of coefficients in the MFCC representation.

If the coefficients are linearly spaced in the mel scale, $$m(f)$$

The first coefficient is $$m(f_1)$$ the last coefficient is $$m(f_N)$$, and the $$i-$$th coefficient is

$$m(f_i) = m(f_1) + \frac{i-1}{N-1} (m(f_N) - m(f_1))$$

Then you replace apply the definition of interest for $$m(f)$$, e.g.

$$m(f) = \frac{1000}{log(2)}log\left(1 + \frac{f}{1000}\right)$$

And solve for $$f_i$$.