# AVR Atmega audio input RMA using FFT Radix-4

audiogetradix4 is a simple library you can use to interface with a ac audio input.

It reads data from an ADC pin and returns the RMS value of the input using **DFT Radix-4** algorithm.

The discrete **Fourier transform (DFT)** converts a finite list of equally-spaced samples of a function into the list of coefficients of a finite combination of complex sinusoids, ordered by their frequencies, that has those same sample values.

I’m not a DSP expert, so if some expert find error in the method i use, explained below, please let me know.

A Radix-4 decimation-in-time (DIT) FFT is a **Cooley–Tukey algorithm** for computing the discrete Fourier transform (DFT).

My radix4 implementation is a port of RADIX-4 FFT library by Anatoly Kuzmenko.

**To obtain RMS value** of the AC current of the audio signal processed, from which you can get the SPL db value, we **use the Parseval’s theorem for FFT** so for fft size of N.

Let’s conside x[i] as the signal, and let’s take N samples, we could take only real part of the signal using our ADC, so we can consider Parseval therorem for DFT.

consider X[k] as the “modified” frequency spectrum, valid from 0 to N/2, and the the first (0) and the last (N/2) element have to be divided by the root square of 2.

We define:

given ReX as the real part, and ImX as the imaginary part of the signal in frequency domain.

So Parseval therorem for DFT became:

Now we know that RMS in time domain is:

Applying the parseval theorem we can compute RMS in frequency domain as:

Because we are computing a discrete signal, we can treat the equation above for DTFT.

So X[k] is not a “modified” frequency spectrum.

So Parseval therorem for DTFT became:

And we can compute RMS in frequency domain as:

Once again as audiogetavarage library here is also implemented a dynamic bias, for the electronic board used during experiment, refer to that project.

This time the TLC272 opamp is used in pre, this ic is a little more sensitive then the TLC27L2 used previously, so a filter caps is added in this pre circuit revision (rev 03).

For more detail: AVR Atmega audio input RMA using FFT Radix-4

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