Implementing Discrete Fourier Transform in Atmega32 to make an audio spectrum analyzer
"All waveforms, no matter what you scribble or observe in the universe, are actually just the sum of simple sinusoids of different frequencies."
DFT is too slow compared to FFT. My lcd spectum analyzer doesn't need a great speed like that of an FFT, now if it is capable of providing a speed of around 30 frame/second, then it is more than enough for visualizing the audio spectrum on an LCD. But any way, in my case I can roughly achieve around 100 frames/second(any way it is too high refresh rate for a 16x2 lcd, not recommended also :-)). My audio sampling rate is 20KHz for 32 point DFT. Since the transform result is symmetric, I need to use only the first half, ie the first 16 results. So, it means, it can display upto 10KHz spectrum. So the resolution is 10KHz/16 = 625Hz.
I have tried to improve the speed of DFT computation. If it is an N point DFT, it needs to find (N^2)/2 sin and cos values. For a 32 point DFT, it needs to find 512 sine and cosine. Before finding the sine and cosine, we need to find the angle(degree) which takes some processor time, so I implemented a lookup table for that. Next two tables are for sine and cosine. I didn't used any float or double since it takes more processing time in 8 bit avr, instead I implemented the sine and cosine lookups as 16bit intiger, by multiplying the real sine and cosine values by 10000. Then after finding the transform, finally I need to divide each result by 10000. This eliminates the need of using float or double and makes it more faster. Now I can calculate 120 32-point DFT operation/sec which is more than enough for my small spectrum analyzer.
Now one of the most important part of this stuff is the audio sampling via an eletret microphone. Special care must be given while designing the pre-amp for the mic. We need to set the zero-level of the ADC input to exactly half of the ADC reference voltage ie to 2.5v. Now it can have positive and negative swing on this 2.5v level according to the input audio signal but it should not cross the limit ie the amplifier gain should be properly adjusted to prevent clipping. I am using an LM324 op-amp for the mic pre-amp to meet the above conditions.
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