Logarithmic fourier transform(LFT) on audio signal I am trying to analyze the musics as possible as precisely.
Of course I tried FFT, but got some problems.

I found low frequencies have very low resolution than human's hearing. I tried very long time FFT to resolve this problem, but even analyzing with 8192 samples/s in 44100Hz sample rate(Means lack of time resolution), I got not enough resolution on low frequencies.

I found there are few solutions.

Firstly, a quadratic interpolation on FFT bins.
But it seems not a perfect way. Problems of this method are:


1. 'If i want to determine freqs between the freq bins, which three bins should I select to do an interpolation?'

2. 'Even I do this, there are no actual additional informations on result. I know interpolations are kind of tricky method.'

Secondly, extracting each freq bins with desired frequency, so I can extract the bins logarithmically.

But have a critical computational cost problem: (maybe over)N^2.

Thirdly, LFT(Logarithmic Fourier Transform).

This requiers logarithmically-spaced samples and gives me result exactly what I looking for with incredibly fast speed; https://stackoverflow.com/questions/1120422/is-there-an-fft-that-uses-a-logarithmic-division-of-frequency

But I have no idea with that algorithm. I tried to understand the paper and implement it, but it was impossible because of lack of my english and mathematical skills.

So, I need a help of implementation of LFT.
The simplest and most pragmatic solution is to use a normal FFT of a sufficiently large size that you get the required resolution at the lowest frequency of interest. E.g. if you want 1 Hz resolution at the lowest frequency of interest then you will need a 1 second FFT window, i.e. the FFT size would need to be equal the sample rate, e.g. 44100.

Note that even if you could implement a logarithmic FFT then it would still be bound by the laws of physics (information theory) and you would still need a similar length sample window - all you would gain would be convenience (not having to aggregate output bins) at the expense of performance.