fft | Python code and wav files | Speech library

Your fitted curve will look like this

What you are seeing is indeed the expected outcome of an FFT (Fourier Transform). The logarithmic f-axis that you're expecting is achieved by the Constant-Q transform.

Now, the implementation of the Constant-Q transform is non-trivial. The Fourier Transform has become popular precisely because there is a fast implementation (the FFT). In practice, the constant-Q transform is often implemented by using an FFT, and combining multiple high-frequency bins. This discards resolution in the higher bins; it doesn't give you more resolution in the lower bins.

To get more frequency resolution in the lower bins of the FFT, just use a longer window. But if you also want to keep the time resolution, you'll have to use a hop size that's smaller than the window size. In other words, your FFT windows will overlap.

I see that the comments of @Cris Luengo have already developed your solution into the right direction. The last thing you're missing now is that the spectrum you obtain from np.fft.fft is composed of the positive frequency components in the first half and the 'mirrored' negative frequency components in the second half.

If you now set all components beyond your bandlimit_index to zero, you're erradicating these negative frequency components. That explains the drop in signal amplitude of 6dB, you're eliminating half the signal power (+ as you already noticed every real signal has to have conjugate symmetric frequency spectrum). The np.fft.ifft function documentation (ifft documentation) explains the expected format quite nicely. It states:

"The input should be ordered in the same way as is returned by fft, i.e.,"

• a[0] should contain the zero frequency term,
• a[1:n//2] should contain the positive-frequency terms,
• a[n//2 + 1:] should contain the negative-frequency terms, in increasing order starting from the most negative frequency.

That's essentially the symmetry you have to preserve. So in order to preserve these components just set the components between bandlimit_index + 1 -> (len(fsig) - bandlimit_index) to zero.

• You need to normalize the fft by 1/N with one of the two following changes (I used the 2nd one):
  S = np.fft.fft(s) --> S = 1/N*np.fft.fft(s)
magnitude = np.abs(S[index[0]]) --> magnitude = 1/N*np.abs(S[index[0]])
• Don't use index, = np.where(np.isclose(frequency, f0, atol=1/(T*N))), the fft is not exact and the highest magnitude may not be at f0, use np.argmax(np.abs(S)) instead which will give you the peak of the signal which will be very close to f0
• np.angle messes up (I think its one of those pi,pi/2 arctan offset things) just do it manually with np.arctan(np.real(x)/np.imag(x))
• use more points (I made N higher) and make T smaller for higher accuracy
• since a DFT (discrete fourier transform) is double sided and has peak signals in both the negative and positive frequencies, the peak in the positive side will only be half the actual magnitude. For an fft you need to multiply every frequency by two except for f=0 to acount for this. I multiplied by 2 in magnitude = np.abs(S[index])*2/N

Yes, it's possible and it works well. There are several advantages of Delta for streaming architecture:

• you don't have a "small files problem" that often arises with streaming workloads - you don't need to list all data files to find new files (as in case of Parquet or other data source) - all data is recorded in the transaction log
• your consumers don't see partial writes because Delta provides transactional capabilities
• streaming workloads are natively supported by Delta
• you can perform DELETE/UPDATE/MERGE even for streaming workloads - it's impossible with Parquet

P.S. you can just use .format("delta") instead of full class name

In your tuner.dart File the _TurnerState extends the Sate but it needs to extend for example class _TunerState extends State.

As was pointed out in comments, you used a positive sign in the computation of omg_n. There are different definitions of the DFT, so it isn't wrong by itself. However this would naturally lead to differences if you compare your results with an implementation that uses a negative sign, as is the case with numpy.fft.fft. Adjusting your implementation to also use a negative sign would cover all forward transform cases (leaving only small roundoff errors on the order of ~10^-16).

For the inverse transform cases, your implementation ends up scaling the result by 1/n at every stage, instead of only the final stage. To correct this, simply remove the scaling from the recursion, and normalize only on the final stage:

There seem to be two points of confusion here:

1. What find_peaks is returning.
2. How to interpret complex values that the FFT is returning.

I will answer them separately.

find_peaks returns the indices in "a" that correspond to peaks, so I believe they ARE values you seek, however you must plot them differently. You can see the first example from the documentation here. But basically "peaks" is the index, or x value, and a[peaks] will be the y value. So to plot all your frequencies, and mark the peaks you could do:

You can use any units you want. Feel free to express your sampling frequency as fs=12 (samples/year), the x-axis will then be 1/year units. Or use fs=1 (sample/month), the units will then be 1/month.

The extra line you spotted comes from the way you plot your data. Look at the output of the np.fft.fftfreq call. The first half of that array contains positive values from 0 to 1.2e6 or so, the other half contain negative values from -1.2e6 to almost 0. By plotting all your data, you get a data line from 0 to the right, then a straight line from the rightmost point to the leftmost point, then the rest of the data line back to zero. Your xlim call makes it so you don’t see half the data plotted.

Typically you’d plot only the first half of your data, just crop the freqs and power_spectrum arrays.