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Imagine someone had a signal of 32 samples. They cut it to two 16 sample pieces, converted to DCT frequencies, using DCT4.

What I get is these two arrays. My goal is to get a x2 higher freq resolution DCT, x2 less time resolution. Of course, I can convert these two pieces back to time domain, concatenate them to get 32 samples, then do DCT4 and get my 32 resolution frequencies.

But is it possible without converting to time domain? In fact, we have all the information in smaller blocks. Feels like there must be some trick to do this!

Thank you.

shal
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Yes. In fact, the FFT works by doing the discrete Fourier transform in stages. If you have $2^N$ points, it does $2^{N-1}$ 2-point transforms, then combines pairs of these into $2^{N-2}$ 4-point transforms, and it proceeds until it combines 2 $2^{N-1}$ transforms into one answer.

So in your case, you're at the penultimate stage of a $2^5$ FFT, and all you have to do is that last combination of the two transforms into one.

The "trick" to this is to find a good reference on the FFT and study it until you understand the process. If it's really a good reference, then it'll mention this very trick.

enter image description here By OhArthits - Own work, CC BY 3.0

TimWescott
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