An Observation about the Discrete Fourier Transforms

Discrete Fourier Transforms, including the Cosine Transforms, tend to have as many elements in the frequency-domain, as the sampling interval had in the time-domain.

Thus, if a sampling interval had 1024 samples, there will be as many frequency-coefficients, numbered from 0 to 1023 inclusively. One way in which these transforms differ from the FFT, is in the possibility of having a number of elements either way, that are not a power of 2. It is possible to have a discrete transform with 11 time-domain samples, that translate into as many frequency-coefficients, numbered from 0 to 10 inclusively.

If it was truly the project to compute an FFT that has one coefficient per octave, then we would include the Nyquist Frequency, which is usually not done. And in that case, we would also ask ourselves, whether the component at F=0 is best computed as the summation over the longest interval, where it would usually be computed, or whether it makes more sense then, just to fold the shortest interval, which consists of 2 samples, one more time, to arrive at 1 sample, the value of which corresponds to F=0 .

Now, if our discrete transform had the frequency-coefficients

 


G(n) = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}


 

Then the fact could be exploited that these transforms tend to act as their own inverse. Therefore I can know, that the same set of samples in the time-domain, would constitute a DC signal, which would therefore have the frequency-coefficients

 


F(n) = {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}


 

If this was taken to be a convolution again, because the discrete transforms are their own inverse, it would correspond to the function

 


F(n) · S(m) == S(m)


 

We would assume that multiplication begins with element (0) and not with element (10). So I have a hint, that maybe I am on the right track. But, because the DCT has an inverse which is not exactly the same, the inverse being the IDCT, the next question I would need to investigate, is whether indeed I should be using the DCT and not the IDCT, to turn an intended set of frequency-coefficients, into a working convolution. And to answer that question, the simple thought does not suffice.

The main advantage with the DCT would be, that we will never need to deal with complex values.

Dirk

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A Thought on SRS

Today, when we buy a laptop, we assume that its internal speakers offer inferior sound by themselves, but that through the use of a feature named ‘SRS’, they are enhanced, so that sound which simply comes from two speakers in front of us, seems to fill the space around us, kind of how surround-sound would work.

The immediate problem with Linux computers is, that they do not offer this enhancement. However, technophiles have known for a long time that this problem can be solved.

The underlying assumption here is, that the stereo being sent to the speakers should act as if each channel was sent to one ear in an isolated way, as if we were using headphones.

The sound that leaves the left speaker, reaches our right ear with a slightly longer time-delay, than the time-delay with which it reaches our left ear, and a converse truth exists for the right speaker.

It has always been possible to time-delay and attenuate the sound that came from the left speaker in total, before subtracting the result from the right speaker-output, and vice-verso. That way, the added signal that reaches the left ear from the left speaker, cancels with the sound that reached it from the right speaker…

The main problem with that effect, is that it will mainly seem to work when the listener is positioned in front of the speakers, in exactly one position.

I have just represented a hypothetical setup in the time-domain. There can exist a corresponding representation in the frequency-domain. The only problem is, that this effect cannot truly be achieved just with one graphical equalizer setting, because it affects (L+R) differently from how it affects (L-R). (L+R) would be receiving some recursive, negative reverb, while (L-R) would be receiving some recursive, positive reverb. But reverb can also be expressed by a frequency-response curve, as long as that has sufficiently fine resolution.

This effect will also work well with MP3-compressed stereo, because with Joint Stereo, an MP3 stream is spectrally complex in its reproduction of the (L-R) component.

I expect that when companies package SRS, they do something similar, except that they may tweak the actual frequency-response curves into something simpler, and they may also incorporate a compensation, for the inferior way the speakers reproduce frequencies.

Simplifying the curves would allow the effect to break down less, when the listener is not perfectly positioned.

We do not have it under Linux.

(Edit 02/24/2017 : A related effect is possible, by which 2 or more speakers are converted into an effectively-directional speaker-system. I.e., the intent could be, that sound which reaches our filter as the (L) channel, should predominantly leave the speaker-set at one angle, while sound which reaches our filter as the (R) channel, should leave the speaker-set at an opposing angle.

In fact, if we have an entire array of speakers – i.e. a speaker-bar – then we can apply the same sort of logic to them, as we would apply to a phased-array radar system.

The main difference with such a system, as opposed to one based on the Inter-Aural Delay, is that this one would absolutely require we know the distance between the speakers. And then we would use that distance, as the basis for our time-delays… )

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