## Comparing two Bose headphones, both of which use active technology.

In this posting I’m going to do something I rarely do, which is, something like a product review. I have purchased the following two headphones within the past few months:

The first set of headphones has an analog 3.5mm stereo input cable, which has a dual-purpose Mike / Headphone Jack, and comes either compatible with Samsung, or with Apple phones, while the second uses Bluetooth to connect to either brand of phone. I should add that the phone I use with either set of headphones is a Samsung Galaxy S9, which supports Bluetooth 5.

The first set of headphones requires a single, AAA alkaline battery to work properly. And this not only fuels its active noise cancelling, but also an equalizer chip that has become standard with many similar middle-price-range headphones. The second has a built-in rechargeable Lithium-Ion Battery, which is rumoured to be good for 10-15 hours of play-time, which I have not yet tested. Like the first, the second has an equalizer chip, but no active noise cancellation.

I think that right off the bat I should point out, that I don’t approve of this use of an equalizer chip, effectively, to compensate for the sound oddities of the internal voice-coils. I think that more properly, the voice-coils should be designed to deliver the best frequency response possible, by themselves. But the reality in the year 2019 is, that many headphones come with an internal equalizer chip instead.

What I’ve found is that the first set of headphones, while having excellent noise cancellation, has two main drawbacks:

• The jack into which the analog cable fits, is poorly designed, and can cause bad connections,
• The single, AAA battery can only deliver a voltage of 1.5V, and if the actual voltage is any lower, either because a Ni-MH battery was used in place of an alkaline cell, or, because the battery is just plain low, the low-voltage equalizer chip will no longer work fully, resulting in sound that reveals the deficiencies in the voice-coil.

The second set of headphones overcomes both these limitations, and I fully expect that its equalizer chips will have uniform behaviour, that my ears will be able to adjust to in the long term, even when I use them for hours or days. Also, I’d tend to say that the way the equalizer arrangement worked in the first set of headphones, was not complete in fulfilling its job, even when the battery was fully charged. Therefore, If I only had the money to buy one of the headphones, I’d choose the second set, which I just received today.

But, having said that, I should also add that I have two 12,000BTU air conditioners running in the Summer months, which really require the noise-cancellation of the first set of headphones, that the second set does not provide.

Also, I have an observation of why the EQ chip in the second set of headphones may work better than the similarly purposed chip in the first set…

(Updated 9/28/2019, 19h05 … )

## 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

## 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… )