Sound quality - what is this?, page 2

I would like to show the very interesting Energy Compaction Property of the Discrete Cosine Transform (DCT).
An MP3 encoder is very complicated to be discussed but at some point of the encoding process it uses the Modified Discrete Cosine Transform (MDCT). I do not know MDCT and that is why I am going to use an example with a DCT found in the book Discrete-Time Signal Processing by Alan V. Oppenheim and Roland W. Schafer.
I have done the calculations for the example so as to be able to show the figures illustrating the results. I would like to point out my results seem a little bit different, so at least one mistake exists somewhere but I do not believe I have it. Otherwise the main result is the same.

The DCT used in the example (DCT-2) is defined by the transform pair:



and β[k] = ½ if k=0 and β[k] = 1 if k≠0.
Here x[n] is the discrete sequence (a discrete signal like a PCM signal) of N points
and X[k] is the discrete sequence of DCT coefficients of N points also calculated for the sequence x[n]..
(Actually, the content of an mp3 file is such MDCT coefficients.)
The signal sequence is represented by the a set of basic (cosine) functions and a set of coefficients X[k] calculated for the specific signal using the same set of basic functions.
The sequences   n = 0, 1, … 31 (N = 32)
and the sequences  X[k] are shown in the figure below.
As it can be seen in the figure the magnitude of the coefficients X[k] decreases very fast which reminds many of them do not contribute scientifically in reconstructing the original sequence x[n].



To demonstrate the Energy Compaction Property of the Discrete Cosine Transform the example shows a sequence    n = 0, 1,…, N-1  p = 1, 2,…, N
Which is reconstructed sequence x[n] using only p (part of the) coefficients.
The figure below shows the original sequence – x[n] and the  sequences X5[n], X6[n], and X10[n] reconstructed by using 5, 6, and 10 coefficients (For perfect reconstruction all 32 coefficients are needed.).
The visual inspection confirms 10 coefficients produce a perfect reconstruction.



Finally, the example shows how the approximation error depend on p by defining

to be the mean squared (ms) approximation error
It is plotted in the figure below along with the X[k] series of coefficients. As it can be seen the max ms approximation error is when using only one coefficient and it is 1000 times less when 9 or more coefficients are used.



The example illustrates how powerful the DCT Energy Compaction Property is and why DCT is widely used for data compression.

One more thing to mention. If we think x[n] was an PCM sound and we need to encode it then this particular example shows that if the criterion for acceptance was an ms error less than 0.0001 then no more than 9 coefficients would be needed to reconstruct the original signal and the reconstruction would be, I would say, perfect. Then every next coefficient to be encoded would only increase the file size without adding anything significant to the quality of the sound.

spasv  
ur uploading xmstreams with two to three times the bandwidth they are streamed originaly  
peeps please read on here


a big fuck to the ajax if the link doesnt work!

and u can post as much funny self test pictures from ur laboratory as u like:

XM-HAS-NO-CD-QUALITY!!!

Let him do his job. As long as people like it (there are enough of them), even the bloated m4a files are not important - as long as people think they downloaded great music it is okay. The quality of XM is very limited (to say the least) but still a few million people are willing to pay for the streams - so why not offering the same music for free?

i appreciate spasvs efforts, no doubt about that!
but the teqnique ...  

I have two problems: one easy and one not so easy.

You have a Digital Audio Source, a Digital Audio Broadcast Channel (DABC), and a Receiver as shown in the figure below:



The DABC transmits the Digital Sound in real time, you are at the receiver site and you are listening the music. You have perfect hearing, perfect sound system but …
What is the sound quality you are listening?

1) The first problem.
The DABC streams a 48 kHz digital sound @ 1536 kbps. You have recorded the sound, you have done a Spectrum Analysis and you have found the sound spectrum bandwidth was 16 kHz.
a) What is the source signal spectrum bandwidth?
b) What would be the minimum channel bit rate if it was to transmit an mp3 stream so as the source signal bandwidth to be preserved?

2) The second problem – the more difficult one.
Neither know you something about DABC nor about the Digital Source.
You have recorded the sound, you have analyzed it using Spectrum Analyzer and you have found the digital sound you have received was 44.1 kHz sampled and its spectrum bandwidth was 22.05 kHz.
a) What was the source spectrum bandwidth?
b) What should have been the DABC bit rate if it would be to broadcast this sound as a PCM signal?
c) What should have been the DABC bit rate if it would be to broadcast this sound as an AAC encoded signal?
d) What should have been the DABC bit rate if it would be to broadcast this sound as an AAC++ version 2 encoded signal?

Notes:
• What could help in solving the problems is:
Shannon Sampling Theorem/the Nyquist  rules,
Some “laboratory” work experience,
maybe Ojay and TomMix also but I am not sure about that.
• part d) is the most difficult part – TomMix couldn’t solve it at all.

oh part d is solved: 34 to 50kbit/s
https://www.tribalmixes.com/#/forums.php%QUaction=viewtopic&topicid=3148&ajaxloader=1&poid=1183751598270
ah i see u havent posted a response there yet ... maybe u missed it.

i dont get ur strange setup, sorry.

todays DABC just send 192kbit 48khz mp2 in my country. ordinry DAB.
and theses streams are just lossy to mp2 encoded audio cds
to me they are transparent also...