Non-integer sampling frequency conversion is no big deal. The worst case error is a fraction of the error which would have existed if the interpolated data had not been calculated and output at the higher rate. |
Phillips used 4 times oversampling in their first CD players so that they could achieve 16 bit accuracy from a 14 bit D/A. At that time, 16 bit D/A, as used by Sony, were lousy, but the 14 bit units that Phillips used were good. The really cool part of the story is that Phillips didn't tell Sony what they were up to until it was too late for Sony to respond, and the Phillips players ran circles around the Sony ones.
In Sean's explanation the second set of 20 dots in set B should not be random. Those dots should lie somewhere between the two dots adjacent to them.
Here is my explanation.
Assume there is a smoothly varying analog waveform with values at uniform time spacing, as follows. (Actually there are an infinite number of in-between points).
..0.. 1.. 2.. 3.. 4.. 5.. 6.. 7.. 8.. 9.. etc
If the waveform is sampled at a frequency 1/4 that of the example, (44.1 KHz perhaps) the data will look like the following:
..0.......... 3.......... 6...........9..... THIS IS ALL THERE IS ON THE DISC.
A D/A reading this data, at however high a frequency, will output an analog "staircase" voltage as follows:
..000000000000333333333333666666666666999999999
But suppose we read the digital data four times faster than it is really changing, add the four values up,
and divide by 4.
First point
..(0+0+0+3)/4 = 0.75
Second point
. (0+0+3+3)/4 = 1.5
Third point
(0+3+3+3)/4 = 2.25
Fourth point
.. (3+3+3+3)/4 = 3.0
Fifth point
. (3+3+3+6)/4 = 3.75
Sixth point
. (3+3+6+6)/4 = 4.5
Seventh point
. (3+6+6+6)/4 = 5.25
Eighth point
(6+6+6+6)/4 = 6
.And so on
Again we have a staircase that only approximates the instantaneous analog voltage gererated by the microphone when the music was recorded and digitized, but the steps of this staircase are much smaller than the staircase obtained when the digital data stream from the disc is only processed at the same rate that it was digitized at. The smaller steps mean that the staircase stays closer to the original analog ramping signal.
Note also that we are now quantized at 0.25, instead of 1, which is the quantization of the data stream obtained from the disc. A factor of 4. Thats like 2 bits of additional resolution. Thats how Phillips got 16 bit performance from a 14 bit D/A. |
The term "Error Correction" applies to a scheme where redundant data is combined with the information in such a way that a decoding algorithm can recover the original information WITHOUT ANY LOSS, provided that the number of transmission errors, and their distribution in time, does not exceed what the encoding algorithm is designed to deal with. This is not a "bandaid" for poor transmission. It is a way to make it possible to run the hardware at much higher bandwidth because errors can be alowed to occur.
"Interpolation" is not "Error Correction". Interpolation is what you can do if the errors do exceed what your algorithm is designed to deal with. Depending on what the signal is doing at the time that transmission glitches occur interpolation may or may not result in significant error in recovery of the information. |
Sean...The sampling (your first set of dots) is at 44.1KHz. The highest audio information that exists at this sampling rate is around 20KHz, and at this frequency the music signal amplitude is very small. Therefore, unless the signal is momentarily a constant (two adjacent points the same) the in-between points will lie between adjacent points. Of course this is all overlaid with random noise that will blur the quantization staircase.
The CD recording protocol has been cited as an everyday example of the application of CRC error correcting technology, and I have seen descriptions of the CD protocol as having interpolation as a "fall back" procedure when the CRC error correction fails. Of course the second "fall back" is to abort playing the disc, and this ought to be the only time that the process is easily heard.
To tell the truth I have never actually read this infamous "Red Book" which defines the CD spec, and so am relying on what others have reported. How would I get a copy? |
Bombaywalla...Your check is in the mail :-) |
Sean....Homework is to read.. http://en.wikipedia.org/wiki/Reed-Solomon_error_correction
Test on Monday :-) |
Pabelson...According to Nyquist, just two (error free) samples per cycle will perfectly recover a sine wave. But, in this error-prone nonsinusoidal world, where I played with digital data stream representations of analog waveforms (non audio), experience taught me that four samples per cycle was worth the trouble. That's why the 96 KHz PCM of DVDA (or 192KHz for stereo) solves the bigest problem with redbook CDs. 24 Bits is nice too. |
Pabelson...I guess you mean that if the sine wave frequency is EXACTLY one half the sampling frequency, a sync situation exists. OK. Change the sampling frequency enough so that the phasing of the sine wave drifts across the sampling interval. Picky, Picky :-)
I personally don't have much of a gripe about CDs, but then my ears are 67 years old, and don't have the HF sensitivity of some of our golden eared friends. Based on my experience, which led me to believe that Nyquist was an optimist, I can believe that HF is a lot better with 96KHz sampling.
Sean...I disagree about the effect on quality of "off the shelf" parts. In the military electronics business, we used to design all our own chips, even microprocessors. However, even at great expense we could never match the research and development effort, propriatary skill, and quantity production, typical of commercial products that were functionally equivalent to our designs. A mature "off the shelf" product has had all its bugs weeded out. |
Since Sean has confessed his error, I will do the same. My explanation actually showed a ramping signal of 3 units in four samples. While this was not incorrect, it is not consistent with the analog signal that I assumed at the beginning. The following is an updated version of my explanation, for posterity.
Phillips used 4 times oversampling in their first CD players so that they could achieve 16 bit accuracy from a 14 bit D/A. At that time, 16 bit D/A, as used by Sony, were lousy, but the 14 bit units that Phillips used were good. The really cool part of the story is that Phillips didn't tell Sony what they were up to until it was too late for Sony to respond, and the Phillips players ran circles around the Sony ones.
In Sean's explanation the second set of 20 dots in set B should not be random. Those dots should lie somewhere between the two dots adjacent to them.
Here is my explanation.
Assume there is a smoothly varying analog waveform with values at uniform time spacing, as follows. (Actually there are an infinite number of in-between points).
..0.. 1.. 2.. 3.. 4.. 5.. 6.. 7.. 8.. 9.. 10. 11. 12 etc.
If the waveform is sampled at a frequency 1/4 that of uniform time spacing of the example, (44.1 KHz perhaps) the data will look like the following:
..0............... 4.............. 8...............12..
THIS IS ALL THERE IS ON THE DISC.
A D/A reading this data, at however high a frequency, will output an analog "staircase" voltage as follows:
..000000000000000004444444444444444488888888888888812
But suppose we read the digital data just four times faster than it is really changing, add the four values up,
and divide by 4.
First point
..(0+0+0+4)/4 = 1
Second point....(0+0+4+4)/4 = 2
Third point.....(0+4+4+4)/4 = 3
Fourth point....(4+4+4+4)/4 = 4
Fifth point.....(4+4+4+8)/4 = 5
Sixth point.....(4+4+8+8)/4 = 6
Seventh point...(4+8+8+8)/4 = 7
Eighth point....(8+8+8+8)/4 = 8
....And so on
Again we have a staircase that only approximates the instantaneous analog voltage gererated by the microphone when the music was recorded and digitized, but the steps of this staircase are much smaller than the staircase obtained when the digital data stream from the disc is only processed at the same rate that it was digitized at. The smaller steps mean that the staircase stays closer to the original analog continuously ramping signal.
Note also that we are now quantized at 1, instead of 4, which is the quantization of the raw data stream obtained from the disc. A factor of 4. Thats like 2 bits of additional resolution. Thats how Phillips got 16 bit performance from a 14 bit D/A. |
More corrections! They don't affect the basic idea, but could easily confuse people. Sorry about that. Hopefully this is it.
If the waveform is sampled at a frequency four times that which corresponds to the uniform time spacing of the example, (44.1 KHz perhaps) the data will look like the following:
Note also that we are now quantized at 1/4, (0+0+0+1)/4 ,instead of 1, which is the quantization of the raw data stream obtained from the disc. A factor of 4. Thats like 2 bits of additional resolution. Thats how Phillips got 16 bit performance from a 14 bit D/A.
OK Sean...Sorry you felt left out because no one jumped all over you. The following is my modification of your statement.
Some digital representations of analog (analogue in England) waveforms are a poor replication of the analog source because they lack the resolution (bits) and sampling rate appropriate for the bandwidth of the signal. Inaccuracy is not inherent to the digital format, but represents a design decision regarding what level of error is acceptable. |
Bombaywalla...It's an interesting question about whether the world is, at its heart, digital or analog. The electrical signal, regarded as analog, actually consists of discrete packets of charge, called electrons. The human sense of hearing is implemented by the "firing" of discrete cells called neurons.
Of course this digital condition exists at so fine a level that it is entirely reasonable to consider the process as being continuous. But it does point out the fact that at some point, and we can debate where that point is, digital audio becomes, for all practical purposes, the same as analog. |
