[okydoky]

Originally Posted by Dan Lavry

OK, let’s leave dither and noise shaping out.

When talking about noise, one should not lump all the noise sources into one. Instead, one should examine the various causes of noise individually. This goes beyond a single post, but a few comments are in order:

Much of the limitations in dynamic range of both “analog only” and converter devices is due to noise generated by components (resistors, semiconductors and even small value caps). Take resistor noise, and you find that it increases with bandwidth (though not linearly, but as a square root of the bandwidth). If one provides more bandwidth, you end up with MORE NOISE. You do not deal with a fixed amount of noise spread over more bandwidth. This notion is plain wrong. The statement that noise goes up by a square of the bandwidth is correct, so double the bandwidth means 3dB more noise (not 6dB), but that is still more noise, not less noise…

One can look up and note that the lowest bandwidth converters (such as for weighing scales) offer the highest signal to noise, and as one increases bandwidth, the SNR and the number of bits drop. At 100MHz, you do not have 16 bits, and at 4GHz, a single bit (a comparator) is not an easy task… One can also look at OPamps and see a similar “curve”.


Another aspect (other then analog noise):
When it comes to converter design, one has to look at a more specific picture – what type of converter. Given that so much of audio conversion is based on sigma delta, there is a basic fundamental reason why more bandwidth means more noise. The factors impacting a “basic block diagram paper design” of a sigma delta are: 1. Number of modulator bits 2. Modulator filter order and 3. modulator oversampling rate. For example, one can have a 5 bit modulator with a 5th order feedback filter operating at 256fs.

Holding all things equal, the “amount of noise shaping” is fixed. The designer gets to remove noise from a selected frequency range (such as the audio range), and move that noise to a frequency range that is not in use. It is analogous to “digging a whole in the ground”. You start with 1000 cubic feet capability, and you may end up with 1000 feet deep hole that is only 1 square foot, or a 1 foot deep hole that has an area of 1000 sq feet. You may also end up with 10 feet deep and 100 square foot area…
The “area of the hole” is analogous to frequency bandwidth. The deeper the hole, the lower the noise floor. So given some fixed resources, you have to trade off bandwidth against noise! That is the ABC of sigma delta.

Note that the noise limitation and the tradeoff between noise and bandwidth take place at the modulator of the converter. The decimator block converts the high rate low bit modulator data (such as 5 bits at 256fs or similar) to a final format, but the trade off is already etched in stone. If a modulator is designed to accommodate near 96KHz audio (192KHz sampling rate), it offers the capability to digitize signals near 96KHz, but at cost to dynamic range. Making such a converter accommodate 96KHz sampling (near 48KHz audio) calls for one more stage of decimation (factor of 2) but the noise limitation is already "built in" into the previous modulator stage.

A converter designed for say 48KHz audio (96KHz sampling rate) offers better SNR then a 192KHz converter used at 96KHz rate. The 192KHz converter offers an extended audio range (48-96KHz) that you do not need, at the expanse of lower SNR at 0-48KHz range (some of that range you do need).

Regards
Dan Lavry

hi,

you are talking about a different thing [and i do not even believe it would be a salient issue in truth].

one does not increase quantization error by sampling faster. why lead people to that erroneous conclusion?

i'll post a more complete quote and reference below, because, respectfully, i think you may have misunderstood what was said.

"A converter where the input signal is sampled much faster than the Nyquist rate is called an oversampling converter. The signal bandwidth of the input signal is denoted by f(sub)b and the Nyquist rate, which is the minimum sampling frequency to avoid aliasing, equals f(sub)N = 2f(sub)b.
........
Compared to a 1x Nyquist-rate converter, the noise in the output signal is reduced by OSR [oversampling ratio].

The formulas to calculate the signal power are identical as for a Nyquist-rate converter. From the ratio of figures 2.6 and 2.11, the peak signal-to-noise ratio of an oversampled converter results as: SNR(sub)p = 3/2 times 2(sup)2B times OSR.

Expressing the equation in dB yields SNR(sub)p = 1.76 + 6.02 B + 10log(OSR) dB.

This clearly illustrates the advantage of oversampled converters. The SNR(sub)p improves by 3dB/octave, or equivalent, by 0.5 bit /octave of oversampling ratio. In other words, the accuracy of an AD converter can be improved simply by oversampling the input signal. This comes of course at the cost of increased sampling rates and the need for faster circuits. This shows that oversampled converters allow to trade speed for accuracy.

This improvement over Nyquist-rate converters can also be explained intuitively. By oversampling, more samples of the input signal are taken. The signal components add linearly, but the quantization noise components add with a square root. So, when the samples are averaged together, the quantization error is reduced.


It is possible to exploit the benefits of oversampling even further by employing noiseshaping. This results in a delta-sigma converter, as explained in the next section.

[Design of Multi-Bit Delta-Sigma A/D converters: by Yyes Geerts, Michiel Steyaert, Willy M. C. Sansen]"



its my understanding that the statement in mr. katz's book is correct.

and that is not even taking into consideration other advantages of higher sampler rates, such as better impulse response [amplitude], better results in processing the files, and, importantly, reduced latency [since we are making music with these things, not running sine wave tests, and timing / rhythm is an essential aspect of music].

frankly, reduced latency alone trumps any miniscule design issues you say you encounter in your designs at higher sample rates.

i also see no sigma delta modulators that would exclude a 192kHz sample rate. therefore, even under the theory that there is some sort of "trade-off" "etched in stone at the modulator", the 192kHz system rate would fare no worse in that regard.

always an interesting discussion.


right.