Sample Rates Revisited

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In the October 6, 2008 "Tips from Titus" column I made an error that would confuse readers. There are two ways to correct the error, which occurs in the section about using a 16-bit converter.

First:  Substitute an 8-bit ADC for the 16-bit ADC used as an example in the column. An 8-bit ADC has about a 48 dB dynamic range. Thanks to Dr. Carl Dreher at Samsung Telecommunications America for pointing out this solution.

Second: Read the revised column below that uses a 12-bit ADC in the example and includes comments from Hans Weedon of Analogic:

In my previous Tips column (Sept. 22, 2008), I noted my rule of thumb for sampling at five to 10 times a signal's bandwidth. That range provides a good starting point, but you can better determine a sample rate for given signal conditions. To start, assume you have a signal that includes unwanted harmonics, noise or other signals of no interest. To remove these signals you use an anti-alias filter - usually a low-pass filter - so your analog-to-digital converter (ADC) "sees" only the signals of interest.

Unfortunately, you cannot make a "brick-wall" filter that, at a specific frequency, instantly goes from passing all signals to blocking all signals. Filters have a characteristic roll-off that attenuates signals at different rates. The roll-off varies by filter type and the number of filter stages or poles. A 2-pole low-pass Butterworth filter, for example, provides a shallow attenuation with frequency whereas an 8-pole Chebychev low-pass filter offers a steep attenuation. Filters also have a characteristic cut-off frequency (fc), specified as the point at which filter attenuation drops to and stays below -3 dB.

I'll use a 5-pole Butterworth filter as an example and you can see its response in the graph, below. This filter starts to roll off at about 650 Hz, the highest frequency component I want to examine in my unknown signal. This low-pass filter reduces the signal power by half (-3 dB) at 1 KHz, and by 1000 fold (-30 dB) at 2 KHz.

In the original column, I erroneously calculated a 48 dB dynamic range for a 16-bit ADC. Hans Weedon of Analogic in Peabody, MA said, "What a minute, you got it wrong. The dynamic range for a 16-bit ADC is more like 96 dB, or even 98 dB for a sine wave."  He's right and I apologize for leading readers astray with my poor attention to math details. Dynamic range = 20 * log (2ⁿ), where n represents the resolution in bits.  In my haste, I used a factor of 10 (which works for power) rather than 20, which applies to voltages (amplitudes).  I have reworked the example with a 12-bit ADC and the revised information follows:

Assume I plan to use a 12-bit ADC to digitize my signals. That converter has a dynamic range of 1 part in 4096, or about 72 dB. Dynamic range = 20 * log (2¹²) = 20 * log (4096). Look at the filter graph and you'll find the -72 dB point at about 5,280 Hz. So, even though I use a filter to try to remove signals above 650 Hz, my ADC must digitize signals out to 5,280 Hz.  As a result, I need to set the ADC to sample at more than 10,560 samples/sec or my digitized data will include aliased signals. So, even though I want to digitize a 650 Hz signal, with this filter configuration I must oversample at more than about 16 times the frequency of the signal I want to measure.

You can change filter characteristics to obtain a sharper attenuation. An 8-pole Butterworth filter with an fc equal to 880 Hz, for example, reaches the -72 dB point at 2,785 Hz. So you would sample at more than 5,570 samples/sec, or over eight times the 650 Hz signal frequency. Both sample rates in these examples come close to my rule of thumb. (In a conversation with Hans Weedon, he recommended sampling at a rate at least four times the Nyquist rate.) If I had a higher-resolution ADC, it would offer a higher dynamic range, so the sample rate would have to increase accordingly. And other considerations become more important at higher resolutions. So, keep in mind  there's more to sampling than just the Nyquist criterion.

There are other reasons to oversample a signal and I'll cover them in another column. If you want to experiment with filter characteristics and anti-alias filters, I recommend the free FilterLab software from Microchip Technology.

Hans Weedon also added some additional helpful information in a second email message:

There is one statement, however, where, although completely correct, there is a point of possible confusion. You state that 30 dB is a factor of 1000 reduction. In power that is correct, but as far as signal integrity that is only a factor of 31.6, not at all much to talk about.

Consider for instance the case of audio digitizing. A normal listening environment require about 50 mW of power from the amplifier, yet we use power amplifiers with 100W maximum output power. That is only 33 dB above 50 mW. Music easily has 20 dB of dynamic range, so an amplifier that normally outputs 50 mW must be able to output at least 5W to cover the dynamic range of the audio program.

Where I want to go with this is that as far as normal audio digitizing is concerned much more than 30 dB is required, actually 30dB is less than the instantaneous dynamic range of a listener with about a 40 dB range. A typical audio digitizer requires about 50 dB instantaneous dynamic range.

As far as instrumentation is concerned, 30dB may not be good enough either. Distortion of 0.5% could be considered reasonable in a digitized signal system and 0.5% distortion is 46 dB down from the signal. Still more than the 30 dB.

We live in a world with requirements of about 1000 to 1 dynamic range in the signal domain, which in the power-domain is 1,000,000 to 1.

This whole story basically states that to accurately digitize a signal we really need very aggressive anti-aliasing filters. These filters are hard to build and require expensive capacitors and inductors to work well. Active filters are not that much simpler than passive filters either.

Thirty years ago, when accurate A/D converters cost more than $100, spending $50 on an anti-aliasing filter was not that outrageous, but today with A/D converters costing less than $5, spending $50 on anti-aliasing filters is uneconomical. The A/D converter technologies has reduced the cost of a unit by a factor of 100, whereas the cost of a (passive) filter has not changed. We must work smarter, not harder.

The much smarter way of digitizing a signal is to over-digitize by a large amount and do the anti-aliasing filter digitally. A sigma-delta converter does exactly that. A sigma-delta converter may over-digitize the signal by 256 times and only require a signal anti-aliasing filter with two poles. A 128-tap digital filter can easily compute a 64 pole filter, something that inductors and capacitors could never do.

A sigma-delta converter that does this can be had for less than $5. In other words, to digitize a single signal, a sigma-delta converter would be the preferred choice.

The high speed A/D converter is much better suited for use in a multiplexed situation, where many signals should be digitized in a sequential manner. Take for instance a CT (Computed Tomography) scanner where you may have 16,000 channels to digitize at a readout rate of 2500 readings per second.. In this case, the multiple multiplexed A/D converters may be the most economical approach. You would design a very inexpensive front end amplifier with some anti-aliasing properties followed by a one A/D converter servicing maybe 128 or 256 front end circuits. The A/D conversion function then becomes a few pennies per channel.

I have been designing this type of systems for the last 35 years, and the choice of a few high-speed A/D converters multiplexed versus a converter per channel is always a design choice mostly governed by the economics of the situation. The multiplexed A/D converter may suffer some signal integrity issues, but the alternative approach of using a converter per channel becomes prohibitively expensive. -- Hans J. Weedon