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Worst case design and the BITX 40 mic amp

The BITX 40’s microphone amp has a flaw that may shorten the lifespan of microphones by overstressing their components. Here’s what I found and how I fixed it so far.

Here’s the BITX 40 mic amp schematic:

The mic bias is supplied by the TX line through 4.9K of resistance. TX is the DC power supply, 12-15V.

Most electret microphones have a maximum voltage of 10 V, though I’ve seen a few datasheets with a 9 V limit. The data sheets also say that the mics draw 0.5 mA maximum. No minimum is given.

Now, applying Ohm’s law,

$(0.5 \mathrm{mA}) * (4.9 \mathrm{k\Omega}) = 2.45 \mathrm{V}$

Subtracting that from 12 V gives us 9.55 V, so one might think that everything is great.

Not really.

From the books of Bob Pease I learned the importance of worst case design. In worst case design, one looks at the datasheet minimum and maximum specifications and designs the circuit to work under the worst combination of specifications. Ignore the datasheet typical ratings, because those describe the best case. No one wants a circuit that only works in the best possible circumstances.

The BITX 40 docs specify a maximum power supply of 15 V, so that is the worst-case TX. The worst case microphone current is the minimum current, but I haven’t seen a mic datasheet that lists a minimum. Therefore, I have to make a guess. I need the guess to be on the safe side without being ridiculous. Knowing that the current is set by a JFET and knowing about both the variability of JFETs and how data sheet maximums are chosen, I could guess that the typical current is maybe half of the maximum and the minimum half of that, so 0.125 mA. If I want to make a truly robust circuit I could assume a minimum current of zero. I know a JFET can’t create current out of nothing, so the lowest possible current it draws is zero.

Let’s look at what voltages that creates:

$(15 V) - (0.125 \mathrm{mA})(4.9 \mathrm{k\Omega}) = 14.388$

Uh-oh. That’s well above the 10 V maximum for the mic.

The zero current case is easier to work out. With zero current through the resistor, the mic sees the full 15 V.

A simple fix is to put a resistor in parallel with MIC1, making a voltage divider. I use a computer headset with my BITX, and I know that computer mics are typically fed with 5 V through a 2.2 kΩ resistor. Knowing that, I picked a 4.3 kΩ resistor across MIC1 to form a voltage divider equivalent to 2.3 kΩ fed by 7 V. That’s close enough, and 7 V is well below the maximum for most electret elements.

Resistor location on the PCB. — The 4.3 kohm resistor on the underside of the BITX 40 PCB. The wires are for the Si5351 BFO mod I added.

Unfortunately, the mic now drives an impedance about 30% lower than before, reducing its output. The RF drive, R136, should be adjusted to compensate.

One could fix the issue a different way, of course. A Zener diode in series with R121 could drop the voltage just as effectively without lowering the amplifier’s input impedance. An LM780x regulator could do the job, too, albeit with more components. I happened to have 1% resistors within reach and Zener diodes on a different floor of the house. I chose the resistor.

Is this a real problem?

No one else seems to have noticed this issue, so it’s worth asking whether it matters. One answer is that it is worthwhile, and pleasing, to do things right. There is a place for bodging a circuit together that works as a one-off, but I enjoy the n-dimensional puzzle of getting the details right.

The other answer is that it might be killing microphones, but not often enough that anyone has noticed. The BITX 20 mailing list, home to BITX 40 discussion, has seen a few comments from hams whose microphone capsules worked for a while, then failed. Some have been able to trace the problems to bad solder joints or physical damage, but I have to wonder if any of the remaining unsolved cases were caused by overvoltage.

Manufacturer maximum specifications are based on an assumption about the device’s expected lifetime. Operating beyond that specification can be expected to shorten the device’s lifetime. Sometimes that shortened lifetime is dramatic, with a flash of light, a puff of smoke, or an outpouring of heat. Other times it is subtle and takes longer. Perhaps mic elements will fail faster than usual. Perhaps their average lifespan will be 2 years instead of 20.

The difference might not be enough to notice, but we can fix it anyway. Worst case design will save the day.

How Delta-Sigma Works, part 3: The controls-system perspective

This post is part of a series on delta-sigma techniques: data converters, modulators, and more. A complete list of posts in the series are in the How Delta-Sigma Works tutorial page.

In the first installment of How Delta-Sigma Works, I presented the basic first-order delta-sigma converter loop. Now it is time to begin digging a little deeper and look at how the loop works. To do this, we will need the mathematical tools of closed-loop control systems. Even without the mathematics, thinking about a delta-sigma modulator as a closed-loop controller can bring insight into how it works.

In engineering, closed-loop control is often used to keep a system working at a setpoint despite environmental disturbances or variations in the system itself. A furnace thermostat is a simple closed-loop controller, turning on the heat when the temperature drops below a setpoint and turning it back off when the temperature is above. Another example is vehicle cruise control. Unlike a thermostat, which usually has a binary output (on/off), cruise control adjusts the engine fuel control to maintain a roughly constant speed. In my car, it does this by physically moving the gas pedal. The environmental variations cruise control can encounter include hills, the quality of the fuel, and headwinds or tailwinds. The closed-loop controller adjusts the gas pedal as needed to maintain constant speed despite these effects.

A simple closed-loop control system can be drawn as in the figure below. The reference input, r, is the command input to the controller. In the thermostat example, r is the setpoint temperature, while for cruise control, it is the set speed. The output, y, is the controlled value. This is not necessarily in the same units as the reference input r. For example, in the cruise control, y might be a direct measurement of speed, or it might be something else related, such as a cumulative count of revolutions of the vehicle’s wheels.

In between the input r and the output y is the control loop. At the bottom of the loop, in its feedback path, a block h processes the output y into a form that is subtracted from the input r to create an error signal e. This error signal is an indication of how far the controller is from the desired operating point. The goal of a controller design is to keep e near, if not at, 0. The feedback block h can represent many different things. In the example of a cruise control system with the output y in units of distance, h might calculate speed by computing the derivative of y. In other systems, h might simply scale the value to convert its units. If y and r are already in the same units, such as in a thermostat example, h can pass through y unchanged ( $h = y$ ).

Now that the error signal has been calculated, it is processed by the controller gc, the output of which goes to the “plant” being controlled. (To a controls engineer, anything being controlled, whether a car, a heating system, or a giant factory, is a plant.) The plant is represented by the box g_p. The processing in the controller g_c is easy to grasp. The controller calculates some function of its input in order to find the command it should give the plant.

What happens in this plant, g_p, may be a little harder to imagine. The function g_p is a mathematical model of the physics of the actual plant. It may be calculated from basic principles (a physicist’s delight!), or it may be an empirical model derived from the inputs and outputs of the actual plant. In any event, a reasonable guess at the function g_p is necessary before one can design a closed-loop controller.

A delta-sigma modulator also has a closed loop, which suggests that perhaps insight can be gained by comparing it to a controller. The first-order delta-sigma analog-to-digital converter from the first installment, is shown again below.

The resemblance to a closed-loop controller is clear when one groups the blocks as in the next figure. The subtractor has the same function in both diagrams, comparing the input to the output. The integrator functions as the controller, g_c, and the analog-to-digital convertor and its register are the plant, g_p, being controlled. The digital-to-analog converter is the feedback path, h.

This grouping gives immediate insight into how the delta-sigma modulator does its magic: It is a linear controller for an analog-to-digital converter, which is the plant. The controller is always trying to drive that plant’s output as close as possible to the setpoint, and does so by adding up (integrating) the error signal. Also, since the integrator is adding up the history of the error signal, it can be seen that although the output at any given moment may not equal the input, the long-term average will be equal. That integrator will try to keep the long-term average of the error, e, equal to 0.

There are many controllers that can control a given plant. PID (proportional-integral-derivative) controllers are simple and very popular, while more sophisticated controllers can be designed using other techniques. If a delta-sigma modulator is a control loop, it is reasonable to ask if controllers other than an single integrator will result in a better-performing modulator. In fact, other control functions can be used in delta-sigma modulators and can give lower noise or more desirable characteristics in the frequency domain.

Finally, one should not get too carried away with a linear control model. Analog-to-digital and digital-to-analog converters are inherently non-linear, while the mathematics of control theory primarily deals with linear systems. Assuming linear behavior will get us a long way towards understanding delta-sigma techniques, but it is important not to take the analogy too far.

Next in this series: Noise shaping, the frequency-domain secret behind delta-sigma data converters.

AK5388 audio ADC breakout board design

A few weeks ago, I picked the AKM Semiconductor AK5388 as the analog-to-digital converter for my receiver design. This high-performance audio DAC has a 24-bit output with up to 123 dB signal-to-noise ratio (A-weighted). I hope that the narrower bandwidths of a communications receiver will beat that. The idea is to do the automatic gain control (AGC) in digital, so that the receiver will not need an analog AGC loop. I hope that a preamp or attenuator at the front end will be enough to make up for the limited dynamic range of the AGC.

Now, I am aware that getting performance like that requires great care in the details of the design and layout, but if this wasn’t a challenge, it wouldn’t be nearly as much fun! The AK5388 datasheet does give me some concern, because it does not make much mention of the techniques needed for a high-performance DAC. Admittedly, datasheets from Japanese manufacturers are often thinner on details than those from the leading US suppliers, and this one is much more complete than some. On the other hand, it may be that the datasheet is glossing over any coddling this chip will need.

There is one way to find out, and that is to start breadboarding. This is a surface-mount part, so I have designed a breakout board very much like the board I built for the ~~Actel~~ Microsemi A3PN250 FPGA. The schematic is below.

The schematic largely follows the “System Design” section of the datasheet and the example of the AKD5388-A evaluation kit (PDF). It’s not strictly a breakout because I included on-board analog and digital power supplies. An ADC like this needs quiet, local power, so I picked some reasonably quiet supplies and put them on-board. I chose 7800-family regulators from TI because they have reasonable noise specifications. I almost went with other choices that were explicitly designed for low noise, but they were quite a bit more expensive for not much less noise. The AKD5388-A kit uses an uA78M05 and claims to be able to obtain the full dynamic range of this ADC, so there is little reason to spend more at this point.

I’m using the analog power supply as the voltage reference. I considered a dedicated reference, which would be more expensive but perhaps lower noise. Here, too, I followed the lead of the evaluation kit.

The PCB layout should be straightforward. I like to build by skywiring components over a groundplane, so this breakout board will be designed to sit directly on a groundplane, with all components and connections on the top side. All external wiring points will be pads on the PCB, ready for soldering, just as with the A3PN250 FPGA board.

I’m excited about trying out this part. What do you think? Comments welcome!

Choosing a high-performance audio ADC

As I discussed two weeks ago, it is time to refocus my efforts on the DSP-based ham radio project that started this blog. Let’s take a look at the architecture I had in mind originally:

Block diagram of a near-zero IF receiver with I and Q paths

This is a popular topology for radios that put their intermediate frequency at or near 0 Hz. It is also very similar to the most popular ham software-defined radio topology. In fact, those ham SDRs simply substitute a PC sound card for the ADC and the PC’s CPU for the FPGA. Here, though, I don’t want to bring a PC into the picture yet. Radios that require a PC don’t feel like “real radios” to me. I want to end up with a self-contained box, though I won’t mind if it optionally integrates with a PC.

For simplicity, I want to implement the receiver’s automatic gain control (AGC) functional digitally, after the ADCs. This means that with a well-designed front end, the ADC dynamic range will become the radio’s dynamic range.

With that in mind, I went looking for the best dynamic range audio ADC I could find, or at least the best one I could afford. I have identified seven manufacturers of 24-bit audio ADCs. Here is the best that they have:

Part	SNR, A-weighted	THD+N	Package	Price
TI PCM4222	123 dB	-108 dB	TQFP-48	$29.99 D*
TI PCM4220	123 dB	-108 dB	TQFP-48	$19.10 M
AKM AK5388	123 dB (Note 1)	-110 dB	LQFP-44	$10.93 D
AKM AK5394A	123 dB	-110 dB	SOP-28	$22.00 D
Cirrus CS5381	120 dB	-110 dB	SOIC-24, TSSOP-24	$32.22 D/M
Wolfson WM8786	111 dB	-102 dB	SSOP-20	$3.48 M
ADI AD1974	105 dB	-96 dB	LQFP-48	$10.03 D
NXP UDA1361	100 dB	-88 dB	SSOP-16	$1.37 M
Notes: Figures are typical values as shown in the manufacturer’s data sheet. All are for two channels active and 24-bit PCM output. Prices are quantity 1 from the cheaper of Digi-Key or Mouser. 1. The AK5388 is a four-channel ADC with 120 dB SNR. The 123 dB SNR requires the use of “mono mode”, which connects each stereo pair in parallel with a single input, effectively creating a two-channel ADC with 3 dB better SNR. * Non-stock item, but a limited number of units are currently in stock.

The AK5388 looks like the price/performance winner, at $10.93 for 123 dB SNR and the best THD+N. One catch is figuring out the mono mode, which didn’t have crystal-clear documentation in the data sheet. On the other hand, even in four-channel mode it has 120 dB SNR, which still puts it at an excellent price/performance point.

Lurking in the SNR specification is A-weighting, which is a specification used for audio that isn’t much seen in the measurement world. The idea behind A-weighting is to reflect the human ear’s varying perception of noise at different frequencies by doing a weighted sum of the noise in the SNR measurement. Thus, frequencies where noise is more audible count worse than frequencies where it isn’t. A-weighting is not quite the right measurement for a near-zero IF ADC, first because a communications receiver’s bandwidth is quite a bit smaller than the 20 kHz used for the A-weighted measurement, and second because the communications signal being digitized is not necessarily at its final audible frequency yet.

I used A-weighting for the comparison because all of the ADCs were specified that way. Some of them also had an SNR in 20 kHz bandwidth specification, which does not weight the noise spectrum. For those ADCs, the 20 kHz SNR was 3 dB lower than the A-weighted SNR. , I decided to compare the A-weighted SNR to put the ADCs on equal ground, even though the 20 kHz bandwidth SNR is closer to what I would like to know for a communications receiver.

It’s worth mentioning that I looked at precision ADCs as well. I couldn’t find any 24-bit precision ADCs from Analog Devices or Linear Technologies that had a high-enough sample rate to be usable. In contrast, TI offers the single-channel ADS1281 and ADS1282, which offer a stunning 130 dB SNR (unweighted) at 250 samples per second. These might be reasonable for a Morse code receiver with a 100 Hz passband. When used for sideband, these ADCs would have to be operated in their 4000 SPS mode, at which their SNR drops to 118 dB. When one considers that filtering the output of the audio ADCs down to 100 Hz would provide an extra 6+ dB of SNR (because less bandwidth means less noise power), the ADS1281/1282 no longer has such an advantage. Worse, a pair of ADS1281 (two are needed because they are single-channel) will set you back $108 at Digi-Key.

In the end, I’ve found a combination of excellent performance and a good price. The AK5388 it is. The next step is to build or buy a board for it so I can start experimenting.

QST performance measurements side-by-side in a table

PA1HR's QST review measurements table Hans Remeeus PA1HR has compiled QST‘s performance measurements for dozens of recent rigs into a single, easy-to-read table. It’s perfect for anyone shopping for a new rig. It’s also great for numerically inclined homebrewers. I have been wishing for something like this so I have an idea of what kind of goals to set for a design. Have a look!

PA1HR’s QST Product Review Measurements Summary

(via Hans Brakob K0HB and the Twin Cities DX Association e-mail list)

Humbled by a sine wave oscillator

The old saw goes, “oscillators don’t, amplifiers do”, but that’s an analog saying. It doesn’t apply to the digital world. After all, digital is easy, while analog is hard, right? (Be careful not to answer too quickly!)

For the last few weeks, I’ve spent my tinkering time on adding sine wave sidetone to the Verilog iambic keyer. As you may recall, sidetone is audio feedback for the keying, in other words, audible Morse code. My thinking was that while this would appear to be a side trip on the way to FPGA-DSP radio nirvana, it would actually be valuable. My vision for the radio will require sine wave synthesis in a couple of places. Also, to have sine wave sidetone, I will need a digital-to-analog converter (DAC), so why not put theory into practice and build a delta-sigma DAC?

I looked around a bit to find a good way to synthesize a sine wave. I didn’t want to do a simple ROM lookup table, because the radio will need memories for storing filter data and coefficients. Instead, I settled on an algorithm that uses two integrators and a multiplier. For some ratios of clock to output frequencies, the multiplication by $-m$ can be reduced to a simple right-shift, which is always appealing to a hardware designer because it can be implemented without any gates. The negation can then be absorbed into the first adder, costing little more than a set of inverters and a carry input.

Two-integrator sine wave oscillator

Then I found this elegant topology in Delta-Sigma Modulators: Modelling, Design, and Applications, by Bourdopoulos et al:

Digital sine wave oscillator with delta-sigma feedback

In this design, the multiplier has been replaced by a delta-sigma modulator and two constants. This is gorgeous! First of all, the multiplier is gone, replaced by a few adders and a mux. Secondly, the delta-sigma bitstream is a built-in output for a DAC. There are a couple of wrinkles. First of all, there needs to be a scaling by $2^{-b}$ before and $2^b$ after the delta-sigma modulator. Otherwise, the modulator will see signals outside of its stable range. (Stability is a topic for a future tutorial in the How Delta-Sigma Works series.) Second, I’m not sure about putting the delay of the delta-sigma converter into the loop. It seems to me that the two integrators should be non-delaying. Third, are the two integrators sufficient filtering to remove the delta-sigma’s high-frequency noise, or is there a risk of high-frequency feedback making the loop unstable?

It sounds great, but when I coded it up in Verilog I couldn’t get it to be stable. I fiddled with constants and bit widths for a while and couldn’t get it going. I also tried two non-delaying integrators in the loop. That didn’t work, either, and I couldn’t figure out why. Rather than do any stability analysis or model it in Octave, I decided to cut my losses and go to a more direct approach.

I took the original sine wave oscillator topology, which I easily got working, and tacked on a delta-sigma modulator to the output. I decided to increase the clock rate in order to get a better signal-to-noise ratio (SNR) out of the DAC. The oscillator blew up – unstable again! After some more fiddling, I realized I was going to have to run with 24 or more bits of precision if I stuck to the design in the first figure. To reduce the precision I needed to carry, I instead split the scaling between both integrators:

Sine wave oscillator followed by delta-sigma DAC

I fiddled with fewer bits, but with the clock-to-output frequency ratio I was working with (1 MHz clock, 800 Hz output), I needed all 16 bits to see a pretty sine wave in the ModelSim waveform viewer.

It looked great in the waveform viewer, so I burned it onto the FPGA and… noise. There is a brief tone buried in the noise, then it turns to all noise, and a little later, to a high-pitched whine without noise. The sine wave buried in the noise might mean I need better filtering between the FPGA and the audio amp (I didn’t use any, relying solely on the audio amp’s rolloff), but the all-noise and whining stages tell me that the oscillator or modulator is unstable. I probably didn’t run the testbench long enough in simulation to see it.

Meanwhile, at work, I’ve been having a great time introducing Test-Driven Development (TDD) into our firmware development process. The process was tedious at first, but now that I’ve been doing it long enough to see the results, I love it! My code is better organized, better tested, and more solid than ever before.

That, combined with my frustration with the sine wave project, led me to think about applying Agile software engineering techniques to Verilog development. I did some Google searches to see if anyone else is applying TDD or full Agile to hardware. They are! (agilesoc.com: Why Agile is a good fit for FPGA and ASIC development)

I learned some technical things while working on the sidetone. More importantly, though, I was reminded that methodical, well-tested development is not only better than seat-of-the-pants hacking, it is faster as well. Hobby projects benefit from discipline just as much as professional ones. I am going to set aside the sidetone project for a while and look at some other things. Meanwhile I am going to think a bit about how to apply TDD and some other Agile concepts to my humble basement tinkering.

How Delta-Sigma Works, part 2: The Anti-Aliasing Advantage

This post is part of a series on delta-sigma techniques: analog-to-digital and digital-to-analog converters, modulators, and more. A complete list of posts in the series are in the How Delta-Sigma Works tutorial page.

Today, let’s take another look at delta-sigma conversion. The first part of this series showed how a one-bit, first-order delta-sigma modulator creates a bitstream, the average value of which equals the input voltage. It turns out that how we find that average value makes a big difference in the performance of a delta-sigma analog-to-digital converter. In fact, if done right, not only does it improve performance, but it greatly simplifies the analog circuitry preceding the A-to-D. Let’s take a look at why this works!

Aliasing Explained

One phenomenon that happens with any analog-to-digital conversion is known as aliasing. A-to-D is inherently a sampling process, in which an analog signal that is continuous in time is converted to a digital signal that exists in discrete chunks, or samples. The rate at which those samples are taken is known as the sampling rate. The figure below shows three analog sine waves, in red, being sampled at the times marked with the blue vertical lines. The top sine wave has a frequency of 1 Hz, and it is being sampled at 7 Hz. (The seventh sample is not obvious, because it is zero and the left- hand or the right-hand edge of the graph. Which edge you choose is unimportant.)

A frequency above the sampling rate results in aliasing to a lower frequency

The second and third plots in the figure show aliasing in action. When any signal with a frequency above one-half the sampling rate is sampled, the signal is <i>aliased</i> down to a frequency between 0 Hz and one-half the sampling rate. The second line shows a 6 Hz sine wave in red, which is being sampled at 7 samples per second (sps). The blue lines show where the samples fall. Because 6 Hz is above 3.5 sps (one half of 7 sps), the sine wave will be aliased. As you can see, the blue samples are identical to those you see in the top trace, except that they have the opposite polarity. The dark blue trace connects them and shows that a 6 Hz sine wave is indistinguishable from a 1 Hz sine wave when both are sampled at 7 sps. That is aliasing in action.

The same process happens for any input frequency above one-half the sampling rate. The third plot in the figure, for example, shows an 8 Hz sine wave. Again, it is sampled at 7 Hz. This time the samples are identical to those in the top line. The sampled waveform (in dark blue once again) is indistinguishable from the sampled version of our original 1 Hz sine wave.

Aliasing is so important that one-half the sampling rate has become known as the “Nyquist frequency” for a system. You may see people refer to signals as being “above Nyquist” or “below Nyquist”. Aliasing happens in a repeated pattern as the frequency rises, and each repetiion of that pattern is called a “Nyquist zone”. All of this is in memory of Harry Nyquist (1889-1976), who with Claude Shannon discovered much of the mathematics behind sampling.

Because of aliasing, analog-to-digital converters are usually preceded by an anti-aliasing filter. This filter removes the frequency content outside of the desired Nyquist zone, so that noise and interfering signals do not alias into the passband of the ADC. Most often, a low-pass filter is used, so that the selected Nyquist zone runs from DC (0 Hz) to 1/2 the sampling frequency. There are some exotic RF applications in which a bandpass filter selects frequencies in a higher Nyquist zone, which the ADC then aliases down, but these are uncommon.

An anti-aliasing filter before an ADC input.

How Oversampling Makes Anti-Aliasing Easier

In order to get good accuracy, delta-sigma converters need to run at a much higher frequency than the input signal. The first part of this series introduced an idea for an ADC built from a delta-sigma modulator followed by a digital counter:

Conceptual delta-sigma analog-to-digital converter

For this to work, the delta-sigma modulator has to <i>oversample</i>. In order to have the counter reflect the input signal accurately, there have to be many 1 and 0 bits in the modulator’s output for the counter to count. In other words, each sample from the counter’s output has to reflect many samples in the modulator. This is called oversampling.

The nice thing about oversampling is that the Nyquist frequency goes up with the sample rate, even when oversampling! The classic example of this happens in CD audio systems. CDs carry audio signals of up to 20 kHz, with a 44.1 ksps sample rate and a Nyquist frequency of 22.05 kHz. Furthermore, CD audio has a dynamic range of about 97 dB. In order to avoid aliasing signals back into the 0 Hz – 20 kHz audio band, the antialiasing filter at the input of a CD audio ADC needs to have a rolloff frequency (-3 dB) at 20 kHz, and should be down to -97 dB by 24.1 kHz. This is an impractical filter to design and build.

However, if an oversampled delta-sigma ADC is used, then the Nyquist frequency goes up to one-half the oversampled sample rate. A typical choice might be 64x oversampling, in which case the ADC will sample at 2.8224 MHz. Then the Nyquist frequency is 1.4112 MHz. The anti-aliasing filter still needs to have its -3 dB rolloff at 20 kHz, but it does not need to be -97 dB down until 2.82 MHz. That is a much easier filter to design. In fact, a 3-pole filter, easily and cheaply implemented with an op amp, is sufficient.

Moving Anti-Aliasing to the Digital Domain

To go from 2.8 Msps to 44.1 ksps requires another round of sampling, this time in the digital domain. Remember the counter? The process of reading its count, then resetting it for another round of counting, is a form of sampling, and aliasing can result. The figure below shows an example. In this case, an 8 Hz sine wave is being sampled at 70 sps by an oversampling ADC. Then, that digital signal is being downsampled, by taking every tenth sample, to 7 sps. The result is that the 8 Hz input is aliased to 1 Hz, just as if it was sampled at 7 sps in the first place.

Aliasing can result from downsampling a digital signal.

Just as in the analog domain, there are times when the aliasing resulting from downsampling can be useful, but often it is not. To prevent it, we need a low-pass filter, but this time the filter can be digital. In the CD-audio example, the filter needs to have the same rolloff characteristics as the challenging analog filter (-3 dB at 20 kHz, and -97 dB at 24.1 kHz). Doing that in a digital filter, though, is much easier than in analog. Digital arithmetic can produce a filter of arbitrarily good performance, without the precision components or careful tuning adjustments that might be required in analog. All it takes is throwing enough logic gates at the problem, and thanks to Moore’s Law, logic gates are cheap.

Since low-pass filters have an averaging effect, the filter will turn the bitstream of 1’s and 0’s into a series of multi-bit samples. The counter becomes unnecessary. Instead, it is enough to keep one sample from the low-pass filter’s output every so often, discarding the rest. The ADC now looks like this. (A simple anti-aliasing filter before the input is needed, but not shown.)

A delta-sigma ADC with a low pass filter and downsampler

The figure below shows the principle. A 1 Hz sine wave is oversampled at 70 Hz in the top graph, then 9 out of every 10 samples are discarded, leaving only the highlighted ones. Those samples are plotted in the bottom graph. The result is identical to the sine wave samples in the first graph at the top of this article.

Downsampling an oversampled signal is equivalent to sampling in the first place.

Technology similar to this, with some additional improvements to reduce the amount of math needed, makes it possible to put CD-quality ADCs in every desktop and laptop computer. Instead of an expensive analog filter, cheap digital gates on an IC provide most of the filtering, reducing the cost of the ADC to only a dollar or two.

Wrapping Up

In this article, we have come full circle to find out how delta-sigma techniques can make analog design easier. I’ve shown how aliasing happens and explained the need for anti-aliasing filters. Then we looked at the oversampling inherent to delta-sigma modulators, and how that permits a simpler analog antialiasing filter, at long as a digital low-pass filter is included after the modulator. This is just one of the reasons why delta-sigma principles are very cool. Coming up in this series: Simulating a delta-sigma modulator and an introduction to noise shaping.

Reference

Bourdopoulos, George I., Aristodemos Pnevmatikakis, Vassilis Anastassopoulos, and Theodore Deliyannis. Delta-Sigma Modulators: Modeling, Design and Applications. London: Imperial College Press, 2003