ADC and DAC

The converter is where the radio stops being analogue. Everything before it is voltage, and everything after it is arithmetic. I used to treat the bit count as the whole specification of that boundary. It is not even the most important part of it, because a clean clock beats extra bits at high frequencies. Each section below takes one property of the conversion, and gives the number that goes with it.

• Executive Summary
• What does sampling actually cost you?
• How do you deliberately sample below the carrier?
• What does one more bit buy?
• Why is ENOB always lower than the bit count on the box?
• Why does clock jitter set a ceiling that more bits cannot beat?
• What is dither for, and why add noise on purpose?
• What does the DAC side add that the ADC side does not?
• How do you choose the bit depth and the sample rate?
• Reference

Executive Summary

The table is a lookup. The ideal SNR column assumes a full-scale sine wave, and every real converter falls short of it.

What does sampling actually cost you?

Sampling replaces a continuous voltage with numbers taken at fixed intervals. Two things are lost in that step, and only one of them is usually discussed. The first is everything between the samples. The second is the ability to tell certain frequencies apart at all.

The second loss is aliasing. A sine wave at frequency f and a sine wave at the sample rate minus f produce identical sample values. Nothing in the numbers distinguishes them afterwards. So a filter has to make the choice before the converter, and that is the anti-alias filter the front end carries.

The usual statement is that the sample rate must exceed twice the highest frequency. The more useful statement is different. The sample rate has to exceed twice the bandwidth, and the signal has to sit inside one Nyquist zone. Those two versions agree only when the signal starts at 0 Hz.

A Nyquist zone is a band half the sample rate wide. The first runs from 0 to half the sample rate. The second runs from there to the full sample rate, and so on upward. Every zone folds down onto the first one, which is why a signal three zones up appears in the digitised output.

One property of the folding is worth memorising, because it catches people out on a spectrum display. Odd-numbered zones fold with their frequency order preserved. Even-numbered zones fold reversed, so the spectrum appears mirrored. A signal that looks like a valid transmission with its sidebands swapped is usually an even-zone alias.

Aliasing is normally described as a defect. It is better understood as a mechanism, because the next section uses it deliberately. What makes it a defect is only that unwanted energy folds down alongside the wanted energy.

Figure 1 draws the zones and the folding. The upper axis is the real frequency axis, and the lower axis is what the converter produces.

Figure 1. The converter cannot tell which zone a signal came from. That is why the anti-alias filter belongs in front of it, and why the same folding becomes useful once the filter is chosen deliberately.

• Two frequencies can give identical samples : A tone at f and one at the sample rate minus f produce the same numbers. No later processing separates them.
• The rule is about bandwidth, not maximum frequency : The sample rate must exceed twice the bandwidth, and the signal has to sit inside one Nyquist zone.
• Zones are half the sample rate wide : The first runs from 0 to half the sample rate. Every zone above it folds down onto that first one.
• Even zones arrive mirrored : Odd zones keep their frequency order. A signal with its sidebands swapped is usually an even-zone alias.
• The filter makes the choice, not the converter : Anti-aliasing has to happen in the analogue domain, because after sampling the ambiguity is already permanent.
• Aliasing is a mechanism, not only a defect : It becomes a defect when unwanted energy folds down. Chosen deliberately, it is how bandpass sampling works.

How do you deliberately sample below the carrier?

A 2.4 GHz signal does not need a 5 GHz converter. That claim sounds wrong, and it follows directly from the previous section. If the sample rate only has to exceed twice the bandwidth, then a narrow signal at a high frequency needs a slow converter.

The technique is called bandpass sampling, or undersampling. The converter samples at a rate far below the carrier, and the signal folds down into the first Nyquist zone. Folding is the point rather than the problem, because the mixer that would otherwise do the translation is no longer needed.

Two conditions have to hold. The sample rate must exceed twice the signal bandwidth, and the signal must fit entirely inside one Nyquist zone. A signal straddling a zone boundary folds onto itself, and that damage cannot be undone.

One property of the converter becomes critical here, and it is not the sample rate. The analogue input bandwidth has to cover the actual carrier frequency. A converter may sample at 100 MS/s and have only 60 MHz of input bandwidth. It cannot see a 2.4 GHz signal at all, whatever the arithmetic says.

The filter requirement also changes shape. A baseband design needs a low-pass filter. A bandpass sampling design needs a band-pass filter that passes exactly one zone. That filter is doing the job the mixer would have done, so the difficulty moves rather than disappearing.

The cost appears in the next section but one, and it is the reason the technique is not universal. Clock jitter degrades the result in proportion to the input frequency, not the sample rate. So undersampling a 2.4 GHz carrier demands a far better clock than sampling the same bandwidth at baseband.

This is why the technique appears in instruments rather than on hobby boards. A general-purpose SDR mixes to baseband, because a mixer is cheaper than a very fast track-and-hold and a very clean clock. Direct RF sampling receivers make the opposite choice, and they pay for it in the clock.

• The sample rate follows the bandwidth : A narrow signal at a high carrier needs a slow converter, because only twice the bandwidth is required.
• Folding is the mechanism : The signal aliases down into the first zone on purpose, which removes the mixer that would otherwise translate it.
• The signal must fit inside one zone : A signal straddling a boundary folds onto itself. That damage cannot be undone in software.
• Input bandwidth is the hidden requirement : The converter's analogue front end has to reach the carrier frequency, whatever its sample rate happens to be.
• The filter becomes a band-pass : It has to pass exactly one zone, so the difficulty moves from the mixer to the filter rather than disappearing.
• Jitter scales with the carrier : Degradation follows the input frequency rather than the sample rate, so undersampling demands a much better clock.

What does one more bit buy?

Rounding a voltage to the nearest step leaves an error. That error behaves like noise, and its size is fixed by the step. So the resolution sets a noise floor, and one equation gives it.

For an ideal converter and a full-scale sine wave, the signal to noise ratio is 6.02 times the number of bits, plus 1.76 dB. Each extra bit is therefore worth about 6 dB. Four extra bits are worth about 24 dB, which is a factor of 250 in power.

The numbers are worth carrying. An 8-bit converter gives 49.9 dB. A 12-bit converter gives 74.0 dB. A 14-bit converter gives 86.0 dB, and a 16-bit converter gives 98.1 dB. Those figures are the best case, and no real part reaches them.

One qualification changes how the numbers are used, and it is the most useful idea on this page. The figure applies across the whole Nyquist band. A receiver rarely uses the whole band, and the quantisation noise outside the channel is filtered away with everything else.

That recovery is called processing gain. It equals 10 log of the sample rate divided by twice the channel bandwidth. Decimating by a factor of a hundred in bandwidth returns 20 dB, which is more than three bits of resolution.

An RTL-SDR makes the point concretely. Its converter is 8-bit, so 49.9 dB across the band. Sampling at 2.4 MS/s and filtering down to a 15 kHz narrowband channel gives 10 log of 80, which is 19 dB. The channel therefore sees about 69 dB, which is why a very cheap converter sounds perfectly good on a narrow signal.

The same arithmetic explains where cheap converters fail. Processing gain helps the wanted channel and does nothing about a strong signal elsewhere in the band. That strong signal consumes the converter's range before any filtering happens, so the weak one is already lost. Bit depth buys the ability to receive weak and strong signals at the same time, and nothing else buys it.

• One bit is about 6 dB : The ideal figure is 6.02 times the bit count plus 1.76 dB. That assumes a full-scale sine wave and a perfect converter.
• Learn four numbers : 8-bit gives 49.9 dB, 12-bit gives 74.0 dB, 14-bit gives 86.0 dB and 16-bit gives 98.1 dB.
• The figure covers the whole Nyquist band : A narrow channel keeps only a fraction of that noise. So the in-channel result beats the datasheet number.
• Processing gain returns the difference : It is 10 log of the sample rate over twice the channel bandwidth, so a hundredfold reduction returns 20 dB.
• An 8-bit dongle reaches about 69 dB in a narrow channel : 49.9 dB plus 19 dB of processing gain at 2.4 MS/s into 15 kHz. That is why it sounds good.
• Bit depth buys simultaneity : Processing gain cannot help when a strong signal elsewhere in the band has already consumed the converter's range.

Why is ENOB always lower than the bit count on the box?

A 14-bit converter does not deliver 14 bits. The bit count describes how finely the output is divided, and it says nothing about how much of that division is meaningful. Three separate mechanisms consume the difference.

The first is that the steps are not equal. A real converter's thresholds sit slightly off their ideal positions, which the datasheet reports as differential and integral nonlinearity. Unequal steps add error beyond the ideal rounding error.

The second is that the converter adds its own noise. Thermal noise in the sampling network and in the comparators appears at the output whether or not a signal is present. This is the part that does not improve with a better clock or a better signal.

The third is distortion. A converter produces harmonics and other spurious tones, and those are not noise, because they sit at specific frequencies related to the input. On a spectrum display they look exactly like real signals.

SINAD combines all of them, since it compares the signal against noise and distortion together. Running SINAD backwards through the ideal equation gives the effective number of bits. So ENOB is the bit count an ideal converter would need in order to perform as well as this real one does.

A 14-bit part delivering 11.5 ENOB is entirely normal, and that is 15 dB of the ideal 86 dB gone. The gap widens as the input frequency rises, which is the subject of the next section.

SFDR deserves separate attention, because it is often the number that actually matters. It is the distance from the wanted signal to the largest spurious tone, and it is not the same as SNR. A converter can have excellent SNR and one strong spur that hides a weak signal completely.

The practical consequence is a habit rather than a calculation. Before believing a weak signal, change the sample rate slightly and look again. A real signal stays where it is, and a converter spur moves, because its frequency is tied to the sampling.

• Bit count describes division, not accuracy : It says how finely the output is split, and nothing about how much of that division carries information.
• Three mechanisms consume the difference : Unequal step sizes, the converter's own thermal noise, and distortion products at specific frequencies.
• ENOB comes from SINAD : Run the measured signal to noise and distortion ratio backwards through the ideal equation. The answer is an equivalent bit count.
• 11.5 ENOB from a 14-bit part is normal : That is 15 dB below the ideal 86 dB, and the gap widens as the input frequency rises.
• SFDR is often the number that matters : One strong spur can hide a weak signal even when the overall noise performance is excellent.
• Move the sample rate to identify a spur : A real signal stays put and a converter artefact moves, because its frequency is tied to the sampling.

Why does clock jitter set a ceiling that more bits cannot beat?

This is the section that changed how I choose converters. Jitter is uncertainty in when a sample is taken, rather than in what value is recorded. On a fast-changing signal those two are the same thing, and the consequence is a hard ceiling.

The mechanism is easier than the formula. A sample taken slightly early or late records the voltage from the wrong instant. On a slowly changing signal the voltage has barely moved, so the error is small. On a fast one it has moved a lot, and the error is large.

So the damage depends on the slope, which depends on the input frequency. The resulting limit is minus 20 log of two pi times the input frequency times the RMS jitter. The sample rate does not appear in it at all.

The numbers make the point sharply. With 1 ps of RMS jitter, a 100 MHz input is capped at 64 dB. The same 1 ps at 1 GHz is capped at 44 dB. At 2.4 GHz it is capped at about 36 dB.

Now compare those against the resolution figures from two sections ago. A 14-bit converter promises 86 dB. Sampling a 1 GHz input with a 1 ps clock caps it at 44 dB, so more than 40 dB of the promise is unreachable. The extra bits are producing numbers that carry no information.

That comparison is the whole argument for direct RF sampling being hard. Undersampling at 2.4 GHz needs jitter well below 0.1 ps to keep a 14-bit converter useful. Baseband sampling of the same bandwidth after a mixer needs a far more ordinary clock.

One distinction matters here, and it connects this page to the timing one. Jitter is short-term, and it is dominated by the close-in phase noise of the clock source and by the noise added on the board. Long-term accuracy is a different property entirely. A GPSDO improves accuracy over hours and does very little for jitter over nanoseconds, which is covered on the Syncrhonization page.

Figure 2 plots the ceiling against input frequency, with the resolution limits drawn across it. Where the sloping line falls below a horizontal one, that resolution has stopped being useful.

Figure 2. Resolution is a horizontal line and jitter is a sloping one, so the input frequency decides which one binds. Buying more bits above the crossing point buys numbers rather than information.

• Jitter is uncertainty in when, not in what : A sample taken early or late records the voltage from the wrong instant. The error grows with the signal slope.
• The limit follows input frequency, not sample rate : It is minus 20 log of two pi times the input frequency times the RMS jitter. The sample rate is absent.
• Carry three numbers : At 1 ps of jitter, 100 MHz caps at 64 dB and 1 GHz caps at 44 dB. At 2.4 GHz the cap is about 36 dB.
• Above the crossing, bits stop helping : A 14-bit converter promises 86 dB, and a 1 ps clock at 1 GHz allows 44 dB of it.
• This is why direct RF sampling is hard : Undersampling at 2.4 GHz needs jitter well below 0.1 ps to keep a high-resolution converter worth having.
• Jitter and accuracy are different properties : A GPSDO fixes long-term accuracy and does almost nothing for short-term jitter, which is what the converter cares about.

What is dither for, and why add noise on purpose?

Adding noise to improve a measurement sounds like a contradiction. It is not, and the reason is that quantisation error is only noise-like when the signal is large. For a small signal it becomes something worse.

Consider a signal that moves across only two or three converter steps. The rounding error is no longer random, because it repeats with the signal itself. A repeating error is periodic, and a periodic error appears on a spectrum display as a discrete tone.

Those tones are the problem. They look exactly like real signals, they sit at frequencies harmonically related to the input, and no amount of averaging removes them. Averaging reduces noise and leaves anything periodic untouched.

Dither breaks the correlation. A small amount of added noise, roughly one step in size, makes the rounding error effectively random again. The error then behaves like noise, and averaging works on it normally.

The trade is explicit and favourable. The noise floor rises slightly, because real noise has been added. In exchange the discrete spurs disappear, so the spurious free dynamic range improves substantially. Trading a fraction of a dB of noise floor for tens of dB of spur suppression is usually worth it.

Some converters include a dither generator that can be switched on. Where none exists, the front end usually supplies enough noise already, which is a useful thing to know. A receiver with a reasonable noise figure and its gain set correctly is self-dithering, because the amplified thermal noise is larger than one converter step.

That observation connects back to the gain discussion on the RF Front End page. A receiver with too little front-end gain presents a signal that barely moves across the converter steps. It produces exactly the correlated spurs described here. The symptom looks like a converter fault, and the cause is a gain setting.

• Small signals make quantisation error periodic : A signal spanning two or three steps produces a repeating error. A repeating error is a tone rather than noise.
• Averaging cannot remove a spur : It reduces random noise and leaves anything periodic exactly where it was, which is why the tones survive long captures.
• Dither restores randomness : About one step of added noise decorrelates the rounding error, so it behaves like noise again.
• The trade is very favourable : A fraction of a dB on the noise floor buys tens of dB of spurious free dynamic range.
• A well-set receiver dithers itself : Amplified thermal noise larger than one converter step does the job, so no dedicated generator is needed.
• Too little gain looks like a converter fault : A signal that barely spans a few steps produces correlated spurs. The fix is in the front end.

What does the DAC side add that the ADC side does not?

Transmission reverses the process, and it does not simply mirror it. Three properties appear on the DAC side that have no receive equivalent. Each one reaches the transmitted spectrum, where other people can see it.

The first is that a DAC does not output points. It holds each value until the next one arrives, which produces a staircase rather than a smooth waveform. That hold operation shapes the output spectrum, and the shape is a sinc function.

The consequence is droop across the band. The response falls to about minus 3.9 dB at the edge of the first Nyquist zone. A wideband transmission therefore comes out tilted, and the correction is a digital pre-emphasis filter applied before the converter.

The second property is images. The staircase output contains copies of the wanted signal around every multiple of the sample rate. Those copies are real emissions, and a reconstruction filter after the DAC is what removes them.

The sinc envelope helps here, since it attenuates the images as well as the wanted signal. It is not enough on its own. A transmitter without an adequate reconstruction filter radiates energy at frequencies nobody configured. That is a regulatory problem rather than a performance one.

The third property is headroom, and it is the one most often set wrongly. A modulated signal has a peak to average power ratio, and an OFDM signal typically has 10 to 12 dB of it. The average level must therefore sit that far below full scale, or the peaks clip.

Clipping on transmit is worse than clipping on receive, because the damage leaves the building. A clipped signal spreads energy outside its own channel, which shows up as adjacent channel leakage. The transmitter still looks fine on its own centre frequency, so the fault is usually reported by somebody else.

Figure 3 shows the output spectrum with the images and the sinc envelope drawn over them.

Figure 3. Everything drawn in red is radiated. A transmitter without a reconstruction filter emits at frequencies nobody configured, which makes this a compliance question rather than a quality one.

• A DAC holds, it does not emit points : The staircase output imposes a sinc shape on the spectrum. That is where the droop and the image attenuation both come from.
• Droop reaches about -3.9 dB : That is the loss at the edge of the first Nyquist zone, and a digital pre-emphasis filter corrects the tilt.
• Images are real emissions : Copies appear around every multiple of the sample rate, and only a reconstruction filter removes them.
• The envelope is not a filter : Sinc attenuates the images without removing them, so an unfiltered transmitter radiates outside its configured band.
• Leave headroom for the peaks : An OFDM signal carries 10 to 12 dB of peak to average ratio. The average has to sit that far below full scale.
• Transmit clipping is other people's problem : It spreads energy into adjacent channels while the centre frequency still looks correct, so somebody else reports it.

How do you choose the bit depth and the sample rate?

The specifications interact, so choosing them one at a time gives the wrong answer. Four questions settle it, and they are worth asking in this order because each one constrains the next.

The first question is about simultaneity. How large is the difference between the strongest and the weakest signal you need at the same moment? That difference sets the bit depth, and nothing else does. A single dominant signal is an 8-bit problem, and a weak signal beside a strong one is a 14-bit problem.

The second question is about bandwidth rather than frequency. How wide is the signal, including any guard band? Twice that number is the minimum sample rate, and the carrier frequency does not enter into it.

The third question is about the front end. Will the signal be mixed to baseband, or sampled where it sits? Mixing keeps the input frequency low and makes the clock requirement ordinary. Sampling in place removes the mixer and moves the difficulty onto the clock.

The fourth question follows from the third and is the one people skip. What jitter does the clock actually have at the input frequency you chose? Run the ceiling calculation before ordering anything, because a converter operating above its jitter crossing is being paid for and not used.

Two habits are worth adopting alongside the arithmetic. Sample faster than the minimum, because a filter with an infinitely steep edge does not exist. The extra rate is what pays for a realisable one. Then set the front-end gain so the strongest signal sits a few dB below full scale. That is where the resolution you bought becomes available.

One summary sentence covers most decisions. Bandwidth chooses the sample rate, simultaneity chooses the bit depth, and the input frequency decides whether the clock or the resolution is the real limit.

• Simultaneity chooses the bit depth : The gap between the strongest and weakest signal you need together is the only thing resolution buys.
• Bandwidth chooses the sample rate : Twice the occupied bandwidth is the minimum, and the carrier frequency does not appear in that calculation.
• The architecture decides where the difficulty sits : Mixing to baseband keeps the clock ordinary, and sampling in place moves the cost onto the clock.
• Check the jitter ceiling before ordering : A converter running above its crossing point delivers fewer usable bits than a cheaper one would.
• Leave room for a real filter : Sampling at exactly twice the bandwidth assumes an infinitely steep filter, so buy some margin instead.
• Resolution only exists at the right level : Set the gain so the strongest signal sits just below full scale. Otherwise the bits you paid for stay unused.

Reference

The list below is where the device figures come from. The vendor documentation is the authority for anything specific to a part, and the ShareTechnote pages carry the stages on either side of the converter.

• AD9361 Data Sheet : Analog Devices - the 12-bit converters, the decimation and FIR chain, and the gain control that keeps the signal in range.
• HackRF Hardware Components : Great Scott Gadgets - the MAX5864 8-bit converter and its maximum rate.
• MT-001, Taking the Mystery out of the Infamous Formula : Analog Devices - the ideal SNR expression, processing gain and the quantisation noise assumptions behind them.
• MT-007, Aperture Time and Aperture Jitter : Analog Devices - the jitter limited SNR relationship used in the ceiling calculation.
• R820T2 tuner : RTL-SDR.com - the low-IF front end and the 8-bit RTL2832U behind it.
• ShareTechnote - RF Front End : The analogue chain that delivers the signal, and the gain setting that decides how much resolution is used.
• ShareTechnote - Digital Front End : The decimation that turns the converter rate into processing gain.
• ShareTechnote - Syncrhonization : Clock sources, and why long-term accuracy is a different property from short-term jitter.