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I have a DAC which is assumed to be nonlinear, such that it produces unwanted harmonic distortions at integer multiples of the input frequencies. (EDIT: Any other nonlinear distortions, such as intermodulation products, are assumed to be negligible).

If the harmonics fall outside the bandwidth of the analog bandpass filter, then they are assumed to be eliminated:

However, if a harmonic falls within the bandpass filter’s passband, then the unwanted distortion remains. In an attempt to avoid this from happening, I can adjust the sample rate of the DAC. (This changes the Nyquist frequency, which changes the frequency of the harmonic after it is aliased into the Nyquist zone of interest).

Example: Assume the following:

  • Baseband signal bandwidth = 1 GHz.
  • Center frequency = 5.5 GHz.
  • Bandpass filter passband is 5 GHz - 6 GHz (exactly covering the signal).
  • 2nd and 3rd order harmonics are nonzero, but all higher order harmonics are negligible.
  • The DAC supports up to 10 GS/s.

If we try sampling at 8 GS/s, then the 2nd order harmonics (HD2) wrap into the signal band:

If we try sampling at 10 GS/s, then the 3rd order harmonics (HD3) wrap into the signal band:

However, 9 GS/s appears to be a perfect choice in this case. Neither HD2 nor HD3 are wrapped into the signal band:

My problem is that I found this result by brute force. It's not obvious to me how to analyze these harmonics more generally because I don't know how to characterize the aliasing in a convenient way. Could anyone offer any suggestions?

My goal would be to answer more general questions such as (for some maximum sampling rate):

  1. If the signal band is from A MHz to B MHz and the filter passband is from X MHz to Y MHz, then how many harmonics (HD2, then HD3, then HD4, etc) can be avoided (and how)?
  2. If I want to avoid the first N harmonics (given a signal band from A MHz to B MHz), then what is the widest possible filter passband?
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2 Answers 2

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Frequency Planning in Radio Design

The OP has clarified in comments that his question is focused on what would I believe would be commonly referred to as frequency planning in the process of radio design. In such frequency planning, all the known (and predictable) possible intermodulation locations are determined due frequencies due to clocks and local oscillators used, and center frequencies of other (relatively narrowband) modulated signals. The intermodulation signals would typically show up as narrow band tones, called "spurs", limiting radio sensitivity if a spur were to fall into a used frequency band for reception. For this reason such frequency planning is often used for receiver design to determine "clean bands" for use of intermediate frequencies (IF) but also can be used to avoid signal degradation in a transmitter by avoiding the possibility of in-channel distortions. With consideration to the intermodulation from a single modulated waveform alone, this really doesn't make sense as I detail in the next section-- the products due to the signal itself will already unavoidably be in band. However if there are multiple different relatively narrow band channels and other sources of strong spurious signals (clocks), then such an analysis of clear frequency bands can be worthwhile if we have the option of choosing alternate frequency bands and within the range of the DAC and the DAC sampling rate itself (frequency planning!).

The intermodulation spurs are predictable, for example for two frequency components all the spurs would as all components would land at $mf_1\pm nf_2$ for all integers $m,n$, where the spurs would be progressively weaker for larger $m$ and $n$ values. Charts called "mixer spur charts" exist for this very case of two frequencies (given the two frequencies going into an analog mixer, which is a multiplier, as the "LO" or local oscillator, and "RF" or radio frequency input.). So in this case $mf_1$ refers to the LO and all its higher harmonics, and $nf_2$ refers to the RF frequency and all its higher harmonics. Due to non-linearities within the mixer, all spurs appearing at the IF (Intermediate Frequency) output would be predictably at $mf_1\pm nf_2$. When the RF port is a modulated signal, the spurs would also be modulated with increasing bandwidth proportional to $m+n$, and thus all locations where additional noise can be introduced is predicted. A DAC and ADC works very much the same as a mixer; multiplying the signal with an impulse instead of a sinusoid (and filtering with staircase reconstruction in the case of a DAC). The point is we have the same effect (even more some) of all the harmonics of the sampling clock as $mf_1$. An example mixer spur chart from Microwaves 101 is shown below, demonstrating its use to determine frequency bands that would be clear of possible intermodulation noise. This particular one is plotted showing the results with just the $n$ values of 0,1,2,3,4 and $m$ values of -2, -1, 0, 1, 2. As we add more coefficients, more lines would appear, and we expect the lower numbers to be more dominant. The actual magnitude would depend on the coefficients of the specific non-linearity introduced, but this at least shows us the frequency locations where the stronger intermodulation spurs can appear so that we can avoid use of them if possible).

mixer spur chart

This chart drawn is with normalized frequency with 1 on the vertical axis representing the frequency of the LO (the sampling clock in the case of a DAC), and 1 on the horizontal axis representing the frequency of the RF input (for a mixer; for the DAC this would be the frequency of the signal to be converted to analog). For the case of a DAC, the first Nyquist zone would be the square region represented by Input Frequency from $0$ to $0.5$ and output frequency from $0$ to $0.5$, but we also see that all the higher output frequencies that will exist can also be predicted if there was interest in that.

With that, the horizontal line that results with a constant 1 independent of RF frequency is the LO or clock feedthrough that we would expect to see. This would refer to component predicted by $m=1$ and $n=0$. The other lines show the results for the other combinations of $m$ and $n$, and ultimately we can draw a vertical line for a given input RF frequency, and for that frequency read directly the clear output frequency bands that don't have any spurs (for the range of $m$ and $n$ used to create this chart).

The link provided also provides an example spreadsheet for creating such charts that will provide further guidance into such analysis. For the case of a mixer where two frequencies are considered, the result is given by $mf_1 \pm nf_2$ as previously described. This is readily expanded for consideration of multiple frequencies by doing $mf_1 \pm nf_2 \pm kf_3 \ldots$. The viability of having free usable areas will go down as the range for each of the integers $m,n,k \ldots$ are increased, so in creating such a chart or analysis keep in mind that the total order of the intermod product is given by $m+n+k + \ldots$ and the total power will both go down as the order goes up and the overall bandwidth (if one or both of the signals is modulated) will spread by the order. Thus the locations of all the lower order results (up to 5th order would be reasonable to review) should be prioritized for avoidance.

An actual "chart" with more than one input frequency (which is what is desired in this case) would become multi-dimensional so a programmatic search rather than graphical solution would make more sense; but if a dynamic 2D chart was desired for further insight, that could be done by sweeping any one of the frequencies as the horizontal axis with all the other frequencies set to fixed locations (as determined from their own sweeps), and this is iterated. I have pursued this programmatically where the search results in the largest free zones within a certain total order, where order is the sum of all the individual harmonic orders and typically with a total sum < 5 has been sufficient to identify the possible spurs but is dependent on severity of the actual non-linearity.

Distortion due to intermodulation in modulated waveforms

Below is my prior answer focused on the intermodulation effects within a modulated waveform itself.

Unless the OP is only ever converting a single tone, the harmonics due to non-linearities can't be avoided. With the more general case of waveforms that occupy bandwidth, the non-linearity will produce harmonics directly in the user's spectrum (as a noise elevation), as well as adjacent to the spectrum that is difficult to filter out due to it's proximity.

The reason for this is best understood from "Two-Tone Third Order Intermodulation Distortion", what this means is if we were to instead present two tones spaced by $\Delta F$ at the input to a device with a 3rd order distortion (produces a 3rd harmonic for a single tone), the non-linearity will produce harmonics that are only $\Delta F$ away from the two tones! This is depicted in the graphic below.

2 tone 3rd order intermod

$N$ order distortion products with two tones include all frequencies $mf_1\pm nf_2$ with $m$, $n$ as integers and $m+n=N$. Thus for third order distortion in particular, $N=3$ and we can have $m=2$, $n=1$ or $m=1$, $n=2$. The result shown in the graphic is for $2f_1-f_2$ and $2f_2-f_1$.

Given a non-linearity producing harmonics of significance with a single tone, that same non-linearity will produce intermodulation products on significance when multiple tones are involved - the 2 tone 3rd order intermod products described above will be at a similar level to that of the 3rd harmonic with a single tone- so cannot be arbitrarily insignificant (we can’t have one case without the other when the harmonics are created by a non -linearity).

I demonstrate this with a simple example of a 3rd order non-linearity below. In the upper plot is the result of a single tone, where a 3rd harmonic is created by the non-linearity:

single tone

I then repeated this with two tones, each 3dB lower (so that the total power is the same) and we see the intermodulation products I refer to at the same level of the third harmonic:

two tone case

Below shows a more practical example of a 16QAM modulated waveform with 3rd order distortion, where we see "spectral regrowth" on each side of the waveform which is not welcome for over the air transmission (without the distortion this spectrum would be much closer to a rectangular shape), but the entire noise floor within the spectrum is also elevated.

3rd order effect on 16QAM

The harmonics can't be avoided, but pre-distortion techniques can be used however to reduce the power in the harmonics. Depending on the source of the harmonic, dithering techniques can also be effective in reducing peak levels of harmonic distortion.

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  • $\begingroup$ //"...the harmonics due to non-linearities can't be avoided."// ------ If the non-linearity is memoryless, a tunable non-linearity processing the output of the DAC could undo some of it. In fact, back in the 70s there were some 12-bit conventional DACs (R-2R) that they were manufacturing that had an EPROM mapping the input word to another value intended to undo the non-linearity. It's how they got 12-bit performance out of an R-2R circuit, that even with laser trimming, couldn't be better than $\frac14$% precise. $\endgroup$ Apr 2, 2023 at 21:27
  • $\begingroup$ @robertbristow-johnson Thanks! That is exactly the "pre-distortion" I was referring to. Even when it has memory we can also do the same with Markov chain techniques (basically put matching memory in the predistortion). This is common in power amplifier pre-distortion where thermal effects contribute to such memory in the distortion. $\endgroup$ Apr 2, 2023 at 21:32
  • $\begingroup$ @DanBoschen My question started with the explicit assumption that the nonlinearity "produces unwanted harmonic distortions at integer multiples of the input frequencies". In other words, there are assumed to be no intermodulation products. Thank you for your comments about intermodulation products. They are correct, but not relevant to this specific question. I have added an edit to clarify my question. $\endgroup$
    – Harry
    Apr 3, 2023 at 5:28
  • $\begingroup$ @Harry- I assert that if you are seeing that, you will have intermodulation products as well - if a non-linearity produces a harmonic of a single tone, it will produce intermodulation products of multiple tones via the same process. Have you tried testing with two tones? Or are you using this in an application where you will always only be using a single tone? $\endgroup$ Apr 3, 2023 at 11:30
  • $\begingroup$ If you have multiple tones and harmonics of single tones due to a non-linearity, then you can't make the assumption that there are no intermodulation products given they go through the same non-linearity. Please try it with two tones, assuming you have a 3rd harmonic, you should find that the 2 tone 3rd order intermod products are around the same level as the third harmonic with one tone- I updated my answer to demonstrate this. $\endgroup$ Apr 3, 2023 at 16:17
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The answer from @DanBoschen is correct, but I did not ask a very clear question, so Dan's answer (while correct) was not exactly what I was looking for.

I have done some research and I think my idea is valid. "Frequency planning" appears to be widely used to improve SFDR (Spurious-Free Dynamic Range). According to most definitions, SFDR considers only a single input tone. Therefore, intermodulation distortions are excluded from this frequency planning. However, I think I was wrong to say these distortions are "negligible" - it would have been more correct to say that they are assumed to be unavoidable and are therefore ignored during frequency planning.

Here are some references related to frequency planning:

Based on the above references, I am reasonably confident that "frequency planning" to avoid harmonic distortions is a legitimate and worthwhile thing to do. This does not help with intermodulation distortions, which must be addressed some other way (e.g. Digital Pre-Distortion).

Based on the various tools ([3], [4] and [6]), it seems that the expectation from the chip manufacturers is that frequency planning will be performed by manual brute force. I was not able to find any elegant/analytical (non-brute-force) solution.

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