# Signal Chain Basics #90: Calculating Harmonic Distortion Frequencies in High-Speed Data Converters

Data converters, which convert digital to analog signals and analog to digital signals, have different characteristics than purely analog circuits. One of the most common misunderstandings is the frequencies of distortion products in data converters.

For a purely analog circuit, the frequency of the distortion products can be calculated from the trigonometric identities. For example, for a tone at frequency ω, the second- and third-order distortion products would be

So second-order distortion produces a frequency at twice the tone frequency, third-order distortion at three times the tone frequency, and, more generally, Nth-order distortion at up to N×ω.

For data converters, the frequencies produced by harmonic distortion are somewhat more complicated to calculate. Although the distortion product frequencies for analog-to-digital converters (ADCs) and digital-to-analog converters (DACs) are similar, it is useful to treat them differently to gain a better fundamental understanding.

At the output of the ADC with a sample rate of fS , all signals are represented in the frequency range of zero. Sample signals above fS /2 are called undersampling and are used quite frequently — with high-speed ADCs in particular. At the output of the ADC, a signal at frequency f is fS /2, indistinguishable from a frequency fOUT at

The conversion of the signal from f to fOUT is called aliasing.

Relating back to harmonic distortion, the frequency of the Nth-order harmonic distortion product of a tone at f at the ADC output is

The following example considers the case of the dual-channel, 16-bit, 370 Msps ADC16DX370, with a 150.25 MHz input tone and sample rate of 370 Msps. Applying equation (5) up through the ninth harmonic in Table 1, we can identify the harmonics in the output spectrum in Figure 1.

Table 1

Harmonic frequencies for 150.25 MHz input and 370 MSPS.

Figure 1

ADC16DX370 output spectrum with 150.25 MHz input and 370 MSPS.

For high-speed DACs, the same formula applies, but at first glance, it is not apparent why that should be the case. Since the DAC output is a continuous analog signal, there is no aliasing — as is the case for ADCs.

To understand why harmonics at the DAC output also have images, we need to understand the output structure of the DAC. Typical high-speed DACs have what is known as a zero-order hold output. Here the output level is changed at the beginning of the sample clock cycle and is held until the next cycle. This creates images of the signal at higher frequencies: a tone at frequency f (f <>S /2) has images at

So how does this impact the harmonic frequencies? Consider a tone at frequency f and the first image at frequency fS – f . These two tones intermodulate with distortion. For second-order distortion, two tones at f1 = f and f2 = fS – f produce frequencies at |f1 + f2 | and |f1 – f2 |.

and

producing the frequency predicted by equation (8). Higher-order harmonics can be calculated similarly.

As an example, Figure 1 shows the output spectrum for the quad-channel, 16-bit, 2.5 Gsps DAC38J84, with output at 480 MHz and a 2.5 Gsps sample rate. The harmonic distortion frequencies agree with the predictions of equation (5) and calculated in Table 2.

Table 2

Harmonic frequencies for 480 MHz output and 2.5 Gsps.

Figure 2

DAC38J84 output spectrum for 480 MHz tone with 2.5 Gsps.

Join us next month, when we discuss understanding the difference between HBM and IEC 61000-4-2 ESD immunity.

References

— Robert Keller is the systems manager for high-speed data converters at Texas Instruments. He has 10 years of experience supporting high-speed products in wireless infrastructure communication, test and measurement, and military systems. He received a BA in physics and mathematics from Washington University, St. Louis, Mo., and a PhD in applied physics from Stanford University. He has 10 US patents in networking and sensor applications. Robert can be reached at ti_robertkeller@list.ti.com.

## 4 comments on “Signal Chain Basics #90: Calculating Harmonic Distortion Frequencies in High-Speed Data Converters”

1. goafrit2
July 6, 2014

>> One of the most common misunderstandings is the frequencies of distortion products in data converters.

Besides the calculations which could be meaningless without great simulators, THD makes sense when you have a CAD that can mimic your process technology. I have made an ADC that seemed good with nice THD in a process only to come back from fab to be totally worthless. Cadence CAD became a good friend and with nice tools, this helps in any design.

2. goafrit2
July 6, 2014

Anyone interesed in this can use this toolbox which is available free in Matlab. It was created by Dr Schreier, a professor in Uni of Toronto. I have used it and it adds a lot of value.

3. green_is_now
October 23, 2014

@Goefrit2

you should write that up for a planet analog article.

Showing where the wheels came off or performance hit a requirement relative to a bad component model or models and interactions.

for those that do not have IC design experience this would be very enlighting.

4. fasmicro
November 3, 2014

>> Showing where the wheels came off or performance hit a requirement relative to a bad component model or models and interactions.

There are many models for that anyway in circuits and systems. In ASIC, it is not seen as a bad component, but rather as a transistor whose peformance deviates from normal.

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