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.
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 |.
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.
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For more information about data converters, visit: www.ti.com/dataconverters-ca.
— 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 email@example.com.