# Undersampling: Applications & Techniques With SAR ADCs, Part 1

Traditionally, undersampling is utilized in communication applications where a range of non-baseband signals sitting above the nyquist frequency are desired to be sampled. The bandwidth of these signals is less than the nyquist rate of the sampling system. Therefore, by bandlimiting the system with a band-pass filter, knowing the nyquist rate of the sampling system and the bandwidth of the signals of interest, one can reconstruct the input signal under these special circumstances with no actual information lost. This is known as the Nyquist-Shannon sampling theorem. Nonetheless, outside of communications systems, there are other applications that can take advantage of ADC undersampling to achieve system performance goals.

Successive Approximation Register (SAR) Analog-to-Digital Converters (ADCs) typically do not offer the bandwidth and sample rates required of the systems described in the paragraph above. However, SAR ADCs do offer significant power, size, dynamic range, and accuracy benefits when compared to other ADC architectures. In applications where exact frequency knowledge is less important and the user is primarily interested in amplitude and/or phase information, the traditional SAR ADC signal chain can be configured as an undersampled system, providing the benefits of power savings, easing of processing requirements, and potential system cost reduction. This article will introduce the theory behind undersampling a discrete time system and present results collected from Analog Devices Inc. SAR ADCs in undersampled configurations.

Sampling a continuous function, x(t), at a constant rate, fs (samples/second), results in an unlimited quantity of additional continuous functions that match the same set of samples. Only one of these functions is bandlimited to ½ fs (Hz). In the ideal case, the Fourier transform, X(f), is 0 for all |f| ≥ ½ fs. The algorithms implemented to reconstruct a continuous function from the samples will generate sufficient approximations to this ideal, infinite transform. If the original function, x(t), meets the nyqiust criterion by being bandlimited to ½ fs, then it's known the reconstruction algorithms must be approximating x(t). With respect to signal bandwidth (B), the Nyquist criterion is identified as fs > 2B. The Nyquist rate for a bandwidth B is 2B. If the Nyquist criterion is not satisfied, that is B > ½ fs, then signal aliasing occurs. Aliasing results in unavoidable divergences between x(t) and the reconstructed function with lower bandwidth. It is this aliasing condition that will be utilized to implement a lower-power, cost-efficient system.

When a signal (fin) sits in the first nyquist zone (fin < ½ fs), then the fundamental frequency location in a discrete Fourier transform (DFT) spectral analysis can easily be found with the following equation: The above equation is a manipulation of the sampling theorem relationship: For signals in the first nyquist zone, using the above equation to determine the location of a fundamental and its first nyquist zone harmonics is straightforward. However, determining the “folded-frequency” location of a signal outside of the first nyquist zone requires a bit more intensive calculation. Signals that violate the Nyquist criterion will be reconstructed as a lower bandwidth signal that aliases (or folds) down to the first nyquist zone. To determine the first nyquist zone location of a signal of any frequency, the following calculations should be applied: Equation 4 is interpreted as follows: if the signal frequency lies in an odd nyquist zone, then the first nyquist zone location is the remainder of the signal frequency divided by the nyquist frequency. Otherwise, the first nyquist zone location is the difference between the nyquist frequency and the remainder of the signal frequency divided by the nyquist frequency.

With the folded-frequency calculation in place, it can easily be observed that a reconstructed continuous time signal will be a first nyquist zone representation of the sampled signal. If a signal is undersampled, the generated samples are indistinguishable from the samples of a low frequency alias of the higher frequency content signal that exceeds the nyquist frequency. Traditionally, in undersampling configurations, a bandpass-filtered signal would be sampled at a rate below the signals nyquist rate of twice the upper cut-off frequency of the bandpass filter. This undersampling technique can be applied to applications such as harmonic sampling, IF sampling, and direct IF-to-digital conversion.

However, in applications such as AC strain gauges or other RMS-to-DC conversion systems where frequency content is less important, the nyquist criterion can be disregarded. The sampled signal will alias to the first nyquist zone and as long as the system continuous time bandwidth does not substantially attenuate the signal of interest, the amplitude and phase information of the signal can still be extracted.

## 9 comments on “Undersampling: Applications & Techniques With SAR ADCs, Part 1”

1. Davidled
December 12, 2014

->as long as the system continuous time bandwidth does not substantially attenuate the signal of interest, the amplitude and phase information of the signal can still be extracted.

Undersampling information causes most likely the loss of the content and also reduce the representation of data in the signal. I would like to see the application of where undersampling contributes actually rather than the precise sampling.

2. Victor Lorenzo
December 14, 2014

The operation of sampling a signal produces a displacement of its spectrum which can be viewed as a folding with respect to fs/2. Take a look at my previous post “An Approximation to the Aliasing Effect, Part 1: The Origin”.

If you select the sampling frequency so the baseband signal's spectrum (the one you are interest in) fits exactly in the 0-fs/2 frequency range you will be able to extract it. But in that case it is important to take into account the frequency of the carrier (or sub-carriers) so it is mapped to a DC level or to a discardable very low frequency component.

One side effect of undersampling is that it is not possible to reconstruct the original signal (for instance, the modulated high frequency carrier) from the sampled signal. But we are interested in the baseband signal anyway.

3. samicksha
December 14, 2014

I believe it depends what kind of signal we are working on, undersampling a real world signal, the sampling circuit must be fast enough to capture the highest signal frequency, else this will hardly make any use.

4. Victor Lorenzo
December 14, 2014

Exactly @Samicksha, I agree with you. Undersampling is useful provided that the input signal meets the general requirements presented here by Mr Ryan Curran and the analog front-end has enough bandwidth.

Furthermore, eventhough we take advantage of the aliasing effect, we must take good care in handling the out of band signals as they will also be aliased and degrade signal quality.

5. VarunReddy
December 14, 2014

I believe it depends what kind of signal we are working on

6. ryancurran
December 15, 2014

Hi Victor,

This is absolutely true. The analog bandwidth of the system must be able to support the signal of interest to be sampled.

Regardless of whether an undersampling condition is present, we must always take care in designing system filters to handle out-of-band signals. For most sampling systems, this is done with a low-pass anti-alias filter. However, many undersampled systems will employ a band pass filter because the first nyquist zone information may not be of interest in the undersampled application.

7. samicksha
December 16, 2014

I agree you Victor, some amount of aliasing always occurs when such functions are sampled.

8. Victor Lorenzo
December 16, 2014

@Ryan >> However, many undersampled systems will employ a band pass filter…

And it could probably be the filter of choice in most applications as we are interested on taking advantage from the aliasing effect, but we don't want to aliase those frequency components which are out of our band of interest.

9. Victor Lorenzo
December 16, 2014

@samicksha, at first sight it could be a little bit difficult to get the point with this kind of digitizing channel setup. The aliasing effect will always exist as Ryan described in his post, the basic principle of operation is exactly that…

…ups! Should not we “always” avoid the aliasing effect?

Well, this is like in diodes and transistors. We hate the way they perseverate in changing their parameters as a function of temperature, that brings a lot of head aches in high precision data acquisition systems… but we take advantage of that for creating very precise and sensitive temperature sensors.

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