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.