Turning Nyquist upside down by undersampling

Bravo to Harry Nyquist and Claude Shannon! In the 1920s, these gentlemen created the now-well-known Nyquist theorem, which states that when sampling a signal at discrete intervals, the sampling must be greater than twice the highest frequency of the input signal. This scenario must be true for you to perfectly reconstruct the original signal from the sampled version.

Their theorem has held up well through the years. In my discussions with engineers, I use the Nyquist theorem to explain the accuracy of sampling systems in which the bandwidth of the signal of interest is less than twice the sampling frequency of the converter. The sampling systems I describe use a lowpass, antialiasing filter in front of the ADC. This situation is usually an engineer's initial exposure to the Nyquist theorem, in which signals with frequencies greater than one-half of the converter's sampling rate can come back to haunt you. Experts fondly call this occurrence a foldback phenomenon (Reference 1). If the sample rate is less than twice the maximum-input frequency, the digitizing system produces a mixture of in-band and out-of-band data. Once this foldback of signal information occurs, you cannot retrieve the original signal that is less than one-half of the sampling frequency. So, for this type of system, always place a lowpass, antialiasing filter in front of your ADC.

Now, try to turn the Nyquist theorem upside down. Alternatively, you can use this theorem and intentionally force a system configuration that aliases or folds back higher frequency signals that occur at values greater than the converter's sampling rate. Engineers call this method undersampling, bandpass sampling, or super-Nyquist sampling. Using the Nyquist theorem in this way suits applications such as wireless-communication receivers, radar instrumentation, infrared instrumentation, and video.

In these systems, the signal of interest's bandwidth, Δf_SIG, has a higher frequency than the converter's sampling frequency, f_SAMPLE. Δf_SIG is also riding on a high-frequency carrier signal, f_CAR. An analog bandpass filter that acts like an antialiasing filter in this system limits Δf_SIG. It is not unusual to implement a simple second-order filter (one zero and one pole) for this purpose. Users define the order and response of this filter. If you design in a higher order filter, the bandwidth of Δf_SIG is smaller.

Two formulas help you determine your system's sampling frequency: f_SAMPLE>2(Δf_SIG), which directly complements the Nyquist theorem, and f_SAMPLE=4f_CAR/(2×Z–1), where Z is a rounded-down whole number. You use this second formula twice as you zero in on the actual sampling frequency.

With the first formula, the sampling frequency should be equal to twice Δf_SIG. Then, by using this calculated value for the sampling frequency and a predetermined carrier frequency, you can calculate the quantity of Z in the second formula. The value of Z is usually not a whole number, and you should round it down. With this new value of Z, use the second formula to recalculate a value for f_SAMPLE.

An example may clarify any questions that you have: If a system has a signal that has a 3.5-MHz bandwidth, Δf_SIG, that has a 70-MHz carrier frequency, f_CAR, you can initially calculate the sampling frequency as 7M samples/sec, f_SAMPLE. With this number for f_SAMPLE, the calculated value for Z equals 20.5. Rounding down, Z equals 20, making the actual sampling frequency, f_SAMPLE, equal to 7.18 MHz.

In addition to selecting your sampling clock, you must think about several other important issues with your undersampling application. You should select an ADC with an input stage that can accept signals with frequencies greater than the converter's sampling rate. An undersampling converter's product data sheet specifies this characteristic. Jitter and phase noise of the converter's sample clock, f_SAMPLE, can degrade system performance. You may also require a high-quality crystal oscillator.

Author Information

Bonnie Baker is the analog/mixed-signal-applications engineering manager for Microchip Technology's microperipherals division. You can reach her at Bonnie.Baker@microchip.com.