# Nyquist Sampling Theorem: Conserving Signal Characteristics

### Key Takeaways

• The Nyquist sampling theorem provides the minimum sampling rate for signal recreation.

• Sampling rates relative to the Nyquist theorem can provide insufficient data or introduce noise to the signal.

• Designers can improve oversampling outcomes by using a few circuit design techniques.

The Nyquist sampling theorem governs part of the operation of a television converter box.

While there is increasingly a shift to digital electronics for performance and size disadvantages, sensors and outputs are still bound to the analog world. Signal conversion has to heed a few basic practices to maintain coherence and quality, perhaps none more defining than the Nyquist sampling theorem. The Nyquist theorem gives circuit designers a critical sampling rate to target for a given frequency, ensuring the capture of enough data to recreate signals throughout conversion processes accurately.

Examples of Aliasing

 Computer screen text Small text can quickly become illegible on computer screens, but a modern OS uses anti-aliasing methods to reduce eye strain. Medical imaging MRI testing includes artifacts from wrap-around on the image when the target is larger than the camera’s FOV. Astronomical imaging Pixel size needs to be half of the minimum period of the image to preserve the resolution. Larger pixels introduce aliasing, while smaller pixels produce noise. Human vision The lens filters out spatial variations finer than 60 cycles/degree; this matches with most people's trichromatic photoreceptor spacing of 120 cycles/degree for most people.

## The Nyquist Sampling Theorem Guides Signal Recreation

Very plainly, the Nyquist-Shannon sampling theorem (often referred to shorthand as the Nyquist sampling theorem) lays out the minimum sampling rate to reconstruct analog signals as digital without introducing distortion: 1/(2B) seconds, where B is the bandwidth of the signal. Sampling a signal at a rate less than the Nyquist does not provide enough data to locate all the peaks and troughs of the signal; since the signal is missing information, there’s a possibility it is misconstrued by the receiver as a different signal entirely, resulting in aliasing distortion. Nyquist can sample more complex signals composed of component sines and cosines (as indicated by the signal’s Fourier series) by sampling at double the rate of the highest component frequency.

Aliasing is one of the more well-known forms of distortion, although most hear it in the context of anti-aliasing measures (i.e., prevention methods). Essentially, aliasing does not allow for sampling at a fast enough rate to capture the signal change. A visual example of this is the stroboscopic effect: imagine a lever with some rotational velocity and a strobe light with the same frequency. Every time the light turns on, the lever’s position appears unchanged. A strobe light frequency slightly greater or less than the rotation period would appear to have the object slowly rotating backward or forward. A more recent variant of the stroboscopic effect matches a camera’s capture rate to a non-zero integer multiple of the rate of water falling out of a faucet, giving the visual illusion of a stationary, standing water column between the faucet head and the sink.

There are three possible cases for Nyquist sampling rates, and it’s critical to understand these apply to all the component frequencies of the waveform:

• Undersampled - A Nyquist rate less than twice that of any signal’s frequency. The sampled waveform cannot be perfectly reconstructed at this sampling rate, making information loss unavoidable.

• Oversampled - A Nyquist rate over twice that of the highest signal’s frequency. The signal can be perfectly reconstructed from the sample because there are more data points than the minimum. While oversampling is generally preferable to undersampling, it can impart additional noise and lack efficiency with poor implementation. The oversampling factor is the multiple by which the sampling rate is that of the Nyquist sampling rate.

• Critically sampled - A band-limited signal equal to the Nyquist rate. While perfect reconstruction is theoretically possible at this sampling rate, this would require a transition bandwidth from passband to stopband of 0 Hz – a much more feasible option is oversampling to expand the range of transition frequencies.

Typically, the sampling rate adjusts to the circuit's needs, but there are cases where the sampling rate remains locked for a particular function. Instead, engineers can devise a lowpass filter to block the high-frequency component portions of the signal that would exceed the Nyquist rate.

## How to Improve Oversampling Outcomes

Oversampling is the only meaningful option for signal processing. However, optimization will require more forethought than just sampling at an excessive factor of the Nyquist rate. Three aspects of oversampling improve signal processing performance:

• Anti-aliasing - Filters for sampling have to perform a tradeoff between bandwidth and anti-aliasing capabilities; immediate transitions from bandpass to bandstop regions are impossible. Some aliasing is usually preferable to increase the bandwidth in these filters, which oversampling can counteract.

• Resolution - Improving the sampling rate increases the performance of analog-to-digital converters (ADCs) and digital-to-analog converters (DAC); the factor of oversampling improves the signal-to-noise ratio by a square root of the factor. Oversampling then allows for additional bit precision; much like anti-aliasing measures, there is a small noise tradeoff, often seen as permissible.

• Noise reduction - As mentioned, oversampling by a factor improves the SNR by a square root of the factor because the signal amplitude grows by the oversampling factor, while the noise grows by a square root. In cases where the noise correlates with the signal, oversampling loses many benefits as the amplitude of the noise grows at the same rate as the signal, canceling any SNR improvements. Dithering or randomized noise intentionally added to a signal can remove the signal and noise correlation; if the noise added is outside the specified frequency range of the sample, filter networks can remove it.

## Further Investigating Series and Parallel Characteristics

The Nyquist sampling theorem gives circuit designers a framework for signal recreation to avoid loss. Additional methods related to signal oversampling ensure that the recreated signals avoid introducing noise to the system that can undermine performance. Establishing signal analysis parameters is crucial, and designers will want to simulate network response, filters, and more to ascertain the reliability of analog or digital conversion. Cadence’s PCB Design and Analysis Software suite gives design teams all the tools necessary to test, simulate, and forward annotate circuit designs for a seamless ECAD workflow. With the powerful and easy-to-use OrCAD PCB Designer, rapid DFM for prototyping and production-level PCBs is available within one convenient package.

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