Nyquist Frequency

Had a great encounter with this recently!<p>In an environment I work there&#x27;s multichannel audio recordings that are archived. The archival recordings all had a perfect 4kHz tone appearing, seemingly out of nowhere. This was happening on every channel, across every room, but only in one building. Nowhere else. Absolutely nothing of the sort showed up on live monitoring. The systems were all the same and yet this behaviour was consistent across all systems only at one location.<p>The full system was reviewed: from processing, recording, signal distribution, audio capture, and in room. Maybe there was a test gen that had accidentally deployed? Nope. Some odd bug in an echo canceller? Also no. Something weird with interference from lighting or power? Slim chance, but also no. Complete mystery.<p>When looking for acoustic sources there was an odd little blip on the RTA at 20kHz. This was traced back to a test tone emitted from the fire safety system (ultrasonic signal for continuous monitoring). It&#x27;s inaudible to most people and will be filtered before any voice-to-text processing so no reason for concern. Anyway 20kHz is nowhere near 4kHz though so the search continued.<p>The dissimilarly of 20kHz and 4kHz is true, until you consider what happens in a non-bandwidth limited signal. The initial capture was taking place at a 48kHz sampling rate. It turns out the archival was downsampling to 24kHz, without applying an anti-aliasing filter. Without filtering, any frequency content above the Nyquist &#x27;folds&#x27; back over the reproducible range. So in this case a clean 24kHz bandwidth signal with a little bit of inaudible ultrasonic background noise was being folded at 12kHz to create a very audible 4kHz tone.<p>It was essentially a capture the flag for signals nerds and a whole lot of fun to trace.

One misconception that many make regarding the Nyquist frequency is thinking that the sampling rate needs to be twice the highest frequency.<p>Your sampling should really really be twice the <i>bandwidth</i>.<p>e.g. your bandwidth is 100 MHz centered at 1 GHz (it needs to actually be bandlimited to 100 MHz**). You do not need to sample at 2.2 GHz. You sample at 200 MSPS (really, you should sample a little more than that, say 210 MSPS, so that the bandwidth of interest doesn&#x27;t butt up against the Nyquist zone edges.)

If you&#x27;re interested in learning more about various DSP topics, I run a blog on over at <a href="https:&#x2F;&#x2F;signalprocessingjobs.com&#x2F;" rel="nofollow">https:&#x2F;&#x2F;signalprocessingjobs.com&#x2F;</a> - a signal processing job board and blog!<p>One of the more popular series is the Journal2Matlab blog about translating academic journal papers into easy to read matlab.

Signals and systems was a tough course for me. It was what crushed my 4.0 GPA. Nyquist frequency was a concept I could not wrap my head around. I’ve improved, but it still doesn’t click as I’d like it to.<p>When I took the course, it made no sense to me that you could sample at twice the frequency of the signal and reconstruct it. Consider a sine wave at 1 Hz. If you sample at 2 Hz, you’d get readings of 0, 1, 0, -1, etc. If you graph that, it’s a perfect triangle wave, not a sine wave! That’s what I couldn’t not get past. I thought you’d need an infinite sampling rate to accurately capture the sine wave.<p>As I type this out, I’m realizing that a critical component of this that I wasn’t taught (or I didn’t grasp) is the need for the signal to be bandlimited. Returning to my sine example from above, what bothered me was, if I don’t sample more points, how do I know that it’s only a sine wave, and nothing more? That only works if you pretend there are no higher frequencies (or filter them out, though an ideal filter is impossible in practice). If there aren’t higher frequencies, there can’t be anything you “can’t capture” by sampling at the Nyquist frequency.

Dan Worrall made a fantastic video which touches on Nyquist. His youtube channel is a tremendous resource: <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=-jCwIsT0X8M">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=-jCwIsT0X8M</a>

I saw the Nyquist Frequency mentioned in the American Cinematographer Magazine. The article illustrate how detailed patterns, like sweaters, can produce a fuzzy jagged artifact called moire. This is because there is too much information for camera&#x27;s sensor to interpret and summarize the details into pixels (ie. surpassing the Nyquist Frequency).<p>Their suggested solutions were to 1) get a wide-angle lens to reduce detail beamed into the sensor 2) use a larger image sensor or 3) remove the object causing moire artifacts.

The coolest nyquist frequency application I&#x27;ve every come across, if you look up how modulation of nerve impulses works in the optic never you can figure up the fastest rate of blinking your eye can perceive, and it checks out in reality.

Beware, there are lots of misconceptions in the comments.

Soothing.

To add another misconception, the Nyquist frequency is a lower bound, below which you necessarily get aliasing. It doesn&#x27;t say anything about whether said sampling rate is sufficient for reconstruction or whatever your intended use is.<p>E.g. sampling a 1hz signal at 2hz still doesn&#x27;t tell you if the signal was a 1hz sin or a 1hz sawtooth (depending on how lucky or unlucky you are).