What is the Nyquist-Shannon sampling theorem and how can it be applied to signal processing?

Suppose you are watching a person bouncing up and down on a trampoline at a certain frequency, say 1Hz. Further suppose, you are looking through a shutter that opens 10 times a second. What would you see? The person at different positions over a one second cycle allowing you to follow the motion. Now imagine, the shutter opens twice a second, what would you see? Most likely, the person in two distinct positions, say up and down, still allowing you to infer the motion. What if the shutter opened only once a second? You would see the person held in a constant position. Worse, if the shutter opened every 3 seconds, you would start to see a impossible slow motion, an artefact. The shutter needs to open at least twice the bouncing frequency.

The Nyquist-Shannon sampling theorem states that to accurately reconstruct a signal, it must be sampled at a rate at least twice its highest frequency component. In signal processing, this principle ensures faithful representation of analog signals in digital form, preventing aliasing and loss of information during sampling and reconstruction.

Another view related to direct sampling of pass-band signals - where the sampling process can be used to map pass-band to baseband - then the Nyquist criterion requires that the sampling rate be twice the bandwidth of the signal of interest and is typically complex data. This sampling rate is usually much lower than twice the highest frequency. Often, many times smaller than the carrier frequency of the pass-band signal.

The simplest thought experiment that I could put forward to illustrate aliasing was to ask the question "why do the wagon wheels appear to go backwards when you watch an action sequence in a Cowboy movie!?" (though this dates me somewhat as viewing habits have moved on considerably...)

There exists also a Nyquist-Shannon theorem for narrowband signals. In this case, the maximum sampling rate is proportional to the bandwidth of the signal rather than the highest frequency component in the signal.

The reader should think of the sinc function as simply the impulse response of a perfect brickwall low-pass filter with cutoff frequency at exactly Nyquist (or half the sample rate). Now practical reconstruction will not have an ideal brickwall filter, so the interpolation function will not be exactly a sinc function. But it will look like one for any decently designed low-pass interpolation FIR.

Actually I disagree with the sentence "However if you want to filter a continuous-time signal you need to sample it first" In order to sample a signal you might need to filter before sampling in order to avoid aliasing. For that one would use an analog filter, which are generally harder to design and more importantly update

I must agree with Shy-El Cohen here. There is something wrong with this section. In fact the statement: "For example, if you want to filter out frequencies above 10 kHz, you need to sample the signal at least at 20 kHz." is just plain wrong. It's ass-backwards. Assuming we're doing baseband sampling (as opposed to passband sampling, which is a little more complicated), the *only* condition that determines the necessary sample rate is the frequency of the highest component *before* any DSP filtering is done. To "filter out frequencies above 10 kHz", first you must determine what the top limit of those frequencies above 10 kHz is. Then double that limit frequency, then increase it a little more and that is the rate you need to same at.

Another voice supporting Robert's comment! To steal a quote from Pauli, some of that "article" is not even wrong. It reeks of ChatGPT... I highly recommend not reading it.

"Signal compression" (the reduction of data) has nothing to do with the Sampling Theorem. This operation is more accurately called "Data compression". That the signal is already sampled and has become digital data, is assumed at the outset, because the data is not analog or continuous-time. It's the result of sampling (that quantizes time and turns the analog signal into a sequence of discrete-time values) and analog-to-digital conversion (that quantizes the discrete-time values into a finite word width) that the operation of data compression is applied to.

The minimum sampling frequency of a signal that it will not distort its underlying information, should be double the frequency of its highest frequency component. The sampling theorem introduces the concept of a sample rate that is sufficient for perfect fidelity for the class of functions that are band-limited to a given bandwidth, such that no actual information is lost in the sampling process. It expresses the sufficient sample rate in terms of the bandwidth for the class of functions. The theorem also leads to a formula for perfectly reconstructing the original continuous-time function from the samples.

I dunno who wrote this (was it Chat GPT?), but I'm not impressed. I believe there is enough misinformation in this article that it should be removed from public consumption. It does more to mislead than to inform.

I'm with Robert here. The assertions in this article range from misleading and half-right, to outright wrong. There are plenty of good discussions of the Nyquist-Shannon theorem of every length, in books, web pages, forums and videos. There's no need to burden the Internet with this.