How do you apply the Nyquist theorem to avoid aliasing and distortion in signal processing?

In essence, it means that if a continuous signal is changing at a rapid pace (meaning it has a high frequency component in it) the sampling frequency needs to be fast enough to capture the specifics of the change.

This is saying that in order to digitally represent an analog cycle, it has to be sampled at least twice per cycle. If a wave is only sampled once per cycle, the resulting digital stream will be a line (if it's sampled at the same point on each cycle) by sampling it twice, the result will more accurately represent the highest and lowest points of the cycle.

The premise that aliasing is distortion is wrong since it is just the higher frequencies of the input folded into the baseband. This can in fact be the signal of interest and intentional in which case it is called undersampling.

Aliasing occurs when the sampling is done below the Nyquist frequency and artifacts of high-frequency signal may be confused as a low-frequency signal. But aliasing is not necessarily a bad-thing and can be worked with. The process of Multi-Resolution Analysis (MRA) using the DWT wavelets takes into account and exploits the properties of aliasing in some of its family of wavelets in its algorithm. The DWT is then able to take those under-sampled signals, and synthesize it into the original using a particular order of filter banks which eliminates aliasing.

I tend to disagree with the premise of the Shannon nyquist sampling theorem . Simply put you need two or more samples per wavelength to correctly recover the wave that you are trying to sample. If you stick your the concept of simply using sample rates as we see in modern a to d converters then your right. Yet oscilloscopes work sampling multi gigahertz signals with 100 mghz a to d converters. Plus you can always cheat it by changing domains ( which I did ) .

This discussion focuses on sampling a single tone located in a higher Nyquist zone, but it is a bit misleading when one needs to sample an entire bandpass signal whose spectrum does not coincide with a Nyquist zone. In this case, if a single ADC is employed, it is necessary to sample at a rate higher than twice the occupied bandwidth of a signal (see Proakis and Manolakis Sec. 6.4). On the other hand it was shown by Kohlenberg in 1953 that Nyquist rate sampling (with respect to the occupied bandwidth) can be achieved with nonuniformly time interleaved ADCS.

I would go back to fundamentals here; the picture theory rather than a load of maths. Sampling at fs creates copies (images) of your spectrum around integer multiples of fs. It is the overlapping spectra that are the source of aliasing. Correct sampling is a sort of spectral "hygiene" keeping the images separated taking into account filter rolloff etc. This logic applies equally to bandpass as well as lowpass sampling.

One can understand the Nyquist frequency with Fourier integrals. The frequency distinction is made clear as we integrate from -∞ to +∞ in time. In other words, it's asymptotic. For any practical use, we need finite integrals or more information. If we know the frequency of a single carrier and are synchronized to sample at the top/bottom of that carrier (extra info) we can transmit bits at 2x the carrier. Barring that, we need far more information (samples) to reasonably reconstruct a signal with finite time integrals.

Signal reconstruction algorithms that either assume a signal pattern (e.g., time shift in spatial domain) or random sampling techniques have shown promise in reducing the effects of aliasing. The advantage is also that the amount of data size is limited by random sampling, when compared to the same signal that was regularly sampled.

Aliasing and distortion can never be fully eliminated because sampled signals are not bandlimited. So the key design question is how much distortion is acceptable.

In the physical world, there is rarely a frequency beyond which there is no content. Instead, there is a range of frequencies in which we may find useful content. This means that picking a sampling frequency is less about some absolute red line and more about where we think the extra content is useless to our system. Given that, we often need to put in analog filters to remove signals a higher frequency. Low-pass anti-alias filters can add a significant amount of phase lag to the signal. This usually has little or no effect on signal processing applications, but can have a severe effect on feedback systems.

These ideas have to be treated carefully. But in fact, RF converters use this idea of Nyquist zones a lot. In lots of our projects, we use fast ADCs and DACs that sample at 4, 5 or up to 10 GSPS. Depending on the signal frequency, we intentionally use the aliasing effect to up or down convert signals and accommodate higher frequencies. Analog filters would allow selecting the desired band. One last detail, the bandwidth of the analog front end of the converter must be high in order to let these signals to go through it.

To put it simple, Nyquist is the amount of analog bandwidth that can be sampled with a converter. The simplest case is if I sample at fs, my frequency range can go from 0 to fs/2, with fs/2 being the Nyquist frequency. In a broader sense, I can sample at fs and recover frequencies from fs/2 to fs by proper analog filtering and letting alias do the magic. In either case, the amount of analog bandwidth that I manipulate is fs/2, the Nyquist frequency.