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I came across this definition of aliasing.

"Aliasing occurs when frequency components are present in the input signal that are higher than half the sampling frequency of the ADC"

From my understanding, the minimum sampling frequency is twice the maximum input frequency. For example, if the max input signal is 600Hz then your sampling frequency is 1.2kHz.

Therefore saying aliasing occurs at frequencies higher than your maximum input frequency is also correct. From the definition, half of the example sample frequency (1.2kHz) is 600Hz which is the input frequency.

If this is correct, then why word it in such a complicated way?

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    \$\begingroup\$ What's complicated with it? Your assumption seems to be that sampling is always double the highest frequency, and no aliasing happens. But what if you have 700 Hz input and sample it with 1200 Hz. You might have components larger than half the sampling rate. \$\endgroup\$
    – Justme
    Commented May 21, 2023 at 19:44
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    \$\begingroup\$ "From my understanding, the sampling frequency is twice the maximum input frequency." No, the minimum sampling frequency is twice the maximum input frequency. In practice it's often much higher. \$\endgroup\$
    – John D
    Commented May 21, 2023 at 19:47
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    \$\begingroup\$ @JoeyB But that's the point of your quote, you get aliasing if you don't sample at high enough frequency. That's what it means. It defines when you get aliasing. When Nyquist theory is violated. \$\endgroup\$
    – Justme
    Commented May 21, 2023 at 20:01
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    \$\begingroup\$ A pet peeve of mine: You really need the sample rate to be greater than twice the highest frequency component. Otherwise people get confused when they sample a 600 Hz sine wave at 1200 Hz and get zero out. \$\endgroup\$
    – pipe
    Commented May 21, 2023 at 20:41
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    \$\begingroup\$ People also get confused when sampling only a little higher than twice the highest frequency component, particularly if that biggest frequency dominates the input signal, and all they see is a very rough, very square, signal. You really need to sample at many times the highest frequency component if you expect your samples to look anything like your input signal. \$\endgroup\$
    – brhans
    Commented May 22, 2023 at 3:07

3 Answers 3

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The quote only tells you when you get aliasing and when you don't. So depending on sampling rate and maximum input frequency, you either get aliasing or not. That's very neutral statement which is true. Note that it it does not say aliasing is bad, or that it must not happen - that's a completey different thing from just the definition of aliasing and when it happens or does not happen.

Your quotes already have an assumption that aliasing must never be allowed to happen as your definitions already avoided it and it's not allowed to happen by setting the sampling rate to twice the maximum input frequency, or limiting the maximum input frequency to half the sampling rate. As if you already had decided that Nyquist limit is some fundamental law that must be obeyed or is somehow impossible to break, or that somehow you always know or assume the frequencies present on signal to set the sampling rate, or can assume the frequencies present based on sampling rate.

But if you don't limit the input signal, or you sample it with too low frequency, you get aliasing. That can happen. If you sample a 700 Hz signal at 1200 Hz rate, and that can happen if the assumption about input frequency is wrong. Such as a curious student wanting to see what happens if you turn a signal generator from 500 Hz to 700 Hz while signal is still sampled at 1200 Hz. Or the student changing the sampling rate to 900 Hz while sampling a 500 Hz signal.

There is no law against that, a perfectly possible scenario to have aliasing, either by accident or even make it by design to achieve something useful, such as downsampling converters.

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It's not worded in a complicated way. There are two items to be related to each other:

  • Input frequency (fin)
  • Sampling frequency (fsamp)

as to when aliasing will happen.

Their definition relates fin directly to fsamp:

  1. Aliasing happens when fin > (fsamp/2)

Your version defines an extra term, fmax, and explains it in two steps:

  1. Maximum input frequency fmax = fsamp/2
  2. Aliasing happens when fin > fmax

Both do the same arithmetic. But their definition covers it in one simple step, which makes it more succinct.

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The mistake lies here :

For example, if the max input signal is 600Hz then your sampling frequency is 1.2kHz.

For a practical system, you will often have to supply an anti-aliasing filter, to cut off frequencies in the input signal that would generate aliasing. If you set the sampling frequency at 1.2kHz per the Nyquist criterion, then your anti-alias filter must pass all frequencies of interest (i.e. up to 600Hz) and reject all frequencies above the Nyquist frequency (also 600Hz). Such a filter does not exist.

The solution is to include the AA filter in the design, instead of as an afterthought, and define how much attenuation you need above Nyquist, and how much frequency response error (rolloff, ripple) you can tolerate in the passband - and secondary considerations such as how much cost you can tolerate in the filter.

The outcome of this may be that to keep the filter cost down, you can only achieve the attenuation you need above 2 kHz (for a clean 600Hz passband).

That sets the Nyquist frequency at 2 kHz, and thus the sampling frequency at 4 kHz.

Now aliasing does NOT occur above 600Hz (above the highest input frequency) but only above 2000Hz (above half the sampling rate) where the filter will have eliminated it.

The CD Audio specification required 20 kHz signal bandwidth, with a sampling rate of 44.1 kHz (Nyquist 22.05 kHz). This required really tight filter designs, which (in analog electronics) were expensive, difficult to make, and often required hand tuning to allow for component tolerances.

Practical systems quickly pushed the ADC sampling rate out to 4*44.1 = 176.4 kHz, allowing much better and cheaper anti-aliasing filters; the sample rate was reduced to 44.1 kHz in the digital domain where the complex filter could be made cheaply and accurately.

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