What sampling frequency should I use if Nyquist is not available?

I have the following homework question that confuses me:

We have an audio emitter that can emit two signals:

It either emits a sine wave at 23 kHz or it emits a sine wave at 25 kHz.

The receiver has the following sampling frequencies available: 16 kHz, 32 kHz and 48 kHz.

The question asks which is the best sampling frequency for the receiver to know when is 23 kHz being transmitted, and when is 25 kHz is.

The Nyquist criterion states that the sampling frequency should be minimum twice the signal frequency. In this case it should be 50 kHz.

Should I select 48 kHz, or is there a trick?

• If the sampling receiver had the proper brick-wall filtering, you wouldn't be able to detect the 25KHz signal with any of your sampling rates - it would be filtered away entirely. Nov 15 '21 at 22:44
• Recall that the Nyquist criterion establishes a requirement for reconstructing the original signal. That's not necessary to detect which of two signals is being sent. Nov 16 '21 at 13:27
• Is there a mixup between Nyquist criterion and Shannon-Nyquist sampling theorem? Also, a small inacurracy that is commonly accepted, but kinda relevant in this case: The sampling theorem does not make a statement about the maximum signal frequency, but about the maximum signal bandwidth. Nov 17 '21 at 12:00
• @MarkRansom, in which case the 48 kHz sampler would work nicely, as it'd be able to detect the 23 kHz signal fine, and if it was missing, the receiver would know the 25 kHz was sent instead. (Given the phrasing in the question, the emitter always emits one of the two, so we can ignore the possibility of the emitter being broken.) Nov 18 '21 at 12:32

HINT

When you sample at below the Nyquist rate, aliasing happens. That means frequencies higher than half the sampling rate get folded back down to below half the sampling rate.

PS: Tell your teacher, that's a really nice question. :-)

• +1 especially for the PS. I hope the OP conveys the comment to his instructor; professors do enjoy being appreciated. Nov 15 '21 at 19:43
• It's not a good question though as it stands. It could be a good question with more clarification, or a good question for a broad discussion (but no unambiguous answer), but it's horrible as a multiple-choice, because depending on how one interprets the theory vs. implementation level all the answers could apply. Nov 16 '21 at 18:09
• @J... Are you sure that would collect the point? Nov 16 '21 at 23:27
• @CrisLuengo No, I worded it awkwardly. The point was that with multiple choice you don't get to see the thought process of the student and this is a question which lends itself particularly well to a bit of exposition.
– J...
Nov 16 '21 at 23:50
• @J... I agree. And maybe this is not a multiple choice question. I can see this being a question where you are expected to show your calculations and your reasoning. There being three options doesn't make it a multiple-choice question. Nov 16 '21 at 23:53

As correctly stated in Peter K.'s answer, this question is about aliasing. Since you can't sample at a rate that is sufficiently high to avoid aliasing - i.e., $$f_s>50\textrm{ kHz}$$ - you have to take aliasing into account.

Now it's your task to figure out the aliased frequencies of the given signals for the different sampling rates. If you understand how aliasing works, you'll see that for two of the three given sampling frequencies, the two sinusoids become indistinguishable after sampling. Only one sampling frequency results in distinct frequencies for the two given signals. That sampling frequency is not necessarily the highest frequency.

The Nyquist criterion states that the sampling frequency should be minimum twice the signal frequency. In this case it should be 50 kHz.

A more precise statement is that the sampling frequency should more than twice the bandwidth. Usually, you want a bandwidth that include zero frequency, making the bandwidth and the maximum frequency the same, but that is not the case here. Here, the bandwidth is only 2kHz, so the best sampling frequency would be slightly more than 4kHz, e.g. 4.1kHz.

• "Here, the bandwidth is only 2kHz, so the best sampling frequency would be slightly more than 2kHz, e.g. 2.1kHz." If the sampling frequency should be more than twice the bandwidth, and the bandwidth is 2kHz, shouldn't the sampling frequency be slightly more than 2·2 kHz = 4 kHz ? Nov 18 '21 at 9:32