How does decimation work without losing samples?

For example, changing the frequency from 32.768Mhz to 2kHz there is a scale down of 16384.

From the algorithms I've seen you have to discard 16384-1 samples and only count the 16384th sample. However, that many samples being lost will lose important parts of the information. How does decimation work then?

EDIT

A signal is sampled at 2kHz and then modulated with a carrier wave of signal frequency 8.192MHz which is sampled at 32.768MHz. I am trying to revert it back to 2kHz, by decimation. I assume pass it through a LPF and apply decimation however, the points sampled at 32.768MHz are then lost?

• You need to clarify what is the signal spectral content, and what is the sampling rate. I assume you mean that "the frequency" is the sampling rate, and you're changing it from 32.768MHz to 2kHz. If this is the case, please edit your question to reflect that. While you're doing that, please comment on the spectral content of the signal you're interested in -- does it have components at frequencies above 1kHz, or is its spectrum limited to 0-1kHz? Again, if that's the case, edit your question. Mar 21 '21 at 4:53
• @TimWescott Hopefully the edit clears it up. Mar 21 '21 at 6:03

Not quite sure why you accepted my other answer seeing your updated question, but here we go:

What you want to do is passband subsampling. You need to understand what aliasing is: For every frequency $$x$$ in the output nyquist bandwidth $$[-f_{sample, out}/2,+f_{sample, out}/2]$$, the decimation "moves a copy" (that's what we call "it aliases") of every frequency component at $$n\cdot f_{sample, out}, \quad n\in \mathbb Z$$ from the original bandwidth.

Usually, in decimation, you want that to not happen, so you block out all but the target bandwidth with a low-pass filter.

In band-pass subsampling, you block out everything but the target band-pass region. So, you're using aliasing to move your signal by an integer multiple of your output sampling rate. Done!

Your understanding is correct. You need to make sure the information content fits into the output rate, and that implies your signal needs to be band-limited to 1/16384. Otherwise you'll lose information.

Making sure that's the case with a filter is called anti-alias filtering.

• In addition to this, I see that now the minimum sampling rate that can reproduce a 32.768Mhz signal is the 16Mhz. Therefore lowpass filter would work. But the sampling rate is 16Mhz, how can decimation now lead to producing a sampling rate of 2khz? Mar 20 '21 at 20:49
• Check your math. The absolute minimum sampling rate that can reproduce signal that is strictly 32.768MHz wide is 65.536MHz -- and in practice, the sampling rate would need to be higher. Mar 20 '21 at 21:14
• @TimWescott Yes you're right. It should be double the max frequency. However, I still don't understand. For decimation 65.546MHz/2kHz signals are now skipped to obtain the new the sampling frequency, and yet to that means more signals are lost? Mar 21 '21 at 0:15
• again, yes, decimation means throwing away samples. That's it's purpose, @kepsek. You can only do that if you know that it's going to work out. Reasearch anti-aliasing, it'll become clearer. Mar 21 '21 at 12:39