# The impact on the frequency of adding zero samples and non zero samples

Can someone give me a brief explanation of what is going on in here? I struggle to understand the full differences between the impact on the frequency of added zero samples and adding samples which are not zeroes. (In between the samples).

I learnt this course 100% from the lectures so if I have some misconception regarding the basics, I'd be glad if you could kindly help me understand.

The key points I know so far:

• Adding zero(ed) samples in between the existing ones in the time domain, squizes the frequency domain and adds height to it.
• The squized frequency domain is squized while it's normalised (once we turn it to be periodic with period of 2π).
• Adding non zero(ed) samples to our time domain, we get an added zeroes vector to the end of the frequency domain's period. (Padding zeroes in frequency). Which means that when it's being normalized into 2π period's length, then it's also being squized.

From that we conclude, that adding zero samples and non zero samples do the same exact thing.

[Question:] where is my mistake in the above logic sequence? Where is the difference between the impact of adding zero samples in the time domain on the frequency vs adding non zero samples in the time domain on the frequency?

• what do you mean, exactly, with "non-zeroed samples"? Can you give a formula that describes your output signal in terms of input signal? Aug 20, 2020 at 14:21
• If input signal is [5,9,4,2,8,2] (N=6, period is of length 6. => [5,9,4,2,8,2,5,9,4,2,8,2,5,9,4,2,8,2,....]) Then output for zero samples is (if we add two zeroes): [5,0,0,9,0,0,4,0,0,2,0,0,8,0,0,2,0,0]. I meant (I guess it's the name of it), upsampling after decimation. And the adding non zero values is just getting a higher resolution. Does it help by any means? I can add this to the question if it makes it clearer. Aug 20, 2020 at 14:46
• no, doesn't help, sorry. You told be how you're inserting zeros. But I was referring to the sentence "Adding non zero(ed) samples to our time domain". What does that mean? "Just getting a higher resolution" doesn't explain it. Aug 20, 2020 at 14:58
• I don't want to mislead, just in case I got it wrong. I'll try to describe the way we do it instead of providing example, since the example I'd give might be wrong as a result of my misunderstanding. So, one way to get better resolution as I understand is to do DFT to the time domain, to add N padding zeroes to the frequency domain and then do IDFT. IDFT will have now 2N values which are (probably....) are not zeroes. Does it make sense now? I saw there is a tag 'padding-zeroes' in here, so maybe it sounds familiar to you. Aug 20, 2020 at 15:10
• It's five o'clock somewhere. You might want to also consider your question in the context of this interpretation of the DFT: dsprelated.com/showarticle/768.php Aug 20, 2020 at 22:21

## 1 Answer

As soon as you sample in the nuiqist rate, sampling faster won’t add information(!). Sampling zeroes can be looked at as approximation mistake(!) It gives the same frequency as the original signal mathematically because it’s multiplying by zero and IF by contradiction, there would have been any other value instead of the original value, which WOULD [again by contradiction] add some information aside from what we have with the niquist samples, THEN we would need a higher niquist rate and that would be a contradiction to our given Max. Niquist frequency.

Yes, indeed sampled zeroes gives the same signal and they are considered as a measure mistake given the Niquist frequency!

• um, no. If your original signal contained zeros between the values you've sampled, then your sampling wasn't done at nyquist rate. Aug 21, 2020 at 12:52
• *at least **original sample Aug 21, 2020 at 12:53
• sorry, don't understand that. Aug 21, 2020 at 12:54
• don’t worry, there are couple of students who struggle with it occasionally and it’s more meant to be a pivot for them when they search for that topic in google, rather than being the perfect question or anything like questions that grab 10-20-50 upvotes. I’d say skip, if anyone who needs this topic urgently will need it, I’ll dive more into explanations. But you don’t sound like a pupil who’s about to drop out tomorrow :). Let the people who need it to read it and conclude their conclusions. Aug 21, 2020 at 12:58
• The way it’s written is indeed not perfect, but the idea is perfectly clear. Trust me that once anyone will get lost in it, it’ll provide the necessary information. Aug 21, 2020 at 13:01