# Homework Help: Time-domain aliasing, frequency resolution, multi-rate

1. May 29, 2012

### sirpsycho85

Hey guys, this isn't an assigned homework question but I'm self-studying basic DSP from the free Smith book (can't post a link, but it's the first link if you google "smith dsp") and I have a question on some material. Also, I've searched the forum and most questions related to this subject appear to be in this directory. I realize I've asked a lot so if anyone can answer even a part of this, I'd appreciate it. I just wanted to have everything in one place.

In the book in the guide, Chapter 10 has a section titled "Periodic Nature of the DFT". In it, there is discussion of time-domain aliasing. I am confused as to what this represents from a physical standpoint. Frequency-domain aliasing I understand as such: when frequencies higher than 1/2 of the sampling frequency (Fs) appear in the signal, samples of that frequency can also be represented by a sinusoid with frequency lower than 1/2 Fs. This causes information at that lower frequency to be lost because it is impossible to tell where it came from.

In that section of the book, there is discussion about performing some operation on the frequency-domain points. Each point in freq domain is supposed to correspond to a sinusoid that fits an integer amount of times into the N points. That is, if you have an N-point time-domain signal, then the 3rd point of the N/2-1 length freq domain is the sinusoid that makes three complete cycles in the N time-domain points.

So my question is this: What can you possibly do to an N/2-1 point freq domain signal such that you could NOT represent it in N time domain points and would thus cause time-domain aliasing?

Secondly, I have what I think are related questions on multi-rate and frequency resolution. As I understand, if Fs is >= twice the highest frequency in the signal, you can fully reconstruct the original. Is the idea that sampling at even higher rates (reducing space between frequencies) gives more resolution in freq domain (so you can see thin spikes) but both a high and low resolution spectrum can be reconstructed to the time domain? There are multiple ways to break down the time domain into sinusoids...but both high and low sample rates are valid (as long as they're >= twice the highest frequency)?

Then perhaps I'll understand what the purpose if inserting zeroes in between samples is. In the same book in chapter 3 page 60 there is a description of multirate and interpolation. I'm very uncertain as to what the purpose is and why it works. What's frustrating is that decimation makes sense and it seems to be the clear opposite. In decimation, you're sampling at a higher rate than necessary on purpose such that you can use a simpler anti-aliasing filter. You can then use a digital filter to get rid of all the extra frequencies higher than your desired signal. Not sure what's happening in interpolation though.

Thanks ahead for your help, it's tremendously appreciated!!!

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