Time-domain aliasing, frequency resolution, multi-rate

[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!

 

[mighty2000]

Hello there,

Thank you for reaching out with your questions about DSP. I am happy to help clarify some of the concepts you are struggling with.

Firstly, let's discuss time-domain aliasing. This occurs when the sampling rate is not high enough to accurately capture the high-frequency components of a signal. In other words, if the sampling rate is not at least twice the highest frequency in the signal, then aliasing can occur. This means that the high-frequency components will be "folded" into the lower frequencies, making it impossible to distinguish between the original and aliased frequencies in the time domain. This can result in a distorted signal.

To answer your question about what can cause time-domain aliasing, it is important to understand that the DFT (Discrete Fourier Transform) is a finite representation of a continuous signal. This means that the DFT assumes the signal to be periodic, with a period equal to the number of samples taken. If the signal is not truly periodic, then aliasing can occur. This can happen if the signal contains frequencies that are not integer multiples of the fundamental frequency, which is equal to the sampling rate divided by the number of samples. In this case, the DFT will try to fit these non-integer frequencies into the periodic representation, resulting in aliasing.

Now, onto multi-rate and frequency resolution. You are correct in understanding that if the sampling rate is at least twice the highest frequency in the signal, then the original signal can be fully reconstructed. However, sampling at a higher rate does not necessarily give more resolution in the frequency domain. Instead, it allows for a more accurate representation of the original signal. In other words, higher sampling rates do not increase the frequency resolution, but rather the accuracy of the frequency representation.

Regarding inserting zeroes in between samples, this is known as interpolation. It is used to increase the sampling rate of a signal without adding any new information. This is commonly done in signal processing to make it easier to apply filters or perform other operations on the signal.

I hope this helps clarify some of the concepts you were struggling with. If you have any further questions or need any additional assistance, please don't hesitate to ask.