Questions and Comments for Fourier Transform of Rep and Comb Functions
Please post your reviews, comments, and questions below.
- Review by Miguel Castellanos
I like how you state the differences between the two operators in your introduction, but then conclude by stating how the two operators are related through the Fourier transform. Your derivations are also concise. I think it would be helpful to some if you explain some of the less intuitive steps, such as when you use the Fourier series representation of an impulse train. Well done, overall.
- Review by Hyungsuk Kim
- I thought that this slecture is well organized in order so it was easy to follow and understand the topic. However it would be better if you put some graph to see how do they change between Rep and Comb functions.
- Review by Soonho Kwon
Going over the equations and the mathematics, it was very clear to understand. However, to explain how the functions look like, it would be better to put some graphs on. Great job!
- Author answer here
- Review by Michel Olvera
- The content of your Slecture is easy to follow. It really helped me review this topic. What I liked most was the step by step explanations and as Miguel said, few steps can be added at the end of the content to explain the Fourier Transform of the rep operator, that would be a nice improvement. Good job!
- Review by Sahil Sanghani
Overall this is a great Slecture! I like how the derivations are concise and clear. Also, the explanations along the way draw attention to important points. I think the one spot where you could explain a little more is in the Fourier Transform of the comb. In the second line, the substitution of pT(t) with $ \sum_{n=-\infty}^{\infty}\frac{1}{T}e^{j{\frac{2 \pi}{T}}nt} $
could be better explained. Great job!
- Review by Evan Stockrahm
Fantastic slecture. I really like how you used the definition of each operator and showed how the impulse train interacts with the rep and the comb. The fact that you placed the definitions back to back personally helped me see the correlations between the two, particularly that one is a convolution and the other is a multiplication.
- Review by Botao Chen
Well done. Everything is put in very good order. The demonstrations for both comb and rep are specific and easy to follow. Maybe you could try to provide some graphs to show your point. Sometimes I feel very useful to just look at the graphs and suddenly get the idea about how everything is going on.
- Review by Randall Cochran
Great slecture. Everything was laid out in a logical and easy to follow order. One thing you could add is how you used the Fourier Series representation of the impulse train in your Fourier Transform of the comb.
- Review by Yijun Han
The work is clear and easy to understand. The duality of the two functions rep and comb is clearly showed. The derivations for both computations are concise. Good Job.
- Review by Yerkebulan Y.
Did not explain how to get FT of impulse train or I did not see it. Showed that comb and rep a dual functions to each other.
- Review by Michael Hayashi
This is a high-quality slecture. Each of the main points was addressed clearly and succinctly. I liked how you developed the rep() and comb() operators in parallel to highlight the Fourier transform pair that links both of them. Though your wording was clear enough not to demand it, the only improvement I see would be including a couple archetypical graphical examples.
- Review by Ryan Johnson
I think this slecture would be better if graphs were added for students who are more visual learners. Good job overall.
- Review by Chloe Kauffman
I continue to get a little confused when I see the sampling of an impulse train, whether it matters if the summation uses an "n" or "k" variable. For example, in your definition of impulse train you used both by mistake:
$ \begin{align} p_T(t) = \sum_{n=-\infty}^{\infty}\delta(t-kT) \end{align} $
It would have been nice to have all the simplification steps you used mentioned in words, for example
$ \begin{align}comb_T(x(t)) :&= x(t)p_T(t) \\ &= x(t)\sum_{k=-\infty}^{\infty}\delta(t-kT) \\ &= \sum_{k=-\infty}^{\infty}x(t)\delta(t-kT) \\ &= \sum_{k=-\infty}^{\infty}x(kt)\delta(t-kT) \end{align} $
In the last line, why x(t) could be moved inside the summation and the corresponding dirac delta property that made it so. There is a small but important simplification mistake in this last line, it should be
$ = \sum_{k=-\infty}^{\infty}x(kT)\delta(t-kT) $
notice the capital T in the change that I made.
It also would have been more clear to relate the rep and comb operations to variables such as $ X_{sampled}(f) $. Which I think falls into the realm of having an example/graphs to help illustrate the point.
Overall, this topic was done well.
- Review by Robert Stein
Good job. I would have loved to see graphics to make some things clearer, but still really good.
- Review by Ben Capano
It's clear to see the result of each FT and that there is definitely a relationship between comb and rep. Good job.
