HW2 discussion, ECE301 Spring 2011, Prof. Boutin

In question 2e

$ x(t)= \sum_{k=-\infty}^\infty \frac{1}{1+(x-7k)^2} \ $

should it be like this?

$ x(t)= \sum_{k=-\infty}^\infty \frac{1}{1+(t-7k)^2} \ $ 

yes, it should be. The correction has been made. -pm

and I was trying to find out what the peak value is for this question but turns out to be very hard to calculate the sum

$ \sum_{t=-\infty}^\infty \frac{1}{1+t^2} \ $            and wolfram said answer is π * coth(π). is there any easier way to do that? Yimin. Jan 20

You do not have to evaluate the sum. In particular, you do not need the peak value of that functions. Try to guess the period directly by looking at the sum. If you have no idea how to do this, read this page first. -pm

Yeah I'm just trying to figure out the infinite sum just for fun. Thanks.

Oh excellent! I think this deserves a page on its own. Let's try to involve the math folks for help.-pm

And for question 4, are we still using the tempo? so my guess is use step functions to cut out the rhythm we want? (Yes, that's the idea. -pm) Then put the whole line in one equation? that will become pretty messy I guess. Yimin

Not too bad, if you think about it carefully. Each note can be written in a somewhat simple form. Then you just add all the notes together. -pm

For question  4 we only need the first second on the song? Right, we don't have to compress the entire song into second. 

No. You need to pick a note lasting one second (your choice what frequency), say we call this note $ y_0(t) $. Then write your tune z(t) as a function of $ y_0(t) $ rescaled and shifted. Hint: you can write z(t) as a summation of $ y_0(a_i t+b_i) $'s multiplied by some step functions $ u(t-c_i)-u(t-d_i) $ (to make the note last the right amount of time). Does that help? -pm

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