Lecture 09

For today you should:

1) Exercises

2) Progress report

Today:

1) Case study ideas

2) Finish the DFT discussion from Lecture 8

3) Self-organized criticality

4) DFT: from synthesis to analysis

For next time:

1) Exercises

2) Case study

Exercise: Write a recursive version of DFT using the Danielson-Lanczos Lemon

1) If the length of ys is 1, DFT(ys) = ys.

2) Otherwise split it into evens and odds, and compute DFT of the two halves.

3) Use the Lemon to stitch the solutions together. Hint: DFT(ys) is periodic.

See also: FFT on Wikipedia

Mathematical generalization

Euler: "Hey, Ralph. You know the exponential function?"

"Yup."

"And you know we've got these complex numbers."

"Yup."

"So, I was thinking..."

"Uh, huh.

"So what would happen..."

"Yeah?"

"If we went ahead and applied the exponential function..."

"Hmm, I think I see where you're going with this."

"...to a complex number. Eh? Eh?"

"No."

Ralph says no, but Euler says "yes, yes, yes."

Euler's formula:

One way to get there is by power series:

Complex signal

Remember that a (real) signal is a quantity that varies in time. So what's a complex signal?

Mathematically, it's a function that maps from time to a complex number.

Computationally, it's a vector of complex.

But what does it mean?

Let's take a look at chap07.ipynb and see if we can figure it out.