Trash Talk - Mathematica Blocks - Tue Oct 16, 2007 4:21 am Post subject: Mathematica
Any other Mathematica-lovers out there? Sorry for the random spam.
Dr Brain - Tue Oct 16, 2007 8:49 am Post subject:
Why plot from 1 to 10? The 10th term is 0. All odd terms of the Fourier series of a square wave are zero.
baseball43v3r - Tue Oct 16, 2007 12:33 pm Post subject:
someone want to explain to the rest of the community who hasnt paid attention through four years of high school math and a quarter of college math what exactly this is? gracias
eggy
Dr Brain - Tue Oct 16, 2007 2:08 pm Post subject:
It's a truncated Fourier series of a repeating square wave centered at 0.5, with amplitude 0.5. One might see something like this after passing a square wave signal through a low pass filter (i.e. removing the high frequency components).
Fourier series and Fourier transforms are two of the fundamental tools in electrical engineering.
Blocks - Tue Oct 16, 2007 3:53 pm Post subject: Heaviside function
It's actually a truncated Fourier series of the Heaviside function. The reason it was plotted to 10 terms was because it was followed by plots to 100 and 1000 to show Gibb's phenomenon.
But that wasn't the point of the topic! Dr Brain - Tue Oct 16, 2007 8:10 pm Post subject:
Uh, the Heaviside function isn't periodic, so there's no Fourier series to truncate.
I don't use Mathematica. I've dabbled with Maple, but I don't usually need a computer to solve things algebraically for me (I'll use my TI-89 as a CAS sometimes), and MATLAB handles all my numerical needs.
Blocks - Tue Oct 16, 2007 9:40 pm Post subject:
Yes there is ... it's right there. Maths war!
I've never tried Maple, but I use MATLAB quite a bit for school and work. And I only have a TI-83+ Dr Brain - Tue Oct 16, 2007 9:50 pm Post subject:
The Heaviside step is defined as:
0 when t < 0
1 when 0 <= t
That's not periodic. Only periodic functions have Fourier series. Aperiodic functions (like the Heaviside step) have continuous Fourier transforms.
The function you truncated to show the Gibbs phenomenon is a periodic signal (one can instantly tell by the summation instead of integration). It *looks* like a square wave, but since it's been truncated to an approximation, I can't mathematically prove that.
Blocks - Tue Oct 16, 2007 10:31 pm Post subject:
Non-periodic functions can have Fourier series.
I can't prove the general case, but that is the Fourier series for the Heaviside function on the given interval. If you do the integrals, you'll come out with those coefficients.
The function shown is periodic with period 2pi, which means it sucks as an approximation to the Heaviside function outside of that interval, but it's still a partial Fourier series.
Dr Brain - Wed Oct 17, 2007 8:52 am Post subject:
How is the Heaviside function when made periodic NOT a square wave?
Wikipedia wrote:
The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions
