Assignment 3
mod
. Look up the Matlab
help entry on mod
. Execute the following code:
x=linspace(-4*pi,4*pi,1000);The point is that the functions
y1=mod(x,2*pi); %x mod 2*pi; values, [0,2*pi]; breaks, 2*n*pi.
y2=mod(x+pi,2*pi); %Graph of y1 shifted pi units to left.
y3=mod(x+pi,2*pi)-pi; %x mod 2*pi; values, [-pi,pi]; breaks, (2*n+1)*pi.
subplot(3,1,1), plot(x,y1,'b'), axis([-4*pi 4*pi -pi 2*pi])
subplot(3,1,2), plot(x,y2,'r',x,y1,':b'), axis([-4*pi 4*pi -pi 2*pi])
subplot(3,1,3), plot(x,y3,'k',x,y1,':b'), axis([-4*pi 4*pi -pi 2*pi])
mod(x,2*pi)
and mod(x+pi,2*pi)-pi
return the angles corresponding to x with values in [0,2π] and [-π,π], respectively. In the following plots, take x to be the same as above. By the way, don't worry about the vertical line joining discontinuous pieces. We'll get rid of it later. Be sure to put titles on all of the plots.
abs(x)
).
sign
.)
f*g(x) = (1/(2π)∫-ππf(t)g(x-t)dt
Show that if the nth complex Fourier coeffficents for f and g are αn and βn, respectively, then the nth complex Fourier coefficent for f*g is the product αnβn. Equivalently, this means that f*g has the Fourier series
f*g(x) ~ ∑n αnβn einx,
where the sum is over all integers. (Hint: replace g(x-t) by its Fourier series, then interchange the integral and the infinite sum. There are other ways to do it, too.)