Randall Low
This is a plot of the line spectra for the square wave function.
Graph of the square wave function (Fourier series approx.)
The Trigonometric Fourier Series Coefficients of x(t) are:
-----------------------------------------------------------------------
n c(n)
-20.0000 -0.0000 + 0.0000i
-19.0000 -0.0335 - 0.0000i
-18.0000 0.0000 - 0.0000i
-17.0000 0.0374 + 0.0000i
-16.0000 -0.0000 - 0.0000i
-15.0000 -0.0424 - 0.0000i
-14.0000 0.0000 + 0.0000i
-13.0000 0.0490 + 0.0000i
-12.0000 -0.0000 + 0.0000i
-11.0000 -0.0579 - 0.0000i
-10.0000 0.0000 - 0.0000i
-9.0000 0.0707 + 0.0000i
-8.0000 -0.0000 + 0.0000i
-7.0000 -0.0909 - 0.0000i
-6.0000 0.0000 + 0.0000i
-5.0000 0.1273 + 0.0000i
-4.0000 -0.0000 + 0.0000i
-3.0000 -0.2122 - 0.0000i
-2.0000 0.0000 - 0.0000i
-1.0000 0.6366 + 0.0000i
0 -0.0000
1.0000 0.6366 - 0.0000i
2.0000 0.0000 + 0.0000i
3.0000 -0.2122 + 0.0000i
4.0000 -0.0000 - 0.0000i
5.0000 0.1273 - 0.0000i
6.0000 0.0000 - 0.0000i
7.0000 -0.0909 + 0.0000i
8.0000 -0.0000 - 0.0000i
9.0000 0.0707 - 0.0000i
10.0000 0.0000 + 0.0000i
11.0000 -0.0579 + 0.0000i
12.0000 -0.0000 - 0.0000i
13.0000 0.0490 - 0.0000i
14.0000 0.0000 - 0.0000i
15.0000 -0.0424 + 0.0000i
16.0000 -0.0000 + 0.0000i
17.0000 0.0374 - 0.0000i
18.0000 0.0000 + 0.0000i
19.0000 -0.0335 + 0.0000i
20.0000 -0.0000 - 0.0000
Now for the less accurate Fourier Series graph and corresponding trigonometric coefficients:
a0=
-5.2947e-010
n a(n) b(n)
1.0000 1.2732 -0.0000
2.0000 0.0000 0.0000
3.0000 -0.4244 0.0000
4.0000 -0.0000 -0.0000
5.0000 0.2546 -0.0000
These are coefficients for triangle function.
The Trigonometric Fourier Series Coefficients of x(t) are:
-----------------------------------------------------------------------
n c(n)
-20.0000 0.0000 + 0.0000i
-19.0000 0.0000 + 0.0011i
-18.0000 0.0000 + 0.0000i
-17.0000 0 - 0.0014i
-16.0000 0.0000 + 0.0000i
-15.0000 0.0000 + 0.0018i
-14.0000 0.0000 + 0.0000i
-13.0000 0.0000 - 0.0024i
-12.0000 0.0000 - 0.0000i
-11.0000 -0.0000 + 0.0033i
-10.0000 0 - 0.0000i
-9.0000 -0.0000 - 0.0050i
-8.0000 0.0000 - 0.0000i
-7.0000 -0.0000 + 0.0083i
-6.0000 0 + 0.0000i
-5.0000 -0.0000 - 0.0162i
-4.0000 0.0000 + 0.0000i
-3.0000 -0.0000 + 0.0450i
-2.0000 0.0000 - 0.0000i
-1.0000 0 - 0.4053i
0 0
1.0000 0 + 0.4053i
2.0000 0.0000 + 0.0000i
3.0000 -0.0000 - 0.0450i
4.0000 0.0000 - 0.0000i
5.0000 -0.0000 + 0.0162i
6.0000 0 - 0.0000i
7.0000 -0.0000 - 0.0083i
8.0000 0.0000 + 0.0000i
9.0000 -0.0000 + 0.0050i
10.0000 0 + 0.0000i
11.0000 -0.0000 - 0.0033i
12.0000 0.0000 + 0.0000i
13.0000 0.0000 + 0.0024i
14.0000 0.0000 - 0.0000i
15.0000 0.0000 - 0.0018i
16.0000 0.0000 - 0.0000i
17.0000 0 + 0.0014i
18.0000 0.0000 - 0.0000i
19.0000 0.0000 - 0.0011i
20.0000 0.0000 - 0.0000i
Below is a graph of the line amplitude spectrum for the triangle function.
This is the graph of the triangle function’s Fourier Series approximations.
Here are the trigonometric approximations of Fourier Series approximations corresponding to the graph of x(t), the triangle function above.
a0=
0
n a(n) b(n)
1.0000 0 0.8106
2.0000 0.0000 -0.0000
3.0000 0 -0.0901
4.0000 0 0.0000
5.0000 -0.0000 0.0324