Differences

This shows you the differences between two versions of the page.

--- math121a-f23:october_20_friday [2023/10/20 06:10]
pzhou created
+++ math121a-f23:october_20_friday [2026/02/21 14:41] (current)
@@ Line 44: / Line 44: @@
 Hint: Is there a 'planewave' that solves the equation, i.e. somethign like $e^{ct}$ for some $c$?
-==== Ex 4 ====
+==== Ex 4: Discrete Fourier transformation ====
-You received a periodic sequence of numbers, of period 10: a period is of the following
+You received the following a periodic sequence of numbers,
+  * sequence of period 9: 0 0 1 0 0 2 0 0 3
+Use discrete Fourier transformation to analyse it.
+  * You can use this input number to as values of $f(x)$ for $x=0,1, \cdots, 8$.
+  * you know $f(x)$ can be written as
+$$ f(x) = \sum_{p=0}^8 F(p) e^{2\pi i (xp/9)} $$
+Can you find $F(p)$?
+  * These $f(x)$ are all real valued, what does that say about $F(p)$? (try to take complex conjugate of the equation)
+  * Can you guess, which $|F(p)|$s are largest?
+If you have done the above exercise, try this one
+  * sequence of period 10: 2 5 3 8 2 7 1 9 0 4
-5 3 8 2 7 1 9 0 4
-What can you say about it?
-  * the average? the standard deviation?
-  * I have a high-low-high-low pattern in it, can you use Fourier decomposition to detect that pattern? Try to write a computer program to see if that works.

Lecture Notes