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--- math121a-f23:october_13_friday [2023/10/14 07:28]
pzhou [FT Conventions]
+++ math121a-f23:october_13_friday [2026/02/21 14:41] (current)
@@ Line 8: / Line 8: @@
 $$ F(p) = (1/2\pi) \int_\R f(x) e^{-ipx} dx. $$
-Discrete Fourier transformation (OK, I switched to Boas convention)
+Discrete Fourier transformation
-Fix a positive integer $N$. $x,p$ are valued in $\Z / N\Z \cong \{0,1,\cdots, N-1\}$.
-$$ f(x) = \sum_{p \in \Z / N\Z} F(p) F(p) e^{2\pi i \cdot px/N}. $$
+Fix a positive integer $N$. $x,p$ are valued in the 'discretized circle'
-$$ F(p) = (1/N) \sum_{p \in \Z / N\Z} f(x)  e^{-2\pi i \cdot px/N}. $$
+$$ \Z / N\Z \cong \{0,1,\cdots, N-1\}.$$
+$$ f(x) = \sum_{p \in \Z / N\Z} F(p) e^{2\pi i \cdot px/N}. $$
+$$ F(p) = (1/N) \sum_{x \in \Z / N\Z} f(x)  e^{-2\pi i \cdot px/N}. $$
+==== Norm in the Continous Fourier transformation ====
+Let $f(x)$ be a complex valued function on $x \in \R$, we define
+$$ \| f\|_x^2 := (1/2\pi) \int_\R |f(x)|^2 dx $$
+Let $F(p)$ be a complex valued function on $p \in \R$, we define
+$$ \| F\|_p^2 := \int_\R |F(p)|^2 dp $$
+==== Norm in the Discrete Fourier transformation ====
+$$ \| f\|_x^2 := (1/N) \sum_{x=0}^{N-1} |f(x)|^2  $$
+Let $F(p)$ be a complex valued function on $p \in \R$, we define
+$$ \| F\|_p^2 := \sum_{p=0}^{N-1} |F(p)|^2  $$
+==== Parseval Equality ====
+If $F(p)$ is the Fourier transformation of $f(x)$, then $\|F\|^2_p = \|f\|^2_x. $
+We proved in class the discrete case. The continuous case is similar in spirit, but harder to prove.
+===== Convolution =====
+Consider two people, call them Alice and Bob, they each say an integer number, call it a and b. Suppose $a$ and $b$ both have equal probability of taking value within $\{1,2,\cdots, 6\}$, we can ask what is the probabity distribution of $a+b$?
+We know $P(a=i) = 1/6$, $P(b=i) = 1/6$ for any $i=1,\cdots, 6$, otherwise the probabilit is 0.  Then
+$$ P(a+b = k) = \sum_{i+j=k} P(a=i) P(b=j). $$
+This is an instance of convolution.
+==== convlution in $x$ space ====
+Convolution is usually denoted as $\star$.
+If $f$ and $g$ are functions on the $x$ space, then we define
+$$ (f \star g)(x) =  \int_{x_1} f(x_1) g(x-x_1) dx_1 $$
+If $F$ and $G$ are functions on the $p$ space, then we define
+$$ (F \star G)(p) =  \int_{p_1} F(p_1) G(p-p_1) dp_1 $$
+Fourier transformation sends convolution of functions on one side to simply multiplication on the other side.
+$$ (1/2\pi) FT(f \star g) = F \cdot G. $$
+$$ FT(f \cdot g) = F \star G. $$

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