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main.Rpres

@@ -51,7 +51,7 @@ $f:\mathbb{R}\rightarrow\mathbb{C}$
 $$
   \begin{aligned}
     f(x) &= \int_{-\infty}^\infty F(s) e^{2\pi i s x} ds \\
-    F(s) &= \int_{-\infty}^\infty f(x) e^{-2\pi i s x} ds \\
+    F(s) &= \int_{-\infty}^\infty f(x) e^{-2\pi i s x} dx \\
   \end{aligned}
 $$
 
@@ -327,17 +327,241 @@ plot(freqs / w1,Pv, type="b",
 par(mfrow = c(1,1))
 ```
 
+Example: Time-varying signal 2
+========================================================
+
+```{r, echo=FALSE}
+N <- 1024*256
+w1 <- 5
+w2 <- 0.7
+w3 <- 1
+a1 <- 15
+a2 <- 3
+a3 <- 1
+mu <- 5
+sigma <- 0.5
+noise <- 3
+
+x <- seq(from=0, to=10,length.out = N)
+
+y0 <- a1*sin(w1*x*2*pi)*dnorm(x, mu, sigma)
+y1 <- a2*sin(w2*x*2*pi) + a3*sin(w3*x*2*pi)
+y <- y0 + y1 + rnorm(n = N, mean = 0, sd = noise)
+par(mfrow = c(2,1))
+plot(x,y1,type="l", ylim = c(-10, 10),
+     xlab = "t",
+     ylab = "f(t)")
+lines(x,y0, col = "blue", lwd=2)
+plot(x,y,type="l",
+     xlab = "t",
+     ylab = "f(t)")
+par(mfrow = c(1,1))
+```
+
+***
+
+```{r, echo=FALSE}
+Dt <- (max(x) - min(x)) / N
+Dw <- 1 / (N * Dt)
+Y  <- fft(y)/N
+P  <- Mod(Y)^2
+freqs <- (seq(from=0, to=N-1) * Dw)
+plot(freqs / w1,P, type="b", 
+     xlim=c(0,1.2),
+     xlab = expression("w/w"[1]),
+     ylab = "P")
+```
+
+
+Short-time Fourier Transform (STFT)
+========================================================
+incremental: true
+
+Make Fourier Transform of *short* segments in the timeseries using a *window* function centered at $\tau$, $w(t-\tau)$
+
+$$\hat{F}(\omega, \tau) = 
+\int_{-\infty}^\infty f(t) w(t-\tau)  e^{-2\pi i \omega t} dt$$
+
+**Problem:** Choose an appropriate window width 
+
+Frequency vs. time resolution
+
+
+Wavelet transforms
+========================================================
+incremental: true
+
+**Idea:** 
+
+- Narrow windows for large frequencies (good time resolution)
+- Wide windows for small frequencies (good frequency resolution)
+
+Choose a *localized wave* (mother wavelet) $\psi(t)$ and define
+
+$$\psi_{a,b}(t)=\frac{1}{\sqrt{a}}\psi\left(\frac{t-b}{a}\right)$$
 
 Wavelet transforms
 ========================================================
+incremental: true
+
+Example: Mexican-hat wavelet
+
+$$\psi(t) = \left(1-t^2\right)e^{-t^2/2}$$
+$$\psi_{a,b}(t)=\frac{1}{\sqrt{a}}\psi\left(\frac{t-b}{a}\right)$$
+
+***
+
+```{r,echo=FALSE}
+mhw <- function(t, a, b) {
+  return((1/sqrt(a)) * (1-((t-b)/a)^2) * exp(-((t-b)/a)^2/2))
+}
+x <- seq(from=-5, to=5, length.out = 1000)
+y0 <- mhw(x,1.2,0)
+y1 <- mhw(x,0.2,0)
+plot(x,y1, type="l", xlab = "t", ylab = expression(psi))
+text(0.6,2, labels = "a=0.2")
+lines(x,y0, col="blue")
+text(1,0.8, labels = "a=1.2", col="blue")
+```
+
+
+Definition of wavelet transforms
+========================================================
+
+Given $f\in L^2(\mathbb{R})$, we define its *continuous wavelet transform* with respect to the wavelet $\psi$ as
+
+$$\mathcal{W}_{\psi}[f] (a,b) = \int_{-\infty}^\infty f(t)\overline{\psi_{a,b}(t)}dt$$
+
+For $\mathcal{W}_\psi[f]$ to be invertible we require 
+
+$$ 0 < C_{\psi}\equiv\int_{-\infty}^\infty
+\frac{\left|\hat{\psi}(\omega)\right|^2}{\left|\omega\right|} d\omega
+<\infty $$
+
+Example: Time-varying signal CWT
+========================================================
+
+```{r, echo=FALSE}
+N <- 1024*256
+w1 <- 5
+w2 <- 0.7
+w3 <- 1
+a1 <- 15
+a2 <- 3
+a3 <- 1
+mu <- 5
+sigma <- 0.5
+noise <- 3
+
+x <- seq(from=0, to=10,length.out = N)
+
+y0 <- a1*sin(w1*x*2*pi)*dnorm(x, mu, sigma)
+y1 <- a2*sin(w2*x*2*pi) + a3*sin(w3*x*2*pi)
+y <- y0 + y1 + rnorm(n = N, mean = 0, sd = noise)
+par(mfrow = c(2,1))
+plot(x,y,type="l",
+     xlab = "t",
+     ylab = "f(t)")
+Dt <- (max(x) - min(x)) / N
+Dw <- 1 / (N * Dt)
+Y  <- fft(y)/N
+P  <- Mod(Y)^2
+freqs <- (seq(from=0, to=N-1) * Dw)
+plot(freqs / w1,P, type="b", 
+     xlim=c(0,1.2),
+     xlab = expression("w/w"[1]),
+     ylab = "P")
+par(mfrow=c(1,1))
+```
+
+***
+
+Mexican hat CWT
+
+![](tvs1.png)
+
+Example: Time-varying signal CWT 2
+========================================================
+
+Haar CWT
+
+![](tvs2.png)
+
+***
+
+Morlet CWT
+
+![](tvs3.png)
+
+
+Example: Time-varying signal CWT 3
+========================================================
+
+Morlet
+
+![](tvs4.png)
+
+***
+
+Mexican-hat
+
+![](tvs5.png)
+
+
+Parseval's relation for wavelets
+========================================================
+
+$$
+\int_{-\infty}^\infty\int_{-\infty}^\infty \mathcal{W}_\psi f (a,b)\overline{\mathcal{W}_\psi g (a,b)} \frac{da db}{a^2} = C_{\psi}\langle f,g\rangle
+$$
+
+where
+
+$$
+C_\psi \equiv \int_{-\infty}^\infty \frac{|\hat{\psi}(\omega)|^2}{|\omega|}d\omega 
+$$
+
+
+Inverse of a wavelet transform
+========================================================
+
+$$f(t) = \frac{1}{C_\psi}\int_{-\infty}^\infty\int_{-\infty}^\infty \mathcal{W}_\psi[f](a,b) \psi_{a,b}(t) \frac{da\ db}{a^2}$$
+
+
+Discrete wavelet transform
+========================================================
+incremental: true
+
+Change the continuous version
+
+$$\psi_{a,b}(t) = \frac{1}{\sqrt{a}} \psi\left(\frac{t-b}{a}\right),\ \ a,b\in\mathbb{R},\ a\neq 0 $$
+
+to a discrete version
+
+$$\psi_{m,n}(t)=2^{-m/2} \psi \left( 2^{-m} t - n \right),\ \ \ n,m \in \mathbb{Z}$$
+
+When can we recover $f(t)$ from $\mathcal{W}_\psi[f](m,n)$ ?
+
+
+Discrete wavelet transform 2
+========================================================
+
+When $\psi_{m,n}$ is complete in and orthonormal in $L^2(\mathbb{R})$:
+
+$$f(t) = \sum_{m,n=-\infty}^\infty \langle f,\psi_{m,n}\rangle\ \psi_{m,n}(t)$$
+
+
+Multiresolution analysis (MRA)
+========================================================
+
+> MRA is really an effective mathematical framework for hierarchical decomposition of an image (or signal) into components of different scales (or frequencies).
 
-Used to make *time-frequency* analysis
 
 
 Orthogonality relations
 ========================================================
 
-$$\int_{-\infty}^\infty e^{2\pi i (x-x')}dx' = \delta(x) $$
+$$\int_{-\infty}^\infty e^{2\pi i (x-x')}dx' = \delta(x)$$
 
 $$\int_0^p e^{2\pi i x (k-l)/p}dx =
   \begin{cases}