Finish Chapter 10

ch10.r

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+#############
+# Chapter 10.2 Mean and correlation functions
+
+## Example 10.5
+
+# R code for Example 10.5
+x = matrix(0, 1000, 20)
+t = seq(-2, 2, (2+2)/999)
+
+for (i in 1:20) {
+    x[,i] = runif(1) * cos(2 * pi * t)
+}
+
+matplot(t, x, lwd=2, col="gray")
+grid()
+lines(t, 0.5*cos(2*pi*t), lwd=4, col="darkred")
+
+## Example 10.6
+
+# R code for Example 10.6
+x = matrix(0, 1000, 20)
+t = seq(-2, 2, (2+2)/999)
+
+for (i in 1:20) {
+    x[,i] = cos(2*pi*t+2*pi*runif(1))
+}
+
+matplot(t, x, lwd=2, col="gray")
+grid()
+lines(t, 0*cos(2*pi*t), lwd=4, col="darkred")
+
+## Example 10.8
+
+# R code for Example 10.7
+x = matrix(0, 21, 20)
+t = 0:20
+
+for (i in 1:20) {
+    x[,i] = runif(1) ^ t
+}
+
+stem <- function(x,y,pch=1,...){
+    if (missing(y)){
+        y = x
+        x = 1:length(x) 
+    }
+    for (i in 1:length(x)){
+        lines(c(x[i],x[i]), c(0,y[i]), ...)
+    }
+    lines(c(x[1]-2,x[length(x)]+2), c(0,0), ...)
+}
+
+matplot(t, x, pch=1, col="gray", lwd=2)
+grid()
+
+for (i in 1:ncol(x)) {
+    stem(t, x[,i], col="gray", lwd=2)
+}
+stem(t, 1/(t+1), col="darkred", lwd=2, pch=1, type="o")
+
+## Example 10.11
+
+# R code for Example 10.11: Plot the time function
+t = seq(-2, 2, len = 1000)
+x = matrix(0, 1000, 20)
+
+for (i in 1:20) {
+    x[,i] = runif(1) * cos(2*pi*t)
+}
+
+matplot(t, x, lwd=2, col="gray", type="l", lty="solid", xaxp=c(-2,2, 8))
+grid()
+lines(t, 0.5*cos(2*pi*t), lwd=5, col="darkred")
+points(numeric(20), x[501,], lwd=2, col="darkorange")
+points(numeric(20) + 0.5, x[626,], lwd=2, col="blue", pch=4)
+
+# R code for Example 10.11: Plot the autocorrelation function
+t = seq(-1, 1, len=1000)
+R = (1/3)*outer(cos(2*pi*t), cos(2*pi*t))
+image(t, t, R, col=topo.colors(255), xlab="t_1", ylab="t_2")
+
+## Example 10.12
+
+# R code for Example 10.11: Plot the time function
+t = seq(-2, 2, len = 1000)
+x = matrix(0, 1000, 20)
+
+for (i in 1:20) {
+    x[,i] = cos((2*pi*t) + (2*pi*runif(1)))
+}
+
+matplot(t, x, lwd=2, col="gray", type="l", lty="solid", xaxp=c(-2,2, 8))
+grid()
+lines(t, 0*cos(2*pi*t), lwd=5, col="darkred")
+points(numeric(20), x[501,], lwd=2, col="darkorange")
+points(numeric(20) + 0.5, x[626,], lwd=2, col="blue", pch=4)
+
+# R code for Example 10.12: Plot the autocorrelation function
+t = seq(-1, 1, len=1000)
+R = toeplitz(0.5*(cos(2*pi*t)))
+image(t, t, R, col=topo.colors(255), ylim=c(1,-1), xlab="t_1", ylab="t_2")
+
+#############
+# Chapter 10.3 Wide sense stationary processes
+
+## Example 10.14
+
+# R code to demonstrate autocorrelation
+
+# Figure 1
+Xa = rnorm(1000)
+Xa2 = rnorm(1000)
+
+plot(Xa, type="l", lwd=2, col="blue")
+grid()
+lines(Xa2, lwd=2)
+
+# Figure 2
+N = 1000
+T = 1000
+X = matrix(rnorm(N*T), N, T)
+xc = matrix(0, N, 2*T-1)
+
+for (i in 1:N) {
+    xc[i,] = ccf(X[i,], X[i,], lag.max=2*T-1, pl=FALSE)$acf/T
+}
+
+plot(xc[1,], type="l", lwd=2, col="darkblue")
+lines(xc[2,], lwd=2)
+
+#############
+# Chapter 10.5 Wide sense stationary processes
+
+## Example 10.15
+
+# R code for Example 10.15
+t = seq(-10, 10, by=0.001)
+L = length(t)
+X = rnorm(L)
+h = 10 * sapply((1 - abs(t)), max, 0) / 1000
+Y = convolve(X, h, type=c("circular"))
+
+# Figure 1
+plot(t, X, lwd=1, col="gray", type="l")
+grid()
+legend(6, 3.8, legend=c("X(t)", "μ_x(t)", "Y(t)", "μ_y(t)"), col=c("gray", "black", "darkorange", "yellow"), lty=c(1, 1, 1, 3), lwd=c(1, 4, 3, 4), bg="white")
+abline(h=0, lwd=4, lty=1, col="yellow")
+lines(t, Y, lwd=3, col="darkorange")
+abline(h=0, lwd=4, lty=3)
+
+# Figure 2
+h2 = convolve(h, h, type="open")
+Rx = numeric(40001)
+Rx[20001] = 0.2
+plot(seq(-20, 20, by=0.001), Rx, lwd=2, col="gray", type="l", xlim=c(-2,2), ylim=c(-0.05, 0.2))
+grid()
+legend(-2, 0.19, legend=c("R_x(t)", "R_y(t)"), col=c("gray", "darkorange"), lty=1:1, lwd=2)
+lines(seq(-20, 20, by=0.001), h2, lwd=2, col="darkorange")
+
+#############
+# Chapter 10.6 Optimal linear filter
+
+## Solve the Yule Walker equation
+
+# R code to solve the Yule Walker Equation
+
+y = scan("./ch10_LPC_data.txt")
+K = 10
+N = 320
+y_corr = ccf(y, y, lag.max=2*length(y)-1, pl=FALSE)$acf
+R = toeplitz(y_corr[N + 1:K-1])
+lhs = y_corr[N + 1:K]
+h = solve(R, lhs)
+
+# Figure 1
+plot(y, lwd = 4, col="blue", type="l")
+grid()
+legend(10, 0.1, legend=c("Y[n]"), col=c("blue"), lty=1:1, lwd=4)
+
+# Figure 2
+plot(y_corr, lwd = 4, type="l")
+grid()
+legend(10, 0.9, legend=c("R_y[k]"), col=c("black"), lty=1:1, lwd=4)
+
+## Predict sample
+
+# R code to predict the samples
+y = scan("./ch10_LPC_data_02.txt")
+K = 10
+N = length(y)
+
+y_corr = ccf(y, y, lag.max=2*N-1)$acf[,,1]
+R = toeplitz(y_corr[N + 1:K-1])
+lhs = y_corr[N + 1:K]
+h = solve(R, lhs)
+
+z = y[311:320]
+yhat = numeric(340)
+yhat[1:320] = y
+
+for (t in 1:20) {
+    predict = t(z) * h
+    z = c(z[2:10], predict)
+    yhat[320+t] = predict
+}
+
+plot(yhat, lwd=3, col="red", type="l")
+grid()
+legend(10, 0.9, legend=c("Prediction", "Input"), col=c("red", "gray"), lty=c(1, 2), lwd=2)
+lines(y, lwd=4, col="gray", lty=3)
+
+## Wiener filter 
+
+# R code for Wiener filtering
+library(pracma)
+y = scan("./ch10_LPC_data_02.txt")
+w = 0.05 * rnorm(320)
+x = y + w
+
+Ry = ccf(y, y, lag.max=2*length(y)-1, pl=FALSE)$acf
+Rw = ccf(w, w, lag.max=2*length(w)-1, pl=FALSE)$acf
+Sy = fft(Ry)
+Sw = fft(Rw)
+H = Sy / (Sy + Sw)
+
+a = x[1:639]
+a[is.na(a)] = 0
+Yhat = H * fft(a[1:639])
+yhat = Re(ifft(as.vector(Yhat)))
+
+# Figure 1
+plot(x, lwd=5, col="gray", type="l")
+grid()
+legend(10, -0.7, legend=c("Noisy Input X[n]", "Wiener Filtered Yhat[n]", "Ground Truth Y[n]"), col=c("gray", "red", "black"), lty=c(1, 1, 2), lwd=2)
+lines(yhat[1:320], lwd=2, col="red")
+lines(y, lwd=2, lty=3)
+
+# Figure 2
+plot(Rw, lwd=4, col="blue", label="h[n]", type="l")
+legend(500, 0.85, legend=c("h[n]"), col=c("blue"), lty=c(1), lwd=c(4))
+grid()
+
+## Wiener deblurring
+
+# R code to solve the Wiener deconvolution problem
+library(pracma)
+
+conv_same = function(x, y) {
+    s = length(y) / 2
+    e = length(x) + s - 1
+    return (convolve(x, y, type="open")[s:e])
+}
+
+y = scan("./ch10_wiener_deblur_data.txt")
+g = ones(32, 1)/32
+w = 0.02 * rnorm(320)
+s = length(y)/2
+e = length(g) + s - 1
+x = conv_same(y, g) + w
+
+Ry = ccf(y, y, lag.max=2*length(y)-1, pl=FALSE)$acf
+Rw = ccf(w, w, lag.max=2*length(w)-1, pl=FALSE)$acf
+Sy= fft(Ry)
+Sw = fft(Rw)
+a = g[1:639]
+a[is.na(a)] = 0
+G = fft(a[1:639])
+
+H = (Conj(G) * Sy) / (abs(G) ^ 2 * Sy + Sw)
+b = x[1:639]
+b[is.na(b)] = 0
+Yhat = H * fft(b[1:639])
+yhat = Re(ifft(as.vector(Yhat)))
+
+plot(x, lwd=4, col="gray", type="l", ylim=c(-0.7, 0.7))
+grid()
+legend(150, -0.45, legend=c("Noisy Input X[n]", "Wiener Filtered Yhat[n]", "Ground Truth Y[n]"), col=c("gray", "red", "black"), lty=c(1, 1, 2), lwd=2)
+lines(16:(320+15), yhat[1:320], col="red", lwd=2)
+lines(1:320, y, lwd=2, lty=3)
+
+#############
+