Chap 36 Exercises

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Activities

Exercise 1 Here is a square-wave function:

sq_wave <- makeFun(ifelse(abs(t) <= 0.5, 1, 0) ~ t)
slice_plot(sq_wave(t) ~ t, domain(t=-1:1))

Find the projection of the square wave onto each of these functions. Use the domain $-1\le t\le 1$.

• $c_0\equiv 1$
• $c_1\equiv \cos (1\pi t)$
• $c_2\equiv \cos (2\pi t)$
• $c_3\equiv \cos (3\pi t)$
• $c_4\equiv \cos (4\pi t)$
• $c_5\equiv \cos (5\pi t)$
• $c_6\equiv \cos (6\pi t)$
• $c_7\equiv \cos (7\pi t)$

Hint: To find the scalar multiplier of projecting $g(t)$ onto $f(t)$, use $${\int }_{-1}^{1}g(t)f(t)dt/{\int }_{-1}^{1}f(t)f(t)dt$$ or, in R/mosaic

A <- Integrate(g(t)*f(t) ~ t, domain(t=-1:1)) / 
  Integrate(f(t)*f(t) ~ t, domain(t=-1:1))

Then the projection of $g()$ onto $f()$ is $Af(t)$.

Write down the scalar multiplier on each of the 8 functions above.

Answer

If you calculated things correctly, this is the linear combination of the 8 functions that best matches the square wave.

Loading required package: cubature

Exercise 2 Consider these functions/vectors on the domain $0\le t\le 1$:

• $s_1(t)\equiv \sin (2\pi t)$
• $s_2(t)\equiv \sin (2\pi 2t)$ (that is, $\omega =2$)
• $s_3(t)\equiv \sin (2\pi 3t)$ (that is, $\omega =3$)
• $c_0(t)\equiv \cos (\pi t)$
• $c_1(t)\equiv \cos (2\pi t)$
• $c_2(t)\equiv \cos (2\pi 2t)$

1. Plot out each of the functions on the domain. How many complete cycles does each function complete as $t$ goes from 0 to 1?

2. What is the length of each function?

3. All of the functions are mutually orthogonal except one. Which is the odd one out? (Hint: If the dot product is zero, the vectors are orthgonal.)

Exercise 3  

Suppose we are interested in a domain $0\le t\le 10$ and a set of sinusoid functions:

$$\begin{gathered} s_1(t)\equiv \sin (2\pi t/10) \\ {s}_2(t)\equiv \sin (4\pi t/10) \\ {s}_3(t)\equiv \sin (6\pi t/10) \\ {c}_1(t)\equiv \cos (2\pi t/10) \\ {c}_2(t)\equiv \cos (4\pi t/10) \\ {c}_3(t)\equiv \cos (6\pi t/10) \end{gathered}$$

Each of these functions goes through an integer number of cycles over $0\le t\le 10$, as you can confirm by graphing them, e.g.

s1 <- makeFun(sin(2*pi*t/10) ~ t)
s2 <- makeFun(sin(4*pi*t/10) ~ t)
# and so on
c2 <- makeFun(cos(4*pi*t/10) ~ t)
c3 <- makeFun(cos(6*pi*t/10) ~ t)
slice_plot(s1(t) ~ t, domain(t=0:10)) |>
  slice_plot(c2(t) ~ t, color="blue")

Using the definition of the dot product between two functions as $$f(t)\bullet g(t)\equiv {\int }_0^{10}f(t)g(t)dt,$$

1. Calculate the “length” of each of the functions $s_1(t),\dots ,c_3(t)$.
2. Calculate the angle between each pair of functions.

To simplify your calculations, you might want to make use of these “helper” functions:

fdot <- function(tilde1, tilde2, domain) {
  f <- makeFun(tilde1)
  g <- makeFun(tilde2)
  Integrate(f(t)*g(t) ~ t, domain)
}
fangle <- function(tilde1, tilde2, domain) {
  fdot(tilde1, tilde2, domain) /
    sqrt(fdot(tilde1, tilde1, domain) * 
         fdot(tilde2, tilde2, domain))
}

3. To judge from the set of angles you calculated, what about the functions $s_1(t),\dots ,c_3(t)$ would make it easy to project a function $h(t)$ onto the subspace spanned by the functions?

Exercise 4 (Still in draft. Figure out a nice activity based on this.)  

---

id: anchorage-tides

Anchorage, AK

Components: - M2 12.42 hours - S2 12 hours - N2 12.658 hours - K1 23.935 hours

hour <- with(Anchorage_tide, hour)
b    <- with(Anchorage_tide, level)
sin1 <- sin(2*pi*hour/12.42)
cos1 <- cos(2*pi*hour/12.42)
sin2 <- sin(2*pi*hour/23.935)
cos2 <- cos(2*pi*hour/23.935)
sin3 <- sin(2*pi*hour/12)
cos3 <- cos(2*pi*hour/12)
sin4 <- sin(2*pi*hour/12.658)
cos4 <- cos(2*pi*hour/12.658)
A <- cbind(1, sin1, cos1, sin2, cos2, sin3, cos3, sin4, cos4)
mod1 <- b %onto% cbind(1, sin1, cos1)
mod2 <- b %onto% cbind(1, sin1, cos1, sin2, cos2)
x <- qr.solve(A, b)
mod3 <- A %*% x
resid <- b - mod3
gf_line(level ~ hour, data = Anchorage_tide)  %>%
  gf_line(mod3 ~ hour, color="magenta") %>%
  gf_lims(x = c(0,1000))

Warning: Removed 75114 rows containing missing values (`geom_line()`).
Removed 75114 rows containing missing values (`geom_line()`).

gf_line(resid ~ hour)

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36-exercises.rmarkdown

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