Module 2: Prelimaries

• Note: \(2\pi\) is approximately \(6.29\). Why is it OK to define a value for \(\sin(17.5)\) or \(\sin(-156.32)\)? How are these values defined?
• Download PlotSin.java, Function.java, and SimplePlotPanel.java. Then compile and execute PlotSin.java to plot the function sin and cos in the range [0,20]. How many times does the function "repeat" in this range?

Next, let's understand the concept of frequency:

• In English: it refers to "how often"
        \(\rhd\) There is an implied notion of time.

• For a sinusoid like \(\sin(t)\), it means "how often the function repeats itself per unit time".

• To understand this better, let's see what a cycle means:
  • Since \(\sin(t + 2\pi) = \sin(t)\), the crest-and-trough shape repeats.

    

  • A single crest-and-trough is called a cycle.

• The frequency is the number of cycles per unit time.
        \(\rhd\) The unit of measurement is called a Hz (Hertz), the number of cycles per unit time.

• Let's examine the function \(\sin(2\pi\omega t)\).

  In-Class Exercise 13: Compile and execute PlotSin2.java to plot \(\sin(2\pi\omega t)\) in the range \([0,1]\) with \(\omega = 1, 2, 3\). What is the relationship between the three curves?

• Thus, \(\omega\) is the frequency: the number of cycles per second.

• Note: \(\omega\) can be a real number.
        \(\rhd\) You can have a frequency of 3.356 cycles per second.

• The function \(\sin(2\pi k\omega t)\), for integer \(k\), is called a harmonic of \(\sin(2\pi \omega t)\)
        \(\rhd\) \(k\) cycles of \(\sin(2\pi k\omega t)\) fit into one single cycle of \(\sin(2\pi \omega t)\)

• The inverse of frequency is called the period, the length of time for one complete cycle.

• You can shift a sin wave by adding a fixed constant to the argument: \(\sin(2\pi\omega t + \phi)\)

In-Class Exercise 14: Modify PlotSin3.java to plot \(\sin(2\pi t + \phi)\) in the range \([0,1]\). Plot four separate curves with \(\phi=0, \frac{\pi}{4}, \frac{\pi}{2}, \pi\). What can you say about \(\sin(2\pi t + \frac{\pi}{2})\)? Similarly, plot \(\cos(2\pi t + \frac{\pi}{2})\) and comment on the function.

Finally, the third parameter to a sinusoid is its amplitude:

• First, note that multiplying a sinusoid by a constant, e.g., \(\alpha \sin(t)\) does not change the period.

  In-Class Exercise 15: Why?

• The amplitude is the peak height of a sinusoid.
        \(\rhd\) Since the maximum value of \(\sin(t)\) is 1.0, the amplitude of \(\alpha \sin(t)\) is \(\alpha\).

• Thus, a complete \(\sin\) function is written as \(\alpha \sin(2\pi\omega t + \phi)\)
        \(\rhd\) with amplitude \(\alpha\), frequency \(\omega\), phase \(\phi\).

Trigonometric "polynomials"

• Just like \(x^n\) is a "basic" polynomial, one can consider \(\sin(n x)\) and \(\cos(nx)\) to be "basic" sin/cos building blocks.
        \(\rhd\) Sometimes called the trigonmetric polynomials

• It turns out that the trig polynomials are dense in the space of continuous functions
        \(\rhd\) This the celebrated Fourier theorem: $$ f(t) \;\; = \;\; a_0 + \sum_{k=1}^\infty a_k \sin(2\pi kt/T) + \sum_{k=1}^\infty b_k \cos(2\pi kt/T) $$ for any continuous function \(f(t)\) over an interval \([0,T]\).

• Thus, a linear combination of the basic trig's can be used to approximate a function as closely as desired.

• As it turns out, the calculation of the constants \(a_i, b_i\) arises from linear algebra.

	int n = 5;                      // Array size.
	int[] A = new int [n];          // Create the space.
	for (int k=0; k<n; k++) {
	    A[k] = 2*k + 1;             // Populate the array.
	}
  

	int[] B = new int [n];
	B[0] = A[0];
	for (int k=1; k<n; k++) {
	    B[k] = B[k-1] + A[k];
	}
  

	double[][] A = new double [4][6];
	A[1][4] = 0.5;
	double[][] B = A;
	B[1][4] = 0;
	System.out.println (A[1][4]);       // What gets printed out?