2 Periodic functions

A function $f\left(t\right)$ is periodic if the function values repeat at regular intervals of the independent variable $t$ . The regular interval is referred to as the period . See Figure 1.

Figure 1

If $P$ denotes the period we have

$f\left(t+P\right)=f\left(t\right)$

for any value of $t$ . The most obvious examples of periodic functions are the trigonometric functions $sint$ and $cost$ , both of which have period $2\pi$ (using radian measure as we shall do throughout this Workbook) (Figure 2). This follows since

$sin\left(t+2\pi \right)=sint\text{ and}cos\left(t+2\pi \right)=cost$

Figure 2

The amplitude of these sinusoidal functions is the maximum displacement from $y=0$ and is clearly 1. (Note that we use the term sinusoidal to include cosine as well as sine functions.)

More generally we can consider a sinusoid

$y=Asinnt$

which has maximum value, or amplitude, $A$ and where $n$ is usually a positive integer.

For example

$y=sin2t$

is a sinusoid of amplitude 1 and period $\frac{2\pi }{2}=\pi$ (Figure 3). The fact that the period is $\pi$ follows because

$sin2\left(t+\pi \right)=sin\left(2t+2\pi \right)=sin2t$

for any value of $t$ .

Figure 3

We see that $y=sin2t$ has half the period of $sint$ , $\pi$ as opposed to $2\pi$ (Figure 4). This can alternatively be phrased by stating that $sin2t$ oscillates twice as rapidly (or has twice the frequency) of $sint$ .

Figure 4

In general $y=Asinnt$  has amplitude $A$ , period $\frac{2\pi }{n}$ and completes $n$ oscillations when $t$ changes by $2\pi$ . Formally, we define the frequency of a sinusoid as the reciprocal of the period:

$\text{ frequency}=\frac{1}{\text{ period}}$

and the angular frequency , often denoted the Greek Letter $\omega$ (omega) as

Thus $y=Asinnt$   has frequency $\frac{n}{2\pi }$ and angular frequency $n$ .

Task!

State the amplitude, period, frequency and angular frequency of

1. $y=5cos4t$
2. $y=6sin\frac{2t}{3}.$

Answer

Answer

2.1 Harmonics

In representing a non-sinusoidal function of period $2\pi$ by a Fourier series we shall see shortly that only certain sinusoids will be required:

1. $A_{1}cost$ (and $B_{1}sint$ )

   These also have period $2\pi$ and together are referred to as the first harmonic (or

   fundamental harmonic ).

2. $A_{2}cos2t$ (and $B_{2}sin2t$ )

   These have half the period, and double the frequency, of the first harmonic and are referred to as the second harmonic .

3. $A_{3}cos3t$ (and $B_{3}sin3t$ )

   These have period $\frac{2\pi }{3}$ and constitute the third harmonic .

In general the Fourier series of a function of period $2\pi$ will require harmonics of the type

$A_{n}cosnt\left(\text{ and}{B}_{n}sinnt\right)\text{ where}n=1,2,3,\dots$

2.2 Non-sinusoidal periodic functions

The following are examples of non-sinusoidal periodic functions (they are often called “waves”).

Square wave

Figure 5

Analytically we can describe this function as follows:

$f\left(t\right)=\left\{\begin{array}{ccc}\hfill -1& \hfill & \hfill -\pi <t<0\\ \hfill +1& \hfill & \hfill 0<t<\pi \end{array}\right.$ (which gives the definition over one period)

$f\left(t+2\pi \right)=f\left(t\right)$ (which tells us that the function has period $2\pi$ )

Saw-tooth wave

Figure 6

In this case we can describe the function as follows:

$f\left(t\right)=2t0<t<2f\left(t+2\right)=f\left(t\right)$

Here the period is 2, the frequency is $\frac{1}{2}$ and the angular frequency is $\frac{2\pi }{2}=\pi$ . Triangular wave

Figure 7

Here we can conveniently define the function using $-\pi <t<\pi$ as the “basic period”:

$f\left(t\right)=\left\{\begin{array}{ccc}\hfill -t& \hfill & \hfill -\pi <t<0\\ \hfill t& \hfill & \hfill 0<t<\pi \end{array}\right.$

or, more concisely,

$f\left(t\right)=\left|t\right|-\pi <t<\pi$

together with the usual statement on periodicity

$f\left(t+2\pi \right)=f\left(t\right).$

Task!

Write down an analytic definition for the following periodic function:

Answer

Sketch the graphs of the following periodic functions showing all relevant values:

1. $f\left(t\right)=\left\{\begin{array}{ccc}\hfill t^2∕2& \hfill & \hfill 0<t<4\\ \hfill 8& \hfill & \hfill 4<t<6\\ \hfill 0& \hfill & \hfill 6<t<8\end{array}\right.f\left(t+8\right)=f\left(t\right)$
2. $f\left(t\right)=2t-t^{2}0<t<2f\left(t+2\right)=f\left(t\right)$

Answer