2 Periodic functions

A function f ( t ) 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

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If P denotes the period we have

f ( t + P ) = f ( t )

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

sin ( t + 2 π ) = sin t  and cos ( t + 2 π ) = cos t

Figure 2

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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 = A sin n t

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

For example

y = sin 2 t

is a sinusoid of amplitude 1 and period 2 π 2 = π (Figure 3). The fact that the period is π follows because

sin 2 ( t + π ) = sin ( 2 t + 2 π ) = sin 2 t

for any value of t .

Figure 3

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

Figure 4

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In general y = A sin n t  has amplitude A , period 2 π n and completes n oscillations when t changes by 2 π . Formally, we define the frequency of a sinusoid as the reciprocal of the period:

 frequency = 1  period

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

 angular frequency = 2 π ×  frequency = 2 π  period

Thus y = A sin n t   has frequency n 2 π and angular frequency n .

Task!

State the amplitude, period, frequency and angular frequency of

  1. y = 5 cos 4 t
  2. y = 6 sin 2 t 3 .

amplitude 5, period 2 π 4 = π 2 , frequency 2 π , angular frequency 4

amplitude 6, period 3 π , frequency 1 3 π , angular frequency 2 3

2.1 Harmonics

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

  1. A 1 cos t (and B 1 sin t )

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

    fundamental harmonic ).

  2. A 2 cos 2 t (and B 2 sin 2 t )

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

  3. A 3 cos 3 t (and B 3 sin 3 t )

    These have period 2 π 3 and constitute the third harmonic .

In general the Fourier series of a function of period 2 π will require harmonics of the type

A n cos n t (  and B n sin n t )  where n = 1 , 2 , 3 ,

2.2 Non-sinusoidal periodic functions

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

Square wave

Figure 5

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Analytically we can describe this function as follows:

f ( t ) = 1 π < t < 0 + 1 0 < t < π (which gives the definition over one period)

f ( t + 2 π ) = f ( t ) (which tells us that the function has period 2 π )

Saw-tooth wave

Figure 6

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In this case we can describe the function as follows:

f ( t ) = 2 t 0 < t < 2 f ( t + 2 ) = f ( t )

Here the period is 2, the frequency is 1 2 and the angular frequency is 2 π 2 = π . Triangular wave

Figure 7

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Here we can conveniently define the function using π < t < π as the “basic period”:

f ( t ) = t π < t < 0 t 0 < t < π

or, more concisely,

f ( t ) = t π < t < π

together with the usual statement on periodicity

f ( t + 2 π ) = f ( t ) .

Task!

Write down an analytic definition for the following periodic function:

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f ( t ) = 2 t 0 < t < 3 1 3 < t < 5 f ( t + 5 ) = f ( t )
Task!

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

  1. f ( t ) = t 2 2 0 < t < 4 8 4 < t < 6 0 6 < t < 8 f ( t + 8 ) = f ( t )
  2. f ( t ) = 2 t t 2 0 < t < 2 f ( t + 2 ) = f ( t )

Figure 9

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