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Periodic Signals

Module by: Christopher Schmitz. E-mail the author

Summary: This module presents just a few mathematical basics of periodic signals.

Definition 1: Periodic Signals
Signals that exactly repeat themselves after some amount of time.

Mathematically, this means that y(t+T) = y(t) for some value of T. The period of the signal is the minimum value of T for which this equation holds.

Definition 2: Period
The shortest segment of time that fully defines a periodic signal.

Commonly encountered periodic signals include the square wave. Observe the figure below. What is the period of this square wave? Be careful, you may be tempted to say something around 0.25 seconds, but this only defines the first (positive-going) half of the square wave and does not fully define the periodic signal. One needs to also include the second (negative-going) half in order to pass along everything one needs to reconstruct the periodic signal! The period is correctly given as 0.5 seconds.

Figure 1: A square wave example.
Square Wave
Square Wave (squarewaveEx1.png)

Other periodic signals include the sinusoid, the triangular wave, and the sawtooth wave.

Exercise 1

See if you can identify the period, T, of each of the following examples.

Solution

The triangle wave graph shows 2 and one-half cycles across the 1 second display and therefore has a period of 1/2.5 or 0.4 seconds. The sawtooth wave and the sinusoid wave each have a period of 0.2 seconds.

Figure 2: A triangle wave example.
Triangle Wave
Triangle Wave (trianglewaveEx1.png)
Figure 3: A sawtooth wave example.
Sawtooth Wave
Sawtooth Wave (sawtoothwaveEx1.png)
Figure 4: A sinusoidal wave example.
Sinusoidal Wave
Sinusoidal Wave (periodicEx1.png)

Composites (or summations) of periodic signals sometimes results in a new waveform that is also periodic.

Figure 5: A composite of two sinusoids.
Sinusoidal Wave Composite
Sinusoidal Wave Composite (periodicEx2.png)
Figure 6: A composite of many sinusoids.
Sinusoidal Wave Composite 2
Sinusoidal Wave Composite 2 (periodicEx3.png)

Exercise 2

Determine the period of the two sinusoidal-composite waveforms pictured above.

Solution

The composite of two sinusoids has a period of 0.2 seconds. The composite of many sinusoids looks a bit confusing, but we can simplify things by noting that it appears to complete about 4 cycles in 1 second, thus indicating a period of 1/4 or 0.25 seconds. This technique of averaging across many cycles is very useful in many applications such as determining heart rate from a noisy measurement!

The "fundamental frequency," fo, is the smallest frequency component that might exist in the periodic signal. If the signal's period is T, then the fundamental frequency is given by

fo = 1/T
(1)

Exercise 3

Identify the fundamental frequency for each of the examples given above.

Solution

Simply inverting the period gives the fundamental frequency, but you can also just count the fractional number of cycles completed in the 1 second interval. For the six figures above, they are 2 Hz, 2.5 Hz, 5 Hz, 5 Hz, 5 Hz and 4 Hz from top to bottom, respectively.

Although the periodic signal might not have any frequency content at fo, it is still a very important parameter as it specifies exactly which frequencies may be contained in that periodic signal. The fundamental frequency is also very closely related to the "pitch" of a sound.

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