Signals and Systems/Periodic Signals

Periodic Signals

Periodic Signals are signals that repeat themselves after a certain amount of time. More formally, a function f(t) is periodic if f(t + T) = f(t) for some T and all t. The classic example of a periodic function is sin(x) since sin(x + 2 π) = sin(x). However, we do not restrict attention to sinusoidal functions.

Terminology

We will discuss here some of the common terminology that pertains to a periodic function. Let g(t) be a periodic function satisfying g(t + T) = g(t) for all t.

Period

The period, also known as the wavelength is the smallest value of T satisfying g(t + T) = g(t) for all t. The period is defined so because if g(t + T) = g(t) for all t, it can be verified that g(t + T') = g(t) for all t where T' = 2T, 3T, 4T, ... In essence, it's the smallest amount of time it takes for the function to repeat itself. If the period of a function is finite, the function is called "periodic". Functions with an infinite period (they never repeat themselves) are known as "aperiodic functions."

The period of a periodic waveform will be denoted with a capital T. The period is measured in seconds.

Frequency

The frequency of a periodic function is the number of complete cycles that can occur per second. Frequency is denoted with a lower-case f. It is defined in terms of the period, as follows:

$f=\frac{1}{T}$

Frequency has units of hertz.

Radial Frequency

The radial frequency is the frequency in terms of radians. it is defined as follows:

$\omega =2\pi f$

Amplitude

The amplitude of a given wave is the value of the wave at that point. Amplitude is also known as the "Magnitude" of the wave at that particular point. There is no particular variable that is used with amplitude, although capital A, capital M and capital R are common.

The amplitude can be measured in different units, depending on the signal we are studying. In an electric signal the amplitude will typically be measured in volts. In a building or other such structure, the amplitude of a vibration could be measured in meters.

Continuous Signal

A continuous signal is a "smooth" signal, where the signal is defined over a certain range. For example, a sine function is a continuous sample, as is an exponential function or a constant function. A portion of a sine signal over a range of time 0 to 6 seconds is also continuous. Examples of functions that are not continuous would be any discrete signal, where the value of the signal is only defined at certain intervals.

DC Offset

A DC Offset is an amount by which the average value of the periodic function is not centered around the x-axis.

A periodic signal has a DC offset component if it is not centered about the x-axis. In general, the DC value is the amount that must be subtracted from the signal to center it on the x-axis. by definition:

$A_0=\underset{-T/2}{\overset{T/2}{\int }}f(x)dx$

With A₀ being the DC offset. If A₀ = 0, the function is centered and has no offset.

Half-wave Symmetry

Another property of Periodic Signals is the property of half-wave symmetry.

To determine if a signal has half-wave symmetry, we need to examine a single period of the signal. If each individual period is symmetrical about its center axis (that is, about the point T/2 from either end of the period), the wave is said to exhibit half-wave symmetry.

It is important to note that one must choose the proper period of the signal in order to characterize it properly. Even a half-wave symmetric signal can appear to not have half-wave symmetry when a certain, wrong section is chosen for evaluation.

Discontinuities

Discontinuities are an artifact of some signals that make them difficult to manipulate for a variety of reasons.

In a graphical sense, a periodic signal has discontinuities whenever there is a vertical line connecting two adjacent values of the signal. In a more mathematical sense, a periodic signal has discontinuities anywhere that the function has an undefined (or an infinite) derivative. These are also places where the function does not have a limit, because the values of the limit from both directions are not equal.

Common Periodic Signals

There are some common periodic signals that are given names of their own. We will list those signals here, and discuss them.

Sinusoid

The quintessential periodic waveform. These can be either Sine functions, or Cosine Functions.

Square Wave

The square wave is exactly what it sounds like: a series of rectangular pulses spaced equidistant from each other, each with the same amplitude.

Triangle Wave

The triangle wave is also exactly what it sounds like: a series of triangles. These triangles may touch each other, or there may be some space in between each wavelength.

Classifications

Periodic functions can be classified in a number of ways. one of the ways that they can be classified is according to their symmetry. A function may be Odd, Even, or Even & Odd. All periodic functions can be classified in this way.

Even

Functions are even if they are symetrical about the y-axis.

$f(x)=f(-x)$

For instance, a cosine function is an even function.

Odd

A function is odd if it is inversely symmetrical about the y-axis.

$f(x)=-f(-x)$

The Sine function is an odd function.

Neither Even nor Odd

Some functions are neither even nor odd. However, such functions can be written as a sum of even and odd functions. Any function f(x) can be expressed as a sum of an odd function and an even function:

$f(x)=1/2\{f(x)+f(-x)\}+1/2\{f(x)-f(-x)\}$

We leave it as an exercise to the reader to verify that the first component is even and that the second component is odd. Note that the first term is zero for odd functions and that the second term is zero for even functions.