Signals and Systems/Periodic Signals

Periodic Signals
A signal is a periodic signal if it completes a pattern within a measurable time frame, called a period and repeats that pattern over identical subsequent periods. The completion of a full pattern is called a cycle. A period is defined as the amount of time (expressed in seconds) required to complete one full cycle. The duration of a period represented by T, may be different for each signal but it is constant for any given periodic signal.

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 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 that never repeat themselves have an infinite period, and 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 or cycle per second.

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 signal, 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:


 * $$A_0 = (1/T)*\int_{-T/2}^{T/2} f(x)dx$$

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

Half-wave Symmetry
To determine if a signal with period 2L has half-wave symmetry, we need to examine a single period of the signal. If, when shifted by half the period, the signal is found to be the negative of the original signal, then the signal has half-wave symmetry. That is, the following property is satisfied:


 * $$f (t-L) = - f (t)\,$$



Half-wave symmetry implies that the second half of the wave is exactly opposite to the first half. A function with half-wave symmetry does not have to be even or odd, as this property requires only that the shifted signal is opposite, and this can occur for any temporal offset. However, it does require that the DC offset is zero, as one half must exactly cancel out the other. If the whole signal has a DC offset, this cannot occur, as when one half is added to the other, the offsets will add, not cancel.

Note that if a signal is symmetric about the half-period point, it is not necessarily half-wave symmetric. An example of this is the function t3, periodic on [-1,1), which has no DC offset and odd symmetry about t=0. However, when shifted by 1, the signal is not opposite to the original signal.

Half Wave Symmetric signals don't have even "sine and cosine" harmonics.

Quarter-Wave Symmetry
If a signal has the following properties, it is said to quarter-wave symmetric:
 * It is half-wave symmetric.
 * It has symmetry (odd or even) about the quarter-period point (i.e. at a distance of L/2 from an end or the centre).

Any quarter-wave symmetric signal can be made even or odd by shifting it up or down the time axis. A signal does not have to be odd or even to be quarter-wave symmetric, but in order to find the quarter-period point, the signal will need to be shifted up or down to make it so. Below is an example of a quarter-wave symmetric signal (red) that does not show this property without first being shifted along the time axis (green, dashed):

An equivalent operation is shifting the interval the function is defined in. This may be easier to reconcile with the formulae for Fourier series. In this case, the function would be redefined to be periodic on (-L+&Delta;,L+&Delta;), where &Delta; is the shift distance.

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.

Sinusoidal wave
The quintessential periodic waveform. These can be either Sine functions, or Cosine Function.

$$ V_{p} \, \sin ( 2 \pi f \, t ) $$

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 Neither Even nor Odd. All periodic functions can be classified in this way.

Even
Functions are even if they are symmetrical 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) = f_{even}(x) + f_{odd}(x) $  (Note that the first term is zero for odd functions and that the second term is zero for even functions.)

Using the above mentioned equations for even and odd signals in $$ f(-x) $$ we get: $$ f(-x) = f_{even}(-x) + f_{odd}(-x) = f_{even}(x) - f_{odd}(x) $$


 * Hence, $$ f_{even}(x)= \frac{f(x)+f(-x)}{2}

$$   and    $$ f_{odd}(x)= \frac{f(x)-f(-x)}{2} $$