# Signals and systems, determining if signals are periodic.

#### chickwolf

Joined Mar 9, 2014
3
Greetings everyone, studying some signals and systems at the moment and getting stumped by a part in one of the questions I was hoping you would be able to help me with. For the following question I am trying to determine whether or not it is periodic, and if it is, what its fundamental period is.

I wasn't sure how to do use the AAC formula editor tool so have quickly knocked it up and attached it, apologies for the inconvenience.

I have determined the fundamental period but am unsure how to progress to show if the signal if periodic or not mathematically. If anyone has any pointers as to what the next step may be I would be very grateful.

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#### Papabravo

Joined Feb 24, 2006
12,767
The key to showing periodicity for either a continuous or discrete function is to demonstrate that the value of the function is the same for any constant offset equal to any integer multiple of the period. For example you can use the following identity to prove that a continuous sinewave is periodic,

sin(α + β) = sin(α)cos(β) + cos(α)sin(β)

$$sin(x + 2n\pi) = sin(x)cos(2n\pi)+cos(x)sin(2n\pi) = sin(x)\cdot (1)+cos(x)\cdot (0)=sin(x)$$

#### chickwolf

Joined Mar 9, 2014
3
The key to showing periodicity for either a continuous or discrete function is to demonstrate that the value of the function is the same for any constant offset equal to any integer multiple of the period. For example you can use the following identity to prove that a continuous sinewave is periodic,

sin(α + β) = sin(α)cos(β) + cos(α)sin(β)

$$sin(x + 2n\pi) = sin(x)cos(2n\pi)+cos(x)sin(2n\pi) = sin(x)\cdot (1)+cos(x)\cdot (0)=sin(x)$$
So in terms of my questions, α = (2π/3)t and β = 3? Or have I got that completely wrong?

#### Papabravo

Joined Feb 24, 2006
12,767
Close. The argument to the sin function

$$\text is \math\frac{2\pi}{3}\cdot(t+3)$$

so

$$\alpha=\frac{2\pi}{3}t,\;\text and\math\;\beta=\left(\frac{2\pi}{3}\cdot3\right)=2\pi$$

Last edited:

#### chickwolf

Joined Mar 9, 2014
3
Awesome! Thank you very much, put that all in and got the starting expression back proving that it is indeed, periodic.

#### Papabravo

Joined Feb 24, 2006
12,767

#### dannyf

Joined Sep 13, 2015
2,197
It is a sine function. How can it not be periodic?

#### WBahn

Joined Mar 31, 2012
25,062
It is a sine function. How can it not be periodic?
Because the sampling may make it a periodic.

The same basic rule for periodicity applies to discrete-time systems:

x[n + aP] = x[n]

where P is the integer, in samples, and a is an integer.

If the signal period is not rational, then no integer multiple of the sampling period will ever coincide with an integer multiple of the signal period and the sample stream will be aperiodic (remember, the samples are all that exist, the system neither knows nor cares what is going on between samples).