A Periodic Function Looks Like This Formula?

In summary, the conversation is discussing periodic functions and how they can take the form of a sinusoidal function with an offset. The equation mentioned, EQ 17.1, is shown to match the bolded formula by using a specific example and explaining the meaning of the variables involved. The use of phi is not necessary because the waveform is assumed to start at zero phase.
  • #1
yosimba2000
206
9
My book says this:

upload_2016-3-7_19-17-13.png


I don't understand how this works. I learned that the usual sunisoidal function looks like
sin(wt+phi), where w is frequency, t is time, and phi is some offset.

EQ 17.1 doesn't match the bolded formula above. How does this work?
 
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  • #2
They are talking about periodic functions that are not necessarily sinusoidal. A square wave for example,
 
  • #3
f(t) = f(t + nT)

Where 0 <= T < infinity the period of one cycle
0 <= t < T the time interval between samples within a cycle
n=0 1 2 3... which cycle you are evaluating

Looking at the RHS, the part f(t) would be one cycle of a waveform
and the full RHS, f(t + nT) is the waveform of the n th cycle.

All that is saying is that f(t) stays the same regardless of which cycle you look at, which is the definition of a periodic function.

phi isn't used here because the waveform is assumed to start at zero phase, and (t) can be any function of t , such as wt .
 
  • #4
yosimba2000 said:
My book says this:

View attachment 97012

I don't understand how this works. I learned that the usual sunisoidal function looks like
sin(wt+phi), where w is frequency, t is time, and phi is some offset.

EQ 17.1 doesn't match the bolded formula above. How does this work?
Yes, it does match.

Let ##f(t) = sin(wt + \phi)##

Then ##f(t) = f(t + nT)##, where ##T = 2\pi / \omega##
 

1. What is a periodic function?

A periodic function is a type of mathematical function where its values repeat in a predictable pattern over a specific interval. This interval is known as the period of the function.

2. How can I identify a periodic function?

A periodic function can be identified by observing its graph, which will have a repeating pattern over the interval of its period. Additionally, a periodic function will have a specific formula that represents its pattern of repetition.

3. What does the formula for a periodic function tell me?

The formula for a periodic function tells us the mathematical relationship between the input values (x) and the output values (y). It also indicates the period and amplitude of the function, which are important characteristics that define its behavior.

4. Can a periodic function look different from the typical sine or cosine wave?

Yes, a periodic function can take on many different forms and still be considered periodic. While the sine and cosine waves are the most common examples of periodic functions, other shapes such as square waves, triangle waves, and sawtooth waves can also be periodic.

5. Are there real-world applications of periodic functions?

Yes, periodic functions have a wide range of real-world applications in fields such as physics, engineering, and economics. They are used to model and analyze phenomena that exhibit repetitive patterns, such as sound waves, electric currents, and stock market trends.

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