A Periodic Function Looks Like This Formula?

[yosimba2000]

My book says this:

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?

 

[anorlunda]

They are talking about periodic functions that are not necessarily sinusoidal. A square wave for example,

 

[Tom.G]

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 .

 

[PeroK]

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 $T = 2\pi / \omega$