Peroid of a summation of sinusoids

[jegues]

Homework Statement

I have the following function,

$$f(t) = 2048 + 700cos(31.25*2\pi t) - 1100sin(125*2 \pi t)$$

and I need to find 48 data points spread evenly between one peroid of the waveform.

How would I go about doing this?

Homework Equations

The Attempt at a Solution

I know how to solve the individual peroid of each of the sinusoids,

$$T_{cos} = \frac{1}{f_{cos}} = \frac{1}{31.25}$$

$$T_{sin} = \frac{1}{f_{sin}} = \frac{1}{125}$$

But I can't figure the resulting peroid of two added together. Any ideas?

Also, I don't know how to collect 48 evenly spaced data points within the peroid once I find it. How would I go about achieveing such a task?

 

[LCKurtz]

Hint: Write your cosine period as a fraction with no decimals.

 

[jegues]

Hint: Write your cosine period as a fraction with no decimals.

$$T_{cos} = \frac{1}{f_{cos}} = \frac{4}{125}$$

I'm still a little confused how to find the resulting peroid though.

 

[LCKurtz]

$$T_{cos} = \frac{1}{f_{cos}} = \frac{4}{125}$$

I'm still a little confused how to find the resulting peroid though.

Think about the fact that one period is a multiple of the other.

 

[jegues]

Think about the fact that one period is a multiple of the other.

I need more help...

I still don't know where to start...

Can I just add them?

 

[LCKurtz]

I need more help...

I still don't know where to start...

Can I just add them?

If a function has period P can you tell whether or not 2P or 3P or 4P is also a period?

 

[jegues]

If a function has period P can you tell whether or not 2P or 3P or 4P is also a period?

I'm confused with the question.

A function would only have one peroid, wouldn't it?

2P would be twice the peroid, I could tell because it would be twice as long.

The waveform would stretched out further horizontally due to it having to take more time to elapse.

I can tell I'm missing the fundamental insight that should arise from this question, can you be more blunt?

 

[LCKurtz]

I'm confused with the question.

A function would only have one peroid, wouldn't it?

2P would be twice the peroid, I could tell because it would be twice as long.

The waveform would stretched out further horizontally due to it having to take more time to elapse.

I can tell I'm missing the fundamental insight that should arise from this question, can you be more blunt?

When we say "the" period of a periodic function is P, we usually mean that P is the smallest positive value such that f(x+P) = f(x) for all x. However it is correct to say any Q > 0 that has the property that f(x+Q) = f(x) is a period of the function also. It just may not be the smallest period. After all, if a function repeats every 2 units, wouldn't also repeat every 4 units? If f has period P, what do you get if you calculate $f(x+2P) = f((x+P)+P)=?$

 

[NascentOxygen]

2P would be twice the peroid, I could tell because it would be twice as long.

Neatly sketch a large sinusoid across a sheet of squared paper. Now, on the same axis, sketch another sinusoid, but draw this one of smaller amplitude and show it having exactly 4 cycles within the time of the first one having just one cycle.

Underneath these, sketch their sum.

Try some more sketches. Try it again, but this time make the first one of smaller amplitude and the second one the larger.

In each case, what is the period of the waveform that is their sum?