Period of added/multiplied sines/cosines

[mklein]

Dear all

I am looking to write my own computer program for a Fourier transform. However my maths is very rusty and I will need to recap some things.

I am struggling a little bit to recall the rules for the period of sines (or cosines) when they are MULTIPLIED. I haven't had much luck Googling this.

Am I right in thinking that for two waves ADDED their common period is their two periods multiplied (the lowest common multiple) ?

e.g.

y=cos(2.pi.2x) + cos(2.pi.3x) Fundamental period = 1/2 x 1/3 = 1/6?

But what if the two sines (or cosines) are multiplied?

e.g.

y=cos(2.pi.2x).cos(2.pi.3x)

Can anybody explain the rules for the addition and multiplication cases, without going into heavy maths?

Thank you

Matt

 

[micromass]

Hi mklein!

Dear all

I am looking to write my own computer program for a Fourier transform. However my maths is very rusty and I will need to recap some things.

I am struggling a little bit to recall the rules for the period of sines (or cosines) when they are MULTIPLIED. I haven't had much luck Googling this.

Am I right in thinking that for two waves ADDED their common period is their two periods multiplied (the lowest common multiple) ?

e.g.

y=cos(2.pi.2x) + cos(2.pi.3x) Fundamental period = 1/2 x 1/3 = 1/6?

But what if the two sines (or cosines) are multiplied?

e.g.

y=cos(2.pi.2x).cos(2.pi.3x)

Can anybody explain the rules for the addition and multiplication cases, without going into heavy maths?

Thank you

Matt

You can always apply the product-to-sum formula's http://www.sosmath.com/trig/prodform/prodform.html This reduces the case of products to the case of sums!

 

[pmsrw3]

The easy way to deal with this is not to use sines and cosines, but instead to use complex exponentials. I don't know if that comes under the category of heavy maths in your book, but regardless I would really recommend it as worth your time to learn. It'll make the subsequent math much simpler.

Using complex exponentials, the elements of the Fourier transform are not sines and cosines, but exponential functions of the form $e^{i \omega x}$; with $i=\sqrt{-1}$. These are easy to multiply: $e^{i \omega x}e^{i \phi x}=e^{i \left(\omega + \phi \right)x}$.

For addition, the rule for exponentials is the same as for sines as cosines. $e^{2 \pi i m x}+e^{2 \pi i n x}$ has period 1/GCM(n,m), GCM being the greatest common multiple. In your example, the period of $e^{2 \pi i 2 x}+e^{2 \pi i 3 x}$ would be 1.

 

[mklein]

Dear pmsrw3 and micromass

These are two very good suggestions, and I shall look into them, thank you. I can just about handle this level of maths!

Thanks

Matt

 

[CellCoree]

Hello Matt,

Thank you for reaching out and sharing your question. I understand that it can be challenging to remember all the mathematical rules and concepts, especially when they are not used frequently in our work. Let me help you out by providing a brief explanation of the period of added/multiplied sines and cosines.

First, let's start with the period of added sines or cosines. When two waves are added, their common period is indeed the lowest common multiple of their individual periods. This means that the period of the resulting wave will be the smallest time interval in which both waves complete a full cycle simultaneously. Your example of cos(2.pi.2x) + cos(2.pi.3x) is correct, and its fundamental period is indeed 1/6.

Now, when it comes to multiplied sines or cosines, the situation is a bit different. In this case, the resulting wave will have a period that is the same as the individual periods of the two waves. So, in your example of cos(2.pi.2x).cos(2.pi.3x), the period will be 1/2 and 1/3, respectively. This is because when two waves are multiplied, their frequencies are multiplied, but their periods remain the same.

I hope this explanation helps you in your program coding. In case you need further clarification, please feel free to reach out. All the best in your work!

Best regards,