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