• mklein
In summary, the conversation discusses the rules for the period of sines and cosines when they are multiplied or added. The concept of using complex exponentials as an alternative is also mentioned. The participants suggest using product-to-sum formulas or learning about complex exponentials to make the math simpler.
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

Hi mklein!

mklein said:
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!

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.

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

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,

## 1. What is the period of an added sine/cosine function?

The period of an added sine/cosine function is the distance on the x-axis between two consecutive peaks or troughs of the graph. This period is the same for both the sine and cosine functions and is determined by the coefficient of the variable inside the sine or cosine function.

## 2. How can I find the period of a multiplied sine/cosine function?

The period of a multiplied sine/cosine function is determined by the period of the individual sine/cosine functions being multiplied together. To find the period, you can take the smallest common multiple of the periods of the individual functions.

## 3. What happens to the period when I add or multiply sine/cosine functions?

When you add or multiply sine/cosine functions, the period remains the same. Adding or multiplying the functions does not change the distance between consecutive peaks or troughs on the graph, it only affects the amplitude and phase shift.

## 4. Can the period of an added/multiplied sine/cosine function be negative?

No, the period of an added/multiplied sine/cosine function cannot be negative. The period is always a positive value, representing the distance on the x-axis between two consecutive peaks or troughs of the graph.

## 5. How does the period of added/multiplied sine/cosine functions affect the graph?

The period of added/multiplied sine/cosine functions affects the graph by determining the frequency of the waves. A smaller period means the waves will appear more frequently and the graph will appear more compressed, while a larger period means the waves will appear less frequently and the graph will appear more spread out.

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