Where do "Other" Frequency's Come From when Multiplying Cos Waves?

[CraigH]

When you multiply two cos waves together, where do all the "other" frequency's come f

When you multiply two cos waves with similar frequency's you get a graph that looks like this:

http://www.wolframalpha.com/input/?i=(cos(2*pi*0.8*x))*(cos(2*pi*0.9*x))

With equation y=(cos(2*pi*0.8*x))*(cos(2*pi*0.9*x))

Now if you analyse this wave with a Fourier transform to see the the frequency domain there are lots of other frequency's as well as the 0.8HZ and the 0.9HZ that seem to have come from no where.

Can someone please show me the maths that shows where these frequency's come from?

Thanks

 

[Vargo]

The frequencies you see are .8+.9=1.7 Hz and .9-.8=.1 Hz. See the product to sum trig identities:
http://www.cliffsnotes.com/study_guide/ProductSum-and-SumProduct-Identities.topicArticleId-11658,articleId-11616.html

 

[CraigH]

ahhh okay, thanks, that's helped allot. so just to confirm, when you add 2 waves you get a wave which has 2 frequency's, and these 2 frequency's are the frequency's of the original 2 waves. And when you multiply two waves you use trig identity's to see the equivalent solution as a sum.
Is this correct?

 

[Vargo]

Yes, that is correct (as long as you are talking about sine/cosine waves)

 

[CellCoree]

for your question. I can explain the origin of these "other" frequencies when multiplying cos waves using the Fourier transform. The Fourier transform is a mathematical tool that allows us to analyze a signal in the frequency domain, breaking it down into its individual frequency components. In simple terms, it takes a complex signal and decomposes it into simpler components.

When you multiply two cos waves together, you are essentially adding their individual frequency components. Let's take the example you provided of y=(cos(2*pi*0.8*x))*(cos(2*pi*0.9*x)). The individual frequency components of this signal can be represented as cos(2*pi*0.8*x) and cos(2*pi*0.9*x). When we multiply these two signals, we get:

y = (cos(2*pi*0.8*x))* (cos(2*pi*0.9*x))

= (1/2)(cos(2*pi*(0.8+0.9)*x)) + (1/2)(cos(2*pi*(0.8-0.9)*x))

= (1/2)(cos(2*pi*1.7*x)) + (1/2)(cos(2*pi*0.1*x))

This means that when we multiply two cos waves with similar frequencies, we get additional frequency components at the sum and difference of the two original frequencies. In this case, we get a frequency component at 1.7Hz (0.8+0.9) and at 0.1Hz (0.8-0.9). These are the "other" frequencies that seem to have come from nowhere.

In general, when multiplying two signals with different frequencies, the resulting signal will contain frequency components at the sum and difference of the original frequencies. This is known as the "beat frequency" phenomenon. The frequency components at the sum and difference frequencies are often referred to as "sidebands" and can be seen in the frequency spectrum of the multiplied signal.

I hope this explanation helps to clarify where these "other" frequencies come from when multiplying cos waves. The Fourier transform is a powerful tool in understanding the frequency components of a signal, and it can help us to better understand the complex nature of wave interactions.