# Converting Sinosoids to complex exponentials

• Apr 17th 2009, 05:15 AM
briwel
Converting Sinosoids to complex exponentials
I am doing some revision exercises and have come across this question:

Write the following sinusoid as a complex exponential and plot both in time and frequency domains.

x(t) = 2sin(200 pi t)+ 3sin(400 pi t)

I'm not sure how to do this. Is it something to do with the complex exponential version of Fourier transform?
• Apr 17th 2009, 06:28 AM
Mush
Quote:

Originally Posted by briwel
I am doing some revision exercises and have come across this question:

Write the following sinusoid as a complex exponential and plot both in time and frequency domains.

x(t) = 2sin(200 pi t)+ 3sin(400 pi t)

I'm not sure how to do this. Is it something to do with the complex exponential version of Fourier transform?

No it's more to do with the fact that $\displaystyle \sin(x) = \frac{1}{2i}(e^{ix} - e^{-ix})$
• Apr 17th 2009, 07:04 AM
briwel
Yeah I thought Fourier was a bit complicated.

In the example I have given is there anything else that can be done to further simplify the formula after substituting x for 2sin(200pi t) ?
• Apr 17th 2009, 07:08 AM
Mush
Quote:

Originally Posted by briwel
Yeah I thought Fourier was a bit complicated.

In the example I have given is there anything else that can be done to further simplify the formula after substituting x for 2sin(200pi t) ?

$\displaystyle x(t) = 2\sin(200 \pi t)+ 3\sin(400 \pi t) = \frac{2}{2i}(e^{i 200 \pi t} - e^{-i 200 \pi t}) + \frac{3}{2i}(e^{i 400\pi t} - e^{-i 400 \pi t})$

Now... $\displaystyle e^{i 200 \pi t} = \bigg(e^{i \pi}\bigg)^{200t}$

And $\displaystyle e^{i \pi} = -1$

so $\displaystyle e^{i 200 \pi t} = \bigg(-1\bigg)^{200t}$

The same logic can be applied to the other 3 terms.
• Apr 17th 2009, 07:14 AM
Mush
Actually. The LaPlace transform of this would be a good idea.
• Apr 17th 2009, 07:45 AM
briwel
Thanks Mush.

Any idea about the plotting of the graphs? I'm assuming the time domain is pretty simple, just plotting the formula against increasing values of t. I'm not sure about frequency though...
• Apr 17th 2009, 07:59 AM
Mush
Quote:

Originally Posted by briwel
Thanks Mush.

Any idea about the plotting of the graphs? I'm assuming the time domain is pretty simple, just plotting the formula against increasing values of t. I'm not sure about frequency though...

When you perform the LaPlace transform you will be given the function in terms of the complex variable s.
• Apr 17th 2009, 08:40 AM
briwel
I didn't actually use laplace transform as its not something that I have been taught so I assume I'm not meant to do it that way.

Is there another way to plot the frequency graph?
• Apr 17th 2009, 12:59 PM
briwel
also, the equaion always seems to equat to 0. Is that correct?