# Thread: Converting Sinosoids to complex exponentials

1. ## 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?

2. 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 $\sin(x) = \frac{1}{2i}(e^{ix} - e^{-ix})$

3. 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) ?

4. 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) ?

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

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

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

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

The same logic can be applied to the other 3 terms.

5. Actually. The LaPlace transform of this would be a good idea.

6. 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...

7. 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.

8. 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?

9. also, the equaion always seems to equat to 0. Is that correct?