Converting Sinosoids to complex exponentials

**briwel** · April 17th 2009, 05:15 AM

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?

**Mush** · April 17th 2009, 06:28 AM

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

**briwel** · April 17th 2009, 07:04 AM

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) ?

**Mush** · April 17th 2009, 07:08 AM

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.

**Mush** · April 17th 2009, 07:14 AM

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

**briwel** · April 17th 2009, 07:45 AM

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...

**Mush** · April 17th 2009, 07:59 AM

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.

**briwel** · April 17th 2009, 08:40 AM

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?

**briwel** · April 17th 2009, 12:59 PM

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

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