# Finding a Fourier representation of a signal

Given the following signal, find the Fourier representation, ##V(jf)= \mathfrak{F}\left \{ v(t) \right \}##:

##
v(t)=\left\{\begin{matrix}
A, & 0\leqslant t\leqslant \frac{T}{3}\\
2A, & \frac{T}{3}\leqslant t\leqslant T\\
0, & Else
\end{matrix}\right.
##

Then sketch ##V(jf)##.

## Homework Equations

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I don't know whether in such a given form of a signal (apparently looks quite simple, involving constants only) I should use a Fourier SERIES representation or a Fourier TRANSFORM representation. If it's a Fourier Transform representation, then I don't know how would the signal look like in the frequency domain. I tried to Calculate the Transform according to its definition, yet couldn't quite get to anything "sketchable", at least not something I can see it.

## The Attempt at a Solution

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Given below:

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Dr. Courtney
Gold Member
Hint: Is the function periodic in the time domain? What implications does that have for the answer?

Hint: Is the function periodic in the time domain? What implications does that have for the answer?
It is periodic in the sense that it has a period ##T##. However, It has a 0 value for time values larger than T, so that's a bit tricky to determine the signal's periodicity. But that's the first "mine" in finding my way here.

rude man
Homework Helper
Gold Member
It is periodic in the sense that it has a period ##T##. However, It has a 0 value for time values larger than T, so that's a bit tricky to determine the signal's periodicity. But that's the first "mine" in finding my way here.
Either it's periodic or it's zero for t > T. Can't have it both ways.
It sure looks like you're supposed to find the Fourier transform (integral).
Don't know about "graphing" V(jf). It may have real and imaginary parts; I don't have the time to look at your math in detail but if you did it right there are indeed real and imaginary parts to the transform. Which means I don't know about graphing the function. But maybe you messed up somewhere & there are only real or imaginary parts so then you could graph that.
Ther is an expression for the energy spectral density between two frequencies f1 and f2, given the Fourier integral of a pulse, but it's kind of advanced. But that can be graphed since it's a real number.