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Jncik
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Homework Statement
Suppose we are given the following information about a continuous-time periodic signal with period 3 and Fourier coefficients [tex]a_{k}[/tex]
1. [tex]a_{k} = a_{k+2}[/tex]
2. [tex]a_{k} = a_{-k}[/tex]
3. [tex]\int_{-0.5}^{0.5}x(t)dt = 1[/tex]
4. [tex]\int_{0.5}^{1.5}x(t)dt = 2[/tex]
Determine x(t)
Homework Equations
if x(t) is a periodic input signal it can be expressed as
[itex]x(t) = \sum_{k=-\infty}^{+\infty}a_{k}e^{jk\omega_{0} t}[/itex]
which is called the Fourier series of x(t), and [tex]\omega_{0}[/tex] is the fundamental frequency of x(t)
also [tex]a_{k} = \int_{-\infty}^{+\infty} x(t) e^{-j \omega_{0} k t} dt [/tex]
The Attempt at a Solution
I've tried to solve this problem many times, when I was reading about it in the Fourier series chapter.. I thought if I moved onto the next chapter which was Fourier transform it would help me but it didn't, so here I am again..
what I first found is that [tex]\omega_{0} = \frac{2\pi}{3}[/tex]
hence [tex]x(t) = \sum_{k=-\infty}^{+\infty}a_{k}e^{jk\frac{2\pi}{3} t}[/tex]
now from the second property we can understand that x(t) is an even signal... hence
if
x(t) <-> ak
x(-t) <-> a-k
x(t) = x(-t) => ak = a-k
now from this we can simplify the summation
[tex]x(t) = \sum_{k=0}^{+\infty}a_{k}(e^{jk\frac{2\pi}{3} t}+e^{-jk\frac{2\pi}{3} t})=\sum_{k=0}^{+\infty}a_{k} 2cos(\frac{k 2\pi t}{3})[/tex]
now from the first property since [tex]a_{k} = a_{k+2}[/tex]
we see from the frequency shifting property that
[tex] x(t) = x(t) e^{\frac{-j 4\pi t}{3}} [/tex]
good, we have 2 other properties
from the third property since [tex]\int_{-0.5}^{0.5}x(t)dt = 1[/tex] then x(t) must be the dirac delta function in this interval, hence [tex] x(t) = \delta (t)[/tex] for [tex] -0.5<=t<=0.5 [/tex]
and from the fourth property, again it's a shifted dirac delta with amplitude changed hence
[tex] x(t) = 2 \delta (t-1)[/tex] for [tex] 0.5 <= t <= 1.5 [/tex]
well I have these results which I hope to be correct, how can I use them in order to find the final x(t)?
I've solved many such exercises but they were a lot easier.. there was parseval involved, some other properties, for example it was stated that x(t) was real and even etc.. this is the only exercise that I have difficulty to find a solution and I need to learn how to solve it because our professor likes these kind of exercises
thanks in advance
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