Finding the signal x(t) when given properties of fourier series coefficients

[Jncik]

Homework Statement

Suppose we are given the following information about a continuous-time periodic signal with period 3 and Fourier coefficients $$a_{k}$$

1. $$a_{k} = a_{k+2}$$
2. $$a_{k} = a_{-k}$$
3. $$\int_{-0.5}^{0.5}x(t)dt = 1$$
4. $$\int_{0.5}^{1.5}x(t)dt = 2$$

Determine x(t)

Homework Equations

if x(t) is a periodic input signal it can be expressed as

$x(t) = \sum_{k=-\infty}^{+\infty}a_{k}e^{jk\omega_{0} t}$

which is called the Fourier series of x(t), and $$\omega_{0}$$ is the fundamental frequency of x(t)

also $$a_{k} = \int_{-\infty}^{+\infty} x(t) e^{-j \omega_{0} k t} dt$$

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 $$\omega_{0} = \frac{2\pi}{3}$$

hence $$x(t) = \sum_{k=-\infty}^{+\infty}a_{k}e^{jk\frac{2\pi}{3} t}$$

now from the second property we can understand that x(t) is an even signal... hence

if

x(t) <-> a_k
x(-t) <-> a_-k
x(t) = x(-t) => a_k = a_-k

now from this we can simplify the summation

$$x(t) = \sum_{k=0}^{+\infty}a_{k}(e^{jk\frac{2\pi}{3} t}+e^{-jk\frac{2\pi}{3} t})=\sum_{k=0}^{+\infty}a_{k} 2cos(\frac{k 2\pi t}{3})$$

now from the first property since $$a_{k} = a_{k+2}$$

we see from the frequency shifting property that

$$x(t) = x(t) e^{\frac{-j 4\pi t}{3}}$$

good, we have 2 other properties

from the third property since $$\int_{-0.5}^{0.5}x(t)dt = 1$$ then x(t) must be the dirac delta function in this interval, hence $$x(t) = \delta (t)$$ for $$-0.5<=t<=0.5$$

and from the fourth property, again it's a shifted dirac delta with amplitude changed hence

$$x(t) = 2 \delta (t-1)$$ for $$0.5 <= t <= 1.5$$

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

 

[Eynstone]

This is more of an algebra problem. As $$a_{k} = a_{k+2}$$
and $$a_{k} = a_{-k}$$ , two constants determine the series. (3) & (4) are precisely the equations which make this a problem of two equations in two unknowns.

 

[Jncik]

which are the equations?

I mean, from the first two, we have equations about $$a_{k}$$ but these just tell us some properties about the frequency shifting and also about the symmetry of the signal..

about the 3 and 4 I'm not sure at all, I could only figure out what the x(t) would be in a specific range.. now T is 3 and we know from -0.5 to 0.5 and from 0.5 to 1.5

if I knew what happened from 1.5 to 2.5 then I would have the signal, since it's periodic with fundamental period 3