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

In summary, given a continuous-time periodic signal with period 3 and Fourier coefficients a_{k}, we are given four pieces of information: a_{k} = a_{k+2}, a_{k} = a_{-k}, \int_{-0.5}^{0.5}x(t)dt = 1, and \int_{0.5}^{1.5}x(t)dt = 2. Using the Fourier series representation of a periodic signal, we can determine that \omega_{0} = \frac{2\pi}{3} and that x(t) is an even signal. From there, we can simplify the summation and use the properties of frequency shifting and symmetry to
  • #1
Jncik
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0

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

I mean, from the first two, we have equations about [tex]a_{k}[/tex] 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
 

What is the Fourier series?

The Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine functions with different amplitudes and frequencies. It is commonly used in signal processing and allows us to decompose a complex signal into simpler components.

What are Fourier series coefficients?

Fourier series coefficients are the amplitudes of the sine and cosine functions that make up the Fourier series. These coefficients are used to determine the frequency and amplitude of each component in the signal.

How do I find the signal x(t) from Fourier series coefficients?

To find the signal x(t) from Fourier series coefficients, you can use the inverse Fourier transform. This involves summing the individual sine and cosine functions with their corresponding coefficients, resulting in the original signal.

What properties do I need to know to find the Fourier series coefficients?

In order to find the Fourier series coefficients, you will need to know the period of the signal, as well as the amplitude and frequency of each component in the signal. These properties can be determined from the signal itself or from a given function.

Can Fourier series be used for non-periodic signals?

No, Fourier series can only be used for periodic signals. However, there are other methods, such as the Fourier transform, that can be used to analyze non-periodic signals.

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