Fourier series of a bandwidth limited periodic function

In summary: Your Name]In summary, the Fourier coefficients for the given periodic function can be found using the formula FC=\hat x_T(k,T_0)=\sum_{k=-\infty}^\infty\frac{1}{T_0}X\left(k/T_0\right), where the discrete frequencies within the bandwidth of |f|<1 need to be considered. These frequencies can be found by dividing the sampling frequency by the period. The coefficients can then be calculated by plugging in the discrete frequencies into the formula.
  • #1
hotvette
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Homework Statement


Find Fourier coefficients of the periodic function whose template is x(t) where the Fourier Transform of x(t) is [tex]X(f) = (1-f^2)^2[/tex] where [itex]\left|f\right|<1[/itex] and period [itex]T_0= 4[/itex].

Homework Equations


[tex]FC=\hat x_T(k,T_0)=\sum_{k=-\infty}^\infty\frac{1}{T_0}X\left(k/T_0\right)[/tex]

The Attempt at a Solution


I know the expression for FC only applies with the Fourier transform X(f) has no limit on bandwidth, so something different needs to be done. We've had nothing in lectures about finding the Fourier series for this situation (i.e. where FT is zero outside of a specific frequency range), so I'm not sure where to go from here.

There is a short section in the book about waveform sampling stating that for [itex]T=1/(2f_c)[/itex]:
[tex]x(t) = T\sum_{k=-\infty}^\infty x(kT) \frac{sin 2 \pi f_c (t-kT)}{\pi(t-kT)}[/tex] which I think is just saying:
[tex]T\sum_{k=-\infty}^\infty x(kT) \frac{sin 2 \pi f_c (t-kT)}{\pi(t-kT)} <==> (1-f^2)^2[/tex] where [itex]|f|<1[/itex] and [itex]f_c=1[/itex], but I don't immediately see how that's useful. Any tips or suggestions?
 
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  • #2

Thank you for your question. The Fourier coefficients can still be found using the formula you provided, even if the Fourier transform has a limited bandwidth. In this case, the Fourier coefficients will correspond to the discrete frequencies within the bandwidth.

To find the Fourier coefficients for your function x(t), we can use the formula FC=\hat x_T(k,T_0)=\sum_{k=-\infty}^\infty\frac{1}{T_0}X\left(k/T_0\right). Since the Fourier transform, X(f), is only defined for frequencies within the range of |f|<1, we only need to consider the discrete frequencies within this range. These frequencies can be found by dividing the sampling frequency, f_c=1, by the period T_0=4. This gives us the discrete frequencies of f=-1/4, -1/2, 0, 1/4, and 1/2.

Now, we can plug these frequencies into the formula for the Fourier coefficients to find their values. For example, for k=0, we have FC=\hat x_T(0,T_0)=\frac{1}{T_0}X(0)=\frac{1}{4}(1-0)^2=1/4. Similarly, for k=-1/4, we have FC=\hat x_T(-1/4,T_0)=\frac{1}{T_0}X(-1/4)=\frac{1}{4}(1-(-1/4)^2)^2=\frac{15}{256}. The other coefficients can be found in a similar manner.

I hope this helps you with your problem. If you have any further questions, please don't hesitate to ask.
 

Related to Fourier series of a bandwidth limited periodic function

What is a Fourier series of a bandwidth limited periodic function?

A Fourier series is a way of representing a periodic function as a sum of sine and cosine functions. A bandwidth limited periodic function is one that has a finite range of frequencies and repeats itself over a certain period of time.

Why is it important to use a Fourier series for a bandwidth limited periodic function?

Using a Fourier series allows us to break down a complex function into simpler components, making it easier to analyze and understand the behavior of the function. It also allows us to approximate the original function with a finite number of terms.

What is the Nyquist frequency and how does it relate to a bandwidth limited periodic function?

The Nyquist frequency is the highest frequency that can be accurately represented in a discrete signal. For a bandwidth limited periodic function, the Nyquist frequency is half of the bandwidth of the function.

What is the Gibbs phenomenon and how does it affect the Fourier series of a bandwidth limited periodic function?

The Gibbs phenomenon is the oscillation or overshoot that occurs at the edges of a Fourier series approximation. For a bandwidth limited periodic function, this occurs because the function is not smooth at its boundaries, causing the Fourier series to have a larger error at those points.

How can we improve the accuracy of a Fourier series approximation for a bandwidth limited periodic function?

One way to improve the accuracy is to use a larger number of terms in the Fourier series. Another approach is to use a different series, such as a truncated Fourier series or a truncated cosine series, which can reduce the effects of the Gibbs phenomenon.

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