Coefficient ck of the fourier series

In summary, the coefficient ck in the Fourier series represents the amplitude and phase shift of a specific frequency component within a periodic signal. It is calculated by taking the inner product of the signal with a complex exponential function and its magnitude and phase have a significant role in describing the frequency component. The accuracy of the Fourier series representation increases with more coefficients, and it can only be used for periodic signals. Non-periodic signals can be analyzed using the Fourier transform.
  • #1
Luongo
120
0
1. find the complex Fourier series and the real Fourier series
f(t) = 0 , -pi <= t < 0, f(t)= t, 0<=t< pi



Homework Equations





3. to find the coefficient ck i got 2 different answers one is if the index k is even and another if the index is odd, how am i supposed to represent this in the Fourier series?
 
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  • #2
Hi Luongo! :smile:

(have a pi: π :wink:)

c2k = f(2k)

c2k+1 = g(2k+1) :wink:​
 

1. What is the purpose of the coefficient ck in the Fourier series?

The coefficient ck in the Fourier series represents the amplitude and phase shift of a specific frequency component within a periodic signal. It allows for the decomposition of a complex signal into simpler components, making it easier to analyze and understand.

2. How is the coefficient ck calculated in the Fourier series?

The coefficient ck is calculated by taking the inner product of the periodic signal with a complex exponential function of the corresponding frequency. This involves integration over one period of the signal.

3. What is the significance of the magnitude and phase of the coefficient ck?

The magnitude of the coefficient ck represents the strength or amplitude of the specific frequency component in the signal, while the phase represents the shift in the sine or cosine function used to reconstruct the signal. Together, they provide a complete description of the frequency component.

4. How does the number of coefficients affect the accuracy of the Fourier series representation?

The more coefficients included in the Fourier series, the more accurate the representation of the original signal. As the number of coefficients increases, smaller and more complex frequency components can be captured, resulting in a more precise reconstruction of the signal.

5. Can the Fourier series be used for non-periodic signals?

No, the Fourier series is only applicable to periodic signals. Non-periodic signals can be analyzed using the Fourier transform, which is a generalization of the Fourier series for non-periodic signals.

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