Complex representation of fourier series

In summary, the conversation is discussing finding the Fourier coefficients of a periodic function using the complex representation of Fourier series. The speaker is having trouble indicating the spectral lines and calculating the Cn coefficient. They have attempted a solution but are unsure of what to do next. Another person is having a similar problem with their signal and is unsure why their complex Fourier series is a scaled amplitude version of their trigonometric Fourier series. They are seeking guidance on how to proceed.
  • #1
tronxo
7
0

Homework Statement


Using the complex representation of Fourier series, find the Fourier coefficients of the periodic function shown below. Hence, sketch the magnitude and phase spectra for the first five terms of the series, indicating clearly the spectral lines and their magnitudes


Homework Equations


Firstable, I don't know how indicate the spectral lines.
The other problem that i have is when i try to calculate the Cn coefficient and, therefore, the final serie. I don't know if it is right or not, and in case of it is right, I am not able to rewrite "my final function" into the correct answer, which i have it in one of my books.


The Attempt at a Solution



what i have done is:
Cn= 1/T∫(from 0 to T) f(t)*e^(-j*n*omega*t) dt
Cn=1/T ( ∫(from 0 to d) Vm*e^(-j*n*omega*t) dt + ∫(from d to T) 0 *e^(-j*n*omega*t) dt )
The second part of the integral is equal to 0, therefore:
Cn=1/T ∫(from 0 to d) Vm*e^(-j*n*omega*t) dt where omega= (2*pi/T)
Cn=1/T ∫(from 0 to d) Vm*e^(-j*n*(2*pi/T)*t) dt
Cn= Vm/T ∫(from 0 to d) e^(-j*n*(2*pi/T)*t) dt
Cn= Vm/(-j*n*(2*pi/T)*T) (limits of the resulting integral from 0 to d)[e^(-j*n*(2*pi/T)*t)]
Cn= Vm/ (-j*n*2*pi) [e^(-j*n*(2*pi/T)*d) - 1]
what is next?
 

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  • #2
bump...
i am having a similar problem...
i am given a signal which can be written as:
s(t) = -1 {-1 < t < 0}, 1 {0 < t < 1}, 0 {1 < t < 2} [it's a pulse train]
the period, T, is 3.
i have calculated the trig. Fourier series representation, which in MATLAB turns out to be correct, yet when i calculate the exponentical fsr, i get a version of the trig. fsr which has its amplitude halved.

for the trig fsr:

s(t) = 2/(pi * n) * (1 - cos((2 * pi * n)/3)) * sin((2 * pi * n * t)/3);


for the exp fsr:

s(t) = -1/(i * pi * n) * (cos((2 * pi * n)/3) - 1) * exp((i * 2 * pi * n * t)/3)


i also tried

c_n = 0.5 (a_n - i * b_n) = -0.5 * i * ( 2/(pi * n) * (1 - cos((2 * pi * n)/3))



either case, my complex fsr was a scaled amplitude version of my trig fsr




any guidance would be much appreciated
 
  • #3


Your approach to finding the Fourier coefficients using the complex representation is correct. To indicate the spectral lines, you can plot the magnitude and phase as a function of frequency, with each coefficient representing a spectral line at a specific frequency. The magnitude of each coefficient will determine the height of the spectral line, while the phase will determine its position on the frequency axis.

To find the final function, you can use the inverse Fourier transform, which is given by:

f(t) = ∑(from n = -∞ to ∞) Cn*e^(j*n*omega*t)

In this case, you can plug in the values of Cn that you have calculated and simplify the equation to get the final function. It is important to remember that the Fourier series is an infinite sum, so using only the first five terms will result in an approximation of the original function.
 

1. What is a complex representation of Fourier series?

The complex representation of Fourier series is a mathematical technique used to approximate a periodic function by representing it as a sum of complex exponential functions. These exponential functions have different frequencies and amplitudes, and when combined, they can create an accurate representation of the original periodic function.

2. How is a complex representation of Fourier series different from a real representation?

The main difference between a complex representation of Fourier series and a real representation is that the complex representation uses complex exponential functions, while the real representation uses sine and cosine functions. The complex representation is often more convenient for mathematical calculations and can produce more accurate results for certain types of functions.

3. What is the relationship between the complex representation and the Fourier transform?

The complex representation of Fourier series is closely related to the Fourier transform. The Fourier transform is a mathematical operation that decomposes a function into its frequency components, while the complex representation of Fourier series decomposes a periodic function into its complex exponential components. Both techniques are used to analyze and approximate functions in the frequency domain.

4. How is the complex representation of Fourier series used in signal processing?

The complex representation of Fourier series is widely used in signal processing, particularly in the areas of telecommunications and digital signal processing. It allows for efficient analysis and manipulation of signals in the frequency domain, which is essential for tasks such as filtering, compression, and noise reduction.

5. What are some applications of the complex representation of Fourier series in physics and engineering?

The complex representation of Fourier series has many applications in physics and engineering. It is used in fields such as acoustics, optics, electromagnetism, and quantum mechanics to analyze and model periodic phenomena. It is also used in image and signal processing, control systems, and data compression.

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