Problems part 3 : Fourier series

Problems part 3 : Fourier series

Prob. 1) Consider a square wave of length . Over the range , this can

be written as

where is the Heaviside function .

FourierSeriesSquareWave

Since , the function is odd, so , and

reduces to

The Fourier series is therefore

Prob. 2) Find Fourier series of a saw tooth wave described by the following equation

The components of the Fourier series are therefore given by

The Fourier series is therefore given by

Prob. 3 ) Find Fourier Series of the following function of period 2.

Answer

We have T = 2. Lets find its coefficients :

a₀ we have to calculate of the following , because function does not have definition for n = 0.

The fourier series is:

Prob 4) Find Fourier series of the following function

a) Find trigonometric Fourier series of f(t).

b) Graph in the interval [-4p; 4p].

Answer

We can extend the period as 2p. We can obtain the coefficients :

The a₀ coefficient is:

Thus the Fourier series is :

b) Graph of the function:

The point shown values of the function in its discontinuity .

Prob. 5) Find the complex Fourier series of the following f(t) = e^-^a^t, where -p < a < p. With this series we find .

Answer

First we fin its coefficients:

Then:

That function is continuous for all values . If we evaluate this in 0 is the same that f(0), that is , 1. If we evaluate e^int, in 0, we get 1.

Positive terms of in will cancel , thus we can write the function as :