Fourier series coefficients

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whoozit

Joined Feb 25, 2008
1
Here is a problem I'm having trouble with. (Please remember that I am new at this.) Any help is much appreciated.

Consider the following three continuous-time signals with a fundamental period of T = 1/2
x(t) = cos(4*pi*t)
y(t) = cos(4*pi*t)
z(t) = x(t)y(t)

a) Determine the Fourier series coefficients of x(t).

b) Determine the Fourier series coefficients of y(t).

c) Use the results of parts a and b along with the multiplication property of the CT Fourier series to determine the Fourier coefficients of z(t).

d) Determine the Fourier series coefficients of z(t) through direct expansion of z(t) in trig form and compare that with the results of part c. (I'm guessing they will be equal.)

What I have so far:
I think I have to use these trig identities.



Also, I know the definition of the fourier series coefficients a0, an, and bn.
x(t) = a0 + \[\sum\]an\[\times\]cos(2\[\times\]pi\[\times\]n\[\times\]t/T) + \[\sum\]bn\[\times\]sin(2\[\times\]pi\[\times\]n\[\times\]t/T)

Aside from that, I'm pretty much lost. Any suggestions?
 

scubasteve_911

Joined Dec 27, 2007
1,202
You just use Euler's identity and factor out the coefficient. So, your Cn value is of magnitude 1/2 at + and - 4pi.

For the next part, you can add the exponential coefficients together.

Steve

attached is an old assignment i had to do lastyear, which may help you understand, or confuse you, either way.
 

Dave

Joined Nov 17, 2003
6,960
x(t) = y(t):

x(t) = cos(4*pi*t)
y(t) = cos(4*pi*t)
Is that correct given that your question asks:

a) Determine the Fourier series coefficients of x(t).

b) Determine the Fourier series coefficients of y(t).
As written they would obviously have the same coefficients. This has implications for later in the question so it is worth clarifying this point now.

Dave
 
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