# A lot of questions on the fourier series

#### hitmen

Joined Sep 21, 2008
159
1) I am supposed to verify that this IS orthogonal:
x1(t) = ej∏t
x2(t) = e-j∏t on the interval [0,1]

I know that I am suppose to ∫x1(t)x2(t) oner 0,1

However, when I multipy x1 and x2 they become 1. Integrating 1 over [0,1] gives the value of 1.
// I know I am supposed to get zero.
So where have I done wrong?

Q2) I have difficulty find the period of this function
x(t) = 0.5 + cos∏t + 2 sin[1.5∏t + ∏/4 ]

I know that I am suppose to use T= 2∏/ω to find period
Thus the period of cos ∏t is 2
and that of sin[1.5∏t + ∏/4] is 4/3
I assume that 0.5 does nothing to the equation.

My prof mentioned that the period is 4. How can this be?

Q3) Can anyone explain how to START doing this q :
find the TFS of x(t) = Ʃ[δ(t-3n) + δ(t-1-3n)] //from n=- ∞ to ∞
Just tell me how to integrate the the delta function with an n.
I didnt understand my prof..

Q4) x(t) = ∞ n=-∞ Ʃy(t-3n) where y(t) = e-t, 0<=t<=3, 0 otherwise

a) Find complex fourier series of x(t)
b) Plot double sided amplitude and phase spectrum of x(t)

For a I got the answer as ck = (1-e^-3(1+j(2∏/3))) / 3(1+j(2∏k/3))

However, I dont know why my professor threw away the imaginery part

Thanks.

Last edited:

#### studiot

Joined Nov 9, 2007
5,003
I will kick you off with hints on the first two for now.

1) The indefinite integral comes to 1 (plus a constant), as you correctly state.

However the definite integral does not. Review your definite integral.

2) Nobody said the fundamental was present, and yes neither the leading constant, nor the phase angle play any part in determining the period.

so let the cosine term be cos(pπwt) and the sine term be 2sin(qπwt+π/4)
Where p, q are integers to be determined and w is angular frequency.

Then comparing coefficients

pw = 1
qw = 1.5

If w = 0.5 then p = 2, q = 3 works, and you can find the period.

#### hitmen

Joined Sep 21, 2008
159
I will kick you off with hints on the first two for now.

1) The indefinite integral comes to 1 (plus a constant), as you correctly state.

However the definite integral does not. Review your definite integral.

2) Nobody said the fundamental was present, and yes neither the leading constant, nor the phase angle play any part in determining the period.

so let the cosine term be cos(pπwt) and the sine term be 2sin(qπwt+π/4)
Where p, q are integers to be determined and w is angular frequency.

Then comparing coefficients

pw = 1
qw = 1.5

If w = 0.5 then p = 2, q = 3 works, and you can find the period.
Thanks! I tried Q1 again and I brought down 1/j∏ but i am still stuck..

With regards to Q2, where did n disappear to? Also how did you know the values of p and q ?
Is there a systematic way of guessing it?

#### studiot

Joined Nov 9, 2007
5,003
exp(x).exp(-x) = exp(0) for all x

In particular exp(x=0) = exp(x=1) = 1

So ∫(x=0) = ∫(x=1) = 1

So ∫[(x=1) - (x=0) ] = 0

#### studiot

Joined Nov 9, 2007
5,003
where did n disappear to?
I didn't use n because it looks too much like π So I used p and q instead.

'Guessing' p, q and w wasn't so very hard was it?

Did you understand what I said about the fundamental?

#### hitmen

Joined Sep 21, 2008
159
I understand Q2 fundamentals in that ω must be a constant for the two equations. p and q are just arbitrary constants to find the common factor.

However I still dont understand Q1. How can ∫(x=1) be 1

Sorry for being so troublesome but thanks for helping.

#### studiot

Joined Nov 9, 2007
5,003
Embarrassed expressions.

I was totally wrong in what I said about Q1.

I put it down to being too early in my morning when I wrote it.

You were quite correct about the integral being 1.

Don't forget that the condition for orthogonality is not that the integral is zero.

It is that for any two functions, when each multiplied by an arbitrary integer constant (p, q) the integral of their product is zero if p ≠q.

(I prefer to keep p and q, rather than m and n as previously stated)
(the integral may be zero or non zero when p = q)

In your case p = q = 1 so the integral =1 is OK. What you have to do is introduce p and q and show the integral equals zero if p ≠ q.

For parts 3 and 4 what does the square box symbol stand for please.

#### hitmen

Joined Sep 21, 2008
159
Q3) Can anyone explain how to START doing this q :
find the TFS of x(t) = summation [ delta(t-3n) + delta(t-1-3n) ] from n=- ∞ to ∞
Just tell me how to integrate the the delta function with an n.
I didnt understand my prof..

Q4) x(t) = ∞ n=-∞ summation y(t-3n) where y(t) = e-t, 0<=t<=3, 0 otherwise

a) Find complex fourier series of x(t)
b) Plot double sided amplitude and phase spectrum of x(t)

For a I got the answer as ck = (1-e^-3(1+j(2∏/3))) / 3(1+j(2∏k/3))

However, I dont know why my professor threw away the imaginery part

Here it is. I dont understand what u mean by a square box.

#### hitmen

Joined Sep 21, 2008
159
Here are the cut questons:

I dont know what is the integral of a delta function. I just need an example.

In the next question, I got another answer separate from my professor. He threw away the imaginery part or something. I dont get it

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