Signal filtering in digital transmission

[LorDjidane]

Homework Statement

Let's suppose a set of numerical values {a_n} sent with a rectangular function, period T and with amplitude a_n.
The signal can be expressed as follows:
$$x(t)= \sum a_n \Pi_T (t-nT)$$
To optimize detection at reception, the signal x(t) is processed through a filter whose impulse response is:
$$h(t)= sin(\frac{\pi t}{T} \Pi_T (t-\frac{T}{2})$$
We write y(t) the output signal

1/ If we only have a₀=A, calculate y(t) and illustrate the computed elements graphically.

2/ Express y(t) in the general case. Plot x(t) and y(t) if the values to be transmitted are ...,0,0,0,3,5,-2,1,-3,0,0,0,... What are the optimal moments to detect {a_n} from y(t) ?

3/ Express h(t) as a convolution:
$$h(t)=h_0(t)*\delta(t-\frac{T}{2})$$
Determine h₀(t), calculate H₀(f) and give a graphical representation. What are the modulus and phase of the transfer function H(f) ?

4/ Let’s suppose the {a_n} are obtained after an audio signal sampling at Fe=1/T, we have a_n = s(nT). Give the spectrum of x(t), X(f) with a graphical representation. Express the mean value of x(t) with respect to the one of s(t), P_s. Is the filtering of x(t) by h(t) a good way to get back s(t) from x(t)? Why?

Homework Equations

$$x(t)= \sum a_n \Pi_T (t-nT)$$

$$h(t)= sin(\frac{\pi t}{T} \Pi_T (t-\frac{T}{2})$$

$$y(t)=x(t)*h(t)$$

The Attempt at a Solution

Well, this is not exactly for me but for one of my students who’s having a hard time. Except he’s my student in electronics laboratories not in signal processing and since he begged me to help him, here i am getting back in signal processing. :(

My ideas:

h(t) is the multiplication of a sine with a rectangular function translated by T/2 so I can have their common area, plot it and do my convolution graphically. But I can’t carefully write the mathematical operation.
By writing the convolution :
$$y(t)=\int sin(\frac{\pi \tau}{T}) \Pi_T (\tau-\frac{T}{2}) A \Pi_T (t - \tau) d\tau$$
But this doesn’t seem easy to compute. I thought of coming through it with a Fourier transform to get two sinc with the rectangular functions and a dirac with the sine. But I’m not sure about the carefullness of my operation.

Actually once I understand this question and the math behind it, I may be able to go on by myself. The second question is a generalization of the first and the optimal detection instants are trivial.

For the third question the dirac makes me think of a T/2 delay so h(0) would be for me the product of a sine and something doing the rectangular function but I’m not sure. Again, once started I should be able to go on by myself.

I hope this thread doesn’t seem too improper with respect to the policy here, but I’m starting to see my student’s face decomposed.

 

[kurt.physics]

I’d be very grateful for an explanation of the first question that would help me going through this. Thank you for your time.