Discussion For Homework6

[Question]: Does anybody know the number of the corresponding problems in 4th edition?

[Student]: Yes. I didn't check to see if any changes were made to these questions between the 4th and 5th editions (for example: there could be some mistake in the 4th edition that was corrected in the 5th), but here are the problems. As far as I know they are correct.
Problem 1 - 4.36 in 5th Edition is 5.42 in 4th Edition
Problem 2 - 4.37 in 5th Edition is 5.3 in 4th Edition
Problem 3 - 3.19 in 5th Edition is 4.34 in 4th Edition
Problem 4 (3.22 in 5th Edition) does not have a matching problem in the 4th Edition. I don't know how to post equations, but I'll do my best below:
A digital signaling scheme is defined as
x(t) is the sum from n = -inf to inf of [a_n*u(t-nT)*cos(2*pi*fc*t) - b_n*u(t-nT)*sin(2*pi*fc*t)]

where u(t) = tri(t/(2T))

tri(t) = {t+1 for -1<= t <= 0
{-t+1 for 0 <= t <= 1
{0 otherwise

and each (a_n,b_n) pair is independent from the others and is equally likely to take any of the three values, (0,1) , (sqrt(3)/2,-1/2) , (-sqrt(3)/2,-1/2).
Part 1:
Determine the lowpass equivalent of the modulated signal. Determine the in-phase and quadrature components.
Part 2:
Determine the power spectral density of the lowpass equivalent signal; from this determine the power spectral density of the modulated signal.

Question: For Problem 4, if $u(t)=\Lambda(t/2T)$ then the period of $u(t)$ will be $4T$ and thus there will be aliasing. Should not that be$u(t)=\Lambda(2t/T)$?

Comment: [student]: No. I checked the book and $u(t)=\Lambda(t/2T)$ is correct.

Comment: [student]: I know the book has this form. I just doubt if this form is correct for us to solve the problem.

—I think that the correct form shoulb be tri(2t/T) otherwise there will be aliasing and that doesnot make sense!

Comment: [student]: It is true that there will be overlapping in the time domain which will cause Intersymbol Interference (ISI) and that these signals will add together to form some sort of jagged sawtooth waveform. If you think about it, though, expanding the time waveform used to transmit a signal will shrink the waveform in the frequency domain. There will in fact be some aliasing in the frequency domain from these signals because the triangle function has a sinc^2 for a fourier transform.
We are first asked to find the lowpass equivalent to the modulated signal. Since we have ISI in our channel, the time waveform at any time may depend on data from multiple symbols making s(t) be a function of say a_(n-1), a_n, a_(n+1), b_(n-1),… We need to figure out what form this takes.
The second part of the question asks what does the power spectral density look like. We know from our equally signal set that the average value of the a_n's and b_n's is zero so the mean of the information sequence (the Mu_i's) should not have an effect. It should be helpful to think about the problems in these terms.
I agree that this is a pretty bad signaling scheme and would not be very practical.

Question: In problem 5: The simulation of the DPSK system.
1. Do we have to simulate the system in the base band, i.e the pulse shape is just a rectangular pulse or should we make that in the passband?
2. How can we model the phase introduced by the channel?

Question: In problem 6 of the simulation of the truncated raised cosine. Doea anyone know how to take the avaerage of many PSD plots using MATLAB in order to have a non noisy plots?

use Pxx=preiodogram and store it in a matrix like (PSDtotal) do the whole program several times 1000 times and then average the PSDtotal.
Answer[Krishna]: Generally speaking, there is no need to store the periodogram in a matrix and then average. Why not just maintain a running average of the periodogram inside the loop? This will be more efficient from a storage point of view.

Question: In problem 4, that is question 3.22 from Proakis, will there be any change in the way we calculate the PSD because the symbol duration is from -2T to 2T? If so, where will the change be and why? (because in the derivation of the PSD formula, we are not assuming anything about g(t) or the symbols I_n.

Answer[Krishna]: There will be no change to the way in the PSD is calculated. However, I agree that it would be a bad signalling scheme and tri(2t/T) would be a better signalling scheme from a error probability perspective. You will get fulll credit if you used tri(2t/T) or tri(t/2T).

Interestingly, this supposed typo in the book has brought out an issue which seems to confuse some students. Note that we do not need the baseband pulse $g(t)$ to be time limited to $[0,T]$ in order for us to communicate at a symbol rate of $1/T$ symbols/sec or to compute the power spectral density. For example, we have used sinc pulses and shown that they dont result in ISI. sinc pulse is not time limited. Please do not get confused by the fact that one symbol is transmitted every $T$ seconds and the duration of $g(t)$ being $T$ seconds.

I probably chose a bad term for $T$ by calling it the symbol duration. We should really think of $1/T$ as the symbol rate, that is it.

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