1
$\begingroup$

I have three versions of a 16-QAM modulated signal, each showing the time curve of the real and imaginary parts, the spectrum and the symbol constellation. These three versions differ by the respective pulse shaping method (sinc, rect, raised-cosine or root-raised-cosine).

Is there any possibility to assign the corresponding pulse shaping method to each of these three versions and determine the parameters bandwith $B$, symbol duration $T$ and roll-off factor $\beta\:$?

The respective diagrams: The constellation diagram of the second signal looks exactly the same as the constellation diagram of the first one.

enter image description here enter image description here enter image description here enter image description here enter image description here [![enter image description here][6]][6] enter image description here enter image description here enter image description here

$\endgroup$
9
  • $\begingroup$ This is a typical homework assignment or quiz/test question so we can’t bypass the learning opportunity here. All three are easy to recognize. Please clarify where you are stuck and what you’ve already done and we’ll see if we can help without simply providing the answer. $\endgroup$ Jul 16, 2023 at 11:01
  • $\begingroup$ @DanBoschen It's an exercise I would like to understand to prepare myself for an exam. I wasn't sure if I should mention this because I was under the impression that this site wasn't the best place for questions like this und maybe long 'explanation dialogues' between me and a potential helper. However, I couldn't find any other suitable forum and so I at least tried it here because I don't have any similar examples with solutions. I also tried to find some information in the internet or in books but it seems like I don't have a sufficient $\endgroup$
    – lyx
    Jul 16, 2023 at 13:01
  • $\begingroup$ overview at the moment. Before I try do depict my current knowledge I would like to mention that I have to understand this exercise in the course of an relatively broad-ranging introduction course concerning communication technology. The focus wasn't on (digital) modulation and pulse shaping and especially not on a detailed mathematical analysis of these methods. I know that the bandwith of the spectrum of the 16-QAM modulated signal isn't limited and that an appropriate pulse shaping filter can limit the bandwidth so that it can be transmitted over a bandwith-limited channel for example. $\endgroup$
    – lyx
    Jul 16, 2023 at 13:03
  • $\begingroup$ Moreover, pulse shaping is necessary to prevent ISI since mere rectangle signals are susceptible to due to their spectrum when they are transmitted over a low-pass channel. We talked about the sinc-, the raised-cosine- and the root-raised-cosine-pulse as possible pulses for pulse-shaping but I'm not sure how to recognise which one was used and how to determine the parameters $B$, $T$ and $\alpha$. I see that the first signal doesn't have a bandwith-limited spectrum. So I would suppose that there wasn't used a raised-cosine- or a root-raised-cosine-pulse but a rectangle-pulse. $\endgroup$
    – lyx
    Jul 16, 2023 at 13:03
  • $\begingroup$ So I have todetermine the bandwith now. However, the whole spectrum looks more or less arbitrary to me and not as specific as the spectrum of a bandpass for example. I'm also not sure what the bandwith in this context is. I know the x dB bandwith but how am I supposed to read something from this diagram which looks like noise to me? Additionally, I'm not sure how I'm supposed to know about the spectrum for $f>50~\mathrm{Hz}$ or $f<-50~\mathrm{Hz}$. I also tried to read the symbol duration of the first signal from the first diagram. But I don't see clear ones and zeros. $\endgroup$
    – lyx
    Jul 16, 2023 at 13:04

1 Answer 1

1
$\begingroup$

Yes each of the cases would be easily recognized from the constellations and spectrums shown. We don't provide direct answers to homework questions so as to not bypass the learning opportunity, so I will try and provide some hints which should help "connect the dots" (pun intended).

This is an exercise to recognize the purpose of pulse shaping which ultimately is for spectral efficiency. A rectangular pulse which is simplest has the widest spectral occupancy. This is intuitive when we consider a location on the constellation switching instantly from one symbol to the next (in the extreme unrealizable case, infinite bandwidth is required to go from one dot to another on the constellation diagram in zero transition time).

We slow the transitions from one symbol the next, which serves the purpose of restricting the bandwidth (thus better spectral efficiency: less bandwidth required to send the same amount of data).

The Raised Cosine Pulse Shaping Filter is a commonly used implementation and is part of the broader class of Nyquist Pulse shapes which have the attractive property of restricting the bandwidth without introducing inter-symbol interference (ISI). Because of this, it is a common misconception that the purpose of Nyquist Pulse Shaping is to not have ISI - to be clear, the purpose is solely to restrict bandwidth, and the Nyquist Pulse allows us to do that without introducing ISI.

Another feature of optimum radio design is to use a matched filter in the receiver. To do this, it is common to factor the Raised Cosine Pulse Shaping Filter into two filters, each as a Root-Raised Cosine Filter. One of the two filters will be in the transmitter and the other in the receiver. After the transmit filter, there WILL be ISI in the waveform, but this is eliminated after being passed through the second (matched) filter in the receiver, so that ISI is of no consequence- but we would see it if observed between transmit and receive. Ultimately we only care that all ISI is eliminated before making "decisions" on which symbol was transmitted in the receiver, and this would occur after that matched filter (and after all timing, frequency, phase and amplitude offsets have been removed).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.