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Fig from "Digital Modulations Using Matlab", p 95 (pdf) enter image description here

but it is also described here : Bandwidth-Efficient Digital Modulation with Application to Deep-Space Communications

My question is regarding this part

nrz

My understanding of c(t)- NRZ pulse train:

Given a random vector of +1 and -1 ($a_n$). It is multiplied with $p(t-nTb)$ and we get $c(t)$

Assume $a_n = [1 -1 -1 1]$, then $c(t) = [1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1]$.

$c(t)$ represent Kronecker product between $a_n$ and a vector of ones [1 1 1 1]

Could someone explain what $p(t-nTb)$ is?

Should I take a standard multiplication $a_n$ with $p(t-nTb)$ or Kronecker?

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    $\begingroup$ I have no accessible source for this figure that I can access, but I'm almost certain that if you read the text preceding that figure, the will define $T_b$, and $p$, as "Time of a single bit", and "time-domain pulse". I mean, you've been dealing with pulse shaping for more than a month, what did you expect this to be? Also, no offense, but why do you keep on coming here and asking about definitions of symbols from literature you don't link to? that is very inefficient: we have to guess, unless we had the literature you have, and then we'd just have to read the very same text you have to read. $\endgroup$ May 5, 2022 at 8:29
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    $\begingroup$ Should I take a standard multiplication $a_n$ with $p(t-nTb)$ or Kronecker? The text in the figure say that they are exactly the same! $\endgroup$ May 5, 2022 at 8:30
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    $\begingroup$ @MarcusMüller Kronecker product and standard multiplication are the same? p is a rectungle pulse. $a_n$ and $p$ have different length, it can not be multiplication, rigth? $\endgroup$
    – FrimHart64
    May 5, 2022 at 11:00
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    $\begingroup$ no, not right, as I said in my last comment. $\endgroup$ May 5, 2022 at 12:03

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The difference has to do with the interpretation of the input symbols. If we assume the input symbols are from a set of values with each value being either value 1 or -1, then the implementation would be as shown which is mapping each symbol to the time domain by multiplying the symbol times a rectangular pulse given by the generalized pulse shape:

$p(t) = 1$ for $0 \le t \le 1$ and $0$ elsewhere.

And sampling and offsetting this pulse shape in time appropriate to the symbol timing and sampling rate. Thus $p(t-nT_b)$ simply maps each input symbol to the signed and sampled pulse at the appropriate location on the time line.

I prefer to represent data symbols as a stream of impulses in time, in which case the block diagram would show a convolution of the impulses with $p(t)$. (This form maps well to pulse shaping implementations for non-GMSK modulations when they aren't a simple rectangular pulse). Either form is correct in producing a stream of rectangular pulses as the OP has given and just needs to be defined.

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    $\begingroup$ if i want to describe it, I will write $ a_n \cdot p(t-nT_b)$, right? ( standard multiplication as $2 \cdot 3 =6$) $\endgroup$
    – FrimHart64
    May 6, 2022 at 6:38
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    $\begingroup$ @FrHart64 Thank you! Yes that is correct. The result will be the expected "NRZ" pulse train. NRZ means "non-return-to-zero", so just a simple rectangular pulse either plus or minus one for each symbol. $p(t-nT_b)$ is simply a rectangular pulse that starts at $nT_b$ in time and is one for duration $T_b$. $\endgroup$ May 9, 2022 at 12:00
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    $\begingroup$ and sum befor this multiplicatioin, right? $\endgroup$
    – FrimHart64
    May 9, 2022 at 13:45
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    $\begingroup$ thank you very much for your explanation in details and help in any time $\endgroup$
    – FrimHart64
    May 9, 2022 at 13:46

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