# How to write time waveform of transmitted sequence with certain chip rate?

I have been studying a book which describes how to write the waveform of sequence that has certain chip rate $T_c$.

The sequence with chip rate $T_c$ is for example

$$X=[-1 -1 +1 +1 +1 +1 +1 +1 +1 -1 +1 -1 -1 +1 +1 -1 -1 -1 +1 +1 -1 -1 -1 -1 +1 -1 +1 -1 +1 -1 -1 +1 +1 +1 -1 -1 -1 -1 -1 -1 -1 +1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 -1 -1 -1 -1 +1 -1 +1 -1 +1 -1 -1 +1 +1 +1 -1 -1 -1 -1 -1 -1 -1 +1 -1 +1 +1 -1 -1 +1 +1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 +1 +1 -1 +1 +1 -1 -1 -1 -1 -1 -1 -1 +1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 -1 -1 -1 -1 +1 -1 +1 -1 +1 -1 -1 +1]$$

Lets assume that $X$ is of length 128.

So the book explains that the waveform can be written as

$$r(nT_c)= X(n ) \underbrace{\exp( (j\pi \frac{n}{2})}_{???}\hspace{1cm} n=0,1,\cdots,127$$

The chip rate $Tc=0.57 ns$...

I don't understand is the part I have ??? under...

Any ideas.

Best Regards.

• A reference to the book could help... – Deve Jun 8 '15 at 7:32

While a reference to where you got this from would be helpful, I have a theory on where the term in question comes from. Note its form for increasing $n$:
$$\exp\left({j\frac{\pi n}{2}}\right) = [1, j, -1, -j, 1, j, -1, -j, \ldots ], \ n = [0, 1, 2, 3, 4, 5, 6, 7, \ldots]$$
In your case, the equation that you gave may express that you have a chip sequence that is modulated onto a complex-valued carrier at one-fourth of the sample rate. Since you don't indicate what the values of the signal are at time instants other than $nT_c$, it's hard to say whether this is actually the case.
• Thanks, I have a question thought why at n=6, you have that the complex exponential should be $j-1$, I get -1.. If I am correct, then the periodicitiy is altered – Henry Jun 8 '15 at 13:42
• Also what do you mean by one fourth of sample rate, are you referring to the oiginal $X$ sequence? As I understand from your answer we need to modulate over this complex sinusoid because of sampling reasons? And what do you mean by " the center of the desired passband lies in the center of single independent sideband that you get when using real sample"? – Henry Jun 8 '15 at 14:07