# 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, 2015 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 $n$:

$$\exp\left({j\frac{\pi n}{2}}\right) = [1, j, -1, -j, 1, j, -1, -j, \ldots ], \ n = [0, 1, 2, 3, 4, 5, 6, 7, \ldots]$$

This is a discrete-time complex sinusoid whose frequency is one-fourth of the sample rate. In digital communications signal processing, this is a very common intermediate frequency selection when using real-valued sampling. It is often chosen because the center of the desired passband lies in the center of the single independent sideband that you get when using real samples. Shifting the resulting signal to complex baseband is often a subsequent processing step.

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 Jun 8, 2015 at 13:42
• Sorry about that, there were a couple typos in the sequence. Jun 8, 2015 at 13:43
• 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"? Jun 8, 2015 at 14:07
• thanks, for answering. Can you please provide more details I would very much appreciate your help in answering the questions or clarifying in few more sentences. Thanks Jun 9, 2015 at 16:57