# how sinc pulses can be used to distinguish different symbols

Let's say we want to transmit a sequence of numbers Using PAM, this sequence is transmitted with the signal $$s(t)$$ The pulses in $$s(t)$$ are transmitted at a rate $$R_p=\frac{1}{T_p}$$. This is called the "pulse rate" or the "symbol rate". We want $$T_p$$ to be short, to transmit information faster. However, this increases the bandwidth. The maximum spectral efficiency can be achieved using sinc pulses. I have the following questions please:

1. How can sinc pulses be used to distinguish the different numbers? If we are using 16-QAM, then we have 16 different symbols. Is it the amplitude of the sinc pulse that will change to distinguish the 16 symbols??
2. The bandwidth of a single sinc pulse is $$\frac{R_p}{2}$$, what is the bandwidth of the whole signal $$s(t)$$ consisting of several pulses, each of width $$\frac{R_p}{2}$$?

How can sinc pulses be used to distinguish the different numbers? If we are using 16-QAM, then we have 16 different symbols. Is it the amplitude of the sinc pulse that will change to distinguish the 16 symbols?

You send one sinc after the other, and each sinc is multiplied with one complex number (which is your 16QAM constellation point)

The bandwidth of a single sinc pulse is Rp/2, what is the bandwidth of the whole signal s(t) consisting of several pulses, each of width Rp/2?

The bandwidth of something is something that can be seen from the power spectral density, which in itself is an average.

Hence, the spectrum of your transmission, assuming uncorrelated data symbols, is exactly the spectrum of your pulse.

Your two questions are the most fundamental statements underlying all modern digital communications:

1. you can transport data by sending a symbol impulse train through your pulse shaping filter
2. the spectrum of the signal is defined only by pulse shape

It feels a bit like you learned about 16QAM before learning the basics of complex baseband and modulation – maybe go a chapter back in your book until you find a block diagram showing a data source, a symbol mapper, a pulse shaper, and a mixer, and then refresh your knowledge of complex baseband, digital comms, if you feel like you're missing basics.

• According to your answer, the spectrum of your transmission, assuming uncorrelated data symbols, is exactly the spectrum of your pulse. I know that a sinc pulse and delayed versions of itself by nT are orthogonal (uncorrelated). Is orthogonality achieved for any number of delayed versions of the sinc pulse? – Noha Dec 3 '20 at 17:32
• also, according to your answer, You send one sinc after the other, and each sinc is multiplied with one complex number (which is your 16QAM constellation point). Could you please send me a figure which illustrates uncorrelated sinc pulses with different amplitudes? – Noha Dec 3 '20 at 17:34
• @Noha your first question can be answered through induction. Your second is: come on, you can draw things yourself. – Marcus Müller Dec 3 '20 at 20:17