# Confusion between samples per second and samples per symbol

My understanding of sampling frequency is when we have an analog signal x(t) we convert it to a digital signal. Assume the sampling frequency is denoted by $f_s$ then the sampling time $T_s$. In my understading this is denoted the number of samples per second.

In MATLAB programs, I have come across generation of base band signal. The authors have upsampled the base band signal and the sampling rate was stated to be 3 samples per symbol.

My question is what does samples per symbol mean.Assuming we have QAM symbols are we repeating it three times?

Thanks

A digital communications signal such as QAM can be generally written as $$s(t) = \sum_{n=0}^{N}a_n p(t-nT),$$ where $a_n$ are (possibly complex) amplitudes taken from a finite alphabet, and the symbol rate is $1/T$. The pulses $p(t-nT)$ are often called "symbols". Matlab commands that ask you to specify "samples per symbol" are asking how many samples you want to use to define $p(t)$.
• The pulse duration is not, in general, the inverse of the symbol rate, because the pulses may overlap. In the equation in my answer, one pulse is transmitted every $T$ seconds, so the symbol rate is $R_s=1/T$. The bandwidth is determined by the spectrum of $p(t)$, but it cannot be less than $1/(2T)$. The pulse with smallest bandwidth for a given symbol rate is the sinc pulse.