Samples per symbol and number of symbols for QAM
Number of symbols


A baseband signal model that is used is: $x(t)=\sum_{k=1}^N a_k p(t-kT)$, where $a_k$ are the symbols (there are many types, PSK, QAM, PAM for example), $p(t)$ is the pulse shaping filter, $T$ is the symbol period, and $N$ is the number of symbols. The pulse shaping filter limits the bandwidth of $x(t)$.


  Samples per symbol


Having multiple samples per symbol is called oversampling. The samples per symbol, $\text{sps}$, is usually not explicitly chosen but is determined from the symbol rate of the signal, $F$, and sample rate, $F_s$, on a particular platform: $\text{sps} = \frac{F_s}{F}$. Your statement that "samples per symbol is sampling rate" is not correct. Visually, having more samples per symbol "fills in" the symbol and will make it look smoother. There are many reasons, besides the pulse shaping filter, to oversample. For example, some algorithms in the receiver are designed to operate on the oversampled received signal, while others expect to work on the received symbols.



Code: https://github.com/B-William/DSPSE/blob/master/spsScript.m