Sampled Sinusoids

• Sampling corresponds to transforming the continuous time variable $t$ into a set of discrete times that are integer multiples of the sampling period $T_s$. That is, sampling involves the substitution
  $$t \longrightarrow nT_s,$$
  where $n$ is an integer corresponding to the index in the sequence.
• Recall that a sinusoid is a function of time having the form
  $$x(t) = A\sin(\omega t + \phi).$$

• In discretizing this equation therefore, we obtain
  $$x(nT_s) = A\sin(\omega nT_s + \phi),$$
  which is a sequence of numbers that may be indexed by the integer $n$.

• Note: $x(nT_s)$ is often shortened to $x(n)$ (and will likely be from now on), though in some litterature you'll see square brackets $x[n]$ to differentiate from the continuous time signal.