• Discrete-time signals have an independent variable (typically represented with $n$) that only takes integers

• Digital signals are a type of discrete-time signal that accepts values from a finite set.
  • This occurs by sampling continuous-time signals
  • Digital and discrete-time can be used interchangeably
• Digital and discrete-time signals can be represented by:
  • Formula: $x[n] = 0.5e^{0.1n}(u[n]-u[n-3])$
  • Weighted sum of discrete-time impulses: $x[n] = 0.5\delta[n]+0.553\delta[n-1]+0.611\delta[n-2]$
    • Is 1 for $n=0$ or at $n=d$ for $\delta[n-d]$ (timeshifted)
  • Number sequence: $x[n]=(0.5, 0.553,0.611,0,\dots)$

Example

• Weighted sum form: $x[n] = 2\delta[n]+4\delta[n-1]-\delta[n-2]+\delta[n-3]+3\delta[n-4]$
• Sequence form: $x[n] = (2, 4, -1, 1, 3, 0, \dots)$

Sampling

• A continuous-time signal can be converted into digital signals by periodically sampling its values at regular intervals, quantizing them
  • $x[n] = x(t) \vert_{t=nT_s} = x(nT_s) = x(n / f_s)$
• The sample rate should be at least twice the highest frequency component in the signal (Shannon/Nyquist rate)

References

• Discrete-time signals (course handout)