1.1. Discrete-time signals#

  • Discrete-time signal \(x : \mathbb{Z} \longrightarrow \mathbb{R}\) or \(\mathbb{C}\).

    Notation

    Use \(x[n]\) to denote a discrete-time signal, where the square brackets indicate “discrete time”, and letters \(n\), \(m\), \(k\), and \(l\) take on integer “time” values.

  • Continuous-time signal \(x : \mathbb{R} \longrightarrow \mathbb{R}\) or \(\mathbb{C}\).

    Notation

    Use \(x(t)\) to denote a continuous-time signal, where the parantheses indicate “continuous time”, and letters \(t\), and \(s\) take on real “time” values.

  • Simple, but important, discrete-time signals:

    • Unit impulse signal \(\delta[n] = \begin{cases} 1, & n=0 \\ 0, & n \neq 0 \end{cases}\)

    • Unit step signal \(u[n] = \begin{cases} 1, & n \geq 0 \\ 0, & n < 0 \end{cases}\)

    • (Complex-valued) sinusoid \(Ae^{j\phi} e^{j\hat\omega n}\)

      Tip

      Euler’s identity implies \(A\cos(\hat\omega n + \phi) = \frac{A}{2}e^{j\phi} e^{j\hat\omega n} + \frac{A}{2}e^{-j\phi} e^{-j\hat\omega n}\).

    • Stepped sinusoid \(Ae^{j\phi} e^{j\hat\omega n} u[n]\)

  • Some important signal properties:

    • Periodicity:

      • \(x[n]\) is periodic with period \(N\) \((N > 0)\) if \(x[n] = x[n+N]\) for all \(n \in \mathbb{Z}\).

      • The smallest such \(N\) is called the fundamental period.

        Notation

        Often, we simply call the fundamental period of \(x[n]\) the period of the signal when there is no ambiguity.

      • If no such \(N\) exists, \(x[n]\) is aperiodic.

    • Signal energy: \(E_x = \sum_{n=-\infty}^{\infty} |x[n]|^2\).

    • Signal power: \(P_x = \lim_{M \rightarrow \infty} \frac{1}{2M+1} \sum_{n=-M}^{M} |x[n]|^2\).

      Tip

      If \(x[n]\) is periodic with period \(N\), \(P_x = \frac{1}{N} \sum_{n=0}^{N-1} |x[n]|^2\).