1.2. Discrete-time systems#

  • A discrete-time system maps an input signal \(x[n]\) to an output signal \(y[n]\).

    Notation

    Denote the system \(\mathcal{T}\) as an operator that acts on \(x[n]\):

    \[\begin{equation*} y[n] = \mathcal{T}(x[n]) \end{equation*}\]

    or with a block diagram:

    \[\begin{equation*} x[n] \xrightarrow{\hspace{30pt}} \boxed{ \begin{array}{ccc} & & \\[-10pt] & \mathcal{\Large T} & \\[-10pt] & & \end{array}} \xrightarrow{\hspace{30pt}} y[n] \end{equation*}\]

  • In most cases, we care only about this I/O mapping, which may be specified by the:

    • difference equation

    • impulse response \(h[n]\)

    • frequency response \(H(e^{j\hat\omega})\)

    • transfer function \(H(z)\)

    • block diagram

      \(\hspace{30pt} \vdots\)

1.2.1. LTI and stability#

  • Linearity:

    A system \(\mathcal{T}\) is linear (L) if \(\mathcal{T} ( \alpha x_1[n] + \beta x_2[n]) = \alpha \mathcal{T} (x_1[n]) + \beta \mathcal{T}(x_2[n])\) for all \(\alpha\), \(\beta\), \(x_1[n]\), and \(x_2[n]\).

  • Time invariance:

    A system \(\mathcal{T}\) is time-invariant (TI) if \(\mathcal{T} ( x[n]) = y[n]\) implies \(\mathcal{T} ( x[n-n_0]) = y[n-n_0]\) for all \(n_0\) and \(x[n]\).

  • Causality:

    A system \(\mathcal{T}\) is causal if \(\mathcal{T} ( x[n]) \) depends only on the current and past input, i.e., \(x[n]\), \(x[n-1]\), \(\ldots\).

  • Bounded Input Bounded Output (BIBO) stability:

    A system \(\mathcal{T}\) is (BIBO) stable if \(|\mathcal{T} (x[n])| \leq M_y < \infty\) for every \(x[n]\) that satisfies \(|x[n]| \leq M_x < \infty\).

1.2.2. Impulse response and convolution#

  • The impulse response \(h[n]\) of a system \(\mathcal{T}\) is the system’s output when the input is the unit impulse signal, i.e., \(h[n] = \mathcal{T}(\delta[n])\).

  • Convolution:

    For any LTI system \(\mathcal{T}\) with impulse response \(h[n]\) and any input \(x[n]\), the output is given by

    \[\begin{equation*} \mathcal{T}(x[n]) = \sum_{k=-\infty}^{\infty} x[k] h[n-k]. \end{equation*}\]

    • “Proof”: Write \(x[n] = \sum_{k=-\infty}^{\infty} x[k] \delta[n-k]\). Then

      \[\begin{align*} \mathcal{T}(x[n]) &= \mathcal{T}\left( \sum_{k=-\infty}^{\infty} x[k] \delta[n-k] \right) \\ &= \sum_{k=-\infty}^{\infty} x[k] \mathcal{T}(\delta[n-k] ) & (\mathcal{T} \text{ is linear})\\ &= \sum_{k=-\infty}^{\infty} x[k] h[n-k] & (\mathcal{T} \text{ is time-invariant}) \end{align*}\]

      Assumption

      In above and also hereafter, implict convergence (in some sense) is assumed when we write an infinite series like \(\sum_{k=-\infty}^{\infty} x[k] h[n-k]\). We may sometimes discuss more about the assumed convergence.

      Notation

      Write the convolution sum using the following notation:

      \[\begin{equation*} \sum_{k=-\infty}^{\infty} x[k] h[n-k] = x[n]*h[n] = (x*h)[n] \end{equation*}\]

      Tip

      \(y[n] = x[n]*h[n]\) can also be viewed as a matrix multiplication:

      \[\begin{equation*} \left[ \begin{array}{c} \vdots \\ y[-1] \\ y[0] \\ y[1] \\ \vdots \end{array} \right] = \left[ \begin{array}{ccccc} & \vdots & \vdots & \vdots \\ \cdots & h[0] & h[-1] & h[-2] & \cdots \\ \cdots & h[1] & h[0] & h[-1] & \cdots \\ \cdots & h[2] & h[1] & h[0] & \cdots \\ & \vdots & \vdots & \vdots \\ \end{array} \right] \left[ \begin{array}{c} \vdots \\ x[-1] \\ x[0] \\ x[1] \\ \vdots \end{array} \right] \end{equation*}\]

  • Algebraic properties of convolution

    • Commutativity: \(x_1[n]*x_2[n] = x_2[n]*x_1[n]\)

    • Associativity: \(x_1[n]*(x_2[n]*x_3[n]) = (x_1[n]*x_2[n])*x_3[n]\)

    • Distributivity: \((\alpha x_1[n]+ \beta x_2[n])*h[n] = \alpha x_1[n]*h[n] + \beta x_2[n]*h[n]\)

    • Identity: \(x[n] * \delta[n-n_0] = x[n-n_0]\)

1.2.3. LTI system, FIR and IIR filters#

  • An LTI system with impulse response \(h[n]\) is causal if and only if \(h[n] = 0\) for all \(n<0\).

    Notation

    It is common to slightly abuse the terminology to call a signal \(x[n]\), which may not be associated with the impulse response of any LTI system, causal if \(x[n]=0\) for all \(n<0\).

  • An LTI system with impulse response \(h[n]\) is (BIBO) stable if and only if \(\sum_{n=-\infty}^{\infty} |h[n]| < \infty\).

  • The cascade of two LTI systems with impulse responses \(h_1[n]\) and \(h_2[n]\) is LTI with impulse response \(h_1[n]*h_2[n]\).

  • FIR filter: A causal finite-impulse-response (FIR) filter is an LTI system that is specified by the difference equation

    \[\begin{equation*} y[n] = \sum_{k=0}^M b_k x[n-k] \end{equation*}\]

    where \(b_0, b_1, \ldots, b_M\) are the filter taps, \(M\) is the filter order, and \(L=M+1\) is the filter length. Clearly, the filter’s impulse response is \(h[n] = \sum_{k=0}^M b_k \delta[n-k]\).

  • IIR filter: A causal infinite-impulse-response (IIR) filter is an LTI system that is specified by the difference equation

    \[\begin{equation*} \sum_{k=0}^N a_k y[n-k] = \sum_{k=0}^M b_k x[n-k] \end{equation*}\]

    where \(a_0=1\), \(a_1,a_2, \ldots, a_N\) are the feedback taps, and \(b_0, b_1, \ldots, b_M\) are the feedforward taps.

    Caution

    \(N\) (not \(M\)) is usually referred to as the order of the IIR filter.