Norm

Also known as the taxicab norm or the Manhattan norm, the 1-norm for Euclidean vectors is the sum of the absolute values of the components:

\begin{align} \| \mathbf{x} \|_1 = \sum_{k=1}^n \left|x_k\right| \end{align}

For DT signals:

\begin{align} \| x \|_1 = \sum_{n=-\infty}^{\infty} \left|x[n]\right| \end{align}

If \(\| x \|_1 < \infty\), we say that the signal is absolutely-summable (also called stable). The set of all absolutely-summable DT signals \(l_1(\mathbb{Z}) = \{x: \mathbb{Z} \rightarrow \mathbb{C} \mid \sum_{n=-\infty}^{\infty} \left| x[n] \right| < \infty\}\) is a vector space.

Also known as the Euclidean norm, the 2-norm for Euclidean vectors is the intuitive length of the vector. Oftentimes, it is also easier to consider the square:

\begin{align} \| \mathbf{x} \|_2 &= \|\mathbf{x}\| = \sqrt{x_1^2 + \cdots + x_n^2} \\ \| \mathbf{x} \|^2 &= \sum_{k=1}^n x_k^2 \end{align}

For DT signals:

\begin{align} \| x \|_2 &= \|x\| = \sqrt{\sum_{n=-\infty}^{\infty}|x[n]|^2} \\ \mathcal{E}_x &= \|\mathbf{x}\|^2 = \sum_{n=-\infty}^{\infty}|x[n]|^2 \end{align}

The set of all square-summable (finite-energy) DT signals \(l_2(\mathbb{Z}) = \{x: \mathbb{Z} \rightarrow \mathbb{C} \mid \sum_{n=-\infty}^{\infty} \left| x[n] \right|^2 < \infty\}\) is also a vector space.

Additionally, the distance between two vectors \(\mathbf{x}, \mathbf{y} \in \mathcal{V}\) is defined as the Euclidean norm of their difference, or \(d(\mathbf{x}, \mathbf{y}) = \|\mathbf{x}-\mathbf{y}\|\).

The \(\mathbf{\infty}\) norm for Euclidean vectors is the maximum of its components:

\begin{align} \| \mathbf{x} \|_{\infty} = \text{max}(|x_1|, \dots, |x_n|) \end{align}

For DT signals, it's the supremum :

\begin{align} \| x \|_{\infty} = \text{sup}(\dots, |x[-1]|, |x[0]|, |x[1]|, \dots) \end{align}

The set of all bounded DT signals \(l_{\infty}(\mathbb{Z}) = \{x: \mathbb{Z} \rightarrow \mathbb{C} \mid \|x\|_{\infty} < \infty\}\) is also a vector space.