Metric spaces

Signal spaces

We start the content of our course by defining its main concepts of a signal and a signal space.

Definition 1 A signal is the value of some quantity as a function of time, space, frequency, etc.; each signal is labeled by a lower-case letter $x$ .

Definition 2 A signal space is a set of signals defined by some criterion, labeled by an upper-case letter $X$ (since it is a set).

Some familiar sets of signals are $X=\mathbb{R}$ , $X=\mathbb{C}$ , and the set of vectors $X={\mathbb{R}}^n$ .

Definition 3 The signal space $L_2[a,b]$ contains all signals $x\left(t\right)$ such that $x\left(t\right)=0$ for all $t<a$ or $t>b$ and ${\int }_a^b{\left|x\left(t\right)\right|}^{2}dt<\infty$ (i.e., at no time the signal is infinite).

Metric spaces

Definition 4 A metric $d:X\times X\to \mathbb{R}$ is a function used to measure distance between pairs of elements of $X$ with the following properties: for all $x,y,z\in X$ ,

1. $d(x,y)=d(y,x)$ (symmetry),
2. $d(x,y)\ge 0$ (non-negativity),
3. $d(x,y)=0\iff x=y$ ,
4. $d(x,z)\le d(x,y)+d(y,z)$ (triangle inequality).

If $d$ is a metric on $X$ , the pair $(X,d)$ is called a metric space . A set $X$ can have multiple metrics, leading to different metric spaces.

Example 1 The following are some initial examples of metric spaces:

• $X=\mathbb{R}$ with $d_0(x,y)=|x-y|$ for all $x,y\in \mathbb{R}$ : it is easy to check properties (1-4).
• $X=\mathbb{R}$ with $d^{\text{'}}(x,y)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}x\ne y\hfill \\ 0\hfill & \mathrm{if}x=y\hfill \end{array}\right)$ for all $x,y\in \mathbb{R}$ : it is easy to check properties (1-3). To verify (4), assume that $d(x,y)+d(y,z)=0$ ; then both $d(x,y)=0$ and $d(y,z)=0$ , which means $x=y$ and $y=z$ ; by transitivity, $x=z$ and $d(x,z)=0\le d(x,y)+d(y,z)$ , as desired. Now assume that $d(x,y)+d(y,z)=1$ ; then we immediately get $d(x,z)\le d(x,y)+d(y,z)$ , as desired.
• $X={\mathbb{R}}^n$ with metric $d_2^n(x,y):={\left({\sum }_{i=1}^n,|,x_i,-,y_i,{|}^2\right)}^{1/2}$ , known as the Euclidean metric.
• $X={\mathbb{R}}^n$ with metric $d_1^n(x,y):=\left({\sum }_{i=1}^n,|,x_i,-,y_i,|\right)$ .
• $X=L_2[a,b]$ with metric $d_2(x,y):={\left({\int }_a^b,{|x\left(t\right)-y\left(t\right)|}^2,d,t\right)}^{1/2}$ . Formally, $d_2$ is a pseudometric on $X$ , since there are signals $x\ne y$ that yield $d_2(x,y)=0$ . However, one can define a new signal space where all signals with $d(x,y)=0$ are equal to each other.
• $X=L_2[a,b]$ with metric $d_p(x,y):={\left({\int }_a^b,{|x\left(t\right)-y\left(t\right)|}^p,d,t\right)}^{1/p}$ , for $1\le p<\infty$ , an extension of the metric $d_2$ .
• $X=L_2[a,b]$ with metric $d_{\infty }(x,y):={sup}_{t\in [a,b]}|x\left(t\right)-y\left(t\right)|$ ; this metric solves the equivalence problem of $d_2$ .

Here, $sup\left(A\right)$ is the supremum of $A$ , i.e., the smallest value $x_0\in \mathbb{R}$ such that $x\le x_0\forall x\in A$ . Similarly, $inf\left(A\right)$ is the infimum of $A$ , i.e., the largest value $x_0$ such that $x_0\le x\forall x\in A$ .