Time Domain

Signal models

Analog VS Numerical

Analog signals are time continuous signals and the “natural” signals recorded by various sensors. We represents such signals by mathematical function of real variables (usually representing the time), with values in $\mathbb{R}$ or $\mathbb{C}$:

$$ \begin{aligned} x : \mathbb{R} \longrightarrow \mathbb{R}\\ t \mapsto x(t) \end{aligned} $$

The time support (or simply the support) of a signal is given by $$ \text{supp}(s) = \lbrace t ,\ s(t)\neq 0 \rbrace $$

We will consider two particular classes of analog signal:

  1. Periodic signals, which will be a model for oscillatory signals. A signal $x$ is periodic with a period $T$ iff $$ \forall t\quad x(t+T) = x(t) $$
  2. Signal with finite energy and summable (we will say “stable”), which represents realistic physical/natural signals. A signal $x$ is stable with a finite energy iff $x\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$, i.e. $$ \int_{\mathbb{R}} |x(t)|^2\mathrm{d} t < +\infty \text{ and } \int_{\mathbb{R}} |x(t)|\mathrm{d} t < +\infty $$

On a computer, we will consider digital signals. Digital signals are obtained from analog signals by sampling and quantization. However, it will be usefull to studies the more general numerical signals instead of digital signals . Numerical signals depends on a discrete variable $t$ (which usually represent a discrete time). We represent such signals by a mathematical sequence with values in $\mathbb{R}$ or $\mathbb{C}$:

$$ \begin{aligned} x : \mathbb{Z} \longrightarrow \mathbb{R}\\ t \mapsto x[t] = x_t \end{aligned} $$ notice that we will sometimes use the classical mathematical sequence subscipt notation $x_t$ and sometimes the notation $x[t]$.

Numerical signals can be infinite sequences (with an infinite time support), and then cannot be manipulated on a computer. Digital signals are finite numerical sequences. A digital signal of length $N$ with real values will then be a vector of $\mathbb{R}^N$

$$ \boldsymbol{x} = \lbrace x[0],\ldots,x[N-1] \rbrace $$

In this course, we will make the abuse of notation/confusion between a function $x$ and its value $x(t)$. This abuse of notation will help us to make the distinction between continuous signals $x(t)$ and discrete signals $x[t]$ as well as to avoid any confusion between a function (or a sequence) and scalar value.

Signal and systems

Notions of signal and systems are very linked together. A signal $x$ is recorded through a “system”, and we have access to the “measured” signal $y$. We will focus here on “Linear Time invariant systems”, also known as “filters”.

If we denote by $\mathcal{X}$ and $\mathcal{Y}$ the respective spaces of the original signal $x$ and its recorded version $y$, a system $\mathcal{S}$ is a functional: $$ \begin{aligned} \mathcal{S} : \mathcal{X} & \longrightarrow \mathcal{Y}\\ x & \mapsto y=\mathcal{S}(x) \end{aligned} $$

A system $\mathcal{S}$ is linear iff $\forall x_1,x_2$ we have $$ \mathcal{S}(\lambda x_1+x_2) = \lambda \mathcal{S}(x_1) + \mathcal{S}(x_2) $$

A convenient way to deal with linear system is to write it with the helps of an integral for analog signals (and a sum for numerical signals): $$ \mathcal{S}(x)(t) = \int_{\mathbb{R}} h(t,u)x(u)\mathrm{d}u $$ where the function of two variables $h$ determines the system.

A time invariant system will depends only on the time difference between two dates: $h(t,u) = H(t-u)$. Hence, linear time-invariant systems read $$ \begin{aligned} \mathcal{S}(x)(t) & = \int_{\mathbb{R}} h(t-u)x(u)\mathrm{d}u\\ & = (h*x)(t) \end{aligned} $$ where we recognize the convolution product .

We summarize everything in the following definitions

filter A filter is a linear time-invariant system. It is perfectly determined by its impulse response $h$.

  1. Analog filter with impulse response $h(t)$. The output $y(t)$ of an analog filter with input (also called excitation) $x(t)$ is obtained by $$ y(t) = (h*x)(t) = \int_{\mathbb{R}} h(t-u)x(u)\mathrm{d}u $$
  2. Numerical filter with impulse response $h[t]$. The output $y[t]$ of an analog filter with input (also called excitation) $x[t]$ is obtained by $$ y[t] = (h*x)[t] = \sum_{k\in\mathbb{Z}} h[t-k]x[k] $$

Vocabulary

We define here some vocabulary usefull for signal processing.

Causality A signal is causal iff its start at (or after) the date $t=0$.

  1. An analog signal $x(t)$ is causal iff $\forall t<0$, $x(t) = 0$
  2. A numerical signal $x[t]$ is causal iff $\forall t<0$, $x[t] = 0$ A system is causal iff its impulse response is causal

Anti-causality A signal is anti-causal iff its stop a before the date $t=0$.

  1. An analog signal $x(t)$ is anti-causal iff $\forall t\geq 0$, $x(t) = 0$
  2. A numerical signal $x[t]$ is anti-causal iff $\forall t\geq 0$, $x[t] = 0$ A system is anti-causal iff its impulse response is anti-causal

A signal, or system, which is not causal or anti-causal is acausal

For systems, causality implies that the output does not depends on the futur of the input. Let $h(t)$ be the impulse response of a causal analog system. Then $$ \begin{aligned} y(t) & = (h*x)(t) = \int_\mathbb{R} h(t)x(t-u)\mathrm{d} u\\ & =\int_{0}^{+\infty} h(t)x(t-u)\mathrm{d} u\quad \text{ because } h \text{ is causal} \end{aligned} $$ Similarly, for numerical system we get $$ y[t] = \sum_{u=0}^{+\infty} h[k]x[t-u] $$ In both cases, only the values of $x$ before the date $t$ are necessary to obtain $y(t)$ (or $y[t]$). If $h$ is not causal, then one needs the knowledge of $x(t+u)$ with $u>0$, ie the futur of $x$ has to known to obtain $y(t)$.

Stability A signal a stable iff it is summable.

  1. An analog signal $x(t)$ is stable iff $\Vert x\Vert_1 = \int_{\mathbb{R}} |x(t)| \mathrm{d}t < +\infty$
  2. A numerical signal $x[t]$ is causal iff $\Vert x\Vert_1 =\sum\limits_{n\in\mathbb{Z}} |x(t)| < +\infty$ A system is stable iff its impulse response is stable.

Energy The energy of a signal is its squared $2$-norm

  1. The energy of an analog signal $x(t)$ is $\Vert x\Vert_2^2 = \int_{\mathbb{R}} |x(t)|^2 \mathrm{d}t$
  2. The energy of a numerical signal $x[t]$ is causal iff $\Vert x\Vert_2^2 =\sum\limits_{n\in\mathbb{Z}} |x(t)|^2$ If the energy is finite, then the signal is said to be of finite energy.

Realizable signal A signal is realizable iff it is causal and realizable. A system is realizable iff its impulse response is realizable.

This notion of “realizable” signal correspond to “realistic/physicaly plausible” analog signals, and is even more important for systems. A non-realizable analog system cannot be physically implemented. For numerical system, the notion of realizable system is synonymous of “can be implemented in real time”.