In describing systems, we typically will use the notation
$$x \to \fbox{F} \to y,$$
where $x$ is the input signal, $\fbox{F}$ is the system, and $y$ is the output signal.


The system $F$ can be described as
$$F : X \to Y,$$
where $X$ and $Y$ are sets of functions $x_i$ and $y_i$, where $x_i \in X$ is a function, and $y_i \in Y$ is also a function for some $i \in \Z$.


The domain and codomain of a function $x \in X$ determines the type of signal. The input signal $x$ can be defined by $x : A \to B$. 


When we refer to $x$, we mean a function in entirety. $x(t)$ refers to a specific value of $x$ at $t$.

Systems

The \emph{Kronecker Delta} function, also known as \emph{Discrete-Time Impulse Signal} (aka Unit Impulse).

The delta function can be defined by: 
$$
\delta(n) = \left\{ \begin{array}{rl}
 1 & n = 0 \\
  0 &\mbox{ otherwise}
       \end{array} \right.
$$

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$\delta$ is important because any discrete-time signal can be expressed as a linear combination of \emph{scaled} and \emph{shifted} versions of it. For example, $\hat{\delta}(n)=\delta(n-1)$



Kronecker Delta

Week 1

\begin{definition}
If the function $x \in X$ is defined as $x : A \to B$, and $A$ is uncountable, then we say that the signal $x$ is a \emph{Continuous-Time Signal} (a.k.a. CT Signal).
\end{definition}

An example of a CT signal:
$$x : \mathbb{R} \to \mathbb{R}$$


In describing a system $\fbox{F}$ that uses CT signals, $F$ can be defined as $F : \left[ \mathbb{R} \to \mathbb{R} \right] \to \left[ \mathbb{R} \to \mathbb{R} \right]$, which means the signal is a \emph{real-valued} continuous-time signal. $F$ can also be defined as $F : \left[ \mathbb{R} \to \mathbb{C} \right] \to \left[ \mathbb{R} \to \mathbb{C} \right]$, which means the signal is a \emph{complex-valued} CT signal. Here, $\left[ \mathbb{R} \to \mathbb{R} \right] \cup \left[ \mathbb{R} \to \mathbb{C} \right] $ represents the set of all CT Signals.

Continous-Time Signals

\begin{definition}
If the function $x \in X$ is defined as $x : A \to B$, and $A$ is countable, then we say that the signal $x$ is a \emph{Discrete-Time Signal}(a.k.a. DT signal).
\end{definition}

An example of a DT signal: 
$$x : \mathbb{Z_+} \to \mathbb{R}$$


In describing a system $\fbox{F}$ that uses DT signals, $F$ can be defined as $F : A \to B$, where the sets $X$ and $Y$, for example, can be the set of functions  $\left[ \mathbb{Z} \to \mathbb{C} \right]$, or $\left[ \mathbb{Z} \to \mathbb{R} \right]$.

Discrete-Time Signals

\begin{definition}
If the function $x \in X$ is defined as $x : A \to B$, and both $A$ and $B$ are uncountable, then we say $x$ is an \emph{analog signal}. 
\end{definition}

An example of an analog signal:

$$x : \mathbb{R} \to (0,1)$$

This signal is analog because the interval between 0 and 1 is not countable (which can be proved using Cantor's Diagonalization argument).



Analog Signals

\begin{definition}
If the function $x \in X$ is defined as $x : A \to B$, and $A$ is countable, $B$ is finite, then we say the signal $x$ is a \emph{digital signal}. 
\end{definition}

An example of a digital signal:
$$x : \mathbb{Z} \to \{ 0, 1 \}$$

1.1 Signals