Continous-Time Signals

Discrete-Time Signals

Analog Signals

Digital Signals

Signals

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, $\hat{\delta}(n)=\delta(n-1)$



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