Throughout the course, we will visualize systems using block diagrams, such as the one below.
Block diagram illustrating a generic system
In this diagram, the $x(t)$ is the input, which the system manipulates, and $y(t)$ is the output. For general systems, we represent a system by a function $\mathcal{H}\{ \cdot \}$ that operates on signals. Therefore, the system above can be formally expressed as
$$y(t) = \mathcal{H}\{ x(t) \}.$$
Note that this notation will disappear when we start focusing on linear, time-invariant in the next class.
Connecting Systems
We can connect systems in series, in parallel, or with feedback. A series connection is shown below and it can be mathematically described by
Block diagram illustrating two systems in series
A parallel connection is shown below and it can be mathematically described by
Block diagram illustrating two systems in parallel
A feedback connection is shown below and it can be mathematically described by
Block diagram illustrating two systems with feedback
We can build many new systems by chaining together other systems in series, parallel, and with feedback.
General System Properties
Linear / nonlinear
A system is linear if it obeys the law of superposition. Formally, if $\mathcal{H}\{ \cdot \}$ is a system with input $x_1(t)$ or $x_2(t)$ and output $y_2(t)$ or $y_1(t)$ such that
$$y_1(t) = \mathcal{H}\{ x_1(t) \} \\
y_2(t) = \mathcal{H}\{ x_2(t) \} ,$$
then the system is linear if
$$\begin{eqnarray}
y(t) &=& \mathcal{H}\{ a x_1(t) + b x_2(t) \} \\
&=& a \mathcal{H}\{ x_1(t) \} + b \mathcal{H} \{ x_2(t) \} \\
&=& a y_1(t) + b y_2(t) ,
\end{eqnarray} $$
where $a$ and $b$ are arbitrary scalar numbers.
Block diagrams illustrating the linearity property
Time-invariant / time-varying
A system is time-invariant if the system does not change with time.
Formally, if $\mathcal{H}\{ \cdot \}$ is a system at time $t$ with input $x(t)$ and output $y(t)$,
such that $y(t) = \mathcal{H}\{ x(t)\}$, then the system is time-invariant if
$$y(t + \tau) = \mathcal{H}\{ x(t + \tau) \} $$
for any arbitrary time delay $\tau$. That is, if we delay our input, we expect the output to be delayed by the same amount.
In a time-invariant system, an input delay may not affect the output signal in any other way.
Block diagrams illustrating the time-invariance property
Memoryless / with memory
A system $H \{ \cdot \}$ is instantaneous (or memoryless) if its
$H\{x[n]\}$ is a function of $x[n]$ at only time $n$
This is equivalent to stating that
$H\{x[n]\}$ is not a function of $x[n-N]$ for $N>0$ (past values)
$H\{x[n]\}$ is not a function of $x[n+N]$ for $N>0$ (future values)
A system that is not memoryless is said to be dynamic or have memory. A system with memory must "remember" previous inputs / future inputs.
Causal / anticausal / acausal
A system $H \{ \cdot \}$ is causal (or physical or non-anticipative) if
$H\{x[n]\}$ is only a function of $x[n-N]$ for $N\geq 0$ (past and current values)
A system is anticausal if
$H\{x[n]\}$ is only a function of $x[n-N]$ for $N < 0$ (future values)
A system is acausal if
$H\{x[n]\}$ is a function of $x[n-N]$ for any $N$ (past and future values)
A system is BIBO stable if any amplitude-bounded input (i.e., the signal $x(t)$ never reaches or approaches $\infty$ for all $t$) yields and amplitude-bounded output (i.e., the signal $y(t)$ never reaches or approaches $\infty$ for all $t$).
Mathematically, this is written as
$$ x[n] < \infty \qquad \rightarrow \qquad H \{ x[n] \} < \infty $$
