Input-Output System Models

Motivation

Block diagram illustrating a generic system

Throughout the course, we will visualize systems using block diagrams, such as the one shown here. Block diagrams are a straightforward way to illustrate the input and output of a system. In our diagram, the $x(t)$ is the input, which the system manipulates, and $y(t)$ is the output. Mathematically, we represent a system by a function $H\{ \cdot \}$ that operates on signals. Therefore, the system above can be formally expressed as $y(t)=H\{x(t)\}$ . Note that this notation will disappear when we start focusing on linear, time-invariant in the next class.

Connecting Systems

We can build many new systems by chaining together other systems in series, parallel, and with feedback. Each system is shown here.

When in series, the output of one system becomes the input to the second. For example, applying an “auto-tune” system followed by “reverb” system to an audio signal is an example of systems in series.

When in parallel, we use one input for both systems and then add the outputs together. For example, an audio signals may be applied to “bass boosted” system and “treble boosted” system before being added together to get both enhancements.

When in a feedback configuration, one system modifies the inputs while another system modifies the output of the overall system. The outputs of both systems are then added together and become the output. Placing a speaker near a microphone is an example of a feedback system, where audio is from the microphone is fed back into the microphone.

Block diagram illustrating two systems in series — A series connection is mathematically described by $y[n] = H_2 \{ H_1 \{ x[n] \} \}$

A series connection is mathematically described by $y[n] = H_2 \{ H_1 \{ x[n] \} \}$

Block diagram illustrating two systems in parallel — A parallel connection is mathematically described by $y[n] = H_1 \{ x[n} \} + H_2 \{ x[n] \}$

A parallel connection is mathematically described by $y[n] = H_1 \{ x[n} \} + H_2 \{ x[n] \}$

Block diagram illustrating two systems with feedback — A feedback connection is mathematically described by $y[n] = H_1 \{ x[n} \} + H_2 \{ y[n] \}$

A feedback connection is mathematically described by $y[n] = H_1 \{ x[n} \} + H_2 \{ y[n] \}$

General System Properties

Linear / nonlinear

Block diagrams illustrating the linearity property

A system is linear if it obeys the law of superposition. Formally, if $H\{ \cdot \}$ is a system with inputs $x_1(t)$ or $x_2(t)$ and outputs $y_2(t)$ or $y_1(t)$ , such that $y_1(t)=H{x_1(t)}$ and $y_2(t)=H{x_2(t)}$ , then the system is linear if

$y(t)=H\{ax_1(t)+bx_2(t)\} = a H\{x_1(t)\} + b H\{x_2(t)\} = ay_1(t)+by_2(t) \; ,$

where $a$ and $b$ are arbitrary scalar numbers. The diagram on the right illustrates equivalence with block diagrams.

Linearity allows us to separate the systems from their individual inputs. A linear system is like the brightness slider on your phone. Every pixel gets brighter by exactly the same amount, regardless of what’s in the picture. A nonlinear system is like an AI cartoon filter. It doesn’t treat every pixel the same—it recognizes faces, eyes, and hair, then changes them differently. The output depends on the content of the image, not just the value of each pixel.

Time-invariant / time-varying

Block diagrams illustrating the time-invariance property

A system is time-invariant if the system does not change with time. Formally, if $H\{ \cdot \}$ is a system at time $t$ with input $x(t)$ and output $y(t)$ , such that $y(t)=H\{x(t)\}$ , then the system is time-invariant if

$y(t+\tau) = H\{x(t+\tau)\}$

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. The stock market could you considered time-varying system. Putting money in the stock market today versus tomorrow could be the difference between making or losing money.

Memoryless / with memory

A system $H\{ \cdot \}$ is instantaneous (or memoryless) if $H\{x[n]\}$ 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{⋅} is causal (or physical or non-anticipative) if

$H\{x[n]\}$ is only a function of $x[n−N]$ for $N\ge 0$ (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)

All real-time systems are causal systems.

Bounded-input, bounded-output (BIBO) stable / unstable

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

$x[n] < \infty \rightarrow H\{ x[n] \} < \infty$