Signals, Systems and Models

Signals are generally real or complex functions of some independent variables (almost always time/or a variable denoting the outcome of a probabilistic experiment). Signals can be:

• $1$-dimensional or multi-dimensional
• continuous-time (CT) or discrete-time (DT)
• deterministic or stochastic

A DT deterministic time-signal can be denoted by a function $x[n]$ of the integer time (or clock) variable $n$

Systems are collections of software or hardware elements, components, subsystems. A system can be viewed as mapping a set of input signals to a set of output or response signals. A more general view is that a system is an entity imposing constraints on a designated set of signals, where the signals are not necessarily labeled as inputs or outputs. Any specific set of signals that satisfies the constraints is termed a behavior of the system.

Models are (usually approximate) mathematical or software or hardware or linguistic or other representations of the constraints imposed on designated set of signals by a system. We can consider models as an abstraction of physical systems

System / Model Properties

For a system or model specified as a mapping, we have the following definitions of various properties. They are stated here for the DT case but easily modified for the CT case.

• Linear: The response to an arbitrary linear combination (or "superposition") of inputs signals is always the same linear combination of the individual responses to these signals: $T\left\{a{x}_A[n]+b{x}_B[n]\right\}=aT\left\{x_A[n]\right\}+bT\left\{x_B[n]\right\}$
• Time-Invariant: The response to an arbitrarily translated set of inputs is always the response to the original set, but translated by the same amount: If $x[n]\to y[n]$ then $x[n-n_0]\to y[n-n_0]$ for all $x$ and $x_0$

Note

A time-invariant system is one where the system's behavior and properties do not change over time. Intuitively, this means that if you apply a specific input to the system now or at any other time, the output will respond in the same way, just shifted in time.

• Linear and Time-Invariant (LTI): The system, model or mapping is both linear and time-invariant. See LTI systems
• Memoryless or Algebraic or Non-Dynamic: The outputs at any instant do not depend on values of the inputs at any other instant: $y[n_0]=T\left\{x[n_0]\right\}$ for all $n_0$

Note

For a LTI system, the memoryless property means that its unit impulse response $h[n]=0$ for $n\ne 0$ or $h(t)\ne 0$ for $t\ne 0$. That is, $h[n]=K\delta [n]$ or $h(t)=K\delta (t)$

• Causal: The output at any instant does not depend on future inputs: for all $n_0$, $y[n_0]$ does not depend on $x[n]$ for $n>n_0$.

Note

For a LTI system, if it's causal, then for all $t<0$, its unit impulse response $h(t)=0$

• BIBO Stable: The response to a bounded input is always bounded: $\left|x_n\right|\le M_x<\infty$ for all $n$ implies that $\left|y[n]\right|\le M_y<\infty$ for all $n$;

Note

For a LTI system, this means that its unit pulse response $h[t]$ or $h(t)$ is absolutely summable or absolutely integrable over $(-\infty ,\infty )$

• Invertible: A system is called invertible if it produces distinct output signals for distinct input signals.
• Initial Rest: The system's output is zero before any input is applied.