Signals & Systems Classification

Flashcards from Signals & Systems lecture notes.

Continuous-time signal

A signal that is specified for a continuum of values of time t

Discrete-time signal

A signal that is specified only at discrete time values

Analogue signal

A signal whose amplitude can take on any value in a continuous range

Digital signal

A signal whose amplitude can take on only a finite number of values

Periodic signal

A signal 𝑓(𝑡) is said to be periodic if for some positive constant 𝑇0, 𝑓(𝑡) = 𝑓(𝑡 + 𝑇0) for all t. By definition, it must start at 𝑡 = −∞ and continue forever.

Aperiodic signal

A signal is not periodic.

Energy signal

A signal with finite energy.

Power signal

A signal with finite and nonzero power.

Deterministic signal

A signal is deterministic if there is no uncertainty with respect to its value at any instant and can be defined exactly by a mathematical formula.

Non-deterministic signal

A signal is non-deterministic if there is uncertainty with respect to its value and can only be represented by probabilistic description.

Even signal

A signal is said to be even when it satisfies the condition 𝑓(𝑡) = 𝑓(−𝑡)

Odd signal

A signal is said to be odd when it satisfies the condition 𝑓(𝑡) = −𝑓(−𝑡)

Unit Step Function

Defined as 𝑢(𝑡) = 1 for 𝑡 ≥ 0, and 0 for 𝑡 < 0. Useful for describing causal signals.

Unit Impulse Function

Also called the Dirac delta function, defined as 𝛿(𝑡) = 0 for 𝑡 ≠ 0, and the integral from −∞ to ∞ of 𝛿(𝑡) 𝑑𝑡 = 1.

Multiplication by j

Moves sample real to imaginary register, or if already imaginary, moves sample back to real register and flips the sign.

Division by j

Similar to multiplication by j, but also flips the sign when moving from real to imaginary.

Complex Conjugate

Two complex numbers whose product is entirely real.

Real Exponential Signal

Continuous-time (real) exponential signal defined as 𝑥(𝑡) = 𝐴𝑒^(𝜔𝑡), where A is the initial value and 𝜔 defines the rate of decay or growth.

Complex Exponential Signals

Using Euler’s relation, 𝐴𝑒𝑗𝜔𝑡 = 𝐴 cos 𝜔𝑡 + 𝑗𝐴 sin 𝜔𝑡

System Model F(x)

The rules of operation that describe its behavior as a “system”.

Linear System - Homogeneity/Scaling

A system where, given that 𝑥(𝑡) produces 𝑦(𝑡), then the scaled input 𝑎 ⋅ 𝑥(𝑡) must produce the scaled output 𝑎 ⋅ 𝑦(𝑡) for an arbitrary 𝑥(𝑡) and 𝑎.

Linear System - Additivity

Given input 𝑥1(𝑡) produces output 𝑦1(𝑡) and input 𝑥2(𝑡) produces output 𝑦2(𝑡), then the input 𝑥1(𝑡) + 𝑥2(𝑡) must produce the output 𝑦1(𝑡) + 𝑦2(𝑡) for arbitrary 𝑥1(𝑡) and 𝑥2(𝑡).

Principle of Superposition

Linear systems obey the principle of superposition, i.e., they obey associative and distributive rules; combines scaling and additivity.

Time-invariant System

The system does not change over time; a time shift in the input causes an identical time shift in the output.

Systems with Memory

A system is said to have memory if the output at an arbitrary time 𝑡 = 𝑡∗ depends on input values other than, or in addition to, 𝑥(𝑡∗).

Causal System

A system for which the output at any instant 𝑡0 depends only on the value of the input 𝑥(𝑡) for 𝑡 ≤ 𝑡0.

Lumped Parameter Systems

Parameter is constant through the process and can be treated as a “point” in space

Distributed Parameter Systems

Quantities depend on both time and space.

Continuous Time Linear Dynamical System

𝑥ሶ(𝑡) = 𝐴(𝑡)𝑥(𝑡) + 𝐵(𝑡)𝑢(𝑡), 𝑦(𝑡) = 𝐶(𝑡)𝑥(𝑡) + 𝐷(𝑡)𝑢(𝑡)

Linear Time-Invariant Linear Dynamical System

𝑥ሶ(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡), 𝑦(𝑡) = 𝐶𝑥(𝑡) + 𝐷𝑢(𝑡)