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signal

A signal is a function of one or more variables that conveys information about some (usually physical) phenomenon.

system

A system is an entity that processes one or more input signals in order to produce one or more output signals.

continuous-time (CT) signal

A signal with continuous independent variables.

discrete-time (DT) signal

A signal with discrete independent variables.

analog signal

A continuous-valued CT signal.

digital signal

A discrete-valued DT signal.

1D signal

A signal with one independent variable is said to be one-dimensional.

multi-dimensional signal

A signal with more than one independent variable is said to be multi-dimensional.

single-input system

A system with one input is said to be single input (SI).

multi-input system

A system with more than one input is said to be multiple input (MI).

Euler's formula

e^(jθ) = cos(θ) + j·sin(θ).

cos in exponential form

cos(θ) = (e^(jθ) + e^(-jθ)) / 2.

sin in exponential form

sin(θ) = (e^(jθ) - e^(-jθ)) / (2j).

complex number (Cartesian form)

z = x + jy, where x = Re(z) and y = Im(z).

complex number (polar form)

z = |z|·e^(jθ), where θ = arg(z).

conjugate of z

For z = x + jy, the conjugate is x - jy.

polynomial function

A function of the form F(z) = a₀ + a₁z + … + aₙzⁿ.

rational function

zero of a function

A point z₀ where F(z₀) = 0.

nth order zero

F(z₀) = F'(z₀) = … = F⁽ⁿ⁻¹⁾(z₀) = 0.

pole

A point z₀ where 1/F(z) has a zero.

even signal

A signal x is even if x(t) = x(−t) for all t.

odd signal

A signal x is odd if x(t) = −x(−t) for all t.

conjugate symmetric signal

A signal x is conjugate symmetric if x(t) = x*(−t) for all t.

periodic signal (continuous)

A signal x is periodic with period T if x(t) = x(t+T) for all t, and T > 0.

memoryless (LTI)

A system H is said to be memoryless if, for every real constant t0, Hx(t0) does not depend on x(t) for some t =! t0.

invertible system

A system H is invertible if there exists H⁻¹ such that H⁻¹(Hx) = x for all x.

causal system (LTI)

An LTI system is causal iff h(t) = 0 for all t < 0.

BIBO stability (LTI)

A system H is said to be bounded-input bounded-output (BIBO)

stable if, for every bounded function x, Hx is bounded

|x(t)| < inf means |Hx(t)| < inf

time invariant (TI)

A system is time-invariant iff H[x(t - t₀)] = Hx(t - t₀) for all t, t₀.

additivity

A system satisfies additivity if H(x₁ + x₂) = Hx₁ + Hx₂.

homogeneity

A system satisfies homogeneity if H(ax) = a·Hx.

linearity (superposition)

A system is linear iff H(a₁x₁ + a₂x₂) = a₁Hx₁ + a₂Hx₂.

eigenfunction

A signal x is an eigenfunction of system H if Hx = λx for some scalar λ.