A system's behavior can be mathematically modeled and is represented in the time domain as h(t) and in the frequency domain as H(s), where s is a complex number in the form of s=a+ib, or s=a+jb in electrical engineering terms (electrical engineers use "j" instead of "i" because current is represented by the variable i).
The Laplace transform is: and the inverse Laplace transform, if all the singularities of X(s) are in the left half of the complex plane, is: Bode plots are plots of magnitude vs. frequency and phase vs. frequency for a system.
All real world signals can be represented as an infinite sum of sinusoidal functions via a Fourier series.
A sinusoidal function can be represented in terms of an exponential by the application of Euler's Formula.
An impulse (Dirac delta function) is defined as a signal that has an infinite magnitude and an infinitesimally narrow width with an area under it of one, centered at zero.
It is not, in reality, possible to generate such a signal, but it can be sufficiently approximated with a large amplitude, narrow pulse, to produce the theoretical impulse response in a network to a high degree of accuracy.
The impulse response defines the system because all possible frequencies are represented in the input A unit step function, also called the Heaviside step function, is a signal that has a magnitude of zero before zero and a magnitude of one after zero.
The step response shows how a system responds to a sudden input, similar to turning on a switch.
The step response can be multiplied with other signals to show how the system responds when an input is suddenly turned on.
Time-invariance means it doesn't matter when you start a system, the same output will result.
All systems have some dependence on things like temperature, signal level or other factors that cause them to be non-linear or non-time-invariant, but most are stable enough to model as LTI.
Linearity and time-invariance are important because they are the only types of systems that can be easily solved using conventional analog signal processing methods.