Chapter 2 Flashcards
(19 cards)
What is a mapping?
A mapping is a relationship involving two sets that associates each element in one set, called the domain, with an element from the other set, called the codomain.
What is a function?
a mapping where the domain is a set that is continuous in nature, such as the real numbers or complex numbers. Functions are also commonly referred to as continuous-time (CT) signals.
What is a sequence?
a mapping where the domain is a set that is discrete in nature, such as the integers, or a subset thereof. Sequences are also commonly referred to as discrete-time (DT) signals.
Notation for functions vs numbers:
f is a function and f(t) is a number.
What is a domain? A codomain?
A domain is the set of possible inputs you have before mapping and a codomain is the set of possible outputs you get after mapping.
What is a rational number?
A number of the form x/y, where x and y are
integers and y ̸= 0 (i.e., a ratio of integers).
What is a system operator?
a mapping used to represent a system. A (single-input single-output) system operator maps a function or sequence representing the input of a system to a function or sequence representing the output of the system.
What are the domain and codomain of systems sets of?
functions or sequences, not sets of numbers
How would you write brackets for an operator H, a function x, and a real number t
Hx instead of the equivalent expression H(x); and Hx(t) instead of the equivalent expression H(x)(t).
What are transforms?
A type of mapping that transforms map functions/sequences to functions/sequences
Definition of even function x:
x(t)=x(-t)
Definition of odd function x:
x(t)=-x(-t)
Symmetry of even functions:
the graph of an even signal is symmetric with respect to the vertical axis
Symmetry of odd functions:
the graph of an odd signal is symmetric with respect to the origin
definition of conjugate symmetry:
x(t) = x^∗(−t)
Definition of a periodic function:
x(t) = x(t +T) for all t (where t is a real number)
What is a non-periodic function called?
aperiodic
What is the formula for frequency and angular frequency?
A T-periodic function x is said to have frequency 1/T and angular frequency 2π/T.
What is the fundamental period and fundamental frequency?
The smallest period with which a signal is periodic and its corresponding frequency
