Capter 3.4-3.5 Flashcards
(17 cards)
What is the equation of a system?
y = Hx
What is a system connected in series (or cascade)?
A series (or cascade) connection ties the output of one system to the input of the other. A series-connected system is described by the equation:
y = H2(H1(x))
What is a system connected in parallel?
A parallel connection ties the inputs of both systems together and sums
their outputs. A parallel-connected system is described by the equation:
y = H1x+H2x
Define a memoryless system
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. In other words, a memoryless system is such that the value of its output at any given point in time can depend on the value of its input at only the same point in time.
A system that is non memoryless is said to have….
memory!
What is a causal system?
A system H is said to be causal if, for every real constant t0, Hx(t0) does not depend on x(t) for some t > t0. In other words, a causal system is such that the value of its output at any given point in time can depend on the value of its input at only the same or earlier points in time (i.e., not later points in time).
How can a system be invertible? What is an inverse system?
A system is said to be invertible if it has a corresponding inverse system (i.e., its inverse exists). The inverse of a system H (if it exists) is another system H^(−1) such that, for every function x, H^(−1)Hx = x
How do you find out if a system is invertible?
To show that a system is invertible, we simply find the inverse system. To show that a system is not invertible, we find two distinct inputs that result in identical outputs (i.e., x1 ̸= x2 and Hx1 = Hx2).
Define a bounded-input bounded-output (BIBO) stable system
A system H is said to be bounded-input bounded-output (BIBO) stable if, for every bounded function x, Hx is bounded (i.e., |x(t)| < ∞ for all t implies that |Hx(t)| < ∞ for all t).
In other words, a BIBO stable system is such that it guarantees to always produce a bounded output as long as its input is bounded.
How do you find if a system is BIBO stable?
To show that a system is not BIBO stable, we only need to find a single bounded input that leads to an unbounded output. In practical terms, a BIBO stable system is well behaved in the sense that, as long as the system input is finite everywhere (in its domain), the output
will also be finite everywhere.
To show that a system is BIBO stable, you must first start with a bounded input |x(t)| ≤ A where A is a finite constant. Then you have to manipulate the inequality to be of the form |Hx(t)| ≤ B where B is a finite constant.
For an example: https://youtu.be/C53uS3lkeJQ?t=2266&list=PLbHYdvrWBMxYGMvQ3QG6paNu7CuIRL5dX
What is time invariance?
a system is time invariant if a time shift (i.e., advance or delay) in the input always results only in an identical time shift in the output. A system H is said to be time invariant (TI) (or shift invariant (SI)) if, for every function x and every real constant t0, the following condition
holds:
Hx(t −t0) = Hx′(t) for all t, where x′(t) = x(t −t0)
(i.e., H commutes with time shifts).
What is a system that is time varying?
A system that is not time invariant
What is an additive system?
A system H is said to be additive if, for all functions x1 and x2, the following condition holds:
H(x1 +x2) = Hx1 +Hx2
(i.e., H commutes with addition).
What is a homogeneous system?
A system H is said to be homogeneous if, for every function x and every complex constant a, the following condition holds:
H(ax) = aHx
(i.e., H commutes with scalar multiplication).
What is a linear system?
A system that is both additive and homogeneous.
In other words, a system H is linear, if for all functions x1 and x2 and all complex constants a1 and a2, the following condition holds:
H(a1x1 +a2x2) = a1Hx1 +a2Hx2
(i.e., H commutes with linear combinations).
What is the superposition property?
Another name for the linearity property.
What is an eigenfunction?
A function x is said to be an eigenfunction of the system H with the eigenvalue λ if
Hx = λx,
where λ is a complex constant.
In other words, the system H acts as an ideal amplifier for each of its eigenfunctions x, where the amplifier gain is given by the corresponding eigenvalue λ.
