System Fundamentals Flashcards
What is a
System?
An entity that processes a set of signals,
called inputs,
to yield another set of signals called outputs.
A system can be physical:
- Electrical, Mechanical, Hydraulic, Chemical
It can also be an algorithm that computes an output from an input signal.
What is Realization?
When a system is actually “real”.
Two basic kinds:
Hardware Realization
The system exists as something physical:
- Electrical, Mechanical, Hydraulic, Chemical
Software Realization
The system exists as an algorithm that computes an output from an input signal
Excitation Signals
Signals that are applied at system inputs
Response Signals
Signals that are produced at system outputs
System Classifications:
Major Classification Categories/Axes (8)
Linearity:
Linear vs Non-Linear
Variance:
Time Invariant Parameter vs Time Varying Parameter
Memory:
Instantaneous vs Dynamic
Causality:
Causal vs Non-Causal
Range:
Continuous Time vs Discrete Time
Domain:
Analog vs Digital
Invertibility:
Invertible vs Non-Invertible
Stability
Stable vs Unstable
System Classifications:
Linear vs Non-Linear
Basic Difference
A system is Linear when it has the property of “Superposition”:
The total response is just the sum of the “component” responses.
A system is Non-Linear if superposition does not hold.
Linear Systems:
Important Properties
Additive Property
- Sum of inputs results in sum of outputs
Homogeneity (Scaling) Property
- Output will be scaled by the same factor as input
Superposition
- Combines the other two properties
- Repsonse is the sum of scaled inputs
Linear Systems:
Additive Property
If a system responds to two different inputs:
X1 → Y1 , X2 → Y2
The response to the input signals added together is the same as adding the individual responses:
X1 + X2 → Y1 + Y2
Linear Systems:
Homogeneity Property
Also called the “Scaling” property
If a system responds to some input:
X → Y
It’s response to the input scaled by some factor is the original output scaled by the same factor
kX → kY
Linear Systems:
Superposition Principle
Defining feature of Linear Systems,
all Linear Systems follow it.
Combines the Additive and Homogeneity Properties.
If a system responds to some inputs:
x1 → y1 and x<span>2</span> → y<span>2</span>
The system reponse to the sum of those inputs, scaled by some factor is equal to the sum of the scaled inputs:
k1x1 + k2x<span>2</span> → k1y1 + k2y<span>2</span>
System Classifications:
Time Invariant
vs
Varying Time Systems
A Time Invariant System
is one whose parameters do NOT change with time.
The response will not be changed depending on when the input is received.
Another way of looking at it:
Delaying either the input signal x(t) or the output signal y(t) by the same amount results in the same output:
x(t) → y(t) → delay by T seconds → y(t - T)
x(t) → delay by T seconds → x(t - T) → y(t - T)
A Time Varying System has parameters that change over time, so the response may change
System Classifications:
Causal
vs
Non-Causal
Systems
Causal Systems
The response is not affected by future values of any signals.
- Output at t0 depends only on x(t) for t ≤ t0
- Any practical, real-time system must be causal
Non-Causal Systems
The response may be different depending on “future” values.
Generally because the independent variable is something other that time.
How can
Non-Causal Systems
be realized?
-
Independent variable is something other than time
- It may be a spatial dimension, such as length
- Data is not in real-time
- Data may be prerecorded and then processed by a non-causal system
What is an important use
for Non-Causal systems?
We might use a non-causal system to study
the upper bound on the performance of a causal system.
This may be done by analyzing the prerecorded data from the causal system, using the non-causal system.
System Classifications:
Continuous Time
vs
Discrete Time
Systems
Continuous Time Systems
Operate with continuous time input and output signals
Discrete Time Systems
Operate with discrete time input and output signals
System Classfications:
Analog
vs
Digital
Systems
Analog Systems
Use analog input/output signals - infinite values in a range
Digital Systems
Use digital input/outuput signals - restricted set of values
System Classifications:
Instantaneous
vs
Finite Memory Systems
vs
Dynamic Systems
Determined by whether the system has “memory”.
Instantaneous Systems
The output is determined solely by the current input
Past input and output is completely irrelevant.
Also called “Memoryless”.
Finite Memory Systems
Response at time t determined by the input of the past T seconds
Dynamic Systems
Systems output at time t depends on the ENTIRE past input
System Classfications:
Finite Memory Systems
In a Finite Memory System
the system’s response is completely determined by the input signals over the past T seconds.
Examples:
Finite State Machine, Networks with induction inductors and capacitors
System Classifications:
Dynamic Systems
In Dynamic Systems
The system’s output depends on the entire past input.
For example: a state machine
Two types of systems that
are ALWAYS
causal
All Analog Circuits
All Memoryless Systems
Systems Classification:
Invertible
vs
Non-Invertible Systems
Invertible Systems
A system is invertible if another system can use its output to produce the initial input.
y(t) can be used to determine x(t)
- There must be a one-to-one mapping
Non-Invertible Systems
Systems where the output cannot be used to determine the input. This may be because multiple inputs map to some outputs.
Linearity:
Show that the system described by the equation:
dy/dt + 3y(t) = x(t)
is Linear
To show that is is Linear, we need to show that Superposition holds.
Let x1(t) → y1(t) and x2(t) → y2(t)
Substitute into equation
dy1/dt + 3y1(t) = x1(t)
dy2/dt + 3y2(t) = x2(t)
Multiply each by a scaling factor
k1dy1/dt + 3 k1y1(t) = k1x1(t)
k2dy2/dt + 3 k2y2(t) = k2x2(t)
Add both equations
d/dt [k1y1(t) + k2y2(t)] + 3[k1y1(t) + k2y2(t)] = k1x1(t) + k2x2(t)
For the combined equation
X(t) = k1x1(t) + k2x2(t)
Y(t) = k1y1(t) + k2y2(t)]
Therefore, Superposition holds and this system is Linear.
Nonlinear Systems:
Small Signal Analysis
- Almost all systems become non-linear when very large signals are applied
- Nonlinear systems can be approximated by linear systems for Small Signal Analysis
- The basic idea is to approximate a nonlinear system as the superposition of MANY small, linear systems
- Once superposition is applied, analyze the system by decomposition into Zero-Input and Zero-State components
System Analysis:
Representing
Continuous Time Systems
vs
Discrete Time Systems
Mathematically
For both systems, inputs are represented with “x”, outputs with “y”.
The Impulse response is represented with a function “h”
In Continuous Time systems, time is represented with “t”:
x(t) → h(t) → y(t)
In Discrete Time Systems, time is represented with “n”, indicating discrete values.
x(n) → h(n) → y(n)
System Classifications:
Stable
vs
Unstable Systems
To analyze stability, we use
BIBO : Bounded Input, Bounded Output
A system is BIBO stable if, and only if,
All bounded inputs: |x(t)| < ∞
result in bounded outputs: |y(t)| < ∞
No finite inputs can produce infinitely large outputs
