System Fundamentals

Question 1

What is a

System?

Answer 1

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.

Question 2

What is Realization?

Answer 2

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

Question 3

Excitation Signals

Answer 3

Signals that are applied at system inputs

Question 4

Response Signals

Answer 4

Signals that are produced at system outputs

Question 5

System Classifications:

Major Classification Categories/Axes (8)

Answer 5

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

Question 6

System Classifications:

Linear vs Non-Linear

Basic Difference

Answer 6

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.

Question 7

Linear Systems:

Important Properties

Answer 7

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

Question 8

Linear Systems:

Additive Property

Answer 8

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

Question 9

Linear Systems:

Homogeneity Property

Answer 9

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

Question 10

Linear Systems:

Superposition Principle

Answer 10

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>}​

Question 11

System Classifications:

Time Invariant

vs

Varying Time Systems

Answer 11

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

Question 12

System Classifications:

Causal

vs

Non-Causal

Systems

Answer 12

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.

Question 13

How can

Non-Causal Systems

be realized?

Answer 13

• 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

Question 14

What is an important use

for Non-Causal systems?

Answer 14

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.

Question 15

System Classifications:

Continuous Time

vs

Discrete Time

Systems

Answer 15

Continuous Time Systems

Operate with continuous time input and output signals

Discrete Time Systems

Operate with discrete time input and output signals

Question 16

System Classfications:

Analog

vs

Digital

Systems

Answer 16

Analog Systems

Use analog input/output signals - infinite values in a range

Digital Systems

Use digital input/outuput signals - restricted set of values

Question 17

System Classifications:

Instantaneous

vs

Finite Memory Systems

vs

Dynamic Systems

Answer 17

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

Question 18

System Classfications:

Finite Memory Systems

Answer 18

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

Question 19

System Classifications:

Dynamic Systems

Answer 19

In Dynamic Systems

The system’s output depends on the entire past input.

For example: a state machine

Question 20

Two types of systems that

are ALWAYS

causal

Answer 20

All Analog Circuits

All Memoryless Systems

Question 21

Systems Classification:

Invertible

vs

Non-Invertible Systems

Answer 21

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.

Question 22

Linearity:

Show that the system described by the equation:

dy/dt + 3y(t) = x(t)

is Linear

Answer 22

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.

Question 23

Nonlinear Systems:

Small Signal Analysis

Answer 23

• 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

Question 24

System Analysis:

Representing

Continuous Time Systems

vs

Discrete Time Systems

Mathematically

Answer 24

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)

Question 25

System Classifications:

Stable

vs

Unstable Systems

Answer 25

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