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Systems Fundamentals Overview Definition Examples Properties Memory Invertibility Causality Stability Time Invariance Linearity J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.06 1

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Systems Fundamentals Overview

  • Definition
  • Examples
  • Properties

– Memory – Invertibility – Causality – Stability – Time Invariance – Linearity

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

1

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Definition of a System h(t)

x(t) y(t)

h[n]

x[n] y[n]

System: a process in which input signals are transformed by the system or cause the system to respond in some way, resulting in other signals as outputs.

  • All of the systems that we will consider have a single input and a

single output

  • All of the signals that we will consider are likewise univariate
  • We will use the notation x(t) → y(t) to mean the input signal

x(t) causes an output signal y(t)

  • h(t) is the impulse response of the continuous-time system:

δ(t) → h(t)

  • h[n] is the impulse response of the discrete-time system:

δ[n] → h[n]

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

2

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Scope of Systems

  • In this class we will primarily work with circuits as systems
  • In most cases a voltage or current will be the input signal to the

system

  • Another current or voltage will be the output signal of the system
  • However, our treatment applies to a much broader class of systems
  • Examples

– Circuits – Motors – Chemical processing plants – Engines – Spring-mass systems

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

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Memory Memoryless: A system is memoryless if and only if the output y(t) at any time t0 depends only on the input x(t) at that same time: x(t0).

  • Memory indicates the system has the means to store information

about the input from the past or future

  • Capacitors and inductors store energy and therefore create systems

with memory

  • Resistors have no such mechanism and are therefore memoryless

systems: v(t) = Ri(t)

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

4

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Example 1: Memoryless Systems Determine whether each of the following systems are memoryless.

  • y[n] = x[n]2
  • y(t) = x(t − 2)
  • y[n] = x[n + 3]
  • y(t) = sin(2πx(t))
  • y(t) =

t

−∞ x(τ) dτ

  • y[n] = n

k=−∞ x[k]

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

5

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Example 1: Workspace

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

6

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Invertibility h[n]

x[n] y[n]

g[n]

x[n]

Invertible: A system is invertible if and only if distinct inputs cause distinct outputs.

  • If the system is invertible, then an inverse system exists
  • When the inverse system is cascaded with the original system, the
  • utput is equal to the input
  • Normally you can test for invertibility by trying to solve for the

inverse system

  • Alternatively, if you can find two input signals, x1(t) = x2(t) that

both generate the same output, the system is not invertible

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

7

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Example 2: Invertible Systems Determine which of the following are invertible systems. If the system has an inverse, state what it is.

  • y[n] = x[n]2
  • y(t) = x(t − 2)
  • y[n] = x[n + 3]
  • y(t) = sin(2πx(t))
  • y(t) =

t

−∞ x(τ) dτ

  • y(t) = dx(t)

dt

  • y[n] = n

k=−∞ x[k]

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

8

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Example 2: Workspace

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

9

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Causality Causal: A system is causal if and only if the output y(t) at any time t0 depends only on values of the input x(t) at the present time and possibly the past, −∞ < t < t0.

  • These systems are sometimes (rarely) called nonanticipative
  • If two inputs to a causal system are identical up to some point in

time, the outputs must also be equal

  • All analog circuits are causal
  • All memoryless systems are causal
  • Not all causal systems are memoryless (very few are)
  • Some discrete-time systems are non-causal
  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

10

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Example 3: Causal Systems Determine which of the following are causal systems.

  • y[n] = x[n]2
  • y(t) = x(t − 2)
  • y[n] = x[n + 3]
  • y(t) = sin(2πx(t))
  • y(t) =

t

−∞ x(τ) dτ

  • y(t) =

∞

t

x(τ) dτ

  • y(t) = dx(t)

dt

  • y[n] =

1 11

5

k=−5 x[n + k]

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

11

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Example 3: Workspace

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

12

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Stability BIBO Stable: A system is bounded-input bounded-output (BIBO) stable if and only if (iff) all bounded inputs (|x(t)| < ∞) result in bounded outputs (|y(t)| < ∞).

  • Informally, stable systems are those in which small inputs do not

lead to outputs that diverge (grow without bound)

  • All physical circuits are technically stable
  • Ideal op amp circuits without negative feedback are usually

unstable

  • Examples: thermostat, cruise control, swing
  • Counter-examples: savings accounts, inverted pendulum

(questionable), chain reactions

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

13

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Example 4: System Stability Determine which of the following are BIBO stable systems. If the system is not BIBO stable, specify an input signal that violates this property.

  • y[n] = x[n]2
  • y(t) = x(t − 2)
  • y[n] = x[n + 3]
  • y(t) = sin(2πx(t))
  • y(t) =

t

−∞ x(τ) dτ

  • y(t) =

∞

t

x(τ) dτ

  • y(t) = dx(t)

dt

  • y[n] =

1 11

5

k=−5 x[n + k]

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

14

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Example 4: Workspace

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

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Time Invariance Time Invariant: A system is time invariant if and only if x[n] → y[n] implies x[n − n0] → y[n − n0].

  • In words, a system is time invariant if a time shift in the input

signal results in an identical time shift in the output signal

  • Circuits that have non-zero energy stored on capacitors or in

inductors at t = 0 are generally not time-invariant

  • Circuits that have no energy stored are time-invariant
  • Memoryless does not imply time-invariant: y(t) = f(t) × x(t)
  • In general, if the independent variable, t or n, is included explicitly

in the system definition, the system is not time-invariant

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

16

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Testing for Time Invariance x(t)

S

→ y(t)

D

→ y(t − t0) x(t)

D

→ x(t − t0)

S

→ yd(t)

  • To test for time invariance, you should calculate two output signals
  • First, calculate the delayed output, y(t − t0) in response to the
  • riginal signal
  • Second, calculate the output due to the delayed input, yd(t).
  • If these are equal for any input signal and delay t0, the system is

time-invariant. Otherwise, it is not.

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

17

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Example 5: Time Invariance Determine which of the following are time-invariant systems. If the system is not time invariant, specify an input signal that violates this property.

  • y[n] = x[n]2
  • y(t) = x(2t)
  • y[n] = x[−n]
  • y[n] = nx[n + 3]
  • y(t) = sin(2πx(t))
  • y(t) =

t

−∞ x(τ) dτ

  • y(t) =

∞

t

x(τ) dτ

  • y(t) = dx(t)

dt

  • y[n] =

1 11

5

k=−5 x[n + k]

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

18

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Example 5: Workspace

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

19

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Linearity h(t)

x(t) y(t)

h[n]

x[n] y[n]

Consider any two bounded input signals x1(t) and x2(t). x1(t) → y1(t) x2(t) → y2(t) Linear: A system is linear if and only if a1x1(t) + a2x2(t) → a1y1(t) + a2y2(t) for any constant complex coefficients a1 and a2.

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

20

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Linearity Continued a1x1(t) + a2x2(t) → a1y1(t) + a2y2(t) a1x1[n] + a2x2[n] → a1y1[n] + a2y2[n]

  • There are two related properties
  • Additive: x1[n] + x2[n] → y1[n] + y2[n]
  • Scaling: ax1[n] → ay1[n]
  • Scaling is also called the homogeneity property
  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

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Linearity Continued

  • k

akxk(t) →

  • k

akyk(t) U1

U0

auxu(t) du → U1

U0

auyu(t) du

  • k

akxk[n] →

  • k

akyk[n]

  • Linear systems enable the application of superposition
  • If the input consists of a linear combination of different inputs, the
  • utput is the same linear combination of the resulting outputs
  • This also works for infinite sums (integrals)
  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

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Example 6: Linearity Determine which of the following are linear systems.

  • y[n] = x[n]2
  • y(t) = x(2t)
  • y[n] = x[−n]
  • y[n] = nx[n + 3]
  • y(t) = sin(2πx(t))
  • y(t) =

t

−∞ x(τ) dτ

  • y(t) =

∞

t

x(τ) dτ

  • y(t) = dx(t)

dt

  • y[n] =

1 11

5

k=−5 x[n + k]

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

23

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Example 6: Workspace

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

24

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Linear Time-Invariant (LTI) Systems h(t)

x(t) y(t)

h[n]

x[n] y[n]

  • A system is said to be linear time invariant (LTI) if it is both

linear and time invariant

  • All of the circuits we will work with are linear
  • The circuits may not be time invariant if there is some initial

energy stored in the circuit

  • Otherwise the circuits are LTI
  • ECE 222 & ECE 223 will focus primarily on the properties,

analysis, and design of LTI systems

  • J. McNames

Portland State University ECE 222 System Fundamentals

  • Ver. 1.06

25