deﬁnition examples properties memory invertibility...

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

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 thesystem or cause the system to respond in some way, resulting in othersignals as outputs.

• All of the systems that we will consider have a single input and asingle 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 signalx(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]


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 thesystem

• 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


Memory

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

• Memory indicates the system has the means to store informationabout the input from the past or future

• Capacitors and inductors store energy and therefore create systemswith memory

• Resistors have no such mechanism and are therefore memorylesssystems: v(t) = Ri(t)


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]


Example 1: Workspace


Invertibility

h[n]x[n]y[n]

g[n] x[n]

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

• If the system is invertible, then an inverse system exists

• When the inverse system is cascaded with the original system, theoutput is equal to the input

• Normally you can test for invertibility by trying to solve for theinverse system

• Alternatively, if you can find two input signals, x1(t) �= x2(t) thatboth generate the same output, the system is not invertible


Example 2: Invertible Systems

Determine which of the following are invertible systems. If the systemhas an inverse, state what it is.

• y[n] = x[n]2

• y(t) = x(t − 2)

• y[n] = x[n + 3]


• y(t) =∫ t

−∞ x(τ) dτ

• y(t) = dx(t)dt

• y[n] =∑n

k=−∞ x[k]


Causality

Causal: A system is causal if and only if the output y(t) at any timet0 depends only on values of the input x(t) at the present time andpossibly the past, −∞ < t < t0.

• These systems are sometimes (rarely) called nonanticipative

• If two inputs to a causal system are identical up to some point intime, 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


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) =∫ t

−∞ x(τ) dτ

• y(t) =∫ ∞

tx(τ) dτ

• y(t) = dx(t)dt

• y[n] = 111

∑5k=−5 x[n + k]


Stability

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

• Informally, stable systems are those in which small inputs do notlead to outputs that diverge (grow without bound)

• All physical circuits are technically stable

• Ideal op amp circuits without negative feedback are usuallyunstable

• Examples: thermostat, cruise control, swing

• Counter-examples: savings accounts, inverted pendulum(questionable), chain reactions


Example 4: System Stability

Determine which of the following are BIBO stable systems. If thesystem is not BIBO stable, specify an input signal that violates thisproperty.

• y[n] = x[n]2

• y(t) = x(t − 2)

• y[n] = x[n + 3]


• y(t) =∫ t

−∞ x(τ) dτ

• y(t) =∫ ∞

tx(τ) dτ

• y(t) = dx(t)dt

• y[n] = 111

∑5k=−5 x[n + k]


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 inputsignal results in an identical time shift in the output signal

• Circuits that have non-zero energy stored on capacitors or ininductors 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 explicitlyin the system definition, the system is not time-invariant


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 theoriginal 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 istime-invariant. Otherwise, it is not.


Example 5: Time Invariance

Determine which of the following are time-invariant systems. If thesystem is not time invariant, specify an input signal that violates thisproperty.

• y[n] = x[n]2

• y(t) = x(2t)

• y[n] = x[−n]

• y[n] = nx[n + 3]


• y(t) =∫ t

−∞ x(τ) dτ

• y(t) =∫ ∞

tx(τ) dτ

• y(t) = dx(t)dt

• y[n] = 111

∑5k=−5 x[n + k]


Linearity


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.


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


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, theoutput is the same linear combination of the resulting outputs

• This also works for infinite sums (integrals)


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) =∫ t

−∞ x(τ) dτ

• y(t) =∫ ∞

tx(τ) dτ

• y(t) = dx(t)dt

• y[n] = 111

∑5k=−5 x[n + k]


Linear Time-Invariant (LTI) Systems


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

• All of the circuits we will work with are linear

• The circuits may not be time invariant if there is some initialenergy 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


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