Systems Fundamentals Overview • Definition • Examples • Properties ...

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 x(t)

h(t)

y(t)

x[n]

h[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

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

3

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

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)) t • y(t) = −∞ x(τ ) dτ n • y[n] = k=−∞ x[k]

J. McNames

Portland State University

ECE 222

System Fundamentals

Ver. 1.06

5

Example 1: Workspace

J. McNames

Portland State University

ECE 222

System Fundamentals

Ver. 1.06

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Invertibility x[n]

h[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 output 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

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)) t • y(t) = −∞ x(τ ) dτ • y(t) = • y[n] =

J. McNames

dx(t) dt n k=−∞

x[k]

Portland State University

ECE 222

System Fundamentals

Ver. 1.06

8

Example 2: Workspace

J. McNames

Portland State University

ECE 222

System Fundamentals

Ver. 1.06

9

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

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)) t • y(t) = −∞ x(τ ) dτ ∞ • y(t) = t x(τ ) dτ • y(t) =

dx(t) dt

• y[n] =

1 11

J. McNames

5 k=−5

x[n + k]

Portland State University

ECE 222

System Fundamentals

Ver. 1.06

11

Example 3: Workspace

J. McNames

Portland State University

ECE 222

System Fundamentals

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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

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)) t • y(t) = −∞ x(τ ) dτ ∞ • y(t) = t x(τ ) dτ • y(t) =

dx(t) dt

• y[n] =

1 11

J. McNames

5 k=−5

x[n + k]

Portland State University

ECE 222

System Fundamentals

Ver. 1.06

14

Example 4: Workspace

J. McNames

Portland State University

ECE 222

System Fundamentals

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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

Testing for Time Invariance S

D

x(t) → y(t) → y(t − t0 ) D

S

x(t) → x(t − t0 ) → 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 original 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

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)) t • y(t) = −∞ x(τ ) dτ ∞ • y(t) = t x(τ ) dτ • y(t) =

dx(t) dt

• y[n] =

1 11

J. McNames

5 k=−5

x[n + k]

Portland State University

ECE 222

System Fundamentals

Ver. 1.06

18

Example 5: Workspace

J. McNames

Portland State University

ECE 222

System Fundamentals

Ver. 1.06

19

Linearity x(t)

h(t)

y(t)

h[n]

x[n]

y[n]

Consider any two bounded input signals x1 (t) and x2 (t). x1 (t) → x2 (t) →

y1 (t) y2 (t)

Linear: A system is linear if and only if a1 x1 (t) + a2 x2 (t) → a1 y1 (t) + a2 y2 (t) for any constant complex coefficients a1 and a2 .

J. McNames

Portland State University

ECE 222

System Fundamentals

Ver. 1.06

20

Linearity Continued a1 x1 (t) + a2 x2 (t)

→ a1 y1 (t) + a2 y2 (t)

a1 x1 [n] + a2 x2 [n]

→ a1 y1 [n] + a2 y2 [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

21

Linearity Continued  

ak xk (t) →

k U1

U0

ak yk (t)

k



au xu (t) du





→

U1

au yu (t) du

U0

ak xk [n] →



k

ak yk [n]

k

• Linear systems enable the application of superposition • If the input consists of a linear combination of different inputs, the output 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

22

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)) t • y(t) = −∞ x(τ ) dτ ∞ • y(t) = t x(τ ) dτ • y(t) =

dx(t) dt

• y[n] =

1 11

J. McNames

5 k=−5

x[n + k]

Portland State University

ECE 222

System Fundamentals

Ver. 1.06

23

Example 6: Workspace

J. McNames

Portland State University

ECE 222

System Fundamentals

Ver. 1.06

24

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

h(t)

y(t)

x[n]

h[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