Systems Fundamentals Overview Definition Examples Properties - PowerPoint PPT Presentation

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 ) h [ n ] x ( t ) y ( t ) 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

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 t 0 depends only on the input x ( t ) at that same time: x ( t 0 ) . • 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(2 πx ( t )) � t • y ( t ) = −∞ x ( τ ) d τ • y [ n ] = � n k = −∞ x [ k ] J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.06 5

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Invertibility y [ n ] h [ n ] g [ n ] x [ 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, x 1 ( t ) � = x 2 ( 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(2 πx ( t )) � t • y ( t ) = −∞ x ( τ ) d τ • y ( t ) = d x ( t ) d t • y [ n ] = � n k = −∞ x [ k ] J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.06 8

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Causality Causal: A system is causal if and only if the output y ( t ) at any time t 0 depends only on values of the input x ( t ) at the present time and possibly the past, −∞ < t < t 0 . • 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(2 πx ( t )) � t • y ( t ) = −∞ x ( τ ) d τ � ∞ • y ( t ) = x ( τ ) d τ t • y ( t ) = d x ( t ) d t � 5 1 • y [ n ] = k = − 5 x [ n + k ] 11 J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.06 11

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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(2 πx ( t )) � t • y ( t ) = −∞ x ( τ ) d τ � ∞ • y ( t ) = x ( τ ) d τ t • y ( t ) = d x ( t ) d t � 5 1 • y [ n ] = k = − 5 x [ n + k ] 11 J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.06 14

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Time Invariance Time Invariant: A system is time invariant if and only if x [ n ] → y [ n ] implies x [ n − n 0 ] → y [ n − n 0 ] . • 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 − t 0 ) D S x ( t ) → x ( t − t 0 ) → y d ( t ) • To test for time invariance, you should calculate two output signals • First, calculate the delayed output, y ( t − t 0 ) in response to the original signal • Second, calculate the output due to the delayed input, y d ( t ) . • If these are equal for any input signal and delay t 0 , 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 (2 t ) • y [ n ] = x [ − n ] • y [ n ] = nx [ n + 3] • y ( t ) = sin(2 πx ( t )) � t • y ( t ) = −∞ x ( τ ) d τ � ∞ • y ( t ) = x ( τ ) d τ t • y ( t ) = d x ( t ) d t � 5 1 • y [ n ] = k = − 5 x [ n + k ] 11 J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.06 18

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Linearity h ( t ) h [ n ] x ( t ) y ( t ) x [ n ] y [ n ] Consider any two bounded input signals x 1 ( t ) and x 2 ( t ) . x 1 ( t ) → y 1 ( t ) x 2 ( t ) → y 2 ( t ) Linear: A system is linear if and only if a 1 x 1 ( t ) + a 2 x 2 ( t ) → a 1 y 1 ( t ) + a 2 y 2 ( t ) for any constant complex coefficients a 1 and a 2 . J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.06 20

Linearity Continued a 1 x 1 ( t ) + a 2 x 2 ( t ) → a 1 y 1 ( t ) + a 2 y 2 ( t ) a 1 x 1 [ n ] + a 2 x 2 [ n ] → a 1 y 1 [ n ] + a 2 y 2 [ n ] • There are two related properties • Additive: x 1 [ n ] + x 2 [ n ] → y 1 [ n ] + y 2 [ n ] • Scaling: ax 1 [ n ] → ay 1 [ n ] • Scaling is also called the homogeneity property J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.06 21

Linearity Continued � � a k x k ( t ) → a k y k ( t ) k k � U 1 � U 1 a u x u ( t ) d u → a u y u ( t ) d u U 0 U 0 � � a k x k [ n ] → a k y k [ n ] k 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