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Systems Fundamentals Overview Definition of a System Definition h ( t ) h [ n ] x ( t ) y ( t ) x [ n ] y [ n ] Examples System : a process in which input signals are transformed by the Properties system or cause the system to respond


  1. Systems Fundamentals Overview Definition of a System • Definition h ( t ) h [ n ] x ( t ) y ( t ) x [ n ] y [ n ] • Examples System : a process in which input signals are transformed by the • Properties system or cause the system to respond in some way, resulting in other – Memory signals as outputs. – Invertibility • All of the systems that we will consider have a single input and a – Causality single output – Stability • All of the signals that we will consider are likewise univariate – Time Invariance • We will use the notation x ( t ) → y ( t ) to mean the input signal – Linearity 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. 1.07 1 J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.07 2 Scope of Systems Memory • In this class we will primarily work with circuits as systems 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 ) . • In most cases a voltage or current will be the input signal to the system • Memory indicates the system has the means to store information • Another current or voltage will be the output signal of the system about the input from the past or future • However, our treatment applies to a much broader class of systems • Capacitors and inductors store energy and therefore create systems • Examples with memory – Circuits • Resistors have no such mechanism and are therefore memoryless – Motors systems: v ( t ) = Ri ( t ) – Chemical processing plants – Engines – Spring-mass systems J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.07 3 J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.07 4

  2. Example 1: Memoryless Systems Example 1: Workspace 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. 1.07 5 J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.07 6 Invertibility Example 2: Invertible Systems Determine which of the following are invertible systems. If the system y [ n ] has an inverse, state what it is. h [ n ] g [ n ] x [ n ] x [ n ] • y [ n ] = x [ n ] 2 Invertible: A system is invertible if and only if distinct inputs cause • y ( t ) = x ( t − 2) distinct outputs. • y [ n ] = x [ n + 3] • If the system is invertible, then an inverse system exists • y ( t ) = sin(2 πx ( t )) • When the inverse system is cascaded with the original system, the � t output is equal to the input • y ( t ) = −∞ x ( τ ) d τ • Normally you can test for invertibility by trying to solve for the • y ( t ) = d x ( t ) d t inverse system • y [ n ] = � n k = −∞ x [ k ] • 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. 1.07 7 J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.07 8

  3. Example 2: Workspace 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. 1.07 9 J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.07 10 Example 3: Causal Systems Example 3: Workspace 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. 1.07 11 J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.07 12

  4. Stability Example 4: System Stability Determine which of the following are BIBO stable systems. If the BIBO Stable: A system is bounded-input bounded-output (BIBO) stable if and only if (iff) all bounded inputs ( | x ( t ) | < ∞ ) result in system is not BIBO stable, specify an input signal that violates this property. bounded outputs ( | y ( t ) | < ∞ ). • y [ n ] = x [ n ] 2 • Informally, stable systems are those in which small inputs do not lead to outputs that diverge (grow without bound) • y ( t ) = x ( t − 2) • All physical circuits are technically stable • y [ n ] = x [ n + 3] • Ideal op amp circuits without negative feedback are usually • y ( t ) = sin(2 πx ( t )) unstable � t • y ( t ) = −∞ x ( τ ) d τ • Examples: thermostat, cruise control, swing � ∞ • y ( t ) = x ( τ ) d τ • Counter-examples: savings accounts, inverted pendulum t (questionable), chain reactions • 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. 1.07 13 J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.07 14 Example 4: Workspace 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. 1.07 15 J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.07 16

  5. Testing for Time Invariance Example 5: Time Invariance Determine which of the following are time-invariant systems. If the S D x ( t ) → y ( t ) → y ( t − t 0 ) system is not time invariant, specify an input signal that violates this property. D S x ( t ) → x ( t − t 0 ) → y d ( t ) • y [ n ] = x [ n ] 2 • To test for time invariance, you should calculate two output signals • y ( t ) = x (2 t ) • First, calculate the delayed output, y ( t − t 0 ) in response to the • y [ n ] = x [ − n ] original signal • y [ n ] = nx [ n + 3] • Second, calculate the output due to the delayed input, y d ( t ) . • y ( t ) = sin(2 πx ( t )) • If these are equal for any input signal and delay t 0 , the system is � t time-invariant. Otherwise, it is not. • 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. 1.07 17 J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.07 18 Example 5: Workspace 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. 1.07 19 J. McNames Portland State University ECE 222 System Fundamentals Ver. 1.07 20

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