Time-Domain Analysis of CT Systems - Signals and Systems | EE 701, Study notes of Electrical and Electronics Engineering

Download Time-Domain Analysis of CT Systems - Signals and Systems | EE 701 and more Study notes Electrical and Electronics Engineering in PDF only on Docsity! 1EE701 - Erik Blasch Signals and Systems Lecture 04 - Time-Domain Analysis of CT Systems (Stability) Erik Blasch (937) 904-9077 erik.blasch@wpafb.af.mil http://www.cs.wright.edu/~eblasch TA - Rehka Bangladore Kalegowda bangalorekalegow.2@wright.edu 2EE701 - Erik Blasch Ch 2 - CT – Time Domain – 2.1 Introduction – 2.2 Sys. Response to Internal Conditions: Zero-Input Response – 2.3 The Unit Impulse response h(t) – 2.4 Sys. Response to External Input: Zero-State Res. – 2.5 Classical Solution of Differential equations – 2.6 System Stability – 2.7 Intuitive Insights into System Behavior – 2.8 Appendix 2.1: Determining The Impulse Response – 2.9 Summary 3EE701 - Erik Blasch Ch 2 – LTI Systems • 2.1 Discrete-Time LTI Systems: The Convolution Sum • Representation of Discrete-Time Signals in Terms of Impulses • Discrete-Time Unit Impulse Response and the Convolution-Sum Representation • 2.2 Continuous-Time LTI Systems: The Convolution Integral • Representation of Continuous-Time Signals in Terms of Impulses • Continuous-Time Unit Impulse Response and the Convolution Integral • 2.3 Properties of Linear Time-Invariant Systems • Commutative Property Distributive Property • Associative Property With and without Memory • Invertibility of LTI Systems Causality • Stability for LTI Systems The Unit Step Response • 2.4 Causal LTI Systems - Differential and Difference Equations • Linear Constant-Coefficient Differential Equations • Linear Constant-Coefficient Difference Equations • Block Diagram Representations of First-Order Systems • 2.5 Singularity Functions • Unit Impulse as an Idealized Short Pulse • Defining the Unit Impulse through Convolution • Unit Doublets and Other Singularity Functions 4EE701 - Erik Blasch ( ) ( ) ( ) ( )tYTtycTtfctF m i ii m i ii =∑ −⇒∑ −= == 1 1 LTI Systems • Recall for LTI system with input f and output y – Homogeneity: c f(t) ⇒ c y(t) – Time-invariance: c f(t - T) ⇒ c y(t - T) – Adding additivity: • If a signal ( F(t) ) can be expressed as a sum of shifted ( t - Ti ) and weighted ( ci ) copies of a simpler signal ( f(t) ), – we easily find a system’s output ( Y(t) ) to that signal if we only know system’s output ( y(t) ) to that simpler signal • A common choice for f(t) is the impulse 5EE701 - Erik Blasch Derivation of Convolution Sum • Because of additivity of LTI systems: • Because of homogeneity of LTI systems: • Because of time-invariance of LTI systems: [ ]∑ ∞ −∞= −= k knkxHny ][][][ δ [ ]∑ ∞ −∞= −= k knHkxny ][][][ δ ∑ ∞ −∞= −= k knhkxny ][][][ 6EE701 - Erik Blasch Derivation of Convolution Integral • Then, output signal y(t) will be: • Because of additivity of LTI systems: • Because of homogeneity of LTI systems: • Because of time-invariance of LTI systems: ∑ ∞ −∞= −= k knhkxny ][][][       ∆∆−∆== ∑ ∞ −∞= ∆ k ktkxHtxHty )(][))(ˆ()(ˆ δ ( )∑ ∞ −∞= ∆ ∆∆−∆= k ktkxHty )(][)(ˆ δ ( )∑ ∞ −∞= ∆ ∆∆−∆= k ktHkxty )(][)(ˆ δ 7EE701 - Erik Blasch Properties of Convolution • Commutative Property: – Switch the order x[n]*y[n] = y[n]*x[n] x(t)*y(t) = y(t)*x(t) • Distributive Property: – Separate out (addition/subtraction) x[n]*(y1[n] + y2[n]) = x[n]*y1[n] + x[n]*y2[n] x(t)*(y1(t) + y2(t)) = x(t)*y1(t) + x(t)*y2(t) • Associative Property: – Reorder the operations (i.e. like matrix algebra) x[n]*(y1[n]*y2[n]) = (x[n]*y1[n])*y2[n] x(t)*(y1(t)*y2(t)) = (x(t)*y1(t))*y2(t) 8EE701 - Erik Blasch Commutative Property • Convolution is a commutative operator (in both discrete and continuous time), i.e.: • For example, in discrete-time: • and similar for continuous time. • Therefore, when calculating the response of a system to an input signal x[n], we can imagine the signal being convolved with the unit impulse response h[n], or vice versa, whichever appears the most straightforward. ∫ ∑ ∞ ∞− ∞ −∞= −== −== τττ dtxhtxththtx knxkhnxnhnhnx k )()()(*)()(*)( ][][][*][][*][ ][*][][][][][][*][ nxnhrhrnxknhkxnhnx rk =−=−= ∑∑ ∞ −∞= ∞ −∞= 17EE701 - Erik Blasch CT Convolution Properties 18EE701 - Erik Blasch LTI System Memory • An LTI system is memoryless if its output depends only on the input value at the same time, i.e. • For an impulse response, this can only be true if • Such systems are extremely simple and the output of dynamic engineering, physical systems depend on: • Preceding values of x[n-1], x[n-2], … • Past values of y[n-1], y[n-2], … • for discrete-time systems, or derivative terms for continuous- time systems )()( ][][ tkxty nkxny = = )()( ][][ tkth nknh δ δ = = 19EE701 - Erik Blasch Causality for LTI Systems • Remember, a causal system only depends on present and past values of the input signal. We do not use knowledge about future information. • For a discrete LTI system, convolution tells us that h[n] = 0 for n < 0 • as y[n] must not depend on x[k] for k>n, as the impulse response must be zero before the pulse! • Both the integrator and its inverse in the previous example are causal • This is strongly related to inverse systems as we generally require our inverse system to be causal. If it is not causal, it is difficult to manufacture! τττ dthxthtx knhkxnhnx t n k ∫ ∑ ∞− −∞= −= −= )()()(*)( ][][][*][ 20EE701 - Erik Blasch Causality (1) • A system is causal if its output depends only on the past and the present values of the input signal, not the future values • Consider the following for a causal LTI system: – Because of causality h[n − k] must be zero for k>n. – So, n − k < 0 for the LTI system to be causal. – Then h[n] = 0 for n < 0. ∑ ∞ −∞= −= k knhkxny ][][][ 21EE701 - Erik Blasch Causality (2) • So the convolution sum for a causal LTI system becomes: • Similarly, the convolution integral for a causal LTI system: • So, if a given system is causal, one can infer that its impulse response is zero for negative time values, and use the above simpler convolution formulas. • Also, can solve which operation is easier (h or x) ∑∑ ∞ =−∞= −=−= 0 ][][][][][ k n k knxkhknhkxny ∫∫ ∞ ∞− −=−= 0 )()()()(][ ττττττ dtxhdthxny t 22EE701 - Erik Blasch Stability and Invertibility • Stability: A system is stable if it results in a bounded output for any bounded input, i.e. bounded-input/bounded-output (BIBO). – If |x(t)| < k1, then |y(t)| < k2. – Example: • Invertibility: A system is invertible if distinct inputs result in distinct outputs. If a system is invertible, then there exists an “inverse” system which converts output of the original system to the original input. – Examples: )( 4 1 )( )(4)( tytw txty = = ]1[][][ ][][ −−= = ∑ −∞= nynynw kxny n k System x(t) Inverse System w(t)=x(t)y(t) dt tdy tw dttxty t )( )( )()( = = ∫ ∞− ∫= t dttxty 0 )()( ][100][ nxny = 23EE701 - Erik Blasch System Invertibility • Does there exist a system with impulse response h1(t) such that y(t)=x(t)? • Widely used concept for: • control of physical systems, where the aim is to calculate a control signal such that the system behaves as specified • filtering out noise from communication systems, where the aim is to recover the original signal x(t) • The aim is to calculate “inverse systems” such that • The resulting serial system is therefore memoryless h(t) x(t) y(t) h1(t) w(t) )()()( ][][][ 1 1 tthth nnhnh δ δ = = 24EE701 - Erik Blasch Ex: Invertible: Accumulator Sys. • Consider a DT LTI system with an impulse response h[n] = u[n] • Using convolution, the response to an arbitrary input x[n]: • As u[n-k] = 0 for n-k<0 and 1 for n-k≥0, this becomes • i.e. it acts as a running sum or accumulator. Therefore an inverse system can be expressed as: • A first difference (differential) operator, which has an impulse response ∑ ∞ −∞= −= k knhkxny ][][][ ∑ −∞= = n k kxny ][][ ]1[][][ −−= nxnxny ]1[][][1 −−= nnnh δδ 25EE701 - Erik Blasch LTI System Stability • A system is stable if every bounded input produces a bounded output • Therefore, consider a bounded input signal • |x[n]| < B for all n • Applying convolution and taking the absolute value: • Using the triangle inequality (magnitude of a sum of a set of numbers is no larger than the sum of the magnitude of the numbers): • Therefore a DT LTI system is stable if and only if its impulse response is absolutely summable, i.e. ∑ ∞ −∞= −= k knxkhny ][][][ ∑ ∑ ∞ −∞= ∞ −∞= ≤ −≤ k k khB knxkhny ][ ][][][ ∞<∑ ∞ −∞=k kh ][ ∞<∫ ∞ ∞− ττ dh )( Continuous-time system 26EE701 - Erik Blasch Ex: System Stability • Are the DT and CT pure time shift systems stable? • Are the discrete and continuous-time integrator systems stable? )()( ][][ 0 0 ttth nnnh −= −= δ δ )()( ][][ 0 0 ttuth nnunh −= −= ∞<=−= ∑∑ ∞ −∞= ∞ −∞= 1][][ 0 kk nkkh δ ∞<=−= ∫∫ ∞ ∞− ∞ ∞− 1)()( 0 ττδττ dtdh ∞==−= ∑∑∑ ∞ = ∞ −∞= ∞ −∞= 0 ][][][ 0 nkkk kunkukh ∞==−= ∫∫∫ ∞∞ ∞− ∞ ∞− 0 )()()( 0 t dudtudh ττττττ Therefore, both the CT and DT systems are stable: all finite input signals produce a finite output signal Therefore, both the CT and DT systems are unstable: at least one finite input causes an infinite output signal 27EE701 - Erik Blasch Bounded-Input Bounded-Output Stability 28EE701 - Erik Blasch Ch 2 - CT – Time Domain – 2.1 Introduction – 2.2 Sys. Response to Internal Conditions: Zero-Input Response – 2.3 The Unit Impulse response h(t) – 2.4 Sys. Response to External Input: Zero-State Res. – 2.5 Classical Solution of Differential equations – 2.6 System Stability – 2.7 Intuitive Insights into System Behavior – 2.8 Appendix 2.1: Determining The Impulse Response – 2.9 Summary