Signals and Systems 1 Lecture 7 Dr. Ali. A. Jalali September 4, 2002

Signals and Systems 1 Lecture # 7 Continuous Systems EE 327 fall 2002

Continuous Systems Definition of system. Examples of systems. Classifications of systems. Linear time invariant systems. Conclusions. EE 327 fall 2002 Signals and Systems 1

Continuous Systems Preview x ( t ) y ( t ) Input Output A system is transforms input signals into output signals. A continuous-time system receives an input signal x(t) and generates an output signals y(t). y(t)=h(t)x(t) means the system h(t) acts on input signal x(t) to produce output signal y(t). We concentrate on systems with one input and one output signal, i.e., Single-input, single output (SISO) systems. Systems often denoted by block diagram. Lines with arrows denote signals (not wires). Arrows show inputs and outputs Continuous-time System h(t) x ( t ) y ( t ) Input Output EE 327 fall 2002 Signals and Systems 1

Continuous Systems Examples of C.S.: A filter to eliminate unwanted signals. An automobile. An algorithm. A circuit. (input voltage, output current) EE 327 (input instructor and student’s effort and output instructor evaluation and student’s grade) Stock Market (input buy orders and sell orders, output IBM stock price and Intel stock price) We have MIMO, MISO, SIMO and SISO systems. Many physical systems have the same mathematical model. EE 327 fall 2002 Signals and Systems 1

Continuous Systems Example: Robot car block diagram This is a subsystem.

Continuous Systems Example: Human speech production system block diagram

Continuous Systems Example: Mechanical free-body diagram

Continuous Systems Example: Electric Network

Continuous Systems The mechanical and electrical systems are dynamically analogous. Thus, understanding one of these systems gives insights into the other.

Continuous Systems using integrations, adders, and gains. Example: Block-diagram using integrations, adders, and gains. This is a subsystem.

Systems Classifications of systems: 1. Linear and nonlinear systems. 2. Time Invariant and time varying systems. 3. Causal, noncausal and anticausal systems. 4. Stable and unstable systems. 5. Memoryless systems and systems with memory. 6. Continuous and Discrete time systems. EE 327 fall 2002 Signals and Systems 1

Continuous Systems Properties of continuous systems: 1- Linearity: One of the most important concepts in system theory is linearity. 2- Using linear system theory, you have easier and more convenient methods of analysis and design of systems. 3- A system is linear if and only if it satisfies the principle of homogeneity and the principle of additively. EE 327 fall 2002 Signals and Systems 1

Continuous Systems Linearity: LS LS LS LS LS LS LS LS principle of homogeneity (c is real constant). principle of additively homogeneity and additively (Principle of superposition) x ( t ) LS y ( t ) x ( t ) =C x ( t ) LS y ( t ) =C y ( t ) 1 1 1 1 1 1 x ( t ) LS y ( t ) 1 1 x ( t ) = x ( t )+ x ( t ) LS y ( t ) = y ( t )+ y ( t ) 1 2 1 2 x ( t ) LS y ( t ) 2 2 x ( t ) y ( t ) 1 LS 1 X(t)=C1x1(t)+C2x2(t) y(t)=C1y1(t)+C2y2(t) LS x ( t ) y ( t ) 2 LS 2 EE 327 fall 2002 Signals and Systems 1

Continuous Systems Linearity example: Let the response of a linear system at rest due to the system input be given by and let the response of the same system at rest due to another system input be Then the response of the same system at rest due to input given by Is simply obtained as: EE 327 fall 2002 Signals and Systems 1

Continuous Systems Time Invariance: x(t) y(t) LS x(t-T) y(t-T) LS 1 t 1 3 t x(t-T) y(t-T) x(t-T) y(t-T) LS T T t Shifted input Shifted output For All value of t and T. 1 t 1 4 EE 327 fall 2002 Signals and Systems 1

Continuous Systems Time invariant example: Let the response of a time-invariant linear system at rest due to be given by Then, the system response due to the shifted system input defined by Is EE 327 fall 2002 Signals and Systems 1

Continuous Systems Linear and Time Invariant systems: Many man-made and naturally occurring systems can be modeled as LTI systems. Powerful techniques have been developed to analyze and to characterize LTI systems. The analysis of LTI systems is an essential precursor to the analysis of more complex systems. EE 327 fall 2002 Signals and Systems 1

Continuous Systems Example for the input signal as an impulse Causality: A causal system is a nonpredictive system in that the output does not precede, or anticipate, the input. Example for the input signal as an impulse Causal System Anticusal System Noncusal System EE 327 fall 2002 Signals and Systems 1

Continuous Systems Causality: For the causal system the output at time depends only on the input for i.e., the system cannot anticipate the input. EE 327 fall 2002 Signals and Systems 1

Continuous Systems Stability: Example: System stability is defined from several points of view. 1- In term of input –output behavior. If the system input is bounded and if the system is stable, then the output must also be bounded. (This test is known as BIBO.) 2- From the characteristic roots of system we measure the stability. 3- From dynamic matrix A of systems (we learn more later). Note: a) If we connect some stable systems together there is no guaranty that overall system becomes stable! b) BIBO test is not an easy job in practical cases. Example: Unstable System Stable System EE 327 fall 2002 Signals and Systems 1

Continuous Systems Stability: Example with the test of BIBO. For the resistor, if i(t) is bounded then so is v(t), but for the capacitor this is not true. Consider i(t)= u(t) then v(t)=tu(t) which is unbounded. EE 327 fall 2002 Signals and Systems 1

Continuous Systems Memoryless systems: The output of a memoryless system at some time Depends only on its input at the same time . Example is resistive divider network. Therefor, depends upon the value of and not on for .

Continuous Systems System with memory: Example: v(t) depends not just on i(t) at one point in time t. Therefor the system that relates v to i exhibits memory.

Continuous Systems System with memory and memoryless. If a system is contained with capacitors, inductors and flip-flop the system is known as system with memory. Otherwise if it is contained with pure resistors, is known as memoryless system. Continuous and Discrete time systems. If input and output of systems are continuous the system is referred as continuous-time systems. If input and output of systems are discrete the system is referred as discrete-time systems.

Continuous Systems Nonlinear example: A continuous system is described by the input-output Equation where K and A are real constants. This systems is not linear because: If input is the output is For the input the output is But And The system is nonlinear. If A = 0, then the system will be linear. EE 327 fall 2002 Signals and Systems 1

Continuous Systems Time invariant example: A continuous system is described by the input-output Equation where K and A are real constants. This systems is time invariant because: An arbitrary input produces an output and the shifted version on this input produces an output For the input the output is But for all so the system is time invariant (or shift invariant). EE 327 fall 2002 Signals and Systems 1

Continuous Systems Conclution: Linear Time Invariant systems arespecial systems for which powerful mathematical methods of description are available. Classification of systems: linear and nonlinear systems, time-invariant and time-varying systems, causal, noncausal and anticausal systems, stable and unstable systems, systems with memory and memoryless systems, continuous and discrete-time systems. Physical and non physical systems. Man-made and natural systems. EE 327 fall 2002 Signals and Systems 1