1.1 Signals and Systems Signals convey information. Systems respond to (or process) information. Engineers desire mathematical models for signals and systems in order to solve design problems efficiently and thoroughly. Signal - encoded information; data; a dynamic (or change) in some quantity that has meaning. In most cases this is modeled as a function of time or space. Noise - irrelevant data; variability in a quantity that has no meaning or significance. In most cases this is modeled as a random variable. System - a mapping between a set of inputs to a set of outputs; an entity that processes signals received at its input and produces another set of signals at the output; a process that responds to actions or events at its input by generating actions or events at its output. Most physical systems are approximately modeled by differential equations or convolution integrals.
1.2 Classifications of Systems 1. Linear vs. Non-linear For all linear systems superposition holds for input-output relationships. Denote a general system in the following manner: y= H [ x] System H operates on input x to produce output y. The system H is linear if and only if (iff) for any input-output pair: y = H [ x ] and y = H [ x 1 1 2 2] the following statement is also true: ay + ay = H [ ax + ax 1 1 2 2 1 1 2 2] where a 1 and a 2 are constants.
1.3 Determine whether or not each system described below is linear. Assume inputs and outputs are functions of time denoted by x and y, and constants are denoted k. y= kx y 2 dx dx = + +x 2 dt dt y= kx+10 2 y = k x + k x 1 2 y x dx 2 = +x 2 dt 2. Constant-parameter (time-invariant) vs. time-varying systems
1.4 A system whose model parameters change with time is considered timevarying. Note that the output and input will be varying with time for both constantparameter and time-varying systems. Systems where outputs differ only by a time shift when the same inputs are applied at corresponding time shifts is a constant parameter system. A system is a constant parameter system iff for any input output pair: yt () = H [ xt ()] the following statement is also true: yt ( τ ) = H xt ( τ ) for all τ [ ] Determine whether the systems below are time varying or time invariant: y() t = kx() t +10 y() t = cos( 2 π t) x( t)
1.5 3. Instantaneous (memoryless) vs. dynamic (with memory) systems For an instantaneous system, the present output value depends only on the present input value. In a dynamic system the present output value depends on the present and past input values. Dynamic systems usually contain some type of energy storage elements. The response of a dynamic system results from two components; the initial condition and the input. The state of the system refers to the information needed along with the present input to determine the present output. Zero-input response - system output due only to system state (or initial condition). Zero-state response - system output due only to the input of the system. In general: Total response = Zero-input response + Zero-state response Determine which systems are instantaneous and which are dynamic. 2 y = k x + k x 1 2 dx y = + x dt 4. Causal vs. Noncausal
1.6 Systems where the output depends only on the present and/or past values of the input are referred to as causal. Note that for a causal system the output cannot depend on future input values. Systems where the output depends on future input value is referred to as a noncausal system. 5. Lumped-Parameter vs. Distributed-Parameter Systems In most real systems the interactions between the signal energy and the system elements happen continuously over space (i.e. resistance over a wire). In modeling these systems the interaction can be considered to occur at one point in space. This is referred to as a lumped-parameter model. This is a reasonable model when the dimensions of the elements are small with respect to the energy wavelength. When this is not done (i.e. transmission lines), the model is referred to as a distributed parameter system.
1.7 6. Continuous-Time vs. Discrete-Time Systems The input and output for a discrete-time system is defined only at discrete points in time: yn ( ) = H xn ( ) for n { -2, -1, 0, 1, 2, }. [ ] If the inputs and outputs are defined over a continuum of time values then the system is a continuous-time system: yt () = H xt () for t [0,+ ]. [ ] 7. Analog vs. Digital Systems A system whose input and output values take on only a set of discrete values is referred to as a digital system. If the values of the input and output can take on a continuum of values then the system is an analog system.
1.8 Elements of a Digital Signal Processing System Analog Signal xa () t Discrete-time Signal Quantizer Digital Signal Coder x ( nt ) a xnt $( ) xn $( ) 11 10 01 00 Computing Processed Digital Signal Interpolator Processed Analog Signal xn $( ) Hardware yn $( ) and smoothing y$ () t a
1.9 Differential Equation Models for Current and Voltage Systems Capacitors: it c dv t t () 1 () = vt () = dt c i ( τ) d τ i(t) v(t) vt L di t t () 1 Inductors: () = it () = dt L v ( τ) d τ i(t) + v(t) - Resistors: v() t = Ri( t) i(t) + v(t) -
1.10 Example, find the input-output equation relating input i s to output v o i s R 1 + L C R 2 v 0 - Ans: CL i t R v 1 RC v R () = && + & + + s o o 1 L 2 1 2 1 2 RR R C v ( R + ) 1 2
1.11 Differential Equation Models for Position and Force Systems Translational Systems - Consider motion (output) and force (input) in one direction denoted by y(t) and f(t), respectively: Mass (M): f () t = My &&() t Linear Spring (Stiffness K): f () t = Ky() t Linear Dashpot (damping coefficient B): f () t = By&( t) Rotational Systems - Consider an angular position (output) and torque (input) denoted by θ(t) and T(t), respectively: Rotational Mass (J): T() t = J&& θ Torsional Spring (K): T()= t Kθ Torsional Dashpot (B): T() t = B& θ Electromechanical Systems - For a DC motor, consider the an angular position (output), and current (input) denoted by θ(t) and i(t), respectively: Motor Constant (K T ): T() t = K it () T
1.12 Example, consider a torsional spring with stiffness K=2 nt-m/rad fastened to the rotor of an armature controlled DC motor with motor constant K r = 5 nt-m/a, and the rotational mass is J=.5 nt-m/(rad/s 2 ). The friction coefficient for the spinning rotor is B =.05 nt-m 2 /(rad/sec). Find the equation that relates the rotor position to the armature current. Assume the polarity of the motor is such that a positive current moves the angular position is a positive direction. Describe the motion of the rotor for a step input going from 0 to.2 A. Ans: Ki = J&& θ + B& θ + Kθ θ( t) =. 5 exp( t / 20) r a (. 5cos( 2t) 0. 0125sin( 2 t) ) In Matlab: >> t = [0:2*pi/20:100]; >> sig =.5 - exp(-t/20).*(.5*cos(2*t) -.0125*sin(2*t)); >> plot(t,sig) rotor response 1 >> title('rotor response') 0.9 >>xlabel('seconds') 0.8 >>ylabel('radians') 0.7 radians 0.6 0.5 0.4 0.3 0.2 0.1 0 0 20 40 60 seconds 80 100