Noise - irrelevant data; variability in a quantity that has no meaning or significance. In most cases this is modeled as a random variable.
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1 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.
2 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 ] the following statement is also true: ay + ay = H [ ax + ax ] where a 1 and a 2 are constants.
3 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
4 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)
5 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
6 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.
7 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.
8 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 $( ) Computing Processed Digital Signal Interpolator Processed Analog Signal xn $( ) Hardware yn $( ) and smoothing y$ () t a
9 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) -
10 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 RR R C v ( R + ) 1 2
11 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
12 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) sin( 2 t) ) In Matlab: >> t = [0:2*pi/20:100]; >> sig =.5 - exp(-t/20).*(.5*cos(2*t) *sin(2*t)); >> plot(t,sig) rotor response 1 >> title('rotor response') 0.9 >>xlabel('seconds') 0.8 >>ylabel('radians') 0.7 radians seconds
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