Chapter 1 Chapter 1 Fundamental Concepts Fundamental Concepts 1 - PDF document

Chapter 1 Chapter 1 Fundamental Concepts Fundamental Concepts 1 Signals Signals • A signal signal is a pattern of variation of a pattern of variation of a physical quantity, often as a function of time physical quantity, (but also space, distance, position, etc). • These quantities are usually the independent independent variables of the function defining the signal variables • A signal encodes information information, which is the variation itself 2

Signal Processing Signal Processing • Signal processing is the discipline concerned with extracting, analyzing, and manipulating extracting, analyzing, and manipulating the information carried by signals the information • The processing method depends on the type of signal and on the nature of the information carried by the signal 3 Characterization and Classification Characterization and Classification of Signals of Signals • The type of signal type of signal depends on the nature of the independent variables and on the value of the function defining the signal • For example, the independent variables can be continuous or discrete continuous or discrete • Likewise, the signal can be a continuous or continuous or discrete function of the independent discrete function variables 4

Characterization and Classification Characterization and Classification of Signals – Cont’d of Signals – Cont’d • Moreover, the signal can be either a real real- - valued function or a complex valued function complex- -valued function valued function • A signal consisting of a single component is called a scalar or one scalar or one- -dimensional (1 dimensional (1- -D) D) signal signal 5 Examples: CT vs. DT Signals Examples: CT vs. DT Signals x t ( ) [ ] x n t n plot(t,x) stem(n,x) 6

Sampling Sampling • Discrete-time signals are often obtained by sampling continuous-time signals . . = = x t ( ) x n [ ] x t ( ) x nT ( ) = t nT 7 Systems Systems • A system system is any device that can process process signals for analysis, synthesis, enhancement, signals format conversion, recording, transmission, etc. • A system is usually mathematically defined by the equation(s) relating input to output signals (I/O characterization I/O characterization) • A system may have single or multiple inputs and single or multiple outputs 8

Block Diagram Representation Block Diagram Representation of Single-Input Single-Output of Single-Input Single-Output (SISO) CT Systems (SISO) CT Systems input signal output signal { } = x t ( ) y t ( ) T x t ( ) T t ∈ � t ∈ � 9 Types of input/ output Types of input/ output representations considered representations considered • Differential equation • Convolution model • Transfer function representation (Fourier transform, Laplace transform) 10

Examples of 1-D, Real-Valued, CT Signals: Examples of 1-D, Real-Valued, CT Signals: Temporal Evolution of Currents and Temporal Evolution of Currents and Voltages in Electrical Circuits Voltages in Electrical Circuits y t ( ) t 11 Examples of 1-D, Real-Valued, CT Signals: Examples of 1-D, Real-Valued, CT Signals: Temporal Evolution of Some Physical Temporal Evolution of Some Physical Quantities in Mechanical Systems Quantities in Mechanical Systems y t ( ) t 12

Continuous-Time (CT) Signals Continuous-Time (CT) Signals ≥ ⎧ 1, t 0 = ⎨ u t ( ) • Unit Unit- -step function step function • < ⎩ 0, t 0 ≥ ⎧ t , t 0 = ⎨ • Unit Unit- -ramp function ramp function ( ) • r t < ⎩ 0, t 0 13 Unit-Ramp and Unit-Step Functions: Unit-Ramp and Unit-Step Functions: Some Properties Some Properties ≥ ⎧ x t ( ), t 0 = ⎨ x t u t ( ) ( ) < ⎩ 0, t 0 = ∫ t λ λ r t ( ) u ( ) d −∞ dr t ( ) t = = 0 u t ( ) (with exception of ) dt 14

The Rectangular Pulse Function The Rectangular Pulse Function = + τ − − τ p t ( ) u t ( / 2) u t ( / 2) τ 15 The Unit Impulse The Unit Impulse • A.k.a. the delta function delta function or Dirac distribution Dirac distribution • It is defined by: It is defined by: • δ = ≠ ( ) t 0, t 0 ε ∫ δ λ λ = ∀ > ε ( ) d 1, 0 − ε δ (0) • The value is not defined, in particular • The value is not defined, in particular δ ≠ ∞ (0) 16

The Unit Impulse: The Unit Impulse: Graphical Interpretation Graphical Interpretation δ = ( ) t lim p A t ( ) →∞ A A is a very large number 17 The Scaled Impulse K δ ( t ) The Scaled Impulse K δ ( t ) K ∈ � δ • If , is the impulse with area , K ( ) t K i.e., δ = ≠ K ( ) t 0, t 0 ε ∫ δ λ λ = ∀ > ε K ( ) d K , 0 − ε 18

Properties of the Delta Function Properties of the Delta Function t = ∫ δ λ λ 1) u t ( ) ( ) d −∞ ∀ except t = t 0 + ε t 0 ∫ δ − = ∀ > ε x t ( ) ( t t dt ) x t ( ) 0 2) 0 0 − ε t 0 (sifting property sifting property) 19 Periodic Signals Periodic Signals x t ( ) • Definition: a signal is said to be periodic with period , if T + = ∀ ∈ � x t ( T ) x t ( ) t ( ) • Notice that is also periodic with period x t q qT where is any positive integer • T is called the fundamental period fundamental period 20

Example: The Sinusoid Example: The Sinusoid = ω + θ ∈ � x t ( ) A cos( t ), t ω ω [ rad / sec] = = f [1/ sec] [ Hz ] π θ 2 [ ] rad 21 Time-Shifted Signals Time-Shifted Signals 22

Points of Discontinuity Points of Discontinuity x t ( ) • A continuous-time signal is said to be ≠ + − x t ( ) x t ( ) t discontinuous at a point if + = − = 0 0 0 + ε − ε ε where and , being a t t t t 0 0 0 0 small positive number x t ( ) t t 0 23 Continuous Signals Continuous Signals t • A signal is continuous at the point if x t ( ) 0 = + − x t ( ) x t ( ) 0 0 • If a signal is continuous at all points t , x t ( ) x t ( ) is said to be a continuous signal continuous signal 24

Example of Continuous Signal: Example of Continuous Signal: The Triangular Pulse Function The Triangular Pulse Function 25 Piecewise-Continuous Signals Piecewise-Continuous Signals x t ( ) • A signal is said to be piecewise t continuous if it is continuous at all except a finite or countably infinite t i = … collection of points , 1,2,3, i 26

Example of Piecewise-Continuous Example of Piecewise-Continuous Signal: The Rectangular Pulse Function Signal: The Rectangular Pulse Function = + τ − − τ p t ( ) u t ( / 2) u t ( / 2) τ 27 Another Example of Piecewise- Another Example of Piecewise- Continuous Signal: Continuous Signal: The Pulse Train Function The Pulse Train Function 28

Derivative of a Continuous-Time Signal Derivative of a Continuous-Time Signal x t ( ) • A signal is said to be differentiable differentiable at a point if the quantity t 0 + − x t ( h ) x t ( ) 0 0 h h → has limit as independent of whether 0 h h > approaches 0 from above or from ( 0) h < ( 0) below ( ) x t t • If the limit exists, has a derivative derivative at 0 + − dx t ( ) x t ( h ) x t ( ) = 0 0 lim = t t → dt h h 0 29 0 Generalized Derivative Generalized Derivative • However, piecewise-continuous signals may have a derivative in a generalized sense t x t ( ) • Suppose that is differentiable at all = t t except 0 x t ( ) • The generalized derivative generalized derivative of is defined to be dx t ( ) + ⎡ − ⎤ δ − + − ⎣ x t ( ) x t ( ) ⎦ ( t t ) 0 0 0 dt = ordinary derivative of at all except x t ( ) t t t 30 0

Example: Generalized Derivative Example: Generalized Derivative of the Step Function of the Step Function K = • Define x t ( ) Ku t ( ) K x t ( ) • The ordinary derivative of is 0 at all t = 0 points except x t ( ) • Therefore, the generalized derivative of is ⎡ − ⎤ δ − = δ + − K u ⎣ (0 ) u (0 ) ⎦ ( t 0) K ( ) t 31 Another Example Another Example of Generalized Derivative of Generalized Derivative • Consider the function defined as + ≤ < ⎧ 2 t 1, 0 t 1 ⎪ ≤ < ⎪ 1, 1 t 2 = ⎨− + x t ( ) ≤ ≤ 3, 2 3 t t ⎪ ⎪ ⎩ 0, all other t 32

Another Example Another Example of Generalized Derivative: Cont’d of Generalized Derivative: Cont’d 33 Example of CT System: Example of CT System: An RC Circuit An RC Circuit + = i t ( ) i t ( ) i t ( ) Kirchhoff’ Kirchhoff ’s current law: s current law: C R 34

RC Circuit: Cont’d RC Circuit: Cont’d • The v-i law for the capacitor is dv t ( ) dy t ( ) = = i t ( ) C C C C dt dt • Whereas for the resistor it is 1 1 = = ( ) ( ) ( ) i t v t y t R C R R 35 RC Circuit: Cont’d RC Circuit: Cont’d • Constant Constant- -coefficient linear differential coefficient linear differential • equation describing the I/O relationship if equation the circuit dy t ( ) 1 + = = C y t ( ) i t ( ) x t ( ) dt R 36