Chapter 1 Fundamental Concepts
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1 Chapter 1 Fundamental Concepts 1
2 Signals A signal is a pattern of variation of a physical quantity, often as a function of time (but also space, distance, position, etc). These quantities are usually the independent variables of the function defining the signal A signal encodes information, which is the variation itself 2
3 Signal Processing Signal processing is the discipline concerned with extracting, analyzing, and manipulating the information carried by signals The processing method depends on the type of signal and on the nature of the information carried by the signal 3
4 Characterization and Classification of Signals The 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 Likewise, the signal can be a continuous or discrete function of the independent variables 4
5 Characterization and Classification of Signals Cont d Moreover, the signal can be either a real- valued function or a complex-valued function A signal consisting of a single component is called a scalar or one-dimensional (1-D) signal 5
6 Examples: CT vs. DT Signals xt () xn [ ] t n plot(t,x) stem(n,x) 6
7 Sampling Discrete-time signals are often obtained by sampling continuous-time signals.. xt () xn [ ] = xt ( ) = xnt ( ) t= nt 7
8 Systems A system is any device that can process signals for analysis, synthesis, enhancement, format conversion, recording, transmission, etc. A system is usually mathematically defined by the equation(s) relating input to output signals (I/O characterization) A system may have single or multiple inputs and single or multiple outputs 8
9 Block Diagram Representation of Single-Input Single-Output (SISO) CT Systems input signal xt () t T output signal y() t = T x() t t { } 9
10 Types of input/output representations considered Differential equation Convolution model Transfer function representation (Fourier transform, Laplace transform) 10
11 Examples of 1-D, Real-Valued, CT Signals: Temporal Evolution of Currents and Voltages in Electrical Circuits y() t t 11
12 Examples of 1-D, Real-Valued, CT Signals: Temporal Evolution of Some Physical Quantities in Mechanical Systems y() t t 12
13 Continuous-Time (CT) Signals Unit-step function Unit-ramp function 1, t 0 ut () = 0, t < 0 t, t 0 rt () = 0, t < 0 13
14 Unit-Ramp and Unit-Step Functions: Some Properties xtut () () xt (), t 0 = 0, t < 0 t rt () = u( λ) dλ ut () dr() t dt t = 0 = (with exception of ) 14
15 The Rectangular Pulse Function pτ ( t) = ut ( + τ / 2) ut ( τ / 2) 15
16 A.k.a. the delta function or Dirac distribution It is defined by: The Unit Impulse δ () t = 0, t 0 ε ε δ( λ) dλ = 1, ε > 0 The value is not defined, in particular (0) δ δ (0) 16
17 The Unit Impulse: Graphical Interpretation δ () t = limp At A () A is a very large number 17
18 The Scaled Impulse Kδ(t) K K () t If, δ is the impulse with area, i.e., Kδ () t = 0, t 0 ε ε Kδ( λ) dλ = K, ε > 0 K 18
19 Properties of the Delta Function 1) t ut () = δ ( λ) dλ t except t = 0 2) t t ε ε xt () δ( t t) dt= xt ( ) ε > (sifting property) 19
20 Periodic Signals xt Definition: a signal () is said to be periodic with period, if T xt ( + T) = xt ( ) t Notice that () is also periodic with period qt where q is any positive integer is called the fundamental period T xt 20
21 Example: The Sinusoid xt () = Acos( ωt+ θ ), t ω θ [ rad / sec] [ rad ] f = ω 2π [1/ sec] = [ Hz] 21
22 Time-Shifted Signals 22
23 Points of Discontinuity xt A continuous-time signal () is said to be + discontinuous at a point t if xt ( 0 0) xt ( 0) where t + and, being a 0 = t0 + ε t 0 = t0 ε ε small positive number xt () t 0 t 23
24 Continuous Signals xt xt A signal () is continuous at the point if + ( ) ( ) xt = 0 0 t 0 xt If a signal () is continuous at all points t, xt () is said to be a continuous signal 24
25 Example of Continuous Signal: The Triangular Pulse Function 25
26 Piecewise-Continuous Signals xt A signal () is said to be piecewise continuous if it is continuous at all t except a finite or countably infinite collection of points, 1,2,3, t i= i 26
27 Example of Piecewise-Continuous Signal: The Rectangular Pulse Function pτ ( t) = ut ( + τ / 2) ut ( τ / 2) 27
28 Another Example of Piecewise- Continuous Signal: The Pulse Train Function 28
29 Derivative of a Continuous-Time Signal xt A signal () is said to be differentiable at a point t 0 if the quantity ( ) ( ) xt + h xt h h 0 0 has limit as 0independent of whether approaches 0 from above ( h > 0) or from below ( h < 0) If the limit exists, () has a derivative at () ( ) ( ) xt t0 dx t x t + h x t 0 0 t t = lim 0 h 0 dt = h h 29
30 Generalized Derivative However, piecewise-continuous signals may have a derivative in a generalized sense Suppose that xt () is differentiable at all t except t = t 0 The generalized derivative of xt () is defined to be dx() t dt + + xt ( 0) xt ( 0) δ ( t t0) ordinary derivative of x() t at all t except t = t0 30
31 Example: Generalized Derivative of the Step Function Definext () = Kut () K K xt The ordinary derivative of () is 0 at all points except t = 0 Therefore, the generalized derivative of () is xt + K u(0 ) u(0 ) δ ( t 0) = Kδ ( t) 31
32 Another Example of Generalized Derivative Consider the function defined as xt () 2t+ 1, 0 t < 1 1, 1 t < 2 = t + 3, 2 t 3 0, all other t 32
33 Another Example of Generalized Derivative: Cont d 33
34 Example of CT System: An RC Circuit Kirchhoff s s current law: i () t + i () t = i() t C R 34
35 RC Circuit: Cont d The v-i law for the capacitor is dv () t dy() t C ic () t = C = C dt dt Whereas for the resistor it is 1 1 ir() t = vc() t = y() t R R 35
36 RC Circuit: Cont d Constant-coefficient linear differential equation describing the I/O relationship if the circuit dy() t 1 C + y () t = i () t = x () t dt R 36
37 RC Circuit: Cont d Step response when R=C=1 37
38 Basic System Properties: Causality A system is said to be causal if, for any time t 1, the output response at time t 1 resulting from input x(t) does not depend on values of the input for t > t 1. A system is said to be noncausal if it is not causal 38
39 Example: The Ideal Predictor y() t = x( t+ 1) 39
40 Example: The Ideal Delay y() t = x( t 1) 40
41 Memoryless Systems and Systems with Memory A causal system is memoryless or static if, for any time t 1, the value of the output at time t 1 depends only on the value of the input at time t 1 A causal system that is not memoryless is said to have memory. A system has memory if the output at time t 1 depends in general on the past values of the input x(t) for some range of values of t up to t = t 1 41
42 Ideal Amplifier/Attenuator RC Circuit y() t = Kx() t 1 y t = e x d t C t (1/ RC )( t τ ) () ( τ) τ, 0 0 Examples 42
43 Basic System Properties: Additive Systems A system is said to be additive if, for any two inputs x 1 (t) and x 2 (t), the response to the sum of inputs x 1 (t) + x 2 (t) is equal to the sum of the responses to the inputs (assuming no initial energy before the application of the inputs) x () t x () t + system 1 2 y () t + y () t
44 Basic System Properties: Homogeneous Systems A system is said to be homogeneous if, for any input x(t) and any scalar a, the response to the input ax(t) is equal to a times the response to x(t), assuming no energy before the application of the input ax() t system ay() t 44
45 Basic System Properties: Linearity A system is said to be linear if it is both additive and homogeneous ax () t () + bx t system ay () t + by () t A system that is not linear is said to be nonlinear 45
46 Example of Nonlinear System: Circuit with a Diode yt () R R + 2 R xt (), whenxt () 0 = 1 2 0, when x( t) 0 46
47 Example of Nonlinear System: Square-Law Device y t 2 () = x () t 47
48 Example of Linear System: The Ideal Amplifier y() t = Kx() t 48
49 Example of Nonlinear System: A Real Amplifier 49
50 Basic System Properties: Time Invariance A system is said to be time invariant if, for any input x(t) and any time t 1, the response to the shifted input x(t t 1 ) is equal to y(t t 1 ) where y(t) is the response to x(t) with zero initial energy xt ( ) t system 1 1 y( t t ) A system that is not time invariant is said to be time varying or time variant 50
51 Examples of Time Varying Systems Amplifier with Time-Varying Gain y() t = tx() t First-Order System y () t + a() t y() t = bx() t 51
52 Basic System Properties: CT Linear Finite-Dimensional Systems If the N-th derivative of a CT system can be written in the form N 1 M ( N) ( i) ( i) = i + i i= 0 i= 0 y () t a () t y () t b() t x () t then the system is both linear and finite dimensional To be time-invariant a () t = a and b () t = b i and t i i i i 52
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