Chapter 1 Fundamental Concepts

Chapter 1 Fundamental Concepts

Signals A signal is a pattern of variation of a physical quantity as a function of time, space, distance, position, temperature, pressure, etc. These quantities are usually the independent variables of the function defining the signal A signal encodes information, which is the variation itself

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

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

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 A signal consisting of multiple components is called a vector or multidimensional (M-D) signal

Definition of Function from Calculus y = f() t f : t y = f( t) independent variable dependent variable f () t y domain: set of values that t can take on range: set of values spanned by y

Plot or Graph of a Function y = f() t range domain t

Continuous-Time (CT) and Discrete-Time (DT) Signals A signal x(t) depending on a continuous temporal variable t will be called a continuous-time time (CT) signal A signal x[n] depending on a discrete temporal variable n will be called a discrete-time time (DT) signal

Examples: CT vs. DT Signals xt () xn [ ] t n plot(t,x) stem(n,x)

CT Signals: 1-D vs. N-D, Real vs. Complex 1-D, real-valued, CT signal: N-D, real-valued, CT signal: 1-D, complex-valued, CT signal: N-D, complex-valued, CT signal: xt (), t N xt (), t xt (), t N xt (), t

DT Signals: 1-D vs. N-D, Real vs. Complex 1-D, real-valued, DT signal: N-D, real-valued, DT signal: 1-D, complex-valued, DT signal: N-D, complex-valued, DT signal: xn [ ], n N xn [ ], n xn [ ], n N xn [ ], n

Digital Signals A DT signal whose values belong to a finite set or alphabet Α = { α } 1, α2,, αn is called a digital signal Since computers work with finite-precision arithmetic, only digital signals can be numerically processed Digital Signal Processing (DSP): ECE 464/564 (Liu) and ECE 567 (Lucchese)

Digital Signals: 1-D vs. N-D 1-D, real-valued, digital signal: N-D, real-valued, digital signal: xn [ ] Α, N xn [ ] Α, n n Α =,,, N { α α α } 1 2 If, the digital signal is real, if instead at i least one of the α, the digital signal is i complex α

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

Block Diagram Representation of Single-Input Single-Output (SISO) CT Systems input signal xt () t T output signal yt () = T xt () t { }

Block Diagram Representation of Single-Input Single-Output (SISO) DT Systems input signal xn [ ] n T output signal yn [ ] = T xn [ ] n { }

A Hybrid SISO System: The Analog to Digital Converter (ADC) Used to convert a CT (analog) signal into a digital signal xt () t ADC yn [ ] n

Block Diagram Representation of Multiple-Input Multiple-Output (MIMO) CT Systems

Example of 1-D, Real-Valued, Digital Signal: Digital Audio Signal xn [ ] n

Example of 1-D, Real-Valued, Digital Signal with a 2-D Domain: A Digital Gray-Level Image n 2 xn [, n] 1 2 [ n, n ] 1 2 2 n 1 + [ n1, n2] pixel coordinates

Digital Gray-Level Image: Cont d xn [ 1, n2] n 2 n 1

Example of 3-D, Real-Valued, Digital Signal with a 2-D Domain: A Digital Color Image n 2 rn [ 1, n2] xn [ 1, n2] = gn [ 1, n2] bn [ 1, n2] 2 [ n, n ] 1 2 n 1

Digital Color Image: Cont d rn [ 1, n2] gn [ 1, n2] bn [ 1, n2]

Example of 3-D, Real-Valued, Digital Signal with a 3-D Domain: A Digital Color Video Sequence k (temporal axis) n 2 rn [ 1, n2, k] x[ n1, n2, k] = g[ n1, n2, k] bn [ 1, n2, k] k 2 [ n1, n2], n 1

Types of input/output representations considered Differential equation (or difference equation) The convolution model The transfer function representation (Fourier transform representation)

Examples of 1-D, Real-Valued, CT Signals: Temporal Evolution of Currents and Voltages in Electrical Circuits yt () t

Examples of 1-D, Real-Valued, CT Signals: Temporal Evolution of Some Physical Quantities in Mechanical Systems yt () t

Continuous-Time (CT) Signals Unit-step function Unit-ramp function 1, t 0 ut () = 0, t < 0 t, t 0 rt () = 0, t < 0

Unit-Ramp and Unit-Step Functions: Some Properties xtut () () xt (), t 0 = 0, t < 0 t rt () = u( λ) dλ ut () dr() t dt = (to the exception of t = 0 )

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)

The Unit Impulse: Graphical Interpretation A is a very large number

The Scaled Impulse Kδ(t) K K () t If, δ is the impulse with area, i.e., K() t = 0, t 0 ε ε Kδ( λ) dλ = K, ε > 0 K

Properties of the Delta Function 1) t ut () = δ ( λ) dλ t except t = 0 2) t t 0 0 + ε ε xt () δ( t t) dt= xt ( ) ε > 0 0 0 (sifting property)

Periodic Signals xt Definition: a signal () is said to be periodic with period, if T xt ( + T) = xt ( ) t xt Notice that () is also periodic with period qt where q is any positive integer is called the fundamental period T

Example: The Sinusoid xt () = Acos( ωt+ θ ), t ω θ [ rad / sec] [ rad ] f = ω 2π [1/ sec] = [ Hz]

Is the Sum of Periodic Signals Periodic? Let x1( t) and x2( t) be two periodic signals with periods T 1 and T 2, respectively x t + x t Then, the sum is periodic only if 1() 2() the ratio T 1 /T 2 can be written as the ratio q/r of two integers q and r In addition, if r and q are coprime, then T=rT 1 is the fundamental period of x () t + x () t 1 2

Time-Shifted Signals

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

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

Example of Continuous Signal: The Triangular Pulse Function

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

Example of Piecewise-Continuous Signal: The Rectangular Pulse Function pτ ( t) = ut ( + τ / 2) ut ( τ / 2)

Another Example of Piecewise- Continuous Signal: The Pulse Train Function

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

Continuity and Differentiability xt In order for () to be differentiable at a point t 0, it is necessary (but not sufficient) that () be continuous at xt t0 Continuous-time signals that are not continuous at all points (piecewise continuity) cannot be differentiable at all points

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 xt () at all t except t = t0

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 + K u(0 ) u(0 ) δ ( t 0) = Kδ ( t) xt

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

dx() t dt Another Example of Generalized Derivative: Cont d The ordinary derivative of, at all except is dx() t dt t = 0,1,2,3 Its generalized derivative is xt () [ ut ut ] [ ut ut ] = 2 ( ) ( 1) ( 2) ( 3) + x(0 ) x(0 ) δ + + ( t) + x(1 ) x(1 ) δ( t 1) 1 2 t

Another Example of Generalized Derivative: Cont d

Signals Defined Interval by Interval Consider the signal x (), t t t < t xt () = x (), t t t< t x (), t t t 1 1 2 2 2 3 3 3 This signal can be expressed in terms of the unit-step function ut () and its time-shifts as [ ] xt () = x() t ut ( t) ut ( t) + 1 1 2 [ ] + x () t u( t t ) u( t t ) + 2 2 3 + x () t u( t t ), t t 3 3 1

Signals Defined Interval by Interval: Cont d By rearranging the terms, we can write xt () = f() tut ( t) + f() tut ( t) + f() tut ( t) 1 1 2 2 3 3 where f () t = x () t 1 1 f () t = x () t x () t 2 2 1 f () t = x () t x () t 3 3 2

Discrete-Time (DT) Signals A discrete-time signal is defined only over integer values We denote such a signal by xn [ ], n = {, 2, 1,0,1,2, }

Example: A Discrete-Time Signal Plotted with Matlab Suppose that x[0] = 1, x[1] = 2, x[2] = 1, x[3] = 0, x[4] = 1 n=-2:6; x=[0 0 1 2 1 0 1 0 0]; stem(n,x) xlabel( n ) ylabel( x[n] )

Sampling Discrete-time signals are usually obtained by sampling continuous-time signals.. xt () xn [ ] = xt ( ) = xnt ( ) t= nt

DT Step and Ramp Functions

DT Unit Pulse δ[ n] 1, n = 0 = 0, n 0

Periodic DT Signals xn [ ] A DT signal is periodic if there exists a positive integer r such that xn [ + r] = xn [ ] n r is called the period of the signal The fundamental period is the smallest value of r for which the signal repeats

Example: Periodic DT Signals Consider the signal The signal is periodic if xn [ ] = Acos( Ω n+ θ ) Acos( Ω ( n+ r) + θ ) = Acos( Ω n+ θ ) Recalling the periodicity of the cosine xn [ ] cos( α) = cos( α + 2 kπ) is periodic if and only if there exists a positive integer r such that Ω r = 2kπ for some integer k or, equivalently, that the DT frequency Ω is such that Ω = 2 kπ / r for some positive integers k and r

xn [ ] = A cos( Ω n + θ ) Ω Example: for different values of Ω= π /3, θ = 0 Ω = 1, θ = 0 periodic signal with period r = 6 aperiodic signal (with periodic envelope)

DT Rectangular Pulse p [ n] L 1, n= ( L 1)/2,, 1,0,1,,( L 1)/2 = 0, all other n (L must be an odd integer)

Digital Signals xn [ ] A digital signal is a DT signal whose values belong to a finite set or alphabet a, a,, an { } 1 2 A CT signal can be converted into a digital signal by cascading the ideal sampler with a quantizer

xn [ ] Time-Shifted Signals If is a DT signal and q is a positive integer xn [ q] xn [ + q] is the q-step right shift of xn [ ] is the q-step left shift of xn [ ]

Example of CT System: An RC Circuit Kirchhoff s current law: i () t + i () t = i() t C R

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

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

RC Circuit: Cont d Step response when R=C=1

Example of CT System: Car on a Level Surface Newton s s second law of motion: 2 d y() t dy() t M + k () 2 f = x t dt dt xt () yt () where is the drive or braking force applied to the car at time t and is the car s position at time t

Car on a Level Surface: Cont d Step response when M=1 and k = 0.1 f

Example of CT System: Mass-Spring-Damper System 2 d y() t dy() t M + D + Ky() t = x() t 2 dt dt

Mass-Spring-Damper System: Cont d Step response when M=1, K=2, and D=0.5

Example of CT System: Simple Pendulum 2 d θ () t I + MgLsin θ ( t) = Lx( t) 2 dt If sin θ ( t) θ ( t) 2 d θ () t I + MgLθ () t = Lx() t 2 dt

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

Example: The Ideal Predictor yt () = xt ( + 1)

Example: The Ideal Delay yt () = xt ( 1)

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

Ideal Amplifier/Attenuator RC Circuit yt () = Kxt () 1 yt = e x d t C t (1/ RC )( t τ ) () ( τ) τ, 0 0 Examples

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 1 2

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

Basic System Properties: Linearity A system is said to be linear if it is both additive and homogeneous ax () t () + bx t system 1 2 1 2 ay () t + by () t A system that is not linear is said to be nonlinear

Example of Nonlinear System: Circuit with a Diode yt () R R + 2 R xt (), whenxt () 0 = 1 2 0, when x( t) 0

Example of Nonlinear System: Square-Law Device yt 2 () = x() t

Example of Linear System: The Ideal Amplifier yt () = Kxt ()

Example of Linear System: A Real Amplifier

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 yt ( t) A system that is not time invariant is said to be time varying or time variant

Examples of Time Varying Systems Amplifier with Time-Varying Gain yt () = txt () First-Order System yt () + at () yt () = bxt ()

Basic System Properties: Finite Dimensionality Let x(t) and y(t) be the input and output of a CT system Let x (i) (t) and y (i) (t) denote their i-th derivatives The system is said to be finite dimensional or lumped if, for some positive integer N the N-th derivative of the output at time t is equal to a function of x (i) (t) and y (i) (t) at time t for 0 i N 1

Basic System Properties: Finite Dimensionality Cont d The N-th derivative of the output at time t may also depend on the i-th derivative of the input at time t for i N ( N) (1) ( N 1) y t f y t y t y t () = ( (), (),, (), xt x t x t t (1) ( M ) ( ), ( ),, ( ), ) The integer N is called the order of the above I/O differential equation as well as the order or dimension of the system described by such equation

Basic System Properties: Finite Dimensionality Cont d A CT system with memory is infinite dimensional if it is not finite dimensional, i.e., if it is not possible to express the N-th derivative of the output in the form indicated above for some positive integer N Example: System with Delay dy() t dt + ay( t 1) = x( t)

DT Finite-Dimensional Systems Let x[n] and y[n] be the input and output of a DT system. The system is finite dimensional if, for some positive integer N and nonnegative integer M, y[n] can be written in the form yn [ ] = f( yn [ 1], yn [ 2],, yn [ N], xn [ ], xn [ 1],, xn [ M], n) N is called the order of the I/O difference equation as well as the order or dimension of the system described by such equation

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

Basic System Properties: DT Linear Finite-Dimensional Systems If the output of a DT system can be written in the form N 1 M = i + i i= 0 i= 0 yn [ ] a( n) yn [ i] b( nxn ) [ i] then the system is both linear and finite dimensional

Basic System Properties: Linear Time-Invariant Finite-Dimensional Systems For a CT system it must be a () t = a and b() t = b iand t i i i i And, similarly, for a DT system a ( n) = a and b( n) = b iand n i i i i

About the Order of Differential and Difference Equations Some authors define the order as N Some as ( M, N) Some others as max( M, N)