CHP 1: CONTINUOS TIME SIGNAL & SYSTEM

CHP 1: CONTINUOS TIME SIGNAL & SYSTEM Definition Signal & SystemClassification Of SignalTransformation Of SignalElementary / Common SignalCT System and Its PropertiesConvolution And Its PropertiesProperties Of Linear Time-invariance (LTI) System

DEFINITION OF SIGNALSIGNAL - modeled as functions of one or more independent variables. { f(t), x(f), etc.} Example : human speech, electrical signal, (voltage & current), temperature, pressure, etc.f(t) dependant variable t independent variable

DEFINITION OF SYSTEMSYSTEM - entity that processes a set of signals SISO and MIMO typeExample - software systems, electronic systems, computer systems, or mechanical systems

CLASSIFICATION OF SIGNALScontinuous-time vs discrete-time even vs odd periodic vs aperiodic (nonperiodic)energy and power signals;deterministic vs random analog vs digital CONTINUOUS-TIME VS DISCRETE-TIME

CT functions of a continuous variable (time).DT functions of a discrete variable (integer values, n) of the independent variable (time steps).

DT SIGNALXn - samples time interval between them sampling interval (Ts ) constant When the sampling intervals are equal (uniform sampling), then:

EXAMPLE 1Discretize the signal below using a sampling interval of T = 0.25 s, and sketch the waveform of the resulting DT sequence for the range 8 k 8.

EVEN vs ODD SIGNALSEven signal [xe(t)] : symmetric y-axis x(t) = x(t)Odd signal [xo(t)] : anti- symmetric y-axis x(t) = x(t)

EVEN vs ODD SIGNALSAny signal x(t) or x[n] can be expressed as a sum of two signals, one of which is even and one of which is odd.x(t) = xe(t) + xo(t)

xe(t) = [x(t) + x(-t)] even part of x(t)xo(t) = [x(t) - x(-t)] odd part of x(t)

EXAMPLE 2Express CT signal as a combination of an even signal and an odd signal

10PERIODIC vs APERIODICSignal is periodic when it repeats itself. x(t)= x(t+T)T = fundamental period (constant)Signal that is not periodic is called an aperiodic or non-periodic signal

ENERGY vs POWER SIGNALEnergy:

Average Power:

Average Power (periodic signal):

A signal x(t ) = energy signal = the total energy Ex has a non-zero finite value, i.e. 0 < Ex < .Power signal - non-zero finite power, i.e. 0 < Px < . Signal cannot be both an energy and a power signal simultaneously.

Example 3Calculate the average power, and energy present in the two signal below. Classify these signals as power or energy signals.

Because x(t ) has finite energy (0 < Ex = 100 < ) it is an energy signal.Ans; z(t) power signal. Prove that !!!SIGNAL TRANSFORMATIONS3 OPERATION/TRANSFORMATION: time scaling; time shifting; time reversalTime scaling = multiplication of the time variable by a real positive constant, . In the CT case, we can write: y(t) = x( t)

Case 0 < < 1: The signal x(t) is slowed down or expanded in time. Think of a tape recording played back at a slower speed than the nominal speed.Case > 1: The signal x(t) is speed up or compressed in time. Think of a tape recording played back at twice the nominal speed.Example 4 (scaling)Consider signal, x(t):

TIME SHIFTINGA time shift delays or advances the signal in time by a continuous-time interval: y(t)=x(t T)

Time advancedTime delayTIME REVERSAL/INVERSIONA time reversal is achieved by multiplying the time variable by 1.At y-axis

COMBINED OPERATIONSy(t) = x(at b) shifting+scaling+inversion2 Step:shifting the signal x(t) by b to get x(t b); time scaling (replace t by at) the shifted signal by a to get x(at b).

Alternate 2 steps:time scale the signal x(t) by a to get x(at); shift (replace t by t b/a) the time-scaled signal by b/a to get x(a(t b/a)) = x(at b). Note that, time reversal operation is a part of the time scaling operation with a negative.

Example 5 Sketch signal for : a) y(t) = x(-4t + 2) ; b) y2(t) = x(0.5t 3) ; c) y3(t) = 2* x(4 2t); d) y4(t) = -2*x(4t + 4)

ELEMENTRY SIGNALElementry/Common Signal:Unit step functionRectangular pulse functionSignum functionRamp functionSinusoidal functionSinc functionExponential functionUnit impulse function / delta function / dirac

ELEMENTRY SIGNAL

Step functionRectangular pulse function Signum function Ramp function Sinusoidal function Sinc functionELEMENTRY SIGNAL

Exponential function Delta functionUnit Step Function (1)

Unit Step Function (Others Form)

Signal as sum of step functionRectangular waveform of as a sum of unit step functions

X(t) = u(t) u(t1)Example 6Express the signal below of as a sum of unit step functions

The Delta Function/ Unit ImpulseThe unit impulse or delta function, denoted as (t) , is the derivative of the unit step, u(t)

Sampling Property :Shifting Property :Example 7Evaluate the following expressions:

SYSTEM PROPERTIESIn this section, we classify systems into 6 basic categories: linear and non-linear systems; time-invariant and time-varying systems; systems with and without memory; causal and non-causal systems; invertible and non-invertible systems; stable and unstable systems.System properties apply equally to CT and DT systems.

Basic System Interconnections

Cascade:Parallel:Feedback:

Linear vs Non-LinearA system S is linear if it has the additivity property and the homogeneity property.Let y1 := Sx1 and y2 := Sx2.Additivity: y1 + y2 = S(x1 + x2)Homogeneity: ay1= S(ax1)Homogeneity means that the response of S to the scaled signal ax1 is a times the response y1 = Sx1. If the input x(t ) to a linear system is zero, then the output y(t ) must also be zero for all time t .Thus, the system y(t) = 2x(t) + 3 is nonlinear because for x(t) = 0, we obtain y(t) = 3

Example 7 Determine whether the CT systems are linear or non-linear:answer:

y(t ) = x2(t ) non-linear

linear

non-linear

linear

non-linear

Time-invariant vs Time-varying SystemsA system is said to be time-invariant (TI) if a time delay or time advance of the input signal leads to an identical time-shift in the output signal. In other words, except for a time-shift in the output, a TI system responds exactly the same way no matter when the input signal is applied.A system S is time-invariant if its response to a time-shifted input signal x[n N] is equal to its original response y[n] to x[n], but also time shifted by N: y[n N].That is, if for y[n] := Sx[n], y1[n] := Sx[n N], the equality y1[n] = y[n N] holds for any integer N, then the system is time-invariant.Exp 1: y(t) = sin(x(t)) is time-invariant since y1(t) = sin (x(t T)) = y(t T)Exp 2: z(t)= t[x(t)] non time-invariant (time varying) since z1(t)= t[x(t - T)] z(t T)

MemoryA system is memoryless if its output y at time t or n depends only on the input at that same time.A system has memory if its output at time t or n depends on input values at some other times (past or future)

memoryless

memory

CausalityA system is causal if its output at time t or n depends only on past or current values of the input.Non-causal - output up to time t depends on future values of the input signals.Note that all memoryless systems are causal systems because the output at any time instant depends only on the input at that time instant. Systems with memory can either be causal or non-causal.

Stability A system is referred to as bounded-input, bounded-output (BIBO) stable if an arbitrary bounded-input signal always produces a bounded-output signal. In other words, if an input signal x(t ) for CT systems, satisfying : {|x(t )| Bx < for t (,);} is applied to a stable, it is always possible to find an finite number By < such that: {|y(t )| By < for t (,);} Bx is a finite number.

Invertible vs Non-invertible SystemsCT system is invertible if the input signal x(t ) can be uniquely determined from the output y(t) produced in response to x(t ) for all time t (,).To be invertible, two different inputs cannot produce the same output since, in such cases, the input signal cannot be uniquely determined from the output signal. invertible

input x(t ) can be uniquely determined from the output signal y(t ).

non invertible (2 possible value not unique)y(t ) = 3x(t ) + 5

LTI SYSTEMImportant subset of CT systems satisfies both the linearity and time-invariance propertiesCT systems are referred to as linear, time-invariant, continuous-time (LTIC) systems or LTIPrimarily interested in calculating the output y(t ) of the LTIC system from the applied input x(t ).The output y(t ) of an LTIC system can be evaluated analytically in the time domain in several ways.Model of LTIC system - linear constant-coefficient differential equation, differential equation, unit impulse response h(t)

Unit Impulse Response h(t )Define the unit impulse response h(t ) as the output of an LTIC system to an unit impulse function (t ) applied at the input.

This development leads to a second approach for calculating the output y(t ) based on convolving the applied input x(t ) with the impulse response h(t ). The resulting integral is referred to as the convolution integral.Because the system is LTIC, it satisfies the linearity and the time-shifting properties. If the input is a scaled and time-shifted impulse function a(t t0), the output of the system is also scaled by the factor of a and is time-shifted by t0

ExampleCalculate the impulse response for system, y(t ) = x(t 1) + 2x(t 3); Solution: The impulse response of a system is the output of the system when the input signal x(t ) = (t ). Therefore, the impulse response h(t ) can be obtained by substituting y(t) by h(t ) and x(t) by (t ) h(t ) = (t 1) + 2(t 3).

Example The impulse response of an LTIC system is given by h(t ) = exp(3t )u(t ). Determine the output of the system for the input signal x(t ) = (t + 1) + 3(t 2) + 2(t 6).

Solution:Because the system is LTIC, it satisfies the linearity and time-shifting properties. Therefore, (t + 1) h(t + 1), 3(t 2) 3h(t 2), 2(t 6) 2h(t 6).Applying the superposition principle, we obtain x(t ) y(t ) = h(t + 1) + 3h(t 2) + 2h(t 6).

Example

h(t ) of the LTIC systemOutput y(t ) of the LTIC systemConvolutionWhen an input signal x(t ) is passed through an LTIC system with impulse response h(t ), the resulting output y(t ) of the system can be calculated by convolving the input signal and the impulse response.

Convolution

Convolution Integral

Example

Step 1: Mirror

At t =0

At t =1At t = 2

Overall Results: Plot at each pointAt t =3Practice

Answer:Property Convolution IntegralCommutative property : the order of the convolution operands does not affect the result of the convolution.

Distributive property : convolution is a linear operation.

Associative property : changing the order of the convolution operands does not affect the result of the convolution integral.

Property Convolution IntegralShift property: if the two operands of the convolution integral are shifted, then the result of the convolution integral is shifted in time by a duration that is the sum of the individual time shifts introduced in the operands.

Duration of convolutionConvolution with impulse functionConvolution with unit step functionScaling property

Property of LTI SystemMany physical processes can be represented by and successfully analyzed with, linear time-invariant (LTI) systems as models. For example, both a DC motor or a liquid mixing tank have constant dynamical behavior (time-invariant) and can be modeled by linear differential equations. Filter circuits designed with operational amplifiers are usually modeled as LTI systems for analysis LTI models are also extremely useful for design. A process control engineer would typically design a level controller for the mixing tank based on a set of linearized, time-invariant differential equations. DC motors are often used in industrial robots and may be controlled using simple LTI controllers designed using LTI models of the motors and the robot.

Property of LTI SystemCommutative property (convolution)

Distributive property (convolution)

Property of LTI SystemAssociative property (convolution)

Memorylessthe output y(t) of a memoryless system depends on only the present input x(t), then, if the system is also linear and time-invariant, this relationship can only be of the form {Y(t) = Kx(t) } where K is a (gain) constant. Thus, the corresponding impulse response h(t) is simply ,{ h(t) = K(t) } Therefore, if h(t0) 0 for t0 0 , the continuous-time LTI system has memory. An LTIC system will be memoryless if and only if its impulse response h(t ) = 0 for t 0.

Property of LTI SystemCausality : LTI system is causal if and only if h(t)=0, t < 0Invertible: For an LTI system with impulse response, h, this is equivalent to the existence of another system with impulse response such that , h*h1=

y1(t)y2(t)

if BIBO StabilityBIBO stable If the impulse response h(t ) of an LTIC system satisfies the following condition:

ExampleDetermine if systems with the following impulse responses: h1(t ) = (t) (t 2) are memoryless, causal and stable.

Solution :Memoryless property: Since h(t ) = 0 for t = 0, system is not memoryless.Causality property. Since h(t ) = 0 for t < 0, system (i) is causal.To verify if system (i) is stable, we compute the following integral:

(Stable) ConclusionA signal was defined as a function of time, either continuous or discrete.A system was defined as a mathematical relationship between an input signal and an output signal.Special types of signals were studied: real and complex exponential signals, sinusoidal signals, impulse and step signals.The main properties of a system were introduced: linearity, memory, causality, time invariance, stability, and invertibility.

ConclusionAn LTI system is completely characterized by its impulse response.The input-output relationship of an LTI continuous-time system is given by the convolution integral of the systems impulse response with the input signal.Given the impulse response of an LTI system and a specific input signal, the convolution giving the output signal can be computed using a graphical approach or a numerical approach.The main properties of an LTI system were derived in terms of its impulse response.