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.