REPRESENTING SIGNALS (Chapter 1) |1
Question: What is signal?
Answer:
Signal: Signal is a detectable physical quantity or impulse (as a voltage,
current or magnetic field strength) by which massages or informations can
be transmitted.
Or, Any set of human information or machine data can be taken as a signal.
Mathematically, signals are represented as a function of one or more
independent variable that contains some information about the behaviour
of a natural or artificial system.
Examples: Human voice, television pictures, teletype data, and
atmospheric temperature.
Question: Define independent and dependent variable.
Answer:
Independent and dependent variable: The independent variable is the
variable representing the value being manipulated or changed and
dependent variables is the observed result of the independent variable
being manipulated.
Question: Why we use electrical signal?
Answer: Electrical signals are used for the following reasons:
1. Electrical signals are the most easily measured and the most simply
represented type of signals. Therefore, many engineers prefer to
transform physical variables to electrical signals.
For example, many physical quantities, such as temperature,
humidity, speech, wind speed and light intense can be transformed
using transducer to time-varying current or voltage signal.
2. Electrical engineers deal with signals that have a broad range of
shapes, amplitudes, durations and perhaps other physical properties.
For example, a radar-system designer analyze high-energy
microwave pulse, a communication-system engineer analyzes
information carrying signals, a power engineer deals with highvoltage signals and a computer engineer deals with millions of pulses
per second.
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Continuous Signal and Linear Systems (EEE-501)
REPRESENTING SIGNALS (Chapter 1) |2
Question: Classify signal from different types of view.
Answer:
Signal
Natural Signal
Synthetic Signal
Analog Signal
Digital Signal
Synthetic signal: The signal which can be generated in the laboratory is
called synthetic signal.
Analog signal: A continuously varying signal (voltage or current) is called
an analog signal.
Example: An alternating voltage varying sinusoidally is an analog signal.
Figure: An analog signal
Digital signal: A signal that can have only two discrete values is called a
digital signal.
Example: A square wave is a digital signal.
Figure: A digital signal
According to the independent variable there are two types of signals:
Prepared By: Hasan Bin Firoz (533)
Continuous Signal and Linear Systems (EEE-501)
REPRESENTING SIGNALS (Chapter 1) |3
i) Continuous time signal
ii) Discrete time signal
Continuous time signal: If the independent variable is continuous the
corresponding signal is called continuous time signal and is defined for a
continuum of values of the independent variable x(i).
Example: Telephone or radio signal.
Discrete time signal: If the independent variable is discretized then the
signal is called discrete time signal and is defined for a continuum as the
sequence x[n] where n is integer.
Example:
Figure: A discrete time signal
Question: Write down the difference between continuous time signal
and discrete time signal.
Answer:
Continuous time signal
Discrete time signal
1. If the independent variable is
continuous then the corresponding
signal is called a continuous time
signal.
1. If the independent variable is
discrete then the corresponding
signal is called a discrete time
signal.
2. Continuous time signal varies 2. Discrete time signal flows at
linearly in a continuous flow.
discrete instant of time.
3. Continuous time signals are 3. Discrete time signals are usually
denoted by x(t).
denoted by x[k] or x[n].
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Continuous Signal and Linear Systems (EEE-501)
REPRESENTING SIGNALS (Chapter 1) |4
4. Laplace transformations are used 4. Z-transformations are used for
for continuous analysis.
discrete analysis.
5.
5.
Question: What do you mean by system?
Answer:
System: A system is any physical set of components that takes a signal and
produces a signal.
A system can be taught to operate on the input to produce output.
The output is related to the input by a certain relationship known as the
system response.
Example: A robot, filter, radar etc.
Question: Discuss the following signal: (i) Periodic signal (ii) Energy
and power signal.
Answer:
Periodic signal: Any continuous time signal that satisfied the following
conditions,
x(t) = x(t+nT); n=1,2,3,..............
where, T>0 is a constant known as fundamental period is classified as a
periodic signal.
A signal x(t) that is not periodic is referred to as an aperiodic signal.
Prepared By: Hasan Bin Firoz (533)
Continuous Signal and Linear Systems (EEE-501)
REPRESENTING SIGNALS (Chapter 1) |5
Figure shows a periodic signal with fundamental period.
Energy and power signal: The signal which have finite energy and zero
power known as energy signal.
The signal which has finite energy and finite power is known as
power signal.
We know, work = Force × displacement
Power = work/time
Energy is work and power is work per time.
In electric signals, P(t) = V(t)I(t)
= . V2(t) = i2(t)R
∴E = ∫
V2(t)dt
In signal processing total energy of x(t) is defined mathematically as,
E(t) =
∫ | ( )|
= ∫ | ( )|
And we can also define the average power as,
P(t) =
=
∫ | ( )|
( )
Figure: Energy signal
Figure: Power signal
Question: Explain the following elementary signal in details: (i) Unit
step function/ Heaviside step function (ii) Unit impulse function/
Dirac delta function (iii) Ramp function (iv) Sampling function.
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Continuous Signal and Linear Systems (EEE-501)
REPRESENTING SIGNALS (Chapter 1) |6
Answer:
(i) Unit step function/Heaviside step function: The continuous time
unit step function is defined as,
H(t) or θ(t) or U(t) = {
and is shown in below figure,
Figure: Unit step function
Unit step function is continuous for all ‘t’ except at t=0. This function is
like an on-off dc switching.
Application: This function is used in control theory and signal
processing to represent signal that switches on at a specified time and
stays switched on indefinitely.
(ii) Unit impulse function or dirac delta function (δ): We define an
ideal or unit or dirac delta function as a function which has zero width
and infinite amplitude.
Normally the dirac delta function is defined as
∫
( ) ( )
= x(0) ; t1<0<t2
and is shown in below figure,
Properties: 1. δ(0) → ∞
2. δ(t) = 0, t ≠ 0
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Continuous Signal and Linear Systems (EEE-501)
REPRESENTING SIGNALS (Chapter 1) |7
3. ∫
δ(t)
4. δ(t) is an even function i.e. δ(t) = δ(-t)
Application: For the modeling of transient signal, the unit impulse
function is used.
(iii) Ramp function: The ramp function is defined as,
r(t) = {
Figure: Ramp function
The ramp function is obtained by integrating the unit step function, i.e.
∫
(τ)dτ
r(t)
Ramp function is applied in DSP theory.
The device that accomplishes this operation is called integrator.
Example: The linear-sweep waveform of a cathode-ray tube.
(iv) Sampling function: A function frequently encountered in spectral
analysis is the sampling function Sa(x) and defined by,
Sa(x) =
Figure: Sampling function
Question: Discuss the transformation of independent variables.
Prepared By: Hasan Bin Firoz (533)
Continuous Signal and Linear Systems (EEE-501)
REPRESENTING SIGNALS (Chapter 1) |8
Answer:
Transformation of the independent variable: A number of important
operations are often performed on signal. Most of these operations involve
transformations of the independent variable. The operations are:
1. Shifting operation
2. Reflection operation
3. Time scaling operation
1. Shifting operation: A signal x(t-to) represents a time-shifted version of
x(t). The shift in time is to.
If to>0 then the signal is delayed by to seconds.
If to<0 then the signal is advanced replica of x(t).
2. Reflection operation: The signal x(-t) is obtained from the signal x(t) by
a reflection about t=0.
Thus if x(t) represents a signal of video recorder then x(-t) is the signal of
video player.
3. Time scaling operation:
Prepared By: Hasan Bin Firoz (533)
Continuous Signal and Linear Systems (EEE-501)
REPRESENTING SIGNALS (Chapter 1) |9
x(3t) is the contracted version of x(t).
x(t/2) is the expanded version of x(t).
In general, x(ηt) then,
η>1
Compressed.
η<1
Expanded.
Question: Discuss the following system: (i) Linear or non-linear
systems (ii) Time-varying and time-invariant systems (iii) Systems
with and without memory (iv) Causal systems (v) Invertibility and
inverse systems (vi) Stable systems./Discuss the classification of
continuous time systems.
Answer:
(i) Linear and non-linear system: A system is considered linear if it
satisfies the condition of superposition. It implies two conditions:
a) Additivity
b) Homogeneity
So we can say that a system is linear if it satisfies the condition of additivity
and homogeneity.
Mathematically the superposition principle can be stated as follows:
Let, y1(t) be the response of a continuous time system to an input of x1(t)
and y2(t) be the response to the input x2(t). Then the system is linear if,
a) The response to {x1(t) + x2(t)} is {y1(t) + y2(t)} and
b) The response to αx1(t) is αy1(t) where α is an arbitrary constant.
The first property is referred to as the additive property. The second is the
homogeneity property. These two properties defining a linear system can
be combined into a single statement asαx1(t)+βx2(t)
αy1(t)+βy2(t)
A system is said to be non-linear if the last equation is not valid for at least
one set of x1(t), x2(t), α and β.
(ii) Time varying and time-invariant system: A system is said to be timeinvariant if a time shift in the system input signal causes an identical time
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Continuous Signal and Linear Systems (EEE-501)
R E P R E S E N T I N G S I G N A L S ( C h a p t e r 1 ) | 10
shift in the output signal. Specifically, y(t) is the output corresponding to
the input x(t). A time invariant system will have y(t-to) as the output when
x(t-to) is the input.
The procedure of testing whatever a system is time invariant is
summarized as follow:
1. Let y1(t) be the output corresponding to x1(t).
2. Consider a second input x2(t) obtained by shifting x1(t),
X2(t) = x1(t-to)
3. From step-1 find y1(t-to) and compare with y2(t).
4. If y2(t)=y1(t-to) then the system is time invariant. Otherwise it is a
time varying system.
(iii) Systems with and without memory: A system is said to have
memory if the output from the system is dependent on past or future
inputs.
A system is memory less if the output is only dependent on the
current input.
For example, a resistor is a memory less system. If we take the equation,
y(t)=Rx(t) where value of y(t) at any instant depends on the value of x(t) at
the instant.
On the other hand a capacitor is a system with memory,
y(t) = ∫
(τ)dτ
output at any time ‘t’ depends on the entire past history of the input.
Systems that have memory are called dynamic system and that do not have
memory are called static system.
(iv) Causal systems: A system is called causal or non-anticipatory (also
known as physically realizable) if it is only dependent on past or current
inputs.
A system is called non-causal or anticipatory if the output of the
system is dependent on future input.
(v) Invertibility and inverse system: A system is invertible if by
observing the output, we can determine its input and the system by which
the invertible system is being done called inverse system.
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Continuous Signal and Linear Systems (EEE-501)
R E P R E S E N T I N G S I G N A L S ( C h a p t e r 1 ) | 11
Figure: Concept of inverse system
(vi) Stable system: Stability is a very important concept in systems, but it
is also one of the hardest function property to proof, there are several
different criteria for system stability, but the most common requirement is
that the system must produce a finite output when subjected to a finite
input.
This type of stability is often known as bounded input bounded output
(BIBO) stability.
Systems can also be categorized by the number of inputs and outputs
the system has.
Consider a television as a system. This system has two inputs- the power
wire and signal cable. It has one output, the video display. Such systems are
known as single input single output or SISO.
A system with multiple inputs and multiple outputs is called multiple
input multiple output (MIMO) system.
Example: Consider the voltage divider circuit shown in figure with
R1=R2. Prove that the system is linear.
Or, Consider the voltage divider circuit shown in figure with R 1=R2x(t).
Prove that the system is non-linear.
Question: Prove that the ideal inductor is a linear system.
Answer: Let, y(t) = k
( )
Consider the input, x(t) = ax1(t) + bx2(t)
The corresponding output is,
y(t) = k [ax1(t) + bx2(t)]
=ak
x1(t) + b k
x2(t)
∴ y(t) = ay1(t) + by2(t)
Hence the system is linear.
So we can say that,
V(t) = L
( )
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Continuous Signal and Linear Systems (EEE-501)
R E P R E S E N T I N G S I G N A L S ( C h a p t e r 1 ) | 12
i.e. Ideal inductor is a linear system.
[Proved]
Question: Prove that the ideal capacitor is a linear system.
Answer: Let, output y(t) = k
()
----------------- (i)
Consider the input, x(t) = ax1(t) + bx2(t)
The corresponding output may be written as,
y(t) = k [ax1(t) + bx2(t)]
=ak
x1(t) + b k
x2(t)
∴ y(t) = ay1(t) + by2(t)
Hence the system is linear.
Equation (i) can be compared to the ideal equation of capacitor, i(t) = C.
()
Thus ideal capacitor is a linear system.
[Proved]
Question: What is LTI? Discuss in brief.
Answer: Linear time invariant theory commonly known as LTI system
theory comes from applied mathematics and has a direct application in
NMR spectroscopy, seismology, circuits, signal processing, control theory
etc.
It investigates the response of a linear and time invariant system to
an arbitrary input signal.
The two defining properties of LTI system are1. Linearity and
2. Time invariance
1. Linearity: There are three requirements for linearity.
i. Additivity
ii. Homogeneity
iii. If x(t)=0, y(t)=0
If y1(t) is the response to a continuous time system to a signal x1(t)
and y2(t) be the response to x2(t) then the system is linear ifa) The response to {x1(t) + x2(t)} is {y1(t) + y2(t)} and
b) The response to αx1(t) is αy1(t) where α is an arbitrary constant.
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Continuous Signal and Linear Systems (EEE-501)
R E P R E S E N T I N G S I G N A L S ( C h a p t e r 1 ) | 13
The first property is referred to as the additive property. The second is the
homogeneity property. The combined representation of the above two
properties is,
αx1(t)+βx2(t)
αy1(t)+βy2(t)
2. Time invariance: A system is said to be time invariance if a time shift in
the input signal causes an identical time shift in the output signal.
For example, if y(t) is the response to an input signal x(t) then the
system is said to be time invariant if the response to the s(t-to) is y(t-to).
Question: Determine whether the system described by the following
equation is time invariant, y(t) = cosx(t).
Answer:
1. Consider input x1(t) then y1(t)=cosx1(t) --------------- (i)
2. Consider a second input x2(t) i.e. x2(t) = x1(t-to)
∴ y2(t) = cosx2(t) = cosx1(t-to)
----------------- (ii)
3. From equation (i) we get,
y1(t-to) = cosx1(t-to)
------------------ (iii)
Comparing equation (ii) and (iii) we get,
y2(t) = y1(t-to)
So, the system is time invariant.
Prepared By: Hasan Bin Firoz (533)
Continuous Signal and Linear Systems (EEE-501)